ABSTRACT Title of dissertation: EFFECTS OF SCALE AND SPATIAL VARIABILITY ON HYDRAULIC GEOMETRY IN THE POTOMAC RIVER BASIN Chung Hye Kim Read, Doctor of Philosophy, 2006 Dissertation directed by: Emeritus Associate Professor, Derek Thompson Department of Geography Scale issues in hydrology arise because different hydrological processes are dominant at different regional scales. Recent hydrological research suggests that the geographic scale (size) of watersheds may influence the behavior of hydraulic geometry exponents (b and f, but not m values) of stream channels. Hence, the working hypothesis of this study is that variations of hydraulic geometry exponents are not random, but that there are systematic changes as a function of geographic scale as well as of water basin and channel physical and environmental characteristics (predictor variables). To support this analysis, 43 subbasins in the Potomac River Basin ranging in size from 0.38 square miles to 1,642 square miles and representing a broad spatial diversity of predictor variables within the watershed were selected for study. Research goals were to attempt, via empirical correlations, to discern relationships between a geographic scale factor and b, f, and m values, to investigate the roles of predictor variables on b, f, and m values, and their statistical significance, and to identify the most influential predictor variables and the complexity of fluvial physical processes via stepwise multi-variable regressions. Statistical evidence was found that there is a relationship between geographic scale and hydraulic geometry exponents. In every selected predictor variable case, investigation of the correlations between b, f, and m with a single selected predictor variable in a scale context resulted in a noticeable improvement over the correlations of the hydraulic exponents with each individual predictor variable alone. The research shows that, under higher discharges, the behavior of b, f, and m mainly result in higher m and f, with a slight increase in cross-sectional area (f with negative b) in a scale context. EFFECTS OF SCALE AND SPATIAL VARIABILITYON HYDRAULIC GEOMETRY IN THE POTOMAC RIVER BASIN by Chung Hye Kim Read Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2006 Advisory Committee: Emeritus Associate Professor Derek Thompson, Chairman/Advisor Professor Ralph Dubayah Professor Glenn E. Moglen Professor Karen L. Prestegaard Professor John R. Townshend ?Copyright by Chung Hye Kim Read 2006 ii ACKNOWLEDGEMENTS Completion of this research work would not have been possible without the active support over several years by my advisor, Dr. Derek Thompson. Although he is an excellent teacher and advisor, I am not the best student. I cannot thank him enough for his patience, encouragement, guidance and support. I also wish to thank my committee members at The University of Maryland who assisted me on an array of technical issues as they arose. Dr. John Townshend for helping me conceive of and shape this research. Dr. Ralph Dubayah for helping me master SPlus to prepare data for numerous analyses. Dr. Glenn Moglen for helping me to develop a more mathematical approach to the geographic scale problem. Dr. Karen Prestegaard for helping me focus on data validity and fluvial geomorphology. Special acknowledgement and thanks go to a number of offices and individuals of the U.S. Geological Survey who provided much of the baseline data, without which, this study would not have been possible. The USGS National Mapping Division provided all the DEM data; Water Resources Division offices in Baltimore and LaVale, Maryland and Richmond and Charlottesville, Virginia provided published and unpublished field notes of discharge measurements data. Mr. John Brakebill, Water Resources Division, Baltimore office provided GIS data. Dr. David Wolock, at Lawrence, Kansas provided numerous suggestions on concepts of topographic index calculations. Also, thanks for hydrographic survey information are due to Mr. Ronny Lawson of the Virginia Department of Environmental Quality at Charlottesville. I also am particularly grateful to my professional colleagues, Mr. Robert Hair and Mr. Michael Thomas, for their considerable hands-on statistical and geospatial support and for their suggestions on handling the gigantic volume of data. My thanks, also, to my husband, Randy, for his unwavering support for my ambition to succeed. Finally, I want to dedicate this study to my Mother and Father who encouraged me that I could do anything if I set my mind to it. iii TABLE OF CONTENTS ACKNOWLEDGEMENTS............................................................................................... ii LIST OF TABLES...........................................................................................................viii LIST OF FIGURES............................................................................................................x CHAPTER 1 INTRODUCTION: CHANNEL PROPERTIES AND HYDRAULIC GEOMETRY......................................................................................................................1 1.1 Research Context..........................................................................................................1 1.2 Research Objectives, Hypotheses, and Scope...............................................................6 1.3 Approaches for the Empirical Investigation...............................................................11 CHAPTER 2 PREVIOUS STUDIES: HOW IMPORTANT IS GEOGRAPHICAL SCALE?............................................................................................................................17 2.1 Introduction.................................................................................................................17 2.2 Automated Drainage Delineation...............................................................................18 2.3 At-A-Station Hydraulic Geometry..............................................................................22 2.4 b, f, and m Computation.............................................................................................28 2.5 Representation and Analyses of b, f, and m................................................................31 2.6 b, f, and m and Disparity.............................................................................................34 2.7 Heterogeneity of Environmental Factors....................................................................36 2.7.1 Environmental Controlling Factors 37 2.7.2 Water Reference Level 39 2.7.3 Scale 43 2.8 Need for Additional Study..........................................................................................46 2.9 Conclusions.................................................................................................................48 CHAPTER 3 STUDY AREA: SELECTED CHARACTERISTICS OF THE POTOMAC RIVER BASIN AND DEMARCATION OF ITS SUBBASINS.................49 iv 3.1 General Study Area.....................................................................................................49 3.1.1 History of the Study Area, Potomac River Basin 50 3.1.2 Present Potomac River Boundaries 51 3.2 Study Area Physiography...........................................................................................56 3.3 Study Area Geology and Lithology............................................................................60 3.4 Study Area Hydrology................................................................................................64 CHAPTER 4 MEASUREMENTS OF SUBBASIN CHARACTERISTICS: COMPUTATION OF DERIVED DATA.........................................................................67 4.1 Gage Stations..............................................................................................................68 4.2 Digital Elevation Data.................................................................................................74 4.3 Hydrographic Survey Data.........................................................................................75 4.4 Field Surveys..............................................................................................................79 4.5 Physiography and Lithology.......................................................................................80 4.6 Landuse.......................................................................................................................80 4.7 Derived Data...............................................................................................................81 4.7.1 Subbasin Boundaries 82 4.7.2 Alternative Method of Drainage Delineation 82 4.7.3 Digital Elevation Model (DEM) Preparation 82 4.7.4 Subbasin Boundary Delineation 83 4.7.5 Subbasin Drainage Area Size Computation 86 4.7.6 Topographic Index 91 CHAPTER 5 HYDRAULIC GEOMETRY AND THE ENVIRONMENTAL SETTING: COMPUTATION OF HYDRAULIC GEOMETRY EXPONENTS...............................96 5.1 Potomac River At-A-Station Hydraulic Geometry.....................................................96 5.2 A Ternary Diagram Portrayal of the Streamflow Exponents....................................105 5.3 Explanation for the b-f-m Values.............................................................................108 5.3.1 Ratio of Width and Depth Exponents (b = f) 108 5.3.2. Ratio of Depth and Velocity Exponents for Competency (m = f) 110 5.3.3 Ratio of Half of Depth and Velocity Exponents (m = f / 2) 112 5.3.4 Rate of Velocity and Depth Exponents for Channel Stability and Flow Resistance (m = b + f or m = 0.50) 113 5.4 Classification of b-f-m Values..................................................................................116 v 5.5 Environmental Settings.............................................................................................118 5.5.1 Channel Pattern 119 5.5.2 Channel Cross-Section Shape 124 5.5.3 Channel Asymmetry 128 CHAPTER 6 INVESTIGATION OF THE ROLES OF SCALE AND OTHER SELECTED WATERSHED VARIABLES...................................................................131 6.1 Hydraulic Geometry and Scale.................................................................................134 6.2 Going Beyond Size: Selected Predictor Variables....................................................142 6.3 The Role of Physiography........................................................................................145 6.3.1 Some Statistical Differences in Hydraulic Geometry Parameters Among Physiographic Provinces 146 6.3.2 Particular Provinces 149 6.3.3 Physiography in a Scale Context 153 6.3.4 Summary 160 6.4 The Role of Lithology...............................................................................................160 6.4.1 Each Lithology Region?s Influence 162 6.4.2 Lithology in a Scale Context 168 6.4.3 Summary 174 6.5 The Role of Landuse.................................................................................................175 6.5.1 Each Landuse Type?s Influence 177 6.5.2 Analyses Using Underlying Rationale 180 6.5.3 b, f, and m and Landuse in a Scale Context 181 6.5.4 Summary 186 6.6 The Role of Channel Pattern.....................................................................................187 6.6.1 Each Channel Pattern?s Influence 188 6.6.2 Channel Pattern in a Scale Context 191 6.6.3 Summary 198 6.7 The Role of Channel Shape......................................................................................199 6.7.1 Each Channel Shape?s Influence (Figure 6.30 and Table 6.21) 201 6.7.2 Channel Shape in Scale Context 206 6.7.3 Summary 213 6.8 The Role of Channel Bed Material...........................................................................213 6.8.1 Each Bed Material Type?s Influence 215 6.8.2 b, f, and m and Channel Bed Material in Scale Context 220 6.8.3 f > m in Sand Bed with Increasing Stream Discharge 225 6.8.4 Empirical Theory, f < m in Gravel Bed with Increasing Stream Discharge 228 6.8.5 Summary 231 vi 6.9 The Role of Channel Bank Material.........................................................................232 6.9.1 Each Bank Material Type?s Influence 233 6.9.2 b, f, and m and Channel Bank Material in Scale Context 236 6.9.3 Summary 242 6.10 The Role of Channel Asymmetry (Figure 6.50 and Table 6.38)............................243 6.10.1 b, f, and m and Channel Asymmetry in a Scale Context (Figure 6.51) 244 6.10.2 Summary 246 6.11 The Role of Topographic Index..............................................................................247 6.11.1 Strength of the Relationship between Topographic Index and b, f, and m 247 6.11.2 b, f, and m and Topographic Index in a Scale Context (Table 6.39) 249 6.11.3 Summary 251 6.12 Which Predictor Variables are most influential on b, f, and m?.............................251 6.12.1 Size and which other Predictor Variables? 251 6.12.2 Multivariate Linear Regression for b 252 6.12.3 Multivariate Linear Regression for f 253 6.12.4 Multivariate Linear Regression for m 254 6.12.5 Conclusion 257 6.13 Constrained b, f, and m Values...............................................................................257 6.13.1 Constrained Hydraulic Geometry and Scale 258 6.13.2 Impact on the Choice of Predictor Variables for b, f, and m 263 6.14 Summary.................................................................................................................266 CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS.................................268 7.1 Summary of Research Objectives and Hypotheses..................................................268 7.2 Findings and Discussion: Scale Influence on b, f, and m.........................................270 7.3 Findings and Discussion: What Are the ?Best? Predictor Watershed Variables for b, f, and m?.............................................................................................................................274 7.3.1 Channel Pattern 279 7.3.2 Bed Material 282 7.3.3 Bank Material 285 7.3.4 Topographic Index 287 7.4 Findings and Discussion: Which Subset of Watershed Predictor Variables Are Most Influential on b, f, and m?...............................................................................................289 7.5 Conclusions in a Broader Context for Future Research............................................293 APPENDICES................................................................................................................300 vii APPENDIX A USGS HYDROGRAPHIC FIELD NOTE/DATA FOR THE POTOMAC RIVER BASIN AREA (STREAM DISCHARGE, CHANNEL WIDTH, MEAN VELOCITY ON STREAM CROSS-SECTIONS..............................................301 APPENDIX B ALGORITHMS FOR SUBBASIN/DRAINAGE AREA DELINEATION..............................................................................................................325 APPENDIX C WATERSHED DATA COLLECTED, CALCULATED AND/OR DERIVED.......................................................................................................................328 BIBLIOGRAPHY...........................................................................................................346 viii LIST OF TABLES Table 3.1 Physiographic Province Distribution of Study Subbasins................................57 Table 3.2 Geological and Lithological Distribution of Study Subbasins.........................63 Table 4.1 Selected Gage Stations......................................................................................71 Table 4.2 USGS Published and Calculated Subbasin Size in Square Miles and Miles....87 Table 4.3 Topographic Index Values for Study Subbasins...............................................94 Table 5.1 b, f, and m Value.............................................................................................100 Table 5.2 Coefficient of Determination for Width Exponent (b)...................................102 Table 5.3 Coefficient of Determination for Depth Exponent (f)....................................103 Table 5.4 Coefficient of Determination for Velocity Exponent (m)..............................104 Table 5.5 Ten Regions Characterized by Five Criteria...................................................118 Table 5.6 Width and Depth Ratio and Channel Pattern..................................................123 Table 5.7 Subbasin Channel Pattern and Channel Shape...............................................127 Table 6.1 Regression of b, f, and m with Subbasin Size................................................139 Table 6.2 Differences of Mean b, f, and m within Each Province..................................147 Table 6.3 Physiography-Based Mean b, f, and m Values and Bank and Bed Erodability .........................................................................................................................................148 Table 6.4 b vs. Log Size, Physiography, and Interaction...............................................154 Table 6.5 f vs. Log Size, Physiography, and Interaction................................................157 Table 6.6 m vs Log Size, Physiography, and Interaction...............................................159 Table 6.7 Differences of Mean b, f, and m within Each Lithology Region....................165 Table 6.8 Lithology-Based Mean b, f, and m Values and Bank and Bed Erodability....166 Table 6.9 b and Subbasin Log Size and Lithology.........................................................169 Table 6.10 f and Subbasin Log Size and Lithology........................................................171 Table 6.11 m and Subbasin Log Size and Lithology......................................................173 Table 6.12 Differences of Mean b, f and m within each Landuse Type.........................179 Table 6.13 Expected Change Orders and Landuse-Based Mean b, f, and m Values......180 Table 6.14 b, Subbasin Log Size, and Landuse..............................................................182 Table 6.15 f, Subbasin Log Size, and Landuse...............................................................183 Table 6.16 m and Subbasin Log Size and Landuse........................................................185 Table 6.17 Differences of Mean b, f, and m within each Channel Pattern Type............190 Table 6.18 b, Subbasin Log Size, and Channel Pattern..................................................193 Table 6.19 f, Subbasin Log Size, and Channel Pattern...................................................195 Table 6.20 m, Subbasin Log Size, and Channel Pattern.................................................197 Table 6.21 Differences of Observed Mean b, f, and m in One Type vs. All Other Channel Shapes.............................................................................................................................203 Table 6.22 Differences of Expected and Observed Mean b, f, and m within Each Channel Shape Type......................................................................................................................204 Table 6.23 b and Subbasin Log Size and Channel Shape...............................................207 Table 6.24 f and Subbasin Log Size and Channel Shape...............................................210 Table 6.25 m and Subbasin Log Size and Channel Shape..............................................211 Table 6.26 Differences of Mean b, f, and m within Each Bed Material.........................217 Table 6.27 Bed Material Based Mean b, f, and m Values and Expected b, f, and m Changes...........................................................................................................................218 ix Table 6.28 b and Subbasin Log Size and Bed Material..................................................221 Table 6.29 f and Subbasin Log Size and Bed Material...................................................222 Table 6.30 m and Subbasin Log Size and Bed Material Types......................................224 Table 6.31 Differences of Mean f - m Values in Sand Beds and all Study Channel Beds .........................................................................................................................................227 Table 6.32 Differences of Mean f - m Values in Gravel and All Non-Gravel Channel Beds.................................................................................................................................230 Table 6.33 Differences of Mean b, f, and m within Each Bank Material.......................235 Table 6.34 Bank Material Based Mean b, f, and m Values and Expected b, f, and m Changes...........................................................................................................................236 Table 6.35 b and Subbasin Log Size and Bank Material................................................237 Table 6.36 f and Subbasin Log Size and Bank Material.................................................239 Table 6.37 m and Subbasin Log Size and Bank Material...............................................241 Table 6.38 Statistical Significance of Asymmetry and b, f, and m Relationships..........245 Table 6.39 Statistical Significance of Topographic Indices and b, f, and m Relationships .........................................................................................................................................250 Table 6.40 Stepwise Regression for b, f, and m with All Predictor Variables...............256 Table 6.41 Distribution of Coefficient of Determination for b, f, and m.......................259 Table 6.42 Regression of the Combined Well and Moderately Constrained Subbasin b, f, and m with Subbasin Size...............................................................................................260 Table 6.43 Regression b, f, and m with Subbasin Size for All and the Combined Well and Moderately Constrained Subbasins.................................................................................263 Table 6.44 Stepwise Regressions for b, f, and m with all Predictor Variables...............264 Table 7.1 Regressions of All and the Well and Moderately Constrained Subbasin b, f, and m with Subbasin Size (in Logarithms)............................................................................272 Table 7.2 Summary of b, f, and m, Scale, and Selected Predictor Variable Relationships......................................???????????????????.275 Table 7.3 Most Significant Predictor Variables from Stepwise Regressions for b, f, and m.....................................................................................................................................290 x LIST OF FIGURES Figure 1.1 Study Area - Potomac River Basin..................................................................11 Figure 3.1 Study Area - Potomac River Basin..................................................................50 Figure 3.2 Physiographical Characteristics of the Study Area.........................................52 Figure 3.3 Study Area Lithology......................................................................................61 Figure 4.1 Subbasin Locations..........................................................................................73 Figure 4.2 Midsection Method of Computing Cross-Section Area for Discharge Measurements...................................................................................................................77 Figure 4.3 (a) Delineated Subbasins with Gage Stations.................................................88 Figure 4.3 (b) Delineated Subbasins with Gage Stations................................................89 Figure 4.4 Study Area Subbasin Groups...........................................................................90 Figure 5.1 Ternary Diagram for Hydraulic Geometry Exponents Values of Potomac River Subbasins..............................................................................................................107 Figure 5.2 Ten Regions Determined by Five Criteria?????????????..108 Figure 5.3 Five Criteria...................................................................................................108 Figure 5.4 b, f, and m Values with 5 Classification Criteria..........................................117 Figure 5.5 Ferguson?s Four Channel Shapes..................................................................125 Figure 5.6 An Example of Cross-section Profiles for Wills Creek near Cumberland, Maryland (m10)..............................................................................................................126 Figure 6.1 Subbasin Size Histogram...............................................................................135 Figure 6.2 b, f, and m Values and Subbasin Size with Linear Regression.....................136 Figure 6.3 b, f, and m Values and Size with Quadratic Regression...............................137 Figure 6.4 b, f, and m Values and Log Size with Linear and Quadratic Regressions... 140 Figure 6.5 b, f, and m Values and Log Size with Linear and Quadratic Regressions....140 Figure 6.6 Physiographical Provinces of the Study Area...............................................145 Figure 6.7 b, f, and m Average Values by Physiographic Province...............................146 Figure 6.8 Mean Differences of b, f, and m in Great Valley and Non-Great Valley and Blue Ridge and Non-Blue Ridge Provinces....................................................................151 Figure 6.9 b and Subbasin Log Size in Physiography....................................................155 Figure 6.10 f and Subbasin in Log Size in Physiography...............................................156 Figure 6.11 m and Subbasin in Log Size in Physiography.............................................159 Figure 6.12 Study Area Lithology..................................................................................161 Figure 6.13 Average b, f, and m Values in Five Lithology Regions..............................162 Figure 6.14 Mean Differences of b, f, and m in the Crystalline and Non-Crystalline and Siliciclastic and Non-Siliciclastic Regions..............................................................164 Figure 6.15 b and Log Size in all Lithology Types........................................................170 Figure 6.16 f and Subbasin Log Size in All Lithology...................................................172 Figure 6.17 m and Subbasin Log Size in All Lithology.................................................174 Figure 6.18 Landuse Types of the Study Area...............................................................176 Figure 6.19 Average b, f, and m Values in Landuse Types............................................177 Figure 6.20 Differences of b, f, and m in Forest vs. Non-Forest and Agriculture vs. Non-Agriculture Landuse Types.....................................................................................179 Figure 6.21 b and Log Size in All Landuse Types.........................................................182 Figure 6.22 f and Subbasin Log Size in All Landuse Types..........................................184 xi Figure 6.23 m and Subbasin Log Size in All Landuse...................................................186 Figure 6.24 Average b, f, and m Values in All Channel Patterns...................................188 Figure 6.25 Mean Differences of b, f, and m in the Braided vs. Non- Braided, Meander vs.Non-Meander, and Straight vs. Non-Straight Channels..............................189 Figure 6.26 b and Subbasin Log Size in all Channel Pattern Types...............................194 Figure 6.27 f and Subbasin Log Size in all Channel Pattern Types...............................196 Figure 6.28 m and Subbasin Log Size in All Channel Pattern Types.............................198 Figure 6.29 Average b, f, and m Values by Channel Shape...........................................200 Figure 6.30 Mean Differences of b, f, and m in All Channel Shapes.............................202 Figure 6.31 Channel Shape of Subbasin m31.................................................................205 Figure 6.32 Channel Shape of Subbasin m311...............................................................206 Figure 6.33 b and Subbasin Log Size in Channel Shape................................................208 Figure 6.34 f and Subbasin Log Size in Channel Shape.................................................210 Figure 6.35 m and Log Size in Channel Shape Type.....................................................212 Figure 6.36 Average b, f, and m Values by Bed Material..............................................215 Figure 6.37 Mean Differences between Sand, Gravel, and Cobble vs. Other Bed Material...........................................................................................................................216 Figure 6.38 b and Subbasin Log Size in all Channel Bed Materials..............................221 Figure 6.39 f and Subbasin Log Size in all Channel Bed Material Types......................223 Figure 6.40 m and Subbasin Log Size in all Channel Bed Material Types....................225 Figure 6.41 f ? m Analysis in Sand Bed Channel...........................................................226 Figure 6.42 f ? m and Subbasin Size in All Sand Bed Channels....................................228 Figure 6.43 f ? m Analysis in Gravel Bed Channel........................................................229 Figure 6.44 f ? m Relation in All Gravel Bed Channels.................................................231 Figure 6.45 Average b, f, and m Values by Bank Material............................................233 Figure 6.46 Mean Differences between Silt and Cobble vs. Other Bank Material........234 Figure 6.47 b and Subbasin Log Size in Channel Bank Materials.................................238 Figure 6.48 f and Subbasin Log Size in Channel Bank Materials..................................240 Figure 6.49 m and Bank Materials on Subbasin Log Size..............................................242 Figure 6.50 b, f, and m and Channel Asymmetry...........................................................244 Figure 6.51 Subbasin Size versus Asymmetry...............................................................246 Figure 6.52 b, f, and m and Topographic Index..............................................................248 Figure 6.53 b, f, and m with Subbasin in Log (Watershed Area in Square Miles).........262 1 CHAPTER 1 INTRODUCTION: Channel Properties and Hydraulic Geometry 1.1 Research Context River channel research published over the last fifty years, since Horton?s (1945) study of the relationships of stream length and stream number to stream order and Hack?s (1957) work on the possible relations between the channel geometry and erosion processes, has focused on understanding the interactions between river mechanics and river morphology. Such research has produced a wealth of empirical information about the hydraulic geometries of different types of river channel systems and has developed a better understanding of stream flow and sediment transport processes in channels (Richards, 1982). The ultimate goal of much of the research in the second half of the twentieth century was to develop a physically based model to explain the morphological evolution of river systems over different space and time-scales. The responses of channels to hydraulic dynamics like stream flow velocity and hydraulic geometry, channel width and depth, to changes of stream discharge are known to share certain characteristics which apply to many natural river cross-sections. These characteristics are important determinants of the shape of a channel at a cross-section and in the progressive changes in cross-section channel shapes downstream. The term hydraulic geometry was introduced by Leopold and Maddock (1953) to describe the ways in which channel characteristics of width, depth, and velocity vary with 2 stream discharge. In essence, hydraulic geometry is an empirical model expressing the changes in channel and hydraulic variables as simple power functions of discharge. The resulting hydraulic geometry equations have the form: w = aQb (1) d = cQf (2) v = kQm (3) where Q is the discharge in cubic feet per second; w is the channel width; d, the channel depth; v is the stream flow velocity; and a, c, k, b, f, and m are numerical constants. The numerical constants, a, c, and k are intercepts for width, depth, and velocity and increasing discharge relationships. Since this research is focused on the rates of change in width, depth, and velocity with changing discharge (not with time), b, f, and m constants are the subject of this research rather than a, c, and k. The intercepts are more dependent on local settings, but b, f, and m slope lines are more indicative of the trend and nature of channel interactions with various predictor variables. It follows that any spatial or temporal change in discharge must be accommodated by a suitable combination of changes in channel width and depth and stream flow velocity. Since the aim of hydraulic geometry research is to describe and predict these changes, and because the exponents b, f, and m provide quantitative descriptions of rates of change, their variations can provide insight into relevant controlling conditions and maybe even the role of geographic scale. The quantitative description of how river width, depth, and velocity vary with changing discharge can be explained by, over time, at one site (at-a-station), or along and between rivers at a comparable discharge frequency (downstream hydrographic geometry). 3 At-a-station hydraulic geometry concerns the response of flow hydraulics at a channel cross-section to varying imposed flows. For a given imposed discharge, the problem is to determine how channel width, depth, and flow velocity vary with, respond to, or accommodate the changed stream discharge volume. Channel adjustments are reflected in the numeric values of the hydraulic exponents, b, f, and m. At any one time and place these variables are interrelated by the mass continuity equation: Q = wdv (4) Throughout this dissertation, the expression of the rates of changes in b, f, and m are with respect to discharge and not to time. That is, d (w, d, v)/dQ not d (w, d, v)/dt. Examining at-a-station hydraulic geometry relationships as a function of drainage area characteristics is different than examining downstream hydraulic geometry relationships. At-a-station hydraulic geometry, which is the focus of this research, and downstream hydraulic geometry, differ in that at-a-station compares flows of vastly different frequencies while downstream hydraulic geometry analyzes variables at the same frequency of Q, even though the absolute values in cubic feet per second (cfs) units are different. The most meaningful discharge for any analysis of channel morphology is the one that forms or maintains the channel. The concept of downstream hydraulic geometry involves spatial variation in channel form and process at a constant frequency of flow. At-a-station hydraulic geometry involves temporal variation, but is spatially invariant (a fixed point on a channel). Although hydraulic geometry deals specifically with only three parameters, many other morphologic and dynamic factors are considered implicitly because they influence these three elements of continuity. Thus, the equations of hydraulic geometry provide a 4 simple summary of the complicated relations among stream channel and flow characteristics. Subbasin morphology and hydrological processes are linked through the geomorphic development of a subbasin. Current morphology often acts as a dominant control on water flow paths and current hydrological processes, resulting in changes in landscapes. Therefore, studies of the relationships between morphology and hydrological processes can be a clue to understanding geomorphological changes in a subbasin. Ferguson (1986) showed that the cross-sectional shape of a channel determines the rate of increase of stream width with depth, whereas the laws of flow resistance determine the rate of increase of mean velocity with depth. In turn, factors that affect channel geometry include vegetation (Hadley, 1961; Zimmerman et al., 1967), bank cohesiveness and sediment size (Schumm, 1960; Wolman and Brush, 1961; Knighton, 1974; Maizels, 1988), changes in suspended load (Leopold and Maddock, 1953; Wolman, 1955; Thornes, 1970), seasonal variation (Thornes, 1970), channel sinuosity (Knighton, 1975), channel roughness or flow resistance (Knighton, 1975), riffle spacing (Harvey, 1975; Prestegaard, 1983), the processes of scour and fill (Foley, 1978; Andrews, 1979), and stream size and order (Thornes, 1970; Miller and Onesti, 1977; Leopold, 1994). All of these stream parameters are candidates as covariates of the hydraulic exponents. However, many of these previous research efforts dealt with the correlations of candidate predictor variables to hydraulic geometry individually, not as collective variables to explain hydraulic geometry synergistically. The fact that hydraulic geometry, specifically at-a-station hydraulic geometry, is notoriously unpredictable in a deterministic sense, but demonstrates clear regularity and a 5 degree of predictability in an aggregate probabilistic sense is described by Leopold and Maddock (1953), Williams (1978), and Phillips (1990, 1991, 1995). Richards (1977) stated that ?geographical patterns in the at-a-station exponent set may occur, but inter- regional differentiation is obscured by the more direct influence of bed and bank material.? Physical geographers and engineers have long been aware that landscapes are comprised of nested hierarchies of geomorphic systems and sub-systems at all scales. Sugden and Hamilton (1971), and O'Neill (1988) suggested that scale may be used as a framework for analysis because "by focusing on systems functioning at particular scales one can minimize 'background noise' emitted by systems at different scales." Kennedy (1977) proposed that establishing geomorphological theory necessitates developing an understanding of the rules linking process and form at different temporal and spatial scales. An appreciation of scale effects was viewed as essential to overcome the disparities in spatial and temporal scales used in the different disciplines concerned with global environmental change by Roswall et al. (1988). The levels of certainty or rules deemed acceptable at one scale are not necessarily appropriate at another scale. This problem emphasizes the need to clearly define the spatial and temporal domains of geomorphic theory. Except for Penning-Rowsell and Townshend's (1978) study of the factors influencing stream channel slope, which found that a change in spatial scale caused a change in relative importance of different factors, and besides Leopold?s (1994) comparison of b, f, and m values in three different river sizes, and Griffith?s work (2003) on downstream hydraulic geometry and hydraulic scaling, few studies have shown the 6 effects of spatial scale on fluvial geomorphology, specifically for at-a-station hydraulic geometry. To understand a geomorphic system one does not have to consider every level of scale since, depending on the scale of the system and the objective of the investigation, certain levels will be dominant whereas others play a secondary role and can be ignored (de Boer, 1992). This statement raises questions such as: what are the significant levels of scale for the investigation of a specific problem; how can knowledge about the functioning of a geomorphic system at one scale be extrapolated to a system at a different scale; or can it be extrapolated to a system at a different scale at all. Since the late 1980?s, numerous stream channel research studies have revealed that hydrological processes and channel parameters exhibit considerable spatial variability, particularly with respect to scale, yet there have been no evaluations of spatial and temporal variability, that this author is aware of, as measured on at-a-station hydraulic geometry and considering the total spectrums of environmental parameters which may influence hydraulic geometry. 1.2 Research Objectives, Hypotheses, and Scope This study investigates the role of watershed variables including geographic scale on the at-a-station hydraulic geometry exponents in selected subbasins of the Potomac River. The main objectives are: a) to determine whether at-a-station hydraulic geometry changes significantly with watershed scale (which is manifested by the size differences of each subbasin area); and b) to determine whether watershed variables such as 7 physiography, topography, lithology, geomorphology, landuse, channel pattern, channel shape, and bed and bank materials, affect the size and shape of stream channels and, thus, the at-a-station hydraulic geometry. The working hypothesis for the first objective is that variations in each of the hydraulic geometry exponents are not random but that there are systematic changes as a function of watershed scale. In other words, there is a geographic scale impact on hydraulic geometry exponents such that b, f, and m in Amazon-size rivers are different from those of smaller rivers. This might explain why hydraulic geometry has different b, f, and m values where all other basin variables are the same or similar. Even though width is the dominant variable in downstream hydraulic geometry and that there are interrelationships between at-a-station and downstream hydraulic geometry, the expected changes in the at-a-station hydraulic geometry as a function of scale is that channel width does not change much, depth increases rapidly, and flow velocity change rates will also increase. A set of working hypotheses for the second objective is that particular watershed variables individually influence channel morphology by affecting stream discharge and sediment characteristics such as the size and amount of sediments delivered to the channels. Most researchers have assumed that watershed variables influence channel morphology by affecting the discharge and the sediment characteristics such as size and amount delivered to the channels. However, most previous work has not examined direct links between channel morphology and watershed variables. Given this, I am not postulating any different level of importance of the particular watershed variables to be used in this research. 8 In brief, previous research and literature that are relevant to this investigation, including hydraulic geometry in light of hydrogeomorphological and hydraulic engineering perspectives as well as in theoretical and empirical approaches are described in Chapter 2. The expected correlations for b, f, and m with watershed physiography and lithology are that in the most easily eroded bank and bed physiography regions, increased rate of changes in width and depth of channels are expected resulting in higher values of b and f. The expected correlations for b, f, and m with watershed landuse are that in agricultural areas the rate of changes in width and depth will increase higher and rate of change in flow velocity will be lower. It is also expected that b and f changes will be greater for agricultural areas than that of forest areas. The rationale is that, in agricultural areas, increased water discharge flows through plowed agricultural fields carrying more suspended and bed load material and has greater erosional impacts on channel beds and banks. On the other hand, in forest landuse areas, increased volumes of discharge occur after ground water saturation; therefore, the discharge would be more or less free of sedimentation compared to agricultural watersheds. That is, it is expected that watersheds in agricultural areas are more susceptible to erosion than those in forest areas. The expected changes in m would be the reverse because the increased bed or suspended load in stream discharge will reduce the flow velocity. The expected correlations for b, f, and m with channel pattern are that in straight channels, b values are lower than those of f; in meandering channels, relatively higher rates of changes in width than those of depth. The rationale for these expectations is that 9 straight channels are typically symmetrical and meandering channels are normally asymmetric in cross-section at the bends. Asymmetry provides for a relatively greater rate of increase in width, but f will probably still be greater than b. Braided channels are typically broad and shallow because of the unstable nature of their banks and the width- depth ratio commonly increases with increasing discharge. Based on Rhodes? research (1977), the expected correlations between channel shape, which is categorized by visual inspection of each subbasin cross-section profile, and b, f, and m are rectangular channels which will have the lowest b values, parabolic channels which will have b = f/2, triangular channels which will have b = f, and braided channels which will have b > f. The expected correlations for channel asymmetry and b, f, and m are that a positive, stronger relationship between b and asymmetry is expected compared to the correlationships of f and asymmetry or m and asymmetry. That is because when the thalweg of channel moves from the center of the channel, the width of the channel will inevitably change. And, since stream power is consumed in the widening of the channel, the m value must be reduced. Based on Bathurst?s research (1993), the expected correlations between bed material and b, f, and m are that the flow velocity increases as the bed material size increases such that, for sand-bed channels, m < 0.40; for gravel-bed channels, m = 0.35 to 0.45; and for boulder-bed channels, m = 0.45 to 0.55. The expected correlations for bank material and b, f, and m are such that the value of the change in b will become larger in sequence from the least to the most resistant bank material: sand, gravel, cobble, and rock. In easily eroded channel banks, 10 with the eroded bank material in the discharge, the rate of depth and flow velocity changes would be much less than b values. So that in easily eroded banks, such as sand banks, a relatively wide shallow channel develops, while in cemented bank materials (silt), the channel becomes deeper and narrower. As streams begin eroding their banks, their channel cross-sections become wide and shallow with consequent reductions of flow velocities. Regarding topography (ln area/tan? indices), expected correlations for topography and b, f, and m are such that b will be higher in areas of higher topographic index values which are reflective of wide floodplains and areas of increased runoff due to soil saturation. Steep valley areas reflect low topographic index values. In area of low topographic index values, the rates of change in depth and flow velocity will be less than the rate of change in width. In order to undertake sufficient analysis for addressing the research questions, the thesis will, as a practical matter: (1) investigate rate of changes in width (b), rate of changes in depth (f) of channel, and rate of changes in velocity of channel flow (m) to changing discharge volumes; (2) determine correlations between scale and b, f, and m; (3) determine correlations between individual watershed controlling variables with b, f, and m and with the scale variable; and (4) investigate the total effects and role of watershed controls on at-a-station hydraulic geometry. 11 Figure 1.1 Study Area - Potomac River Basin Source: US Geological Survey Water Resources Investigations Report 96-4034 1.3 Approaches for the Empirical Investigation In order to accomplish the investigation, the Potomac River Basin was selected as the study area due to its ready access by the author, the variability of spatial watershed settings such as wide ranges of physiography, geology, lithology, etc., and the plethora of information on this area such as high resolution digital elevation data and hydrographic survey data. 12 The study area consists of 43 Potomac River subbasins sampled over a 10-year period, 1985-1995 as shown in Figure 1.1. The primary data are USGS Water Resource data and digital elevation model (DEM) data supplemented by numerous field surveys by the author. Using primary data, many derived measurements data such as drainage delineation, nested subbasins, topographic index, hydraulic geometry, channel asymmetry, channel pattern, channel shape were created by the author for this hydraulic geometry investigation and analyses. The study area is discussed in Chapter 3 and data (primary and derived) are discussed in detail in the Chapters 4 and 5. The first part of the research was to identify Potomac River Basin stream channels most representative of the spectrum of subbasin sizes, the various Potomac River Basin physiographic settings, and the channel patterns and shapes as well as a hierarchy of channels feeding to bigger streams. The channels selected included those with a high number of hydrographic survey measurements at the same location along the channels in the Potomac River Basin. After study stream channels were identified, required primary data types and sources were identified and data were obtained from the USGS Water Resource Division Offices and Virginia Department of Environmental Quality in Baltimore and in La Vale, Maryland, and in Charlottesville and Richmond, Virginia. This data source is described in detail in Chapter 3. In some cases, the author accompanied USGS hydrographers on stream measurement surveys to acquire first hand familiarity with data collection methods and the survey sites. In addition, the author visited study sites and made numerous field observations to validate and verify channel bank and bed material information as well as environmental setting information and other reported channel characteristics. 13 For the preparation of data to perform the investigation and analyses, the hand- written survey measurement data, of USGS hydrographers? such as width, depth, and velocity data across a channel in segments, were digitized. The hydrographic surveys chosen for the study have, on average, width, depth, and velocity data in 30 segments across each channel cross-section per survey. Each of the 43 channels had an average of 30 surveys over the ten-year study period. The digitized data were then used to calculate channel pattern and shape and draw channel cross-section profiles. In addition, with the USGS?s Maryland Index to Topographic and other Map Coverage, and that of Virginia, all topographic maps at 1:24,000 scale for the 43 subbasin areas and the DEMs (7 ? minutes by 7 ? minutes area by 30 meter post spacing data) which fall within these subbasins were identified and acquired from the USGS. The DEMs were merged using ARC/INFO? and IMAGINE? software for the preliminary drainage delineation and topographic index calculation. Due to the large scale of the topographic maps, not all needed DEMs were identified at first; more DEMs were identified and merged during the delineation of the subbasin boundaries. Preparation and processing of DEMs for subbasin boundary delineation are described in detail in Chapter 4. The delineation of drainage boundaries (subbasin boundaries) and calculation of subbasin sizes, hydraulic geometry, and topographic index, which is the key component of the topography-based TOPMODEL and defined as (a / tan ?) where a is the area draining through a point from upslope and tan ? is the local slope angle, mid-point of the cross-section of channel width, width and depth ratio, and channel asymmetry and visualization of hydraulic geometry on an equilateral triangle (ternary diagram) are described in detail in Chapters 4 and 5. 14 To explain the hydraulic geometry in terms of hydraulic processes of b, f, and m, the temporal changes of the study area streamflow were used and these historical stream discharge volume data in histogram form came from downloading from the USGS?s home page, Waterdata site, for each of the 43 gage stations for the study duration (1985- 1995). Hardcopy descriptions of the 43 gage stations were also used to characterize each subbasin. The environmental settings, such as lithology and physiography, and needed primary and derived data are described in detail in Chapters 3 and 4. To address the objectives, prior to the analyses, first the subbasin boundaries were delineated and stream flow directions identified. Also, the sizes of the subbasins were calculated and nested subbasins were identified for the relationships among these 43 subbasins. Secondly, hydraulic geometry exponents for width, depth, and velocity change rates with changing discharge volume were calculated using log regression. Thirdly, to amplify the nature of the width, depth, and velocity exponents, these values were plotted on a ternary diagram to present b, f, and m graphically and the location of the study area hydraulic geometry on the ternary diagram was divided into regions based on hydraulic principles and hydraulic geometry researchers? empirical data. Fifthly, to characterize the subbasins, channel cross-sections by profiles, mid-points of the channel widths, and channel asymmetry were generated. A series of analyses intended to characterize b, f, and m (predicted variables), and show their relationships to predictor variables were conducted. The predictor variables for b, f, and m are continuous and categorical explanatory variables. Continuous explanatory variables used for the analyses are subbasin size (scale), channel asymmetry, topographic indices such as minimum, topographic index maximum, topographic index 15 mean, and topographic index standard deviation, and various width to depth ratios such as the ratio of depth values associated with the widest width, depth value associated with the narrowest width. Categorical predictor variables are physiographical characteristics of the 43 individual subbasin areas, lithology of the 43 individual subbasin area, land use, channel pattern, channel shape, bank material, and bed material. The following statistical analyses were performed (see Chapter 6) to investigate relationships between b, f, and m and the selected predictor variables: (1) Descriptive statistics for b, f, and m and analysis of the b, f, and m values; (2) Correlation between the geographical scale variable and b, f, and m, including hypotheses testing; (3) Relationships between b, f, and m and the predictor variables such as phys1, litho1, landuse1, channel bed material, channel bank material, channel asymmetry, topographic index, channel pattern, and channel shape individually; (4) Correlation between b, f, and m and each predictor variable in the scale context and the probability that correlations are statistically different from zero; (5) Relationships between b, f, and m and predictor variables with interaction between geographical scale and predictor variables; and (6) Linear regression analysis for which each predicted variable (b, f, and m) is regressed onto all predictor variables separately using stepwise regression. Characteristics of the hydraulic geometry of each of the 43 subbasins and a subset of 23 well and moderately constrained subbasins are discussed in terms of hydraulic principles and empirical hydraulic geometry values. In conjunction with these statistical analyses, the roles of each of the predictor variables were investigated without and with 16 scale context. Statistical analyses established the relationships between the rate of changes in channel width, channel depth, and stream flow velocity and predictor variables in the scale context. Also, statistical analyses identified the best predictor variables for estimation of the b, f, and m values. 17 CHAPTER 2 PREVIOUS STUDIES: How Important is Geographical Scale? 2.1 Introduction This chapter provides a brief history of how at-a-station hydraulic geometry became one of the main research themes for understanding channel form and stream processes and describes previous research on at-a-station hydraulic geometry in general. These previous studies are cited to show: pioneering models developed to calculate the rate of changes in width (b), depth (f), and velocity (m) in accordance with the changing discharge; how to present and analyze the exponents of b, f, and m; what parameters were used to explain b, f, and m; as well as what other hydraulic processes might explain b, f, and m values in this study of Potomac River stream channels. At the end of each section, the author?s approach and needed areas for further investigation are discussed. In part because Roswall et al., (1988) stated that an appreciation of scale effect is essential to overcome the disparities in spatial and temporal scales used in different disciplines, the author initially considered investigating the geographical scale effects as a major predictor variable in hydraulic geometry. Even though stream hydraulic geometry, which deals with hydraulic geometry changes as a stream flows throughout the system, traversing many scales, can better investigate scale effects than at-a-station hydraulic geometry, the author hypothesized that there is a systematic geographical scale effect on at-a-station hydraulic geometry exponents. Hence, in this study, each predictor variable is investigated as a singularity and with the scale variable. This way the impacts of predictor variables of notoriously unpredictable at-a-station hydraulic geometry (Phillips, 18 1995), and Williams (1978)) are investigated with equal weight and with a geographical scale factor. 2.2 Automated Drainage Delineation Watershed hydrological studies often begin with the generation and analysis of basin hydrogeomorphological features such as basin boundaries, stream networks, and slope and aspect maps. These hydrogeomorphological features are traditionally derived from topographic maps. Interpretation of drainage boundaries involves physically drawing a dividing line between two or more drainage basins on a topographic map. Basically, this means determining which direction the water flows from the highlands to the lowlands. Where the contour intervals are close together, which indicates appreciable relief, determining the divide is fairly easy. However, when the slope gradients are small as in a flatland area, interpretation becomes more difficult. With the advent of modern computing techniques and data known as digital elevation models (DEMs), it has become possible to develop tools for the extraction and manipulation of components of river network systems (O?Callaghan and Mark, 1984; Band, 1986; Jenson and Domingue, 1988). Snell and Sivapalan (1994) have extended the work on the extraction of the catchment width function, to an automated extraction of the catchment area-distance function, the catchment area convergent to a specific flow distance from the outlet. There currently exists a suite of algorithms that automatically extracts basin hydrogeomorphology from DEMs (O?Callaghan and Mark, 1984; Band, 1986; Jenson and Domingue, 1988; Tarboton et al., 1991; Martz and Garbrecht, 1993; Tarboton, 1997). 19 Many researchers (Mason, 2000; Azagra, 1999; Wolock, 1996) demonstrated that the accuracy and detail of the hydrologic information which is automatically extracted from a DEM is directly related to the quality and resolution of the DEM itself. Since DEM data often contain subtle and striking errors like spikes, these errors can bias subbasin boundary and channel flow direction identification. A depression filling procedure developed by O?Callaghan and Mark (1984) finds watersheds for cells that have no neighbors lower in elevation, identifies cells lower in elevation than the lowest boundary elevation for their watershed, and encodes these cells as being flat for use in the basin delineation process. A flat area is assigned in its entirety to the first watershed that touches it. This procedure does not include logic for finding looping depressions such as might occur when many depressions are located on a surface that is relatively flat. Jenson and Domingue (1988) developed an algorithm which follows depressions and fills the depressions first and treats these areas as flat areas where water can be routed. Their algorithm compares the elevations of its eight neighboring cells, and the steepest down-slope direction is assigned to each cell in the grid. Directions are also assigned to grid cells in flat areas to provide a continuous path from all cells in the watershed to the outlet of the watershed. This is accomplished by finding the neighboring cell with the steepest downslope direction and assigning that direction to the flat area cell. This algorithm is incorporated into ARC/INFO? as part of that software?s Surface Hydrologic Model. Under the guidance of Maidment (1994), ARC/INFO??s Surface hydrologic model is widely used by the Center for Research in Water Resources at the University of Texas at Austin and where more than a dozen research papers are produced a year. In these research papers, there are many discussions 20 of the resolution of the DEM data being used for flow direction identification and flow accumulation. The author used the ARC/INFO??s Surface hydrographic analysis package which is commonly used by most of the hydrographic research workers, and the finest resolution DEM data over the Potomac River Basin area, for the delineation of subbasin boundaries and flow accumulation calculations. Topography is recognized as an important factor in determining the stream flow response of drainage (Kirkby and Chorley, 1967; Dunne et al., 1975; Beven and Wood, 1983). A steeper slope results in greater energy - therefore the shape of a surface determines how water will flow across it. The author considered the topographic index as one of many predictor variables for this hydraulic geometry investigation. Since the inception of topographic index concepts by Beven and Kirkby (1979) and by O'Loughlin (1981), the topographic index has become an important hydrologic modeling component because it reflects the spatial distribution of runoff generation processes (Zhang and Montgomery, 1994). The topographic index facilitates many hydrological simulations, most notably TOPMODEL (Beven and Kirkby, 1979; O'Loughlin, 1981; Moore et al., 1991; Wolock, 1993). The topographic index, which represents ?saturation? potential, is a function of the upstream contributing area and the slope of the landscape. In areas of no slope, the topographic index is obtained by substituting a slope of 0.001. This value is smaller than the smallest slope obtainable from a 30 m data set with a 1 m vertical resolution. The topographic index is calculated as ln (a/tan ?) which is the essence of the upslope area, as divided by surface slope gradient (tan ?). This approach was originally derived independently by Beven and Kirkby (1979) and by O'Loughlin (1981). 21 As in research on stream flow direction determination, elevation data resolution is continuously an issue in topographic index calculations. There has been considerable research on the impact of DEM data resolution for the calculation of the topographic index (Wolock, 1997; Thompson et al., 2001). It has been concluded that the finer horizontal and vertical resolution DEM data result in the more accurate topographic index values (Wolock et al., 2000; Wolock, 1997; Moore et al., 1991; Wolock et. al., 1995). Also, for the calculation of the topographic index in TOPMODEL, single flow direction versus multiple flow direction algorithms are being debated (Wolock, 1995 and Thompson et al., 2001). The single flow direction algorithm is based on Jenson and Domingue (1988); the multiple flow direction algorithm uses one set of computations for sloping areas in the watershed (grid cells with one or more downslope neighboring cells) and another set of computations for flat areas (Wolock et al., 1995). The comparison of the topographic indices computed from a single flow direction versus multiple flow direction algorithm revealed that the multiple flow direction algorithm results in a higher mean value ln (a / tan ?). That means that probably the multiple flow direction algorithm provides higher mean values of the ln (a), since the calculation of the slope of the local angle (tan ?) will be the same in both algorithms (Wolock et al., 2001). However, many hydrological model users assumed that the areas within the basin possessing the same value of the topographic index would behave, hydrologically, the same, regardless of their location on the landscape. After many years of use of topographic indices and hydrological models, many efforts over the last several years have focused more on alterations of the form of the topographic index and on 22 improvements in the methodology for computing the indices (Hornberger and Boyer, 1995). Also, the assumption that the areas within the basin possessing the same topographic index value would behave hydrologically the same, regardless of their location on the landscape is debatable. The author believes that there are many topographic and spatial variabilities which could be a predictor variable for b, f, and m. As an example, Dubayah (1992) linked a radiation transfer algorithm with both satellite reflectance and digital terrain data to model the spatial variability in net solar radiation for FIFE (First ISLSCP Field Experiment, where ISLSCP is the acronym for International Satellite Land Surface Climatology Project). In this prairie environment, Dubayah (1992) concluded that topographic variability was the dominant factor affecting the variability of net incoming radiation. Bl?schl et al. (1991) consider the topographic effects on radiation distribution and use this information to compute spatio-temporal patterns of snow accumulation and melt in an alpine basin. 2.3 At-A-Station Hydraulic Geometry Leopold (1994) stated that one of the surprising characteristics of rivers is that each cross-section, on any river, has been shaped and dimensioned over time to accept a range of flows. He also stated that there is a consistency from one river to another, and from one cross-section to another, in the way in which the hydraulic parameters change from low flow to high flow. It is easy to visualize that water depth increases as discharge in a river increases. But what happens to water velocity or river width is not so intuitively obvious. Hydraulic 23 geometry is a way of describing these changes in quantitative terms. At a river cross- section, as discharge changes, the following generalities usually hold (Leopold, 1994): ? Both depth and velocity increase substantially with increasing discharge, and at about the same rate; ? Width increases slightly with discharge; ? Channel flow resistance or hydraulic roughness, which can be defined as Manning?s roughness equation that is: v = 1.486 d2/3 s1/2/n, where s is water surface slope, n is Manning roughness coefficient, decreases slightly with increasing discharge; and ? Suspended load increases rapidly with discharge, and at a much higher rate than any other parameter. On a given day, for example, a large flood with high w, d, and v values may be occurring in an upstream reach while flow conditions far downstream are normal. A comparison of the hydraulic variables in these two widely divergent frequencies of flow would be misleading. Obviously the frequency of the discharge must be considered for any observations to be valid. Examining at-a-station hydraulic geometry relationships as a function of drainage area characteristics is different than examining downstream hydraulic geometry relationships. At-a-station hydraulic geometry, which is the focus of this research, and downstream hydraulic geometry, differ in that at-a-station compares flows of vastly different frequencies while downstream hydraulic geometry analyzes variables at the same frequency of Q, even though the absolute values in cubic feet per second (cfs) units are different. The most meaningful discharge for any analysis of channel morphology is the one that forms or maintains the channel. The concept of downstream hydraulic 24 geometry involves spatial variation in channel form and process at a constant frequency of flow. At-a-station hydraulic geometry involves temporal variation, but is spatially invariant (a fixed point on a channel). Engineers and hydrologists frequently are required to estimate flow characteristics at un-gaged sites. Conventional techniques have used relations between flow characteristics and physical characteristics of drainage basins, such as drainage area, to transfer information to the un-gaged sites. Understanding the relations between flow characteristics and stream-channel size offer a promising alternative (Wahl, 1984). Methods of quantifying the interrelation between flow characteristics of rivers and channel size have developed only in recent years. The regime concept, as originated by Kennedy (1895) and Lindley (1919) and discussed by Leliavsky (1955), developed empirical relationships for the hydraulic properties of stable canals in India and Pakistan. However, this method was not extended to natural rivers in the United States until half a century later. A channel is considered to be in regime if it can accommodate its flow for one or more years without a net change in its hydraulic characteristics (Blench, 1969). Within that period, scour or deposition may occur, in either the lateral or vertical direction, as long as they are transient phenomena. The morphology of regime canals has been the subject of many investigations since Kennedy stated his empirical equation of non- scouring velocity for canals of the Punjab in 1895 (Leliavsky, 1955). The basic principle generally was not applied to rivers in the United States, however, until Leopold and Maddock (1953) reported their analysis of the relationships between hydraulic properties of channel cross-section and river discharge. They theorized that the hydraulic geometry 25 of river channels in approximate equilibrium could be expressed as exponential functions of discharge such that W = aQb, (1) D = cQf, (2) V = kQm, (3) where W is width, D is mean depth, V is velocity, Q is discharge, and a, c, k, b, f, and m are numerical constants. Since Leopold and Maddock, there has been many empirical studies on hydraulic geometry, notably studies by Wolman (1955), and Hack (1957). Investigations by Leopold and Maddock (1953) determined that the sum of the exponents would be unit one due to the continuity equation for the hydraulic exponents holds for changes in discharge at-a-station and proposed the following equations. Q = area * velocity = wd * v (4) which yields Q = aQb * cQf * kQm = ackQb+f+m (5) Therefore, b + f + m = 1.0, and a * c * k = 1.0. (6) Because the product of width, depth, and velocity must be discharge, the three equations are related to one another and the summation of the slopes has to be one. 26 Exponents of the first-order equations represent the rates of change of the dependent variables with changing discharge. Thus, exponents of the power functions are the slopes of the first-order equations. The slopes represent the rates of change of width, depth, and velocity with changing discharge. Comparison of the numerical values of the exponents indicates which variables show greater rates of change at which scale ranges. As stated above, Leopold and Maddock (1953) were not the first to apply the regime concept to rivers, although their analysis was one of the first to gain wide acceptance. In 1930, Lacey extended his earlier empirical equations for Punjab canals by including limited data for rivers from the United States, Europe, and the Punjab (Mahmood and Shen, 1971); however, he grouped river data by discharge and used averages. Pettis (1937) independently developed similar regime equations based on natural streams in the Miami River basin of Ohio. Pettis? relations were intended for use in river channelization; therefore, his discharge was a flood discharge, apparently near bankfull (Pettis, 1937). Leopold and Maddock?s (1953) paper had an immediate impact on geomorphology, primarily because it and Horton?s (1945) earlier paper on Erosional Development of Streams and Their Drainage Basins demonstrated quantitatively that there is order in landform development, and a series of papers demonstrating similar relations in other areas followed its publication (Brush, 1961; Wolman, 1955; Leopold and Miller, 1956; Nixon, 1959; Miller, 1958). Five decades after Leopold and Maddock (1953) first applied the technique of setting width, depth, and velocity equal to a power function of discharge, there is still 27 debate over the theoretical value of hydraulic geometry exponents. Leopold and Maddock (1953) claimed that the exponents of hydraulic geometry tend to be similar for rivers of different sizes in widely different physiographic settings. They stressed that the similarity of these exponents may be due to the fact that they are characteristic of a steady-state system comprised of force and proportional resistance. However, Mackin (1963) pointed out that the velocity exponents can vary markedly within a single drainage system, and Park (1977) found systematic differences in the width and velocity exponents between humid temperate and tropical streams, and between perennial and ephemeral semi-arid streams. Yang et al., (1981) and Phillips and Harlin (1984) indicated that b, f, and m exponents reflect a tendency for minimum work in the stream channel, and so are similar for all rivers. However, the literature reveals that some researchers take a less restrictive view of controls on hydraulic geometry, and have emphasized variations in exponents as opposed to mean values (Knighton, 1974). While Chang and Toebes (1970) state that rivers over varying environments behave in similar fashion, Park (1977) argues that certain hydraulic exponents could be characteristic of different climatic and environmental regimes. Obviously, the more recent studies are not in good accord with an earlier model which forces discharge into the role of an independent variable. Observed at-a-station hydraulic geometry relationships are quite variable. Exponents for at-a-station hydraulic geometry, reported in the literature, range at least from 0.0 to 0.84 for the width exponent, b; 0.01 to 0.84 for the depth exponent, f; and 0.03 to 0.99 for the velocity exponent, m using 587 sets of exponents (Rhodes, 1978). Park (1977) calculated values for 139 at-a-station data sets for streams in proglacial, 28 humid temperate, semiarid, and tropical regions. Park found that exponent values for width fell within the class 0.0 to 0.1, for depth within the class 0.3 to 0.4, and for velocity within the class 0.4 to 0.5. Knighton (1975) reported average exponents for 206 cross sections in the United States as 0.16 for width, 0.43 for depth, and 0.42 for velocity. The most commonly cited average exponent values are those computed by Leopold and Maddock (1953) for " a large variety of rivers." The values are b = 0.26, f = 0.40, and m = 0.34. There are a wide range of mean b, f, and m values. Categorizing channels based on scale, similar geomorphic and environmental settings would reduce the range of mean b, f, and m values. As data accumulated on the hydrology and morphology of rivers, it was inevitable that the ?regime? approach would be applied to rivers, and this was done by Leopold and Maddock (Wahl, 1984). Leopold (1994) himself stated that due to the variances introduced to a variety of predictor variables, any combination of the b, f, and m values can be possible. 2.4 b, f, and m Computation Basically, there are three methods of computing rate of changes in width, depth, and velocity with regard to changing discharge volume. Statistical and analytical modeling of channel form remains a major theme of current research on channel morphology (Rhoads, 1994). As seen in the Leopold and Maddock?s power law, log transformations of variables allow equations (1) - (3) to be expressed as linear functions, and hence the equations define a log-linear model (LLM). Substitution of equations (1) - (3) into the 29 mass continuity equation, (4), yields equations (5) and (6). The concept of hydraulic geometry has been used extensively to describe the behavior of natural river flows in a variety of physiographic and climatic regions (Leopold and Maddock, 1953; Wolman, 1955; Brush, 1961; Fahnestock, 1963; Carlston, 1969; Knighton, 1975; Park, 1977; Rhodes, 1977; Andrews, 1984). Some of these studies have used values of the exponents in equations (1) - (3) to compare the characteristics of different river sections and basins. Several authors have questioned the validity of the LLM in general to adequately represent hydraulic geometry data for discharges less than channel capacity such as bankfull. Wolman (1955), Lewis (1966), Thornes (1970), Richards (1973), Williams (1978), and Eschner (1983) have noted breaks in the slopes of the hydraulic geometry relations at a number of sites (Bates, 1990). The origin of the breaks was discussed by Ferguson (1986), and Singh and Broeren, et al. (1987). Ferguson (1986) explained these breaks by cross-sectional shapes and frictional characteristics of natural river channel, and Singh, Broeren, et al. (1987) by water reference level. Ferguson (1986) stressed that the LLM will be valid only in situations where the width-depth and mean velocity-depth relationships conform to a power law (Bates, 1990). Thornes (1970), Williams (1978), and Eschner (1983) have fitted piecewise linear relations to log-transformed hydraulic geometry data. Bates (1990) stated that ?Although their work has shown that these relations can give improved fits to the data, there does not appear to be any evidence of a formal piecewise linear model of hydraulic geometry in the relevant literature?. Richards (1973) was strongly against the use of piecewise linear relations on the grounds that the fitting of polynomials is a more appropriate approach. Richards (1973) 30 advocated the fitting of quadratic relations to log transformed data (Bates, 1990). Richard?s log quadratic model (LQM) is defined by Log w = b1 + b2 log Q + b3 (log Q)2 (7) Log d = f1 + f2 log Q + f3 (log Q)2 (8) Log v = m1 + m2 log Q + m3 (log Q)2 (9) Substitution of equations (7) - (9) into equation (4), and differentiation of both sides of the resulting terms with respect to log Q produces: b2 + f2 + m2 =1 (10) b3 + f3 + m3 =0 (11) and b1 + f1 + m1 =0 (12) which in turn becomes to the continuity equation. Bates (1990) proposed a formal log piecewise linear model (LPM) for the computation of at-a-station hydraulic geometry to model non-linear trends, a more sophisticated method of Thornes' (1970) technique of fitting separate regression lines about a 'break point' in the hydraulic geometry data. Bates? (1990) LPM is for at-a- station hydraulic geometry of discharges below bankfull. Bates (1990) compared the results of his LPM computed b, f, and m on 22 stations on various rivers and creeks in Australia with Leopold and Maddock?s (1953) LLM and Richards?s (1973) LQM results and concluded that LPM provided objective means for identifying the breaks in the slopes of hydraulic geometry relations for a particular station. These breaks reflect changes in the physical constraints that control at-a-station variations in width, depth, and velocity with rising discharge. Bates (1990) stated that by using fitted straight lines (LLM) or polynomials (LQM) to log-transformed data, important pieces of information which are manifested by 31 break points are lost. He also concluded that further work is required to fully establish the physical bases of the LPM and application of the model to natural channel hydraulic geometry is necessary to determine unbiased parameter estimates and draw valid inferences on the regression models. The author used LLM which has been employed by a majority of at-a-station hydraulic geometry researchers to compute b, f, and m where the width-depth and mean velocity-depth relationships conform to the power law and most of the 43 gage stations? b + f + m = l. LLM method is employed by this author because there are rich sources of hydraulic geometry data calculated by LLM methods. Comparing b, f, and m calculated by the same method would reduce variability introduced by other methods. The author plans to identify any breaks in slope and to explain these breaks by cross-sectional shapes and frictional characteristics of natural river channels and by a water reference level. The author used LQM for analyses of scale, and b, f, and m for log quadratic relationships. 2.5 Representation and Analyses of b, f, and m The b, f, and m values from equations of hydraulic geometry have been used to describe and compare stream channels formed under the influence of many environments. However, the comparisons have been based almost solely on the numerical similarity of the exponents. Even though the numerical values of slopes and intercepts may not provide any visual picture of a river basin, comparison of the values of these factors among rivers has useful aspects. Park (1977) and Rhodes (1977, 1978) independently proposed a far more extensive analysis of b, f, and m values. Both represented b, f, and m values on an 32 equilateral triangle using a triangular coordinate system via a ternary diagram. The concept of the ternary diagram has been extensively used for soil classification by the American Association of State Highway and Transportation Officials and classification of Maryland coastal sediment sample classification (Shepard, 1954). However, Rhodes (1977) and Park (1977) were the first to portray b, f, and m exponential values on ternary diagram. Rhodes (1977) and Park (1977) both used ternary diagram and classification systems to examine the simultaneous variations of hydraulic geometry exponents. They attempted to interpret at-a-station hydraulic geometry to show certain relationships between hydraulic geometry and channel pattern and group the data according to these relationships. However, Rhodes?s and Park?s analyses differed on the criterion used to classify the b, f, and m data. Park (1977) chose climatic differences such as representative values for regions of proglacial, humid temperate, semi-arid, and tropical environments as classification of b, f, and m data. Park (1977) concluded that his climatic grouping offered ?relatively little explanation? of the variability of the data and concluded that; ?it would seem logical to consider more local factors ? to explain the variability of b, f, and m. Rhodes (1977), Park (1977), and Ferguson (1986) collated available b, f, and m values from various research results such as Leopold and Maddock (1953), Wolman (1955), Leopold and Miller (1956), Leopold and Wolman (1957), Fahnestock (1963) and many more, and found there are huge scatters that cannot be explained by differences in the numerical values of b, f, and m. Histograms do not show the simultaneous values of these three exponents, which are of more interest because these three exponents are 33 interrelated. Consequently ternary diagrams are preferred because almost any combination of exponents can be recorded in the triangle. The implicit assumption in this type of analysis is that the channels, as characterized by a particular set of b, f, and m values, differ only in their rate of response to changing discharge. Rhodes? (1978) investigation challenged the assumption by demonstrating that channel responses may differ not only in degree but also in direction on a ternary diagram. This demonstration is based upon a graphical representation of at- a-station hydraulic geometry data and division of the ternary diagram based on empirical and hydrological principles. Park (1977) used recognition of different channel types which are delineated by the divisions of the diagram. Rhodes (1977) attempted to analyze at-a-station hydraulic geometry exponents on the basis of local factors such as (1) width/depth ratio for stability of the channel ( b = f), (2) competency of the channel (f = m), (3) Froude number (m = f/2), (4) velocity/cross- sectional area ratio (m = b + f), and (5) slope/roughness ratio (m = 2/3f). Rhodes classified the channels according to their dynamic and morphological responses to changing discharge. Using these five criteria, a ternary diagram is divided into ten regions. The first criterion of dividing the ternary diagram was the line by b = f which divides the triangle by two regions where width exponents are bigger or smaller than depth exponents. The second criterion was checking the competency of the channel by comparing the depth and velocity exponents; the third criterion was stream flow condition by Froude numbers; the fourth, channel stability and flow resistance, and the last criterion was roughness in terms of depth and velocity exponents. The author followed Rhodes? criteria to categorize b, f, and m values of the study area and derived 34 categorically similar regions based on the five criteria. The derivations of relationships within the b, f, and m equation for these five criteria are discussed in detail in Chapter 5. This additional information is used to understand the hydraulic geometry relationships and environmental setting for specific hydraulic geometry groups. 2.6 b, f, and m and Disparity Hydraulic geometry is a versatile analytic technique, which can be used to describe and, in part, explain the interactions of measurable parameters in natural streams as discharge changes or as channel form is modified. However, the degree of variation in the hydraulic relations casts doubt on the validity of defining a mean at-a-station hydraulic geometry even for a single stream system or regional group of streams on the basis of existing empirical evidence. Knighton (1975) reported that the variation is apparently not random, but systematically related to channel pattern, straight reaches being distinguishable from meander and braided reaches in terms of the rates of change of width, velocity and resistance, and slope. Variability may be reduced by a systematic selection of stations of similar characteristics of streams flowing through either cohesive or non-cohesive materials and a mean geometry defined for each category. There are many research reports which try to explain the disparity or wide range of the exponents values of at-a-station hydraulic geometry computed from observed width, depth, and velocity with the respect to the changing discharge volume. Even at-a- station interpretations of the hydraulic intercepts and exponents will be misleading for extremely low and high flows, that is, where the deviation from mean discharge is extreme. Rhodes (1978) noted that some researchers have found that exponent values for 35 high-flow conditions can be vastly different than that for low-flows. Richards (1973) and Park (1977) have suggested that simple power functions may not be the best way of describing hydraulic geometry. This study investigates high and low flows in the subbasins where each b, f, and m value shows a wide range. Eschner (1982) also found that on many of the hydraulic geometry plots breaks in the slopes of the width, depth, and velocity discharge relations are evident at a certain discharge in his study of the Platte River. He concluded that the width- and depth- discharge relations at breaks do not appear to follow a single power function model. Power functions fitted separately to the low and high ranges of discharge generally yielded large proportionate reductions in the variances of the dependent variables relative to the variances of the variables alone. This complex hydraulic geometry is not peculiar to the Platte River (Eschner, 1982). Wolman (1955) stated of Brandywine Creek in Pennsylvania: "There is a suggestion in some of the data ... that the depth-discharge and velocity-discharge curves may actually plot as curved rather than straight lines on log log paper. Such a relationship of the at-a-station curves is not uncommon." Richards (1973) noted that non-linear changes of depth and velocity with discharge may result from non-linear changes of roughness with discharge. Richards (1976) also proposed that channel cross- section shape can produce breaks or dissimilarities in the width-discharge relationship. The fact that hydraulic geometry, specifically at-a-station hydraulic geometry, is notoriously unpredictable in a deterministic sense, but demonstrates clear regularity and a degree of predictability in an aggregate probabilistic sense is described by Leopold and Maddock (1953), Williams (1978) and Phillips (1990, 1991, 1995). Richard (1977) 36 stated that geographical patterns in the at-a-station exponent set may occur, but inter- regional differentiation is obscured by the more direct influence of bed and bank material. As the scale of study diminishes, statements of equilibrium can only address ?at- a-station? geometry, as each cross-section of the channel is adjusting to the discharge of water and sediment in a unique subsurface and surface environment (Abrahams, 1984). As the sample expands in space and/or time, the interpretation of hydraulic exponents becomes less and less meaningful. This does not mean that there is no value in establishing regional hydraulic exponents; however, a good deal of caution must be exercised before any assumption of transferability is made. Further analyses will almost always be demanded, as comparisons of coefficients and exponents are made within, or between, regions defined by physical parameters. As stated earlier, using local environmental variability of the subbasin and channel characteristics, the author investigated the nature of b, f, and m values and the relationships among each other; the impact of the predictor variables on the b, f, and m; and the significance of their impact on the b, f, and m. 2.7 Heterogeneity of Environmental Factors Leopold and Maddock (1953) claimed that the exponents of b, f, and m and the discharge relations tend to be similar for rivers of different sizes in widely different physiographic settings. However, Mackin (1963) pointed out that the velocity exponent can vary markedly within a single drainage system, and Park (1977) found systematic difference in the width and velocity exponents between humid temperate and tropical streams, and between perennial and ephemeral semi-arid streams. 37 While furthering the state of knowledge, most of the preceding studies, including the 1953 work of Leopold and Maddock, were of limited practical value because the hydraulic-geometry variables used were those of specific discharges and could not be identified with recognizable channel features. Lane?s paper (1935) demonstrates clearly how empirical equations as well as the qualitative conclusions reached in one part of the world do not necessarily apply elsewhere because of the different climatic, physiographic, and geologic conditions. Wahl (1984), summarized these situations as: ?The controversy involving the validity of equations developed locally but applied generally resembles some geologic controversies that were resolved when the investigators visited their opponents? field area.? As Leopold (1994) stated, stream channel morphology is directly influenced by channel slope, roughness of channel materials, sediment load, and sediment size. A change in any one of these variables sets up a series of channel adjustments which lead to a change in the others such as stream flow processes, resulting in channel pattern alteration and alteration of the hydraulic geometry. Because stream morphology influences hydraulic geometry, any changes in environmental controlling factors need to be investigated for their roles on b, f, and m changes. 2.7.1 Environmental Controlling Factors As Riggs (1978) stated there is an extensive literature on channel morphology that established general relations among channel width, depth, slope, discharge, and velocity, and attempts to explain channel response to changes in streamflow regimen. These 38 general relations among channel width, depth, discharge, and velocity are usually associated to only a few environmental controlling factors. Orsborn and Stypula (1987) developed the interrelationships among the geometric characteristics of stream channels and streamflow from which the natural or modified states of streams could be determined. They suggested and developed a regional hydraulic geometry model for basins in Oregon based on the assumption that if the interactions of streamflows and freely deformable stream boundaries are governed by the same hydraulic forces, then stream channels of different sizes should have comparable dimensionless geometric and streamflow ratios. In this case, the influence of scale, scale defined as the differences in size of stream channel, on hydraulic geometry was completely ignored. Ferguson (1986) showed that the cross-sectional shape of a channel determines the rate of increase of stream channel width with depth, whereas the laws of flow resistance determine the rate of increase of mean velocity with depth. In turn, factors that affect channel geometry include vegetation (Hadley, 1961; Zimmerman et al., 1967), bank cohesiveness and sediment size (Schumm, 1960; Wolman and Brush, 1961; Knighton, 1974; Maizels, 1988), changes in suspended load (Leopold and Maddock, 1953; Wolman, 1955; Thornes, 1970), seasonal variation (Thornes, 1970), channel sinuosity (Knighton, 1975), channel roughness or flow resistance (Knighton, 1975), riffle spacing (Harvey, 1975, Prestegaard, 1983), the processes of scour and fill (Foley, 1978; Andrews, 1979), landuse (Moglen and Berger, 1998), and stream size (scale factor) and order (Thornes, 1970; Miller and Onesti, 1977). All of these stream parameters are candidates as covariates of the hydraulic exponents. 39 Phillips and Harlin?s analysis (1984) of a subalpine stream in a relatively homogeneous environment indicates that hydraulic exponents are not stable over space. Hydraulic exponents may be so influenced by local soil and other surface conditions that stressing pervasive stream behavior through b, f, and m values may prove of limited value. Unfortunately, Phillips did not discuss the specific hydraulic, geomorphic, or pedologic conditions that give rise to these contrasting hydraulic geometries. Consequently, it remains unclear whether comparable contrasts are to be expected in other environments. Early attempts to apply channel size to furnish engineering answers were concentrated on identifying bankfull stage and on verifying the recurrence interval of the corresponding discharge. Among the studies of this type are those by Kilpatrick and Barnes (1964), Woodyer (1968), Potter, et al., (1968), Brown (1971), and Kellerhals, et al., (1972). Such studies generally did not produce a method for estimating a discharge corresponding to bankfull stage. These studies might be considered transitional between research in channel morphology and application of this research to estimating flow characteristics. 2.7.2 Water Reference Level Ideally, a channel feature used as an index to discharge should be a unique recognizable feature of the channel. It should also be active, that is, free to adjust to changes in the flow regime. These considerations led to the attempts to relate an active, within-channel feature to discharge characteristics. According to Wahl?s (1984) account, 40 the approach was apparently first suggested by Langbein (USGS, written communication, 1966). Comparing published results for different physiographic areas is difficult because the areas have morphological and hydrological differences. In addition, three different water flow reference levels could have been used in other studies. Singh et al. (1989), for stream habitat evaluation, developed a basin model that defines the average values of width, depth, and velocity for a given streamflow or for a given flow duration and drainage area. They concluded that width, depth, and velocity parameters increase with drainage area when compared at the same duration. They derived relationships linking a flow parameter to drainage area for a given stream network in a hydrologically homogenous basin: log (VAR) = a + bF + c (logAd), in which VAR: w, d, or v; F: the decimal flow duration; Ad: the drainage area; and a, b, and c are: coefficients. Even this concept is applicable only in a hydraulically homogeneous basin, this is the first step that drainage area and flow duration variables are introduced to determining width, depth, and velocity in a formal form. Most researchers used mean annual discharge as the predictor variable because it provided a discharge of approximately the same frequency throughout the area of investigation, thus permitting comparison between relations. At first it appeared that the values of the exponents b, f, and m were relatively constant, and the average values agreed quite closely with previously defined values for regime canals. The coefficients a, c, and k, however, varied among river systems. Furthermore, flow characteristics in arid and semiarid regions generally are only weakly related to the size of the drainage basin. 41 Wolman's (1955) analysis of the Brandywine Creek drainage in Pennsylvania, in which he related hydraulic geometry to bankfull discharge, was significant. In addition, he analyzed the hydraulic-geometry relationships with flow of 50-, 15-, and 2-percent duration. The recurrence interval of flows exceeding bankfull on Brandywine Creek ranged between 1 and 3 years and averaged 2.2 years. Although simple in concept, bankfull stage may be interpreted in a number of different ways, each associated with different values of width and depth and yielding a different bankfull discharge. Williams (1978) gave a comprehensive review of definitions of bankfull stage. He identified and discussed eleven definitions that have been used by investigators. There are three water reference levels referenced by most researchers. They are: within-channel, active-channel, and main-channel. The first published analysis using the within-channel feature was that by Moore (1968) for streams in Nevada. He graphically related mean annual discharge to the width and average depth of the channel cross- section defined by the tops of the channel bars and gave separate results for perennial and ephemeral streams. The results are of limited practical use as only 34 percent of the sample variance was explained by an equation using width; a relation using only precipitation explained 83 percent of the sample variance. In an earlier study of small drainages in the Sleeper?s River basin of northern Vermont, Zimmerman et al., (1967) found that stream width did not increase in the downstream direction for drainage areas less than 0.8 square miles. They attributed this to the effect of vegetation, mostly tree roots, and to relatively small annual peak discharges. 42 While studying Kansas streams, Hedman (U.S. Geological Survey, written communication, 1972) recognized a channel feature somewhat higher than the in-channel bars that had been used previously. He first referred to this feature as the active flood plain but redefined it as the active channel (Hedman et al., 1974). Osterkamp and Hedman (1977) described the active channel as ". . . a short-term geomorphic feature subject to change by prevailing discharges. The upper limit is defined by a break in the relatively steep bank slope of the active channel to a more gently sloping surface beyond the channel edge. The break in slope normally coincides with the lower limit of permanent vegetation so that the two features, individually or in combination, define the active channel reference level. The section beneath the reference level is that portion of the stream entrenchment in which the channel is actively, if not totally, sculptured by the normal process of water and sediment discharge.? Since then, the active-channel section has been used in numerous studies to define mean annual and flood flows in the Western States and in selected Eastern States. Scott and Kunkler (1976) related the width of the active channel to characteristics of the 2- through 50-year floods in New Mexico. The relations using channel width gave significantly smaller standard errors of estimate than similar relations that used basin and climatic characteristics. Osterkamp and Hedman (1982) expanded on the earlier Missouri River basin study by Hedman and Kastner (1977) by considering the effect of channel - sediment properties, channel gradient, and discharge variability. They concluded "Results show that channel width is best related to variables of discharge, but that significant 43 improvement, or reduction of the standard errors of estimate, can be achieved by considering channel-sediment properties, channel gradient, and discharge variability." The third and highest reference level used is the main channel section (also referred to as the whole-channel section). This section was described by Riggs (1974) as "? variously defined by breaks in bank slope, by the edges of the flood plain, or by lower limits of permanent vegetation.? Harenberg (1980) used the bankfull width to define relations for the 1.5- through 100-year floods for Idaho. Lanham developed a relation between the geometric mean of peak discharges (approximately the 2-year flood) and main-channel width for Wyoming that had an average standard error of estimate of 47 percent (Wahl, 1984). Since at-a-station hydraulic geometry compares flows of vastly different frequencies, the author used all measurements during various water reference levels. However, the author will use the water reference level information of each hydrographic survey data for explanation of b, f, and m values. 2.7.3 Scale The indications of the existence of the scaling factor in hydraulic geometry were not heeded by many geomorphology researchers, even though studying the effects of scale has become popular over the last few decades. Wilcock's (1971) observation on the relationship between increases in discharge and drainage area is a good example of the existence of scaling. He noted that the rate of increase of discharge with drainage area tends to be very much greater for lower than for higher duration flows. However, instead of the relationship between discharge and drainage area as a scale related approach, it is 44 suggested that this is most probably the result of differences in topography and the consequent differences in the roles played by surface runoff and groundwater flow in the different parts of the catchment. As Penning-Rowsell and Townshend (1978) stated, understanding of the effect of scale on hydrographical and geomorphological processes is noteworthy because there is a need to relate spatial patterns and forms to correct processes so that valid process model can be developed. To explain how spatial patterns and form of hydraulic geometry can be expected where we do not have data and understanding which variables have prominent influence on hydrographic geometry at which scale, including spatial and temporal aspects, are not fully understood yet. Over the years, numerous field experiments have revealed that hydrological processes and parameters can show considerable spatial variability. Bloschl, et al. (1995) stated that hydraulic process varies over the size of a particular catchment and with other environmental parameters. It has increasingly been realized that hydraulic variables and processes usually exhibit a large spatial variability. Rosso (1996) stated that there is an increasing awareness that the development of methods enabling transformation of data from one temporal or spatial scale to another can bring important and substantial developments in many hydrological applications. And Leopold (1994) compared the hydraulic geometry of three rivers of different basin sizes: Watts Branch drainage area, 3.7 square miles; Seneca Creek drainage area, 100 square miles; Amazon River at Obidos drainage area, 1.9 million square miles. Visual inspection of the scatter diagrams of discharge versus width, depth, and velocity for these three rivers of different size clearly indicates there are differences in the slopes of the 45 three sets of the lines for each hydraulic exponents. Assuming all predictor variables for these three rivers are same, which might not be the case, it can be inferred that there is clearly scale effects on these three rivers of different drainage size. However, the author has not come-across any investigations of the scale effects on at-a-station hydraulic geometry yet. Another scale related study is Hack?s law. Rigon et al. (1996) reviewed Hack?s law (1957) and explained Hack?s law from scaling aspect such that the length of the longest stream has a power relationship with size of the basin. From the results, Rigon et al. (1996) suggested that a statistical framework referring to the scaling invariance of the entire basin structure should be used in the interpretation of Hack?s law. Hack (1957) demonstrated the applicability of a power function relating length and area for streams of the Shenandoah Valley and adjacent mountains in Virginia. He found the equation L = 1.4A0.6 where L is the length of the longest stream in miles from the outlet to the divide and the A is the corresponding area in square miles. Hack also corroborated this equation through the measurements of Langbein (1947), who had measured L and A for nearly 400 sites in the northeastern United States. Mueller (1973), on the basis of extensive data analysis of several thousand basins, found that the exponent in Hack?s equation was not constant but that it changed from 0.6 for basins less than 8,000 square miles (20,720 km2) to 0.5 for basins between 8,000 and 105 square miles (20,720-259,000km2), which clearly shows a scale factor in Hack?s equation. Clearly, there has been considerable research on scaling factors in various hydrographic processes but the author is, other than Leopold?s (1994) comparison of, b, f, and m in three different sizes of rivers and Griffiths? (2003) work on downstream 46 hydraulic geometry, not aware of research on quantification of scale influence on at-a- station hydraulic geometry. Hence, the author investigated the impact or non-impact of scaling factors which is one of many environmental controlling factors in b, f, and m. The author used many environmental controlling factors such as scale, channel shape, channel pattern, asymmetry of channel, topographic index for slope of study subbasin characteristics, subbasin size for scale influence, physiographic and bed and bank material, and landuse characteristics, as well as Rhodes? (1977) classification of the b, f, and m to synergistically investigate the relationships between b, f, and m and environmental controlling factors. As Moglen and Bras (1995) stated, the presence of heterogeneities does not simply blur and smear analytical relationships. The heterogeneities themselves introduce new structure to the organization of the drainage basin. The author investigated the new structure to the organization of the drainage basin by synergistic analyses using significant variables to explain the relationships between b, f, and m and environmental controlling factors. 2.8 Need for Additional Study A number of features, including the in-channel bar section, active-channel section, and the main-channel or whole-channel section, presently are being used to define flow characteristics. Consequently, it is difficult to make comparisons between studies, and the applicability of individual results is dependent on the ability of a user to identify the water reference level feature used in developing the relation. 47 Additional work is needed to determine the role of other variables. Several investigators (Schumm, 1960; Glazzard, 1981; Osterkamp and Hedman, 1982) have examined the relation between sediments in the bed and banks and channel size. Andrews (1984) examined data for gravel-bed streams in Colorado and separated the data into two groups depending on whether bank vegetation along the study reach was light or thick. He made width, depth, and velocity dimensionless by dividing each by the median particle diameter in the riverbed surface. Regression relations for hydraulic geometry exponents showed no significant difference between data for light and thick bank vegetation. This implies that role of bank vegetation was weak on hydraulic geometry for gravel-bed rivers in that study. The inferences concerning the relationship between roughness and hydraulic geometry are speculative at present. As Richards (1973) pointed out in discussing this problem: "Achieving a point of entry into this system for the purpose of analysis is extremely difficult, since the interaction is complete." Much work remains to be done on the relationship of hydraulic geometry and channel roughness. The inferences derived from the b-f-m diagram indicate some possible lines of inquiry. As Abrahams (1984) pointed out employing the hydraulic geometry model to answer questions pertaining to channel storage, flood potential, or sediment delivery would seem to prove difficult at best without constant re-evaluation in ?steady-state? time. An inherent difficulty, the feedback in the system produces variables which, even in the short run and ?at-a-station?, are never consistently dependent or independent. Even more difficult to evaluate is the relative contribution of each of the factors to the observed changes. At present, each case must be considered individually, and this 48 involves a detailed analysis of many variables. Then interaction among predictor variables to hydraulic geometry needs to be investigated so all predictor variables? roles on hydraulic geometry can be considered synergistically. 2.9 Conclusions The working hypothesis of this study is that variations in hydraulic geometry exponents, (b, f, and m values), the change rates of channel width and depth, and flow velocity, are not random but there are systematic changes in hydraulic geometry exponents as a function of watershed scale and basin characteristics. In other words, there is a geographic scale factor in hydraulic geometry exponents such that b, f, and m values in Amazon-sized rivers are different from those of smaller rivers. This might explain why hydraulic geometry has different b, f, and m values where all other basin variables are the same or similar. Using empirical field data, in addition to the overall scale function to b, f, and m, I will specifically test scale impacts of the four hydraulic and channel geomorphic process principles established by other researchers. They are: a) for sand-beds, roughness usually increases with Q due to bedforms, so velocity exponents are smaller than depth; and b) in gravel-bed channels, roughness decreases with Q due to increased relative submergence, and the velocity exponent is larger than the depth exponent. As stated earlier, statistical analyses of predictor variables? impact on the b, f, and m will be explained with these categories within scale context. 49 CHAPTER 3 STUDY AREA: Selected Characteristics Of The Potomac River Basin And Demarcation Of Its Subbasins 3.1 General Study Area The study area for this investigation of hydraulic geometry in various environmental settings is the Potomac River Basin shown as Figure 3.1. This Basin, covering a range of 38 to 40 degrees North and 76 to 80 degrees West, extends from the North Branch Potomac River near Luke, Maryland to the St. Marys River at Great Mills, Maryland, and from the headwaters of the North River near Burketown, Virginia, to Aquia Creek near Garrisonville, Virginia. The sources of the following description of characteristics of the Potomac River Basin are USGS Professional Papers and Technical Reports of the Interstate Commission on the Potomac River Basin. In the eastern United States, the Potomac River Basin is part of the Chesapeake Bay drainage system. The Potomac River and its tributaries drain 14,670 square miles (mile2), including parts of Maryland, Virginia, West Virginia, Pennsylvania, and the entire District of Columbia: Virginia (5,723 mile2), Maryland (3,818 mile2), West Virginia (3,490 mile2), Pennsylvania (1,570 mile2), and Washington, D.C. (69 square miles) (Gerhart and Brakebill, 1996). The Potomac River flows through six physiographically distinct subunits which will be described later. Because of its large drainage area and diversity, the Potomac River Basin?s environmental setting is complex, consisting of various combinations of natural and human characteristics. Knowledge of selected primary features that comprise the environmental setting is necessary to understand hydraulic geometry and the environmental impacts on hydraulic geometry. 50 Figure 3.1 Study Area - Potomac River Basin Source: US Geological Survey Water Resources Investigations Report 96-4034 3.1.1 History of the Study Area, Potomac River Basin Before the Pleistocene glacial age, many great river systems drained the eastern slopes of long mountain ridges along the eastern North American continent. The greatest of these was the Susquehanna, which had a watershed of thousands of square miles with N 51 boundaries extending as far north as upstate New York and as far west as western Pennsylvania. As the Susquehanna meandered southward to the Atlantic, cutting through the Piedmont foothills, it was joined by waters from hundreds of streams, large and small (U. S. Department of the Army, 1973). The largest of these tributaries was the Potomac River, which drained the southwestern slopes of the mountain system. The Potomac, along with other southern tributaries (the York, James, and Rappahannock rivers of Virginia), cut deep channels across the ancient coastal plain ledges (U. S. Department of the Army, 1973). At the end of the last Pleistocene glacial age, from about 15,000 to about 9,000 years ago, sea levels rose with the melting retreat of the glaciers and inundated the valleys of the coastal plain rivers (Schubel and Meade, 1977) eventually reaching the base of the Piedmont hills at what is now called the fall line. These tidal waters flooded the lower portions of the Susquehanna River basin, drowning the valleys inland for almost 180 statute miles (290 kilometers) (Cronin, 1973). Thus, as seawater intruded into the lower reaches of the Susquehanna basin, the Chesapeake Bay and the estuarine portions of all its tributaries, including the Potomac River, came into being (Lippson et al., 1979). 3.1.2 Present Potomac River Boundaries The Potomac River begins as a small spring in West Virginia, at Fairfax Stone, and flows 383 miles through six physiographic provinces to its mouth at Point Lookout, Maryland, and Smith Point, Virginia, where it enters in the Chesapeake Bay (Figure 3.1). As discussed more fully later, it crosses six distinct physiographic regions on its journey 52 to the bay, including the Appalachian Plateau, the Valley and Ridge Province, the Great Valley, the Blue Ridge, the Piedmont, and the Coastal Plain (Vokes and Edwards, 1974). The Potomac River Basin is bounded by the Susquehanna River basin on the north, the Ohio River basin on the west, the James and Rappahannock River basins on the south, and the Patuxent River basin and Chesapeake Bay drainage on the east. Flow of the Potomac River increases downstream so that when it reaches the Chesapeake Bay, it constitutes about 15 percent of the estimated 49,300 Mgal/d total inflow to the Bay (J.F. Hornlein, U.S. Geological Survey, oral communication, 1991). Figure 3.2 Physiographical Characteristics of the Study Area Source: US Geological Survey Water Resources Investigations Report 96-4034 Major tributaries to the Potomac River (Figure 3.1) include the North Branch Potomac River, South Branch Potomac River, Cacapon River, Shenandoah River, 53 Conococheague Creek, Monocacy River, and Occoquan River. The North Branch Potomac River drains the rugged northwestern part of the Potomac River Basin in Maryland, West Virginia, and Pennsylvania. The South Branch Potomac River and Cacapon River drain the mountainous West Virginia part of the basin. The Shenandoah River, the largest of the Potomac River's tributaries, drains the broad, relatively flat Shenandoah Valley in Virginia. Conococheague Creek and the Monocacy River drain the northern and northeastern parts of the basin in Maryland and Pennsylvania. The largest tributary in the eastern part of the basin is the Occoquan River in Virginia, which enters directly into the freshwater tidal Potomac River south of Washington, D.C. (Federal Energy Regulatory Commission, 1980). The fresh headwaters originate high in the Appalachian mountain system, close to the southwest corner of Maryland. Here, the North Branch flows northeastward for 100 statute miles (161 km), forming a part of the Maryland and West Virginia border. As it passes eastward from Cumberland, many other large rivers, such as the South Branch, the Shenandoah, and the Monocacy, contribute substantially to the combined flow. These tributaries and many other smaller streams flow north through Virginia and West Virginia and south through Pennsylvania and Maryland to meet the Potomac. The riverine segment terminates near the fall line, which is demarked by a series of rapids at Washington, D.C. At Little Falls, 300 statute miles (483 km) from its source, the river comes under the influence of tides, and the estuary begins. The estuary runs another 113 statute miles (182 km) before it meets the Chesapeake Bay. The Potomac River is free flowing and contains freshwater upstream from Washington, D.C. At Washington, below the scenic Great and Little Falls, the more than 54 100 miles of tidewater Potomac begins and gradually changes from a fresh tidal river to a salty estuary en-route to its mouth at the Bay. The total fall from this confluence to sea level at Washington, D.C., is 520 feet. The river contains freshwater and is tidal from Washington, D.C., to near Indian Head, Md., where the water becomes brackish. From Indian Head to Point Lookout, Md., the river water becomes progressively more salty as it approaches the Chesapeake Bay. The basin has a generally temperate climate with extremes more pronounced in areas west of the Shenandoah River. The winters are seldom rigorous and usually result in heavy snowfall in only the northwest portion of the basin. The summers are generally warm. The mean annual temperatures range from 59 degrees Fahrenheit in tidewater areas to 51 degrees Fahrenheit in the Appalachian Highlands (Pang, et al., 1991). The prevailing winds are westerly with a mean velocity of about 10 miles per hour. Average annual precipitation ranges from 30 to 35 inches in the foothills of the Allegheny Mountains, to 50 inches along the western divide, and 45 inches along the crest of the Blue Ridge Mountains (Gerhart and Brakebill, 1996). The watershed collects, on the average, a yearly precipitation of approximately 45 inches (1.1 meter), which results in a yearly average freshwater flow of 11,190 cubic feet (313 cubic meter) per second at the head of its estuary (Pang, et al., 1991). Precipitation is generally greatest from June to September with the lowest monthly total occurring in February or November. Snow falls in all sections of the basin and varies in depth from 10 inches along the Chesapeake to 100 inches along the western divide. The average annual runoff is about 0.95 cubic feet per second per square mile (12.8 inches). March and April are generally the months of greatest runoff, usually ranging from 4 to 8 times 55 above normal, while September and October are generally the months of least runoff, usually ranging from 3 to 10 percent below the norm (Gerhart and Brakebill, 1996). Flood-producing storms occur in all seasons of the year. Summer and fall floods are related to tropical disturbances having intense rainfalls of short duration. Spring and winter floods usually result from sustained rainfall that sometimes contributes to large flood flows (Gerhart and Brakebill, 1996). Mean annual lake evaporation varies from 30 inches in the northern part of the Potomac River Basin to 40 inches in the southern portion. Approximately 70 percent of the evaporation occurs from May through October (Gerhart and Brakebill, 1996). The major population center in the Potomac River Basin is the metropolitan area of Washington, D.C., at the eastern boundary of the basin (Figure 3.1). Other, much smaller, population clusters include Cumberland, Hagerstown, Frederick, Rockville, and Waldorf in Maryland; Staunton, Waynesboro, Harrisonburg, Front Royal, Winchester, Alexandria, Leesburg, and Manassas in Virginia; Petersburg, Moorefield, Harpers Ferry, and Martinsburg in West Virginia; and Chambersburg, Waynesboro, and Gettysburg in Pennsylvania. However, 75 percent of the basin?s 1990 population lived in the Washington, D.C., metropolitan area (Gerhart and Brakebill, 1996). The Potomac and its tributaries are used for many important purposes, such as public drinking water supplies, power plant cooling water, commercial and sports fishing, boating and other water-related recreation (Eastman, 1985). It is important to understand how the Potomac River?s hydraulic processes will interact with changing river discharge volumes along with various physiographic settings such that future urban and hydrographic planners can prepare better urban and hydrologic modeling. 56 3.2 Study Area Physiography The sources of the following description of physiography of the Potomac River Basin are: Fenneman and Johnson (1946); Interstate Commission on the Potomac River Basin (Gruessner, 1997); State of Maryland Department of Natural Resources (Edwards,1981); USGS Water Resource Investigations Report 96-4034 (Gerhart and Brakebill, 1996); and Geology of the Chesapeake and Ohio Canal National Historical Park and Potomac River Corridor, District of Columbia, Maryland, West Virginia, and Virginia by USGS geologist Southworth (2001). The Potomac River Basin contains parts of six distinct physiographic provinces that extend from the southwest to northeast along the Atlantic Coast of the United States (Fenneman and Johnson, 1946). The physiographic provinces include the Appalachian Plateau, Valley and Ridge, Great Valley, Blue Ridge, Piedmont, and Coastal Plain (Figure 3.2). The Valley and Ridge is the largest province in the basin and includes the Great Valley subprovince. The Piedmont is the second largest province and contains the Triassic Lowlands subprovince. The varied topography of these six provinces forms a complex landscape within the Potomac River Basin that includes steep mountains, rolling hills, broad valleys, and plains. Many of the differences among physiographic settings are related to the geology of the Potomac River Basin area. The distribution of study subbasins by physiography is shown in Table 3.1. 57 Table 3.1 Physiographic Province Distribution of Study Subbasins Physiography Number of Subbasins Percentage of Total Area (14,670 square miles) Appalachian Plateau 2 4% Great Valley 7 21% Valley and Ridge 2 34% Blue Ridge 15 6% Piedmont 12 12% Coastal Plain 5 22% Source: USGS Water Resource Investigations Report 96 - 4034 The Appalachian Plateau is the westernmost province and comprises about 4 percent of the basin in Maryland, Pennsylvania, and West Virginia (Interstate Commission on the Potomac River Basin?s The Potomac and the Chesapeake Report). It is characterized by narrow valleys and steep, rugged ridges creating local topographic relief of 500 to 2,000 ft. The Appalachian Plateau contains the highest point in the basin, Spruce Knob, which rises to an altitude of 4,860 ft. The North Branch Potomac River drains the Appalachian Plateau. The Appalachian Plateau is separated from the Valley and Ridge Province to the east by the Allegheny Front, a major escarpment with as much as 3,000 ft of local relief that trends northeast through the basin (Pang, et al., 1991). The Valley and Ridge Province is the most extensive province in the basin and occurs in Virginia, West Virginia, Maryland, and Pennsylvania. This province comprises 34 percent of the basin excluding the Great Valley subprovince (Interstate Commission on the Potomac River Basin?s The Potomac and the Chesapeake Report). The rocks in this province have been intensely folded and faulted, producing long, narrow, northeast- trending structures. Subsequent erosion and weathering have resulted in the distinctive 58 topographic grain of this province, with ridges capped by resistant sandstone and valleys underlain by less-resistant shale and carbonate rocks. Topographic relief in the Valley and Ridge Province is considerable, ranging to as much as 1,800 ft. The trend of the ridges substantially affects surface drainage in the basin, so that the principal tributary streams, the South Branch Potomac and Cacapon Rivers, flow northeast to the Potomac River (Gerhart and Brakebill, 1996). The Great Valley occupies the eastern part of the Valley and Ridge Province and is a broad valley, 15 to 20 miles wide, with minor relief over extensive areas. It covers 21 percent of the basin (Interstate Commission on the Potomac River Basin?s The Potomac and the Chesapeake Report). Shale and siltstone underlie the central part of the Great Valley and are bordered by areas underlain by carbonate rocks. The carbonate rocks that make this subprovince favorable for farming also are susceptible to dissolution, resulting in numerous caves and other karstic features throughout most of the area. The Shenandoah River and Conococheague Creek are the major tributaries to the Potomac River that drain this subprovince. The Great Valley, which is divided into carbonate and non-carbonate subunits, is a part of the Valley and Ridge Physiographic Province. The Great Valley is bounded by the Blue Ridge Mountains to the east and Great North Mountain to the west, and is interrupted by Massanutten Mountain in Virginia. Seventy percent of the Great Valley is underlain by carbonate (limestone and dolomite) bedrock (Denis and Blomquist, 1995), with numerous sinkholes and caverns. The major streams in the Great Valley include the North Fork Shenandoah River, South Fork Shenandoah River, the main stem Shenandoah River, Opequon Creek, Conococheague Creek, and Antietam Creek (Ferrari, 1999). 59 Bordering the Great Valley on the east is the Blue Ridge Province. It covers 6 percent of the basin and consists of a mass of crystalline rocks that rises about 1,500 to 2,000 ft above the lowlands on either side (Southworth, 2001). In Virginia, the Blue Ridge forms the southeastern boundary of the Potomac River Basin. In Maryland and Pennsylvania, it forms a major drainage divide within the basin, with all streamflow from west of the Blue Ridge flowing through the gap at Harpers Ferry, West Virginia. Because of relief and narrowness, most of the streams that drain the Blue Ridge are headwater tributaries of larger streams in the Great Valley or Piedmont (Southworth, et al., 2001). The Piedmont Province lies to the east of the Blue Ridge and comprises about 12 percent (Interstate Commission on the Potomac River Basin?s The Potomac and the Chesapeake Report) of the Potomac River Basin in Virginia, Maryland, Pennsylvania, and Washington, D. C. It is an area of gently rolling terrain with low to moderate relief. The eastern and western parts of the Piedmont are underlain by resistant crystalline rocks, whereas the central part, the Triassic Lowlands subprovince, 7-percent of the basin (Southworth, et al, 2001), is underlain by less-resistant sedimentary rocks of primarily Triassic age. The Triassic Lowlands subprovince is generally flatter than the surrounding Piedmont. The principal tributary to the Potomac River in the Piedmont Province is the Monocacy River, which also drains agricultural lands in the Triassic Lowlands subprovince (Southworth, et al, 2001). The Fall Line separates the Piedmont from the Coastal Plain Province to the east. At the Fall Line, the rolling hills of the Piedmont drop in elevation to meet the gently sloping Coastal plain. Stream gradients abruptly steepen through the Fall Line as they 60 enter the Coastal Plain. The Potomac River drops nearly 150 ft to near sea level as it flows through Great Falls and Little Falls near Washington, D. C. The Fall Line also marks the terminus of the upper Potomac River, which, at this point, drains about 11,670 square miles of the six provinces and subprovinces upstream. Smaller tributaries enter the tidal Potomac River after flowing from the Triassic Lowlands and Piedmont through the Fall Line onto the Coastal Plain. The Coastal Plain Province is fundamentally different from the other physiographic provinces in that it is underlain by unconsolidated sediments that form a gentle seaward-sloping plain of low relief. It is in this province that the Potomac River is tidally affected, eventually becoming a broad estuary before entering the Chesapeake Bay. The estuary is typically flanked by broad lowlands that mark the sinuous and deep course of an ancestral Potomac River Valley now filled by unconsolidated sediments (Interstate Commission on the Potomac River Basin?s The Potomac and the Chesapeake Report). 3.3 Study Area Geology and Lithology The geology of the Potomac River Basin is complex and diverse, ranging from relatively undisturbed, unconsolidated sediments to intensely deformed crystalline rocks (Milici, 1963; Cleaves, 1968). The study subbasins are located in the many different physiographic settings. Four major types of rock are present in the region: Siliciclastic sedimentary (shale and sandstone), carbonate sedimentary (limestone and dolomite), crystalline metamorphic and igneous (gneiss, schist, and diabase dikes), unconsolidated sediments (sand, silt, and clay), and undivided (carbonated and silicrystalline). The most 61 intensely deformed and the oldest rocks in the basin are crystalline rocks in the Piedmont and Blue Ridge Provinces. Predominant rock types are massive granite and layered gneiss, foliated phyllite and schist, quartzite, marble, and metadolomite. The folding and faulting associated with the tectonic stresses have produced very complex rock structures. Figure 3.3 Study Area Lithology Source: US Geological Survey Water Resources Investigations Report 96-4034 Sedimentary rocks of Cambrian through Pennsylvanian age underlie the Valley and Ridge and Appalachian Plateau Provinces. The rocks of the Valley and Ridge have been deformed by folding and thrust faulting into a series of plunging folds. Although 62 folded and faulted, these rocks were farther away from the center of tectonic activity than the crystalline rocks of the Piedmont and Blue Ridge. Therefore, the rocks in the Valley and Ridge were not recrystallized during orogenic events. In the Valley and Ridge Province the rocks can be broadly categorized into siliciclastic and carbonate types. Rocks in the Appalachian Plateau Province are similar lithologically to the siliciclastic types in the Valley and Ridge with the exception that coal-bearing rocks are found in the Appalachian plateau. Although major thrust faults occur at depth, rocks in the Appalachian Plateau Province are relatively flat-lying and less deformed than rocks in the Valley and Ridge (Southworth, et al., 2001). The youngest consolidated rocks in the basin are found to the east of the Blue Ridge Province, in the part of the Piedmont Province underlain by sedimentary rocks. These rocks were formed during the Late Triassic and Early Jurassic by streams discharging sediment into down-faulted rift basins. In the Potomac River Basin, the faulted margin of these basins borders the Blue Ridge, and other faults are present throughout the sequence. The sedimentary sequence is interrupted locally by igneous intrusive and extrusive rocks (Gerhart and Brakebill, 1996). The youngest geologic units in the basin are in the Coastal Plain Province. These units consist of unconsolidated sediment that forms a southeastward-thickening wedge of interbedded sand, silt, and clay. Unconsolidated sediments underlie about 15 percent of the basin; carbonate sedimentary rocks underlie about 17 percent; siliciclastic sedimentary rocks underlie about 42 percent; and crystalline rocks underlie about 19 percent of the basin (Southworth, 2001). The remaining 7 percent consists of geologic units that contain significant proportions of both carbonate and siliciclastic sedimentary 63 rocks (Southworth, et al., 2001). The distribution of the study gage stations over various geological settings is shown in Table 3.2. Table 3.2 Geological and Lithological Distribution of Study Subbasins Geology Number of Subbasins Undetermined 7 Siliclastic Sedimentary rocks 13 Carbonate Sedimentary 5 Unconsolidated Sedimentary 5 Crystalline Metamorphic rocks 13 Source: U.S. Geological Survey Open-File Report 01-188 Physical and geologic characteristics provide the primary factors affecting the hydrologic properties of the ground water and surface water systems and, thus, form the primary natural factors affecting hydraulic geometry within the basin. Because of the diverse physical characteristics of the Potomac River Basin, lithology has been divided into six subunits for the purpose of hydraulic geometry assessment. This subdivision was done using a hierarchical process considering basin physiography and geology (Figure 3.3). Physiographic provinces are used as the primary units of the subdivision because of their structural effects on the hydrologic systems. The six subunits will be referred to throughout this paper and will serve as one of areal comparisons of hydraulic geometry. The Appalachian plateau subunit is composed primarily of siliciclastic rocks. The Valley and Ridge subunit is underlain by both siliciclastic and carbonate rocks. Some of its carbonate rocks are interbedded with siliciclastic rocks and are cemented by carbonate material. This interbedding is characteristic of the region. The Great Valley subprovince 64 is divided into a carbonate subunit and a non-carbonate subunit because carbonate regions have unique hydrologic properties and because the carbonate units are areally contiguous and extensive. The Blue Ridge is formed by crystalline rocks. The Piedmont comprises the crystalline rock. As in the Blue Ridge subunit, the rocks underlying this subunit are crystalline. However, the rolling topography of this subunit provides a distinct hydrologic setting. The Coastal Plain subunit is underlain by unconsolidated sediments. Due to the distinct physiology and lithology the hydrologic factors affecting ground water and surface water flow in this subunit are unique within the Potomac River Basin Source (Interstate Commission on the Potomac River Basin?s The Potomac and the Chesapeake Report). 3.4 Study Area Hydrology The six major tributaries of North Branch Potomac River, South Branch Potomac River, Cacapon River, Conococheauge Creek, Shenandoah River, and Monocacy River, draining the Potomac River Basin are depicted in Figure 3.1. These six tributaries drain to the upper Potomac River upstream from the Fall Line. Additional tributaries drain directly to the tidal Potomac River and include the Anacostia and Occoquan Rivers. Streams upstream from the Fall Line generally have steeper gradients and flow more swiftly than streams downstream from the Fall Line. The main stem Potomac River and major tributaries generally have bedrock bottoms, with alluvial sediments in depositional areas. Stream-bottom materials range from bedrock to small cobbles and gravel in upstream areas, to eroded fine sediments to gravel, sand, and silt in Coastal Plain streams. Several streams, notably Conococheague Creek and the South Fork 65 Shenandoah River, undergo considerable flood-plain meandering in their downstream reaches. Coastal Plain streams, downstream from the Fall Line, have shallow gradients and discharge to tidal creeks or wet-lands, which have considerable effect on stream flow and stream morphology. Trainer and Watkins (1975) found that average base runoff from tributaries in the upper Potomac River Basin was approximately proportional to drainage area. The six major tributaries upstream from the Fall Line represent about 64 percent of the drainage area and contribute about 63 percent of the mean annual streamflow. At low flows, however, Trainer and Watkins (1975) found that the area underlain by carbonate rocks, which are mostly in the Great Valley subprovince, contribute a proportionately larger share of flow to the Potomac River at Washington, D. C. For example, the Shenandoah River contributes about 38 percent of the streamflow during low-flow conditions yet contains only 26 percent of the upper Potomac River Basin (Trainer and Watkins, 1975). The ratio of runoff to precipitation is somewhat higher in the Appalachian Plateau and westernmost parts of the Valley and Ridge Province, where steep slopes, shallow soils, and less-permeable bedrock contribute to the greater total runoff. About 50 percent of precipitation becomes surface runoff in these mountain areas. In the remainder of the Potomac River basin, with generally flatter slopes, deeper soil profiles, and more fractured bedrock, karst terrain, or Coastal Plain deposits, about 35 percent of precipitation contributes to surface runoff. Most streams in the Potomac River Basin generally have good year-round flow experienced notable hydrologic extremes. Rains from Tropical Storm Juan in November and infrequently experience very low or no flow. However, the Potomac River Basin has 66 1985 produced catastrophic flooding in the South Branch Potomac River and parts of the Shenandoah River (Carpenter, 1990). Selected characteristics of environmental setting for the Potomac River Basin such as physiography, geology, lithology, and hydrology are describe in this chapter and these characteristics are used as predictor variables for the investigation of hydraulic geometry. The next chapter will discuss measurable characteristics of the study site and data types for the study. 67 CHAPTER 4 MEASUREMENTS OF SUBBASIN CHARACTERISTICS: Computation of Derived Data In order to perform the empirical research for this investigation of at-a-station hydraulic geometry, the following primary data were required, and from these primary data many other datasets were derived. ? Gage Stations (4.1) ? Digital Elevation Data (4.2) ? Hydrographic Survey Data (4.3) -Water Discharge Volume Data -Channel Width -Channel Depth -Flow Velocity -Field Survey Data (4.4) -Bed and Bank Materials ? Physiography and Lithology (4.5) ? Landuse Information (4.6) ? Derived Data (4.7) The items 4.1 thru 4.6 and portions of the Derived Data items (Section 4.7) are discussed in this Chapter 4, and the rest of the derived data are discussed in Chapter 5, Hydraulic Geometry and the Environmental Setting. These chapters discuss how the data were collected and used to generate more information on the study subbasins, and then 68 analyzed to characterize subbasins via a variety of properties. After the general area, the Potomac River Basin, was selected for study, the specific gage stations and hydrographic survey measurements to be included in the research were selected based on the duration of the hydrographic survey time and the size of each drainage subbasin as well as the highest number of survey measurements on a specific survey location at each gage station. Next, information for other selected environmental settings characteristic of each subbasin, such as physiography, lithology, geology, land use, and the nature of nested subbasins which reflects hierarchies of the streams such as the subbasin of a creek within a river subbasin, was collected. In addition, 43 gage station descriptions and historical streamflow daily values covering 1985-1995 were used for explanation of the results of the impact of the environment context for the hydraulic geometry analyses. 4.1 Gage Stations Since 1896, and in cooperation with State agencies, the USGS has been collecting a large amount of data pertaining to the water resources of the nation, such as volume of surface water, width and depth of channel, velocity of flow, and water quality among others at a given gage station throughout the year and publishes some of the surface water discharge volume data each year. A gage station is identified as a particular site on a stream, canal, lake, or reservoir where systematic observations of hydrologic data are obtained. These data, accumulated during many water years, constitute a valuable database for developing an improved understanding of rivers. 69 The USGS established 178 permanent gage stations in the Potomac River Basin to perform hydrographic surface water discharge measurements. Each gage station is at a particular site on a stream where systematic observations of hydrologic data are made. Ninety-four surface water discharge gage stations are in Maryland and the District of Columbia, and 84 are in Virginia. Of these 178 gage stations in the Potomac River Basin, 43 (18 in Maryland and 25 in Virginia) were selected for study based on the following criteria. The first criterion was the hydrographic survey time. In order to accommodate the impacts of latent hydrographic processes on stream channels, gage stations were selected that have continuous survey measurements performed from just before the ?big flood? of the 1985s, until 1995, a ten-year period. The second criterion was the size of the drainage area subbasins, a necessary criterion for investigation of scale effects on hydraulic geometry. The sizes of the selected drainage areas range from 0.31 square miles (published by the USGS Water Resource Division as 0.38 square miles), for subbasin 01640980 (USGS gage station identification number), Bear Branch near Thurmont, Maryland, to 1632 square miles (published by the USGS Water Resource Division as 1,642 square miles), for subbasin 01629500, South Fork Shenandoah River at Front Royal, Virginia. The selected gage stations are listed in Table 4.1 and are portrayed on Figures 4.3, and 4.4. Table 4.1 shows the USGS gage station identification numbers and sequentially numbered by the author?s identification number, gage station name, location of gage station in geographical coordinates, UTM Zone identification, number of measurements that were used for the calculation of the b-f-m, measurement duration, exact location of 70 the channel for hydrographic survey measurements, and measurement methods (wading or using cables). Table 4.1 Selected Gage Stations ID POTOMAC RIVER BASIN USGS Gage Station Latitude Longitude UTM Zone Number of Surveys Survey Duration Survey location Survey Method North Branch Potomac River at Pinto, MD 1 m10 Wills Creek near Cumberland, MD 1601500 394007 784718 17 29 6/85-9/92 500' a wading 2 m11 North Branch Potomac River near Cumberland, MD 1603000 393716 784624 17 21 2/85-6/93 150' a not wading Potomac River at Hancock, MD 3 m19 Conococheague Creek at Fairview, MD 1614500 394257 774928 18 45 1/85-8/94 at gage not wading 4 m21 Marsh Run at Grimes, MD 1617800 393053 774638 18 47 2/85-2/94 10' b wading 5 m24 Antietam Creek near Sharpsburg, MD 1619500 392701 774352 18 37 1/85-8/93 150' a not wading 6 m26 Catoctin Creek near Middletown, MD 1637500 392535 773325 18 25 8/88-10/92 350' b wading Potomac River at Point of Rocks, MD Monocacy River at Bridgeport, MD 7 m29 Piney Creek near Taneytown, MD 1639140 393938 771316 18 13 1/91-11/92 140' a wading 8 m30 Big Pipe Creek at Bruceville, MD 1639500 393645 771410 18 43 11/88-8/95 150' a wading 9 m31 Hunting Creek near Foxville, MD 1640965 393710 772800 18 38 1/85-5/94 100' b wading 10 m311 Huntington Creek Tributary near Foxville 1640970 393742 772744 18 18 3/85-3/91 100' b wading 11 m32 Bear Branch near Thurmont, MD 1640980 393715 772624 18 17 6/90-11/91 15' b wading 12 m34 Monocacy River at Jug Bridge, MD 1643000 392413 772158 18 43 5/87-8/94 Reichs Ford wading 13 m37 Bennett Creek at Park Mills, MD 1643500 391740 772430 18 26 6/85-7/93 150' a wading 14 m38 Seneca Creek at Dawsonville, MD 1645000 390741 772013 18 28 6/88-3/95 600' b wading Potomac River at Chain Bridge at Washington, DC Northeast Branch Anacostia River at Riverdale, MD m43 Northwest Branch Anacostia River near Hyattsville 1651000 385709 765800 18 56 2/85-8/94 2000' b gage wading 16 m45 Piscataway Creek at Piscataway, MD 1653600 394220 765800 18 23 10/87-4/93 100' a wading 17 m47 St. Clement Creek near Clements, MD 1661050 382000 764331 18 54 6/85-6/94 150' b wading 18 m48 St. Marys River at Great Mills, MD 1661500 381436 763013 18 64 11/84-7/95 100' b wading 19 v07 North River near Burketown 1622000 382025 785450 18 33 8/89-11/94 at gage wading Middle River 20 v12 Christians Creek near Fishersville 1624800 380742 785941 17 36 8/89-11/94 500' b wading 21 v13 Middle River near Grottoes 1625000 381542 785144 17 9 9/89-7/92 300' a wading 22 v15 South River near Waynesboro 1626000 380327 785430 17 13 8/89-12/93 Under DSS bridge wading 23 v17 South River near Dooms 1626850 380519 785238 17 25 9/85-12/94 700' b wading 24 v18 South River at Harriston 1627500 381307 785013 17 18 8/89-2/93 500' b wading South Fork Shenandoah River Madison Run 25 v19 White Oak Run near Grottoes 1628060 381501 784457 17 26 8/90-11/94 300' a wading 26 v22 South Fork Shenandoah River near Lynnwood 1628500 381921 784518 17 9 2/93-8/94 1500' a gage wading 27 v23 South Fork Shenandoah River near Luray 1629500 383846 783206 17 17 8/89-12/94 50' a wading 28 v24 South Fork Shenandoah River at Front Royal 1631000 385450 781240 17 64 2/85-12/94 0.7mi.USS bridge not wading 29 v25 North Fork Shenandoah River at Cootes Store 1632000 383813 785111 17 16 8/90-12/93 1,000' a gage wading 30 v26 Linville Creek at Broadway 1632082 383624 784813 17 17 11/85-12/94 150' b wading 31 v27 Smith Creek near New Market 1632900 384136 783835 17 22 8/89-11/94 150' a wading 32 v28 North Fork Shenandoah River at Mount Jackson 1633000 384443 783821 17 22 11/89-11/94 0.6 mi. b wading 33 v29 North Fork Shenandoah River near Strasburg 1634000 385836 782011 17 45 3/85-10/94 100' a wading 34 v30 Cedar Creek near Winchester 1634500 390452 781947 18 17 9/89-12/94 150' b wading 35 v36 Catoctin Creek at Taylorstown 1638480 391518 773436 18 17 9/87-12/91 100' a wading Potomac River at Point of Rocks, Md 36 v38 Goose Creek near Middleburg 1643700 385911 774749 18 10 10/89-12/94 250' b wading 37 v39 Goose Creek near Leesburg 1644000 390110 773440 18 13 5/91-12/94 1000' b wading 38 v44 Difficult Run near Great Falls 1646000 385833 771446 18 16 9/89-12/94 under bridge wading Potomac River at Chain Bridge at Washington, D. C. 39 v48 Cameron Run at Alexandria 1653000 384823 770636 18 22 12/89-9/94 75' b wading 40 v49 Accotink Creek near Annandale 1654000 384846 771343 18 9 2/85-10/94 100' a wading Occoquan River 41 v52 Cedar Run near Catlett 1656000 383812 773731 18 8 9/89-6/94 150' b wading 42 v65 South Fork Quantico Creek near Independent Hill 1658500 383514 772544 18 10 9/90-8/94 100'D.S bridge from bridge 43 v71 Aquia Creek near Garrisonville 1660400 382925 772602 18 10 8/87-12/94 400'b wading Sources: Water Resources Data Maryland and Delaware Water Year 1993, Volume 1, US Geological Survey Water Data Report MD-DE-93-1 Water Resources Data Virginia Water Year 1993, Volume 1, US Geological Survey Water Data Report VA-93-1 73 Pennsylvania West Virginia Maryland Washington DC Virginia Potomac Ri ver Virgin ia Mary land Potomac River She nan doa h R iver Anacostia River Nor th B ran ch Sou th B ran ch Sou th F orkNort h F ork Sou th For k Nor th F ork Mo noc rac y R ive r SubbasinBoundary N Figure 4.1 Subbasin Locations Source: US Geological Survey Water Resources Investigations Report 96 ? 4034 The geographic coordinates of the 43 gage stations, expressed in degrees, minutes, and seconds of latitude and longitude, were available via the USGS Gage Descriptions and Water-Data Reports published by the USGS for each year covering 1985-1995 (Table 4.1). This information on the gage station description and Water-Data Reports was digitized by the author and converted into decimal degrees. This location data was used as a seed point for subbasin delineation ( Figure 4.1) and flow direction, and flow accumulation calculations for each subbasin. 74 After selecting the 43 gage stations, the next step was to select the specific hydrographic surveys within each subbasin that would best support this study. Because the USGS typically conducts hydrographic surveys at different points along a given channel, the author selected those surveys at the stream location that sustained the highest number of surveys over the 1985 - 1995 study periods. For example, some gage stations have up to 109 measurements and others have considerably less, but most of the selected 43 gage stations had around 30 surveys during the 1985 - 1995 time period. This hydrographic survey information was obtained from the USGS Field Folders, that also contained information about the location of the survey points along the channel for each hydrographic survey measurement, such as ?the survey was made across the channel 100 feet above the gage station or 200 feet below the gage station?. All survey measurements for each gage station were then classified and tallied according to the location of the survey points (Appendix A). The number of survey measurements is shown in Table 4.1 for the selected gage stations. 4.2 Digital Elevation Data The most universally accepted hydrologic models use digital elevation data to calculate flow path, flow accumulation, subbasin boundaries, and a topographic index. The common inputs for these hydrological models are the uniformly spaced digital elevation model (DEM) data that the USGS produces. DEMs are inputs to the Arc/InfoTM Hydrologic Model and topographic index generation AMLTM used by the author. Recently, there have been attempts to use other forms of digital elevation data, rather than the regularly spaced DEM data, such as the Triangulated Irregular Network 75 (TIN) data in hydrological modeling (Azagra, 1999). However, the benefits of using the TIN hydrological model are not conclusive and models using other than DEM data are still maturing. Hence, the Arc/InfoTM Hydrologic Model and topographic index AMLTM were chosen because of their ease of use and because they facilitate easy comparison of study results with numerous published hydrological empirical research that employed regular lattice rather than TIN data structures. For this study, the majority of the DEMs that were used are 7.5 minute by 7.5 minute data with the resolution of 30 meter post spacing. Where there was no 7.5 minute coverage at the 1:24,000 scale DEM, which was for two DEMs, 1:250,000 scale DEMs were used. The 7.5 minute DEMs are derived from stereo-profiling or image correlation and the 30 minute DEMs are derived from Digital Line Graphs of 7.5 minute quadrangle maps produced by the USGS. The finer resolution of DEMs could be better for the delineation and calculation of a topographic index. However, the 30 meter spacing DEMs was sufficient for this research. The rationale is that the smallest subbasin, 0.38 square miles (984,455 square meters), is covered by 31 cells by 31 cells of 30 meter DEMs which is more than the minimum number of cells, 9, needed for a meaningful hydrologic model. 4.3 Hydrographic Survey Data For this study, stream discharge (volume per second), channel width, depth, and flow velocity, as well as cross-section information, were needed. From these data, hydraulic geometry, mid-point of channel, channel asymmetry, channel pattern and channel shape as well as channel profiles were derived. 76 USGS hydrologists observe channel cross-sections from an initial point on one side of the bank at 1 or 2 foot intervals, in most cases, to the other side of the bank to obtain data on cross-section width, mean flow velocity, channel depth, and cross-section area size within a measurement of total discharge. The author converted their field notes to machine readable form. The author accompanied a USGS hydrologist during his field observations to see how the measurements are taken at riffle, pool, and intermediate points. The USGS makes streamflow discharge measurements at gage stations using current-meter measurement methods (Buchanan and Somers, 1969). A current-meter measurement is the summation of the products of the partial areas of the stream cross- section and their respective average velocities. The area extends laterally from half the distance from the preceding meter location to half the distance to the next and vertically, from the water surface to the sounded depth (Figure 4.2). 77 Figure 4.2 Midsection Method of Computing Cross-Section Area for Discharge Measurements Source: Discharge Measurements at Gage Stations: U.S. Geological Survey Techniques, Water Resources Investigation Book 3, Chapter A8 The cross-section is defined by depths at locations 1, 2, 3, 4, ?.., n. At each location the velocities are sampled by a current meter to obtain the mean of the vertical distribution of velocity. The partial discharge is computed for any partial section at location x as qx = vx [(bx ? b(x-1) /2 + (b(x+1) - bx)/2] dx = vx [( b(x+1) ? b(x-1) /2] dx where 78 qx = discharge through partial section x, vx = mean velocity at location x, bx = distance from initial point to location x, b(x-1) = distance from initial point to preceding location, b(x+1) = distance from initial point to next location, dx = depth of water at location x. Thus, for example, the discharge through partial section 4 (heavily outlined in Figure 4.2) is Q4 = v4 [( b5 ? b3 /2] d4. The summation of the discharges for all the partial sections is the total discharge of the stream. Streamflow velocity is measured at a point on each sub-cross-section which consists of 30 - 40 such points per each sub-cross-section measurement. The mean velocity in a vertical section is obtained from observations at many points in that vertical plane. The USGS stores average channel width, depth, and flow velocity as well as flow discharge volume information obtained from each hydrographic survey in digital form as a one-line entry in Electronic 9207. For the 1,132 hydrographic surveys comprising this study (43 subbasins * approximately 30 surveys per subbasin), the author used these data to calculate the hydraulic geometry exponents. As shown in Figure 4.2, an average of 30 segments for each survey were measured from one stream bank to the other and recorded in hand written Discharge Measurement Notes by the USGS. The author obtained and digitized these data and used 79 them to calculate channel mid-point, asymmetry of channel and cross-section profiles. These cross-section profiles were used to characterize channel patterns and channel shapes of each subbasin. The Discharge Measurement Notes data which provide sectional views of the channel are analyzed to identify the channel shape such as shallow and wide or deep and narrow channels, and the asymmetry of the 43 channels. Based on the conditions of flow, such as even or uneven flow, slow flow, cross-section such as rocky or firm, free of leaves or debris, etc., each hydrographic survey measurement was rated by the surveyor as excellent, good, fair, and poor. Excellent is defined as 2 percent error rate; good, 5 percent; fair, 8 percent; and poor, over 8 percent. The majority of the hydrographic surveys that this study used are rated as good or fair. 4.4 Field Surveys To supplement the USGS Water Resource?s Description of Gage Station and Discharge Measurement Notes which describe overall information of bed and bank materials of the subbasin, the author accompanied USGS hydrographers on several surveys and conducted independent field surveys to verify and/or collect bed material and bank material information. These field surveys were visual identifications of the bed and bank material such as clay, sand, silt, gravel, cobble, and rock. Types of bed and bank material for each subbasin were recorded in digital form for statistical characterization and were analyzed in the environmental setting. 80 4.5 Physiography and Lithology The author acquired the physiography and lithology of the study area in the Arc/InfoTM file format from the USGS Geographic Information Scientist, John W. Brakebill, who published a sampling strategy for a water-quality assessment of the Potomac River Basin (Water-Resource Investigations Report 96-4034). Hardcopy forms of physiography and lithology information are readily available through many USGS publications such as Geology of the Chesapeake and Ohio Canal National Historical Park and Potomac River Corridor, District of Columbia, Maryland, West Virginia, and Virginia (Southworth et al., 2001). Now, some of these information are in digital form too. 4.6 Landuse General land use information is readily available through many USGS publications and the Interstate Commission on Potomac River Basin (ICPRB) such as: Potomac River Basin Land Use Data: Evaluation and Methodology to Determine Basin Land Use from Non-Digitized County Land Use Data?ICPRB Report 89-8 (Camacho, 1989); various National Water-Quality Assessment Program reports; and in Geology of the Chesapeake and Ohio Canal National Historical Park and Potomac River Corridor, District of Columbia, Maryland, West Virginia, and Virginia (Southworth, et al., 2001). The land use information was acquired in hardcopy maps from the authors of the USGS Water-Resources Investigations Report 96-4034. The land use information was created by USGS Geographic Information Scientists using mid-1970?s and updated using 81 1990 Census data. The land use type information for the study area was also acquired in Arc/InfoTM file format from the USGS Geographic Information Scientist, John W. Brakebill. 4.7 Derived Data In order to understand the hydraulic geometry values and the effects of subbasins characteristics on the hydraulic geometry, the following data were needed and the author derived them using digital elevation data and hydrographic survey data. They are: (1) subbasin boundaries and sizes, and a (2) topographic index, both of which are covered in this chapter; as well (3) hydraulic geometry, (4) channel pattern, (5) channel shape, (6) cross-section profile, and (7) channel asymmetry, all of which are dealt with in Chapter 5. With DEMs and gage station location information, drainage boundaries for subbasins were delineated. Also flow direction, flow accumulation, and topographic index, also referred to as Wetness Index, a function of upstream contributing area and the slope of the subbasin, were calculated. Using the primary data, specifically hydrographic survey measurement data, hydraulic geometry exponents were calculated and asymmetry of channel, and channel profile (for categorization of channel shape) were derived. 82 4.7.1 Subbasin Boundaries A uniformly and accurately determined drainage area boundary for each of the 43 gage stations is necessary for this hydrologic study, in which the size of the drainage area is considered one of the predictor variables for analyzing hydraulic geometry. A drainage subbasin is an area that drains water and other substances to a common outlet as concentrated drainage. This area is normally defined as the total area flowing to a given outlet, the pour point. The location of a gage station is used as the pour point of each subbasin in this research. This is the lowest point along the boundary of the drainage basin. The USGS Water Resource Division?s annual water year books publish the sizes of the drainage area for each gage station which were derived from digitizing USGS 7 ? -minute topographical maps along divides indicated by contour elevations. 4.7.2 Alternative Method of Drainage Delineation An alternative method, used by this author, is to delineate a drainage subbasin system using algorithms implemented in GIS software like Arc/InfoTM for input of grid- based DEMs. These, an array of elevation values provide an estimate of terrain surfaces useful in computer-based hydrogeomorphological analysis, such as the automated extraction of topographic relief, elevation contours, subbasin stream networks, and subbasin boundaries (O'Callaghan and Mark, 1984; Band, 1986; and Moore et al., 1991). 4.7.3 Digital Elevation Model (DEM) Preparation Two hundred and one DEMs covering 7 ? minute quadrangle areas with 30 meter by 30 meter data spacing, and two DEMs covering 30 minutes by 30 minutes of arc, with 83 100 meter by 100 meter data spacing, were acquired from the USGS EROS Data Center in Sioux Falls, SD. After associated DEMs were manually identified for each gage station?s subbasin boundary, the geographic coordinates of the gage station were converted to UTM Grid coordinates. This was necessary because the computer software required an input of a grid coverage consisting solely of the seed point for the delineation of each subbasin drainage boundary. Many researchers (Mason and Maidment, 2000; Azagra, 1999; and Wolock, 1996) have demonstrated that the accuracy and detail of the hydrologic information which is automatically extracted from a DEM is directly related to the quality and resolution of the DEM itself. DEM data often contain subtle elevation errors which influence watershed delineation in flat areas. In order to resolve this problem, many algorithms condition terrain data. The most common problem with DEMs results from errors in the sampling process which includes false elevations, no data, or artificial peaks and sinks. A sink is a topographic condition in which water collects to a point which has no outfall. For hydrologic analysis, artificial sinks are more worrisome than peaks because they could reduce the number of cells that should be contributing to the drainage area. Therefore, it was necessary to treat the DEMs by filling sinks using the Arc/InfoTM software. 4.7.4 Subbasin Boundary Delineation The author used several tools such as ESRI?s Arc/InfoTM, ERDAS? IMAGINETM, and a couple of commonly used Arc/InfoTM Macro Language routines for viewing and merging DEMs for drainage boundary delineation. These tools provide the capability to 84 determine, for any location in a grid, the upslope area contributing to that point and the down slope path water would follow. The drainage boundary delineation AMLTM routine is based on water flow direction, flow accumulation, and watershed delineation. In addition, the calculation for the slope of the drainage area was added to the Arc/InfoTM hydrologic model via an AMLTM. This was necessary to generate a topographic index (see later) for each subbasin area. Through this AMLTM algorithm, the flow direction, flow accumulation, drainage boundary delineation, topographic index, and size of the drainage areas, were generated for the 43 subbasins in this study of the Potomac River Basin. The following procedures were undertaken for determinations of channel flow direction, flow accumulation, and delineation of subbasin boundaries, and subbasin size. 1. Removed all sinks from the DEM using GRID command FILL, A DEM with depressions (?sinks?) need to be treated by smoothing or filling the depressions. The smoothing approach removes shallow depressions, but deeper depressions remain. A second approach is to ?fill? depressions by increasing the value of cells in each depression to the value of the cell with the lowest value on the depression boundary. 2. Computed flow directions using the GRID command FLOWDIRECTION, Most computer programs that automate the identification of drainage basins use neighborhood functions to determine the direction of flow for each cell in a grid elevation file. One of the complicated problems using DEM is encountering the allocation of an outward drainage direction to the depression points and the drainage basin with no outlet, 85 i.e., the flow lines make a loop or the chain of flow line is broken at the depressed point. Jenson and Domingue (1991) encoded the flow direction for a cell to correspond to the orientation of one of the eight cells that surrounds the cell. Jenson and Domingue?s (1991) approach to delineate watershed boundaries and stream networks from DEM were incorporated through the ArcViewTM extensions Spatial AnalystTM and Watershed DelineatorTM, both distributed by ESRITM (Mason and Maidment, 2000). After filling, the resulting grid was then run through the flow direction process. In this step, the flow direction of each cell in the grid was determined by examination of the elevations in each surrounding cell. Thus, the steepest slope determines the cell?s flow direction. This procedure is to determine the direction of flow from every cell in the grid. The direction of flow is determined by finding the direction of steepest descent from each cell. 3. Computed the flow accumulation (number of cells contributing to a cell ? not actual flow volume or rate) using the GRID command FLOWACCUMULATION, This function uses the newly created flow direction grid to determine the number of upstream cells above each point in the basin. The flow accumulation process counts the number of cells that contribute flow to a cell using the flow direction grid. At any given cell, the drainage area to the cell (but not including the cell) is the product of the flow accumulation value and the cell area. 86 4. Subbasin Boundary Delineation Once the flow accumulation grid was created, the information about each gage site location in latitude and longitude was used as a seed point to delineate watersheds. The seed points must be placed in the proper locations on the flow accumulation grid in order to generate the final results. Considering the accuracies of published gage station locations, snappour 100 meters radius was included in the Process.aml (Appendix B) to allow gage site to be on its own stream. Next, the seed point grid was used as inputs into the watershed function to create the watershed delineations. 4.7.5 Subbasin Drainage Area Size Computation Average area is determined by counting flow accumulation cells. The flow accumulation process counted the number of cells (pixels) that contribute flow to a cell using the flow direction and flow accumulation grid. At any given pixel, the drainage area to the pixel is the product of the flow accumulation value and the grid cell area. Thus, by checking the flow accumulation value at a point, only a simple calculation is needed to find the drainage area. Subsequently, the subbasin drainage area size is calculated as: Drainage Area (mile2) = # of cells * cell size2 (meter2) / 2,589,988 (meter2 / mile2). The USGS published subbasin drainage sizes and the calculated subbasin sizes for the 43 subbasins are listed as Table 4.2. 87 Table 4.2 USGS Published and Calculated Subbasin Size in Square Miles and Miles ____________________________________________________________ ID ID Published (mi2) Calculated (mi2) Perimeter (mi) Difference Percentage 1 m10 247 247.34 108.09 -0.34 99.86 2 m11 875 876.97 285.08 -1.97 99.78 3 m19 494 508.06 248.34 -14.06 97.23 4 m21 18.9 18.56 35.55 0.34 101.83 5 m24 281 281.52 137.23 -0.52 99.82 46 m26 66.9 67.19 59.69 -0.29 99.57 7 m29 31.3 31.31 44.04 -0.01 99.97 8 m30 102 102.63 77.58 -0.63 99.39 9 m31 2.14 2.17 9.2 -0.03 98.62 10 m311 4.01 3.96 12.07 0.05 101.26 11 m32 0.38 0.31 2.94 0.07 122.58 12 m34 817 721.43 204.18 95.57 113.25 13 m37 62.8 63.28 56.52 -0.48 99.24 14 m38 101 102.03 65.54 -1.03 98.99 15 m43 49.4 49.44 59.84 -0.04 99.92 16 m45 39.5 39.49 49.74 0.01 100.03 17 m47 18.5 18.1 29.77 0.4 102.21 18 m48 24 24.58 32.12 -0.58 97.64 19 v07 379 374.01 143.04 4.99 101.33 20 v12 70.1 72.89 63.16 -2.79 96.17 21 v13 375 372.38 138.46 2.62 100.7 22 v15 127 126.56 89.5 0.44 100.35 23 v17 149 150.96 101.31 -1.96 98.7 24 v18 212 211.23 131.27 0.77 100.36 25 v19 1.94 1.96 7.71 -0.02 98.98 26 v22 1084 1077.3 245.21 6.7 100.62 27 v23 1377 1370.3 323.53 6.7 100.49 28 v24 1642 1632.4 394.92 9.6 100.59 29 v25 210 209.97 98.81 0.03 100.01 30 v26 45.5 45.07 50.71 0.43 100.95 31 v27 93.2 94.57 76.76 -1.37 98.55 32 v28 506 508.8 171.28 -2.8 99.45 33 v29 768 771.33 229.56 -3.33 99.57 34 v30 103 101.87 84.24 1.13 101.11 35 v36 89.6 89.69 63.83 -0.09 99.9 36 v38 123 117.88 80.56 5.12 104.34 37 v39 332 325.58 137.9 6.42 101.97 38 v44 57.9 57.75 50.75 0.15 100.26 39 v48 33.7 34.09 41.54 -0.39 98.86 40 v49 23.5 23.82 36.29 -0.32 98.66 41 v52 93.4 93.39 68.78 0.01 100.01 42 v65 7.64 7.65 18.07 -0.01 99.87 43 v71 34.9 35.25 40.8 -0.35 99.01 ______________________________________________________________________________ Source: Water Resources Data Maryland and Delaware Water Year 1993, Volume 1, US Geological Survey Water Data Report MD-DE-93-1, and author calculated 88 Figure 4.3 (a) Delineated Subbasins with Gage Stations Source: USGS Water Data Reports MD-DE-93-1 and VA?93-1 and Author-Delineated Subbasin Boundary 89 Figure 4.3 (b) Delineated Subbasins with Gage Stations Source: USGS Water Data Reports MD-DE-93-1 and VA?93-1 and Author-Delineated Subbasin Boundary 90 Figure 4.4 Study Area Subbasin Groups Note: Not to Scale 91 The comparison of the USGS reported subbasin size values and the author- calculated subbasin size values shows that, except for four subbasins, the differences in size values were within an acceptable range, above 97%. For the subbasin with the largest size discrepancy (122.6% difference), subbasin m32, the actual difference of only 0.07 square miles (published value of 0.38 square miles versus the calculated subbasin size of 0.31 square mile), was considered insignificant to this analysis of subbasin size impacts on hydraulic geometry. The rest of the subbasins? size discrepancies were also treated as insignificant for this analysis. The rationale was that even considering the discrepancies; this does not have an impact on size rankings of the subbasin with respect to one another. An exception is subbasin m34 (discrepancy of 113.3%) where the calculated subbasin size results in a smaller subbasin than subbasin v29. However, if published size is corrected and is used for analysis, m34 is larger than subbasin v29. Therefore, these two subbasins get special attention in the size analysis in Chapter 6. 4.7.6 Topographic Index Topography is recognized as an important factor in determining the stream flow response of drainage (Kirkby and Chorley, 1967; Dunne, et al., 1975; Beven and Wood, 1983; Moore et al., 1991; Wood, et al., 2000). Topography determines the effects of gravity on the movement of water in the drainage, and therefore it influences many aspects of geomorphic and hydrologic systems. The energy of stream flow is determined by the slope of the surface. A steeper slope results in greater energy, and as the energy of a stream increases, its ability to transport more and larger particles also increases. 92 Therefore, steeper slopes result in a greater potential for erosion. Topography has been shown to affect the flow path that precipitation follows before it becomes stream flow (Wolock, et al., 1990). And the shape of a surface determines how water will flow across it. Therefore, characterization of the topography of the subbasins should provide a significant dimension of information about the environmental setting of subbasins. Kirkby and Weyman (1974) first introduced the concept of a topographic index, k = a / tan ?, where a is the area draining through a point from upslope and tan ? is the local slope angle (Beven, 1977). The topographic index has been used (Wolock et al., 1995) to study a range of topics including flood frequency derivation (Beven, 1986, 1987), analysis of spatial scale effects in hydrologic processes (Beven et al., 1988; Famiglietti and Wood, 1991; Wood et al., 1988, 1990); analysis of the topographic effects on streamflow (Beven and Wood, 1983; Beven et al., 1984; Kirkby, 1986). This index is used as an index of hydrological similarity (Wolock and McCabe, 1995) and one of the first parameters directly considering the spatial distribution of topography in hydrological modeling. All points with a similar index value are assumed to respond in a hydrologically similar way to hydrological processes (Beven and Kirkby, 1979; Hornberger et al., 1985; Wolock et al., 1989, 1990, 1995; Wolock, 1997). Simplicity and the possibility of visualizing the predictions of the model in a spatial context enabled a topographic index to become a key component of the TOPMODEL, a topography-based watershed hydrology model (Beven and Kirkby, 1979). In part, because of this wide usage, the author selected topographic index as one of the predictor variables to investigate the impact of environmental settings on hydraulic geometry. 93 Since the topographic index characterizes ability to transport discharge of the subbasin, and the discharge can be equated to the drainage size, the topographic index is an important indicator in this study of environmental setting effects on hydraulic geometry. The modatn.aml algorithm, which was written by Wolock (1995) and modified by the author and used as a part of the inserted in the Arc/InfoTM , produced topographic indices for the study subbasins. Higher values of ln (a/tan ?) indicate greater potential for development of saturation. High values of ln (a/tan ?) occur at locations where large upslope areas are drained (high value of a) and where the local gravitational gradient is low (low value of tan ?). Once the topographic index values are computed, the minimum, maximum, and mean topographic index values with standard deviations are tabulated for each subbasins. The topographic index values, maximum, mean, minimum, and standard deviation of each subbasin are shown in Table 4.3. 94 Table 4.3 Topographic Index Values for Study Subbasins ________________________________________________________________ ID ID USGS ID Topographic Index Minimum Topographic Index Maximum Topographic Index Mean Topographic index stdev 1 m10 1601500 3.165 20.624 6.828 1.991 2 m11 1603000 -3.578 21.89 6.965 2.023 3 m19 1614500 -3.588 21.344 7.588 2.125 4 m21 1617800 2.952 20.754 7.427 2.168 5 m24 1619500 4.404 18.557 7.935 1.994 6 m26 1637500 4.163 15.878 7.112 1.722 7 m29 1639140 4.317 12.207 6.812 1.434 8 m30 1639500 -3.59 21.695 7.644 2.075 9 m31 1640965 3.711 19.261 7.17 2.134 10 m311 1640970 3.219 19.392 7.438 2.018 11 m32 1640980 -2.393 19.014 7.448 2.055 12 m34 1643000 1.475 18.789 7.325 2.181 13 m37 1643500 4.995 18.315 8.212 1.809 14 m38 1645000 3.912 18.035 7.548 2.177 15 m43 1651000 3.47 19.321 7.087 2.076 16 m45 1653600 3.544 19.745 7.463 2.1 17 m47 1661050 3.584 16.491 7.277 1.96 18 m48 1661500 3.807 18.009 7.04 2.148 19 v07 1622000 2.832 21.033 6.868 2.143 20 v12 1624800 3.507 19.403 6.967 2.309 21 v13 1625000 1.2 21.033 6.997 2.287 22 v15 1626000 2.996 19.954 6.923 2.286 23 v17 1626850 2.996 20.13 6.934 2.288 24 v18 1627500 2.974 20.467 6.982 2.285 25 v19 1628060 3.551 15.779 6.048 1.801 26 v22 1628500 1.2 22.093 6.998 2.236 27 v23 1629500 1.2 22.335 6.982 2.236 28 v24 1631000 1.2 22.509 6.972 2.239 29 v25 1632000 2.567 20.46 6.396 2.128 30 v26 1632082 3.667 18.922 7.173 2.18 31 v27 1632900 2.742 19.663 7.158 2.137 32 v28 1633000 2.567 21.251 6.867 2.182 33 v29 1634000 2.567 21.761 6.964 2.186 34 v30 1634500 3.387 19.701 6.885 2.078 35 v36 1638480 3.584 19.61 7.406 2.136 36 v38 1643700 3.501 19.88 7.096 2.087 37 v39 1644000 3.368 20.897 7.284 2.11 38 v44 1646000 4.002 19.17 7.343 2.078 39 v48 1653000 4.099 18.643 7.592 2.035 40 v49 1654000 4.327 18.284 7.567 2.008 41 v52 1656000 3.832 19.65 7.635 2.082 42 v65 1658500 4.3 17.148 7.539 1.966 43 v71 1660400 4.078 18.676 7.551 2.062 Source: Calculated by the author 95 The maximum topographic index value ranges from 22.509 to 12.207 (subbasin m29). The low minimum value occurs on subbasins m11, m19, m30, and m32 as -3.578, -3.588, -3.59, and -2.393. The highest maximum topographic index value occurs on subbasins v23 and v24 as 22.335 and 22.509. These two subbasins, v23 and v24, show low minimum topographic index values. In this chapter, discussion of needed data such as DEM, hydrographic survey data, field survey data, data for physiography and lithology, and land use information, including derived data such as subbasin boundary delineation and topographic index values, for the environmental setting?s impact on hydraulic geometry on Potomac River Basin have been presented. Derived data, particularly the delineation of subbasins and topographic indices, were created using 30-meter spacing DEM data for the study area, are described here. The rest of the derived data such as channel pattern, channel shape, channel cross-section profiles, and channel asymmetry are discussed in the next chapter. 96 CHAPTER 5 HYDRAULIC GEOMETRY AND THE ENVIRONMENTAL SETTING: Computation of Hydraulic Geometry Exponents To understand the at-a-site hydraulic geometry of the study sites in the Potomac River basin, a first step in this analysis was to derive the hydraulic geometry exponents, b, f, and m, for each site from hydraulic survey data. In addition, to augment characterization based on hydrologic principles and empirically tested theory, a categorization of the b, f, and m values was conducted. Secondly, to provide a basis for subsequent investigation of the statistical relationships between watershed (subbasin) and hydrologic variables, an array of variables such as channel pattern, channel shape, cross- section profile, and channel asymmetry was constructed from field-based hydrologic survey data collected during this study. In this chapter, the methodologies of calculation and classification of at-a-station hydraulic geometry of the study area are discussed to identify how channel width, depth, and stream flow velocity change due to increased stream discharge volumes. Thirdly, methodologies to collect and compute all other predictor variables such as channel patterns, channel shapes, channel symmetry, and environmental controls such as bank and bed material, land use, and water reference level, listed in Chapter 4, are discussed in this chapter (Section 5.5). 5.1 Potomac River At-A-Station Hydraulic Geometry Hydraulic geometry is concerned with variations in width, depth, velocity, and other hydraulic and geometry parameters such as channel slope in response to changes in channel discharge volume. The hydraulic geometry exponents, b, f, and m, provide 97 quantitative descriptions and data on the variations of the rate of changes in channel width and depth, and stream flow velocity in different environmental settings. Careful collection and analysis of this information can provide insight into the relevant controlling conditions for specific channels. This research paper deals with at-a-station rather than downstream hydraulic geometry. Downstream hydraulic geometry refers to changes at different locations along a stream channel, or between different channels at discharges of comparable frequencies. At-a-station hydraulic geometry is the response of a given cross-section of a channel to imposed flows of water and sediment regardless of discharge frequencies. So, at-a- station hydraulic geometry predicts the changes in water width, depth, and velocity as discharge varies within a channel of imposed size and shape, whereas downstream hydraulic geometry predicts the adjustment of channel size and shape to an imposed discharge. Fluvial geomorphologists are primarily interested in hydraulic geometry because of its implications for drainage network evolution, stream channel response to environmental change, and mutual adjustments between channels and the flows they carry. River engineers are concerned with hydraulic geometry (or river regime, as it is sometimes called) for its applications in channel design, predictions of channel failure, and river channel management. This research is concerned with natural channels in the Potomac River Basin. The term 'natural' here is used to differentiate stream channels from ditches, canals, or aqueducts completely constructed by man but do not exclude natural channels altered by man. 98 Geometric characteristics of stream channels are usually viewed in the three perspectives of cross-section, profile, and plan. For purposes of study efforts on stream discharge, this study considers only the cross-sectional channel relationships to streamflow. The three parameters of water surface width, W, mean hydraulic depth, D, and average velocity, V, are related to streamflow, Q, through power functions (Leopold and Maddock, 1953). W = aQb (1) D = cQf (2) V = kQm (3) These relations should satisfy the continuity equation (Q = AV) so that Q = WDV (4) Q = (aQb) (cQf) (kQm) = (ack) Q (b+f+m) (5) The exponents sum to unity (b + f + m = 1.0). The constants a, c, and k vary from locality to locality but the exponents, b, f, and m, exhibit a remarkable degree of consistency. The exponents, b, f, and m, indicate rates of changes in width and depth of channel and velocity of the flow with respect to changing stream discharge. This study focuses on only these hydraulic exponents. The exponents b, f, and m were found to vary somewhat between river systems in 'downstream' analyses but normally b > f > m, with typical values of 0.5, 0.4, 0.1. At-a- station exponents are more variable but typically f > m > b, with mean values of 0.40, 0.34, and 0.26 in Leopold and Maddock?s study (1953). 99 For this calculation of at-a-station hydraulic geometry, the author used field survey measurement data which were created by hydrographers from the USGS Water Resource Division, Maryland State Geological Survey, and Virginia's Environmental Quality Department. Field surveying produced data on discharge, mean depth, width of the water surface, mean velocity of the flow, and cross-sectional area size measurements. This location-specific measurement data were used for the calculation of the rate of changes for width, b; depth, f; and flow velocity, m. The relationships between discharge with width, depth, and velocity were estimated using logarithmic regression analysis. Each predicted variable (b, f, and m) is regressed onto the predictor variable, water discharge, individually. The resulting regressions are b to water discharge, f to water discharge, and m to water discharge; therefore there is no guarantee that the summation of estimated b, f, and m would be 1. The coefficients and exponents in equations (1), (2), and (3) for the 43 at-a-station sites are presented in Table 5. 1 which lists the summation of b, f, and m, and the coefficients of determination for b, f, and m as R2b, R2f, and R2m, respectively. 100 Table 5.1 b, f, and m Value Id Sample Size b f m b+f+m R2b R2f R2m Code* m10 29 0.1416 0.3605 0.4981 1.0002 0.9131 0.9219 0.9518 W m11 21 0.1084 0.5349 0.3567 1.0000 0.7935 0.9504 0.9017 W m19 45 0.0989 0.3936 0.5078 1.0003 0.7106 0.9803 0.9863 W m21 47 0.1699 0.3841 0.4467 1.0007 0.6658 0.8829 0.8233 M m24 37 0.2441 0.1891 0.5661 0.9993 0.7892 0.5779 0.9186 M m26 25 0.3477 0.2669 0.3842 0.9988 0.8330 0.8586 0.9366 M m29 13 0.2250 0.2597 0.5119 0.9966 0.8202 0.9504 0.9593 W m30 43 0.1121 0.3941 0.4962 1.0024 0.5477 0.9011 0.9431 W m31 38 0.4607 0.2403 0.2895 0.9905 0.8976 0.6526 0.6337 P m311 18 0.5622 0.1442 0.2976 1.0040 0.8572 0.6398 0.5965 P m32 17 0.2972 0.0906 0.6336 1.0214 0.7895 0.3591 0.9464 P m34 43 -0.0080 0.4667 0.5736 1.0323 0.0430 0.9094 0.9351 W m37 26 0.0400 0.3581 0.6011 0.9992 0.5444 0.6955 0.8422 P m38 28 0.0650 0.4289 0.5113 1.0052 0.2848 0.6401 0.7579 P m43 56 0.2741 0.2663 0.4630 1.0034 0.5398 0.6137 0.6852 P m45 23 0.1868 0.4474 0.3668 1.0010 0.7062 0.9019 0.8661 M m47 54 0.2769 0.2784 0.4488 1.0041 0.7572 0.5859 0.7215 P m48 64 0.1000 0.2326 0.6684 1.0010 0.5985 0.6546 0.9483 P v07 33 0.0340 0.3570 0.6072 0.9982 0.6629 0.9822 0.9937 W v12 36 0.1302 0.4717 0.3992 1.0011 0.1673 0.9202 0.6548 P v13 9 0.2286 0.2081 0.5632 0.9999 0.6505 0.2715 0.8763 P v15 13 0.0906 0.2665 0.6411 0.9982 0.5794 0.5553 0.9231 P v17 25 0.0507 0.3832 0.5697 1.0036 0.2590 0.8192 0.9348 M v18 18 0.1761 0.5368 0.2872 1.0001 0.7248 0.8336 0.4825 P v19 26 0.3773 0.2978 0.3461 1.0212 0.8283 0.8779 0.5866 M v22 9 0.0573 0.3363 0.6063 0.9999 0.5277 0.9558 0.9653 W v23 17 0.2195 0.3008 0.4797 1.0000 0.2939 0.7433 0.6886 P v24 64 0.0839 0.2954 0.6208 1.0001 0.4869 0.9316 0.9428 W v25 16 0.3390 0.4160 0.2443 0.9993 0.8090 0.9053 0.5166 M v26 17 0.1865 0.2601 0.5550 1.0016 0.4982 0.6006 0.8906 P v27 22 0.1437 0.2687 0.5870 0.9994 0.5452 0.4415 0.8253 P v28 22 0.2729 0.1612 0.5681 1.0022 0.7521 0.4234 0.8501 P v29 45 0.0075 0.2406 0.7541 1.0022 0.0744 0.7411 0.9702 P v30 17 0.2061 0.1703 0.6242 1.0006 0.7256 0.7549 0.9596 P v36 17 0.2498 0.2162 0.5363 1.0023 0.8083 0.6485 0.8392 M v38 10 0.4398 0.3903 0.1676 0.9977 0.7857 0.9425 0.2271 P v39 13 0.1930 0.2523 0.5559 1.0012 0.6237 0.9454 0.9340 W v44 16 0.1671 0.3015 0.5314 1.0000 0.8933 0.6869 0.8603 M v48 22 0.1374 0.3277 0.5730 1.0381 0.5222 0.8102 0.8190 M v49 9 0.3637 0.2618 0.3754 1.0009 0.6266 0.7312 0.7599 P v52 8 0.4685 0.3194 0.2123 1.0002 0.9202 0.8726 0.6361 M v65 10 0.3515 0.3744 0.2646 0.9905 0.8479 0.8321 0.4832 M v71 10 0.2404 0.3590 0.4032 1.0026 0.9085 0.9050 0.8491 W Note: Calculated using an SplusTM function. Well (W), Moderately (M), or Poorly (P) constrained are based on at least 2 exponents having R2 of 0.9 (W), 0.8 (M), and lower (P). 101 The fact that the summation of the three slopes (exponents) is a unit was used as a means to check the validity of the derivation of hydraulic geometry relationships as developed by Leopold and Maddock (1953). In addition, the fact that the product of width, depth, and velocity equals to discharge was also used to check the validity of the field-observed hydraulic geometry data. Also, using the R2 values of the b, f, and m, each subbasin is categorized as W - well constrained (at least two exponents have R2 of 0.9); M - moderately constrained (at least two exponents have R2 of 0.8), and P - poorly constrained (other than W or M). Out of the 43 study sites, two sites show the sum of three exponents as 1.038 and 1.032, and two sites show 1.0214 and 1.0212. The rest of the sites are all within ? 0.01 from the unit. So 95% of the total study sites fall in ? 0.03 from the unit. And 93% of the total study sites fall in ? 0.02, 88% within ? 0.01. It is expected that the summation of b, f, and m would be 1, however, for some subbasins the summations for b, f, and m fall in ? 0.04 of 1. These subbasins have low coefficients of determination (R2) in b, f, or m; therefore, their b, f, and m values might not be highly trustworthy. However, overall, the summation of b, f, and m, very close to 1, reasonably well meets the continuity equation of Leopold and Maddock (1953). In addition, the coefficients of determination (R2) for the power relationships between discharge and width (R2b), depth (R2f), and velocity (R2m) were reviewed for the appropriateness of the calculation of each subbasin?s b, f, and m values. The results, in Table 5.1, show the average value of coefficients of determination for b is lowest, and that of f is in the middle, and that of m is highest. It is interesting that b, f, and m values are also in this same order, but further reflection is delayed until Chapter 6. 102 The coefficient of determination for the power relationships between discharge and width (R2b) shows the least correlation among the three coefficients, b, f, and m. The average coefficient of determination for width exponents among the 43 study sites is 0.64. Six subbasins have R2b lower than 0.5 and 8 subbasins have R2b at the 0.5 level, 4 at the 0.6 level, 7 subbasins have R2b at the 0.7 level, 12 at the 0.8 and 6 at 0.9 level. Table 5.2 Coefficient of Determination for Width Exponent (b) R2b Number of Subbasins Accumulated Number of Subbasins < 0.5 6 43 0.5 8 37 0.6 4 29 0.7 7 25 0.8 12 18 0.9 6 6 Source: Calculated and observed by author Note: R2b denotes coefficient of determination for width exponent (b) Cumulatively, 58% of the total study sites (22 out of 43) have R2b at the 0.7 and 68% of the total study sites have R2b at the 0.6 level. From these statistics, we might expect that the wide range of rates of changes in width (low R2b value) requires that the various environmental settings of subbasins needs to be investigated. This is also discussed in Chapter 61. One case is worthy of special note. The gage station m34 has a negative value of b (-0.008) with a coefficient of determination of 0.043, and, hence, m34 falls outside the ternary diagram. It is assumed that the b value of m34 is 0.000 for this 1 One case is worthy of special note. The gage station m34 has a negative value of b (-0.008) with a coefficient of determination of 0.043, and, hence, m34 falls outside the ternary diagram. It is assumed that the b value of m34 is 0.000 for this study. This station has b, f, and m summation of 1.032 which is the second highest discrepancy value from 1. The coefficients of determination for f and m are 0.904 and 0.9351, respectively, which indicates that f and m are highly correlated. 103 study. This station has b, f, and m summation of 1.032 which is the second highest discrepancy value from 1. The coefficients of determination for f and m are 0.904 and 0.9351, respectively, which indicates that f and m are highly correlated. The coefficient of determination for the power relationship between discharge and depth (R2f) shows better correlation than that of width (R2b). The average coefficient of determination for depth exponents among the 43 study sites is 0.76. Four subbasins have R2f lower than 0.3 and three subbasins have R2f at the 0.5 level, nine at 0.6 level, four subbasins have R2f at the 0.7 level, eight at the 0.8, and 15 at the 0.9 levels. Table 5.3 Coefficient of Determination for Depth Exponent (f) R2f Number of Subbasins Accumulated Number of Subbasins < 0.5 4 43 0.5 3 39 0.6 9 36 0.7 4 27 0.8 8 23 0.9 15 15 Source: Calculated and observed by author Note: R2f denotes coefficient of determination for depth exponent (f) Cumulatively, 63 % of the total study sites (27 out of 43) has R2f at 0.7 and 84 % of total study site has R2f at 0.6 level. From the statistics of R2b and R2f we can expect that depth change rate is much better explained than width change rate. The coefficient of determination for depth exponents values are much higher than that of width exponents meaning that variation of depth exponents are much less and proportion of variation left unexplained are smaller than that of width exponents. 104 The coefficient of determination for the power relationship between discharge and velocity (R2m) shows the highest correlation among the three coefficients. The average coefficient of determination for velocity exponents among the 43 study sites is 0.81. One subbasin has R2m lower than 0.5, three subbasins have R2m at the 0.5 level, four for 0.6, four subbasins have R2m at the 0.7 level, eight at the 0.8 level, and 23 at the 0.9 level. Out of 23 subbasins which have R2m higher than 0.91, seven subbasins have a greater than 0.95 correlation coefficient. Table 5.4 Coefficient of Determination for Velocity Exponent (m) R2m Number of Subbasins Accumulated Number of Subbasins < 0.5 1 43 0.5 3 42 0.6 4 39 0.7 4 35 0.8 8 31 0.9 23 23 Source: Calculated and observed by author Note: R2m denotes coefficient of determination for velocity exponent (m) Cumulatively, 81% of total study sites have R2m at the 0.7 level, 91% of total study sites has R2m at the 0.6 level, and 98% of the total study sites (42 out of 43) have R2m at the 0.5 level. The coefficient of determination for velocity exponents values are higher than that of width or depth exponents, meaning that the variation of velocity exponents is much less than that of width or depth exponents. So, we can assume that the ranges of width and depth changes rates are wide and validity of b and f are less than m. The average exponent values calculated by the author for this study are b: 0.207, f: 0.314, and m: 0.481. These average values fall within the wide range of Park (1977), 105 Rhodes (1978), and Leopold (1994) studies. The sample of 139 sets of exponents which Park (1977) used and the 587 sets of exponents computed by Rhodes (1978) indicate that the exponents ranges are b: 0.00 to 0.84; f: 0.01 to 0.84; m: 0.03 to 0.99; and Leopold (1994) indicates b: 0.26; f: 0.40, and m: 0.34 as the most common values for exponents. However, the values to be used in the present analysis are those for individual cross- sections and not the averaged values from a particular channel. The rationale for discarding the average exponent values is that the study channels are in heterogeneous environmental setting and they differ so much in their responses to changing discharge that the idea of an average condition is itself of doubtful utility. Respective coefficients of determination are 0.642, 0.763, and 0.810. There appears to be a relationship between hydraulic geometry exponent values and the coefficients of determination. That is, the lower the hydraulic geometry exponents, the lower the coefficients of determination. 5.2 A Ternary Diagram Portrayal of the Streamflow Exponents While numerical values of b, f, and m provide some sense of the rate of width, depth, and velocity changes with changing discharges, these three numbers at-a-station do not intuitively show the relative proportions of the three components. Following Leopold (1994), Rhodes (1978), and Park (1977), a ternary plot (Figure 5.1) of the exponents provides a means of examining and comparing the simultaneous variations of 106 all three exponents. A ternary diagram2 plot also provides more information than simple numerical comparisons of the values. Since the plotting position of a set of exponents supplies information on the results of hydrological principles on the cross-section, the ternary diagram is a very useful tool to understand how a channel adjusts itself to morphologic and dynamic factors such as changing discharge. Most of the hydraulic geometry exponents are clustered in the lower left area of the ternary diagram which is the area of 0.3 to 0.0 for b, 0.2 to 0.4 of f value, and 0.5 to 0.6 of m value. In order to derive more information from these b, f, and m values, the ternary diagram of the hydrologic geometry is divided on the basis of hydrologic principles and empirically tested theories as follows. Similarity of hydraulic geometry data can be inferred based on the plotted position on the ternary diagram. However, as Rhodes (1977) reported, the usefulness of the plot and the possible inferences derived from it may be increased by meaningful division of the diagram. 2Note: Ternary diagrams are used to represent the relative fraction of the three components. The "composition" of any point plotted on a ternary diagram can be determined (or any point can be plotted). Fractional values for b are read from zero along the lateral line (axis) at the left of the diagram to 1 at the vertex of the triangle. Similarly, values for f are read from zero along the lateral line of right side of the triangle to 1 at the lower right vertex. And, finally, values for m are read from zero along the lower right vertex to 1 at the lower left vertex. At any point in the triangle, the distances to b, f, and m must add up to 1.0. Along the line connecting b and f, f must be zero. At any apex, one component is 1.0 while both others are zero. Often the requirement of the three components have to sum to unit 1 which meant that some sites needed to be normalized them to 1, may have to be met by rounding. However for the values of this study, using 2 decimal places, no normalization was necessary. 107 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.9 0.8 0.6 0.5 0.4 0.3 0.2 0.1 0.7 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 ?10 ? ? ?41 369 ?25 ?42 ?34 ?40 ?5 ?32 ?29?6 ?11 ?17 ?21 ?43?7 ?27 ?16 ?15 ?24? 30 ?35 ?37 ?4 ?31 ?1?38 ?2 ?23 ?8 ?39 ?18 ?20 ?3 ?14 ?13 ?28?22 ?19?33 ?26 b f m Figure 5.1 Ternary Diagram for Hydraulic Geometry Exponents Values of Potomac River Subbasins Accordingly, the author divided the ternary diagram for the Potomac River subbasin into ten regions (Figure 5.2). This follows Shepard (1954) who used ten regions for classification of the Maryland coastal sediment samples, based on Rhodes? (1978) five criteria, which are themselves based on their dynamic and morphological responses to changing discharge. The dividing lines (Figure 5.3) for the five criteria are: (1) constant values of width-depth ratio (b = f), (2) competence (m = f), (3) Froude number (m = f/2), (4) velocity-cross-sectional area ratio (related to the Darcy-Weisbach friction factor) (m = b + f), and (5) slope-roughness ratio which is related to the Manning equation (m/f = 2/3). 108 A B C D E F G H I J 0.5 0.5 0.5 0.4 0.33 Figure 5.2 Ten Regions Determined by Five Criteria Figure 5.3 Five Criteria 5.3 Explanation for the b-f-m Values Understanding of the ramification of b, f, and m values is facilitated by an examination of the classification of b, f, and m values, which reveals similarities of response to changes in discharge. Each exponent is presented separately for the five situations shown in Figure 5.2. The overall classification follows as section 5.4. 5.3.1 Ratio of Width and Depth Exponents (b = f) The b-to-f ratio gives an indication of the change in shape of the channel cross- section with changing discharge. The width-to-depth ratio of a channel varies with 109 increasing discharge, manifested as channel stability. Rhodes (1977) rationalized that since w = aQb, and d = cQf, therefore w/d = aQb/ cQf = a/c Qb-f. Eliminating the constants, it becomes w/d ?Qb-f. Consequently, the ternary diagram is divided by the line where b = f line, as shown on Figure 5.3 (1), resulting in b > f and b < f regions. The b? to?f ratio indicates changes in the width-depth relationship and not the absolute values of the parameters. A cross-section may have a relatively large width-to-depth ratio at all stages even though f > b. Changes in width and depth with increasing discharge may occur either by erosion of banks and bed or by progressive filling of a stable channel. Interpretation of this ratio provides information about (1) relative stability of channel via bed and bank material erodability; (2) channel shape, and (3) channel adjustment to transport of bed load (Rhodes, 1977). Channel shape and the relative stability of bed and bank materials are interrelated (Schumm, 1960). Where b > f, the stream is considered that the banks of the stream are more readily eroded than the bed, and where f > b, the stream is implied that bed is more eroded than the banks of the stream. If the channel is stable under normal discharge conditions, then the situation where f > b indicates a tendency for the channel cross- section to become proportionally deeper and narrower with respect to increasing discharge (lower w/d). If b > f, then width-to-depth ratio increases with discharge, and the tendency is toward a wide shallow channel. For stable channels, only those with the shape of an equilateral triangle will have a constant width-to-depth ratio (b = f) with increasing discharge (Rhodes, 1977). Rectangular and parabolic channels will have decreased 110 width-to-depth ratios extreme case b = 0 and f and m make unit 1. The rate of decrease is greater for rectangular channels than for the other shapes (Leopold, 1994). The change of width-to-depth ratio with discharge is an important consideration in fluvial sediment transport. Rhodes (1977) stated that several workers have noted that the wide-shallow cross-section is the type best adapted to the transport of a large bed load (Lane, 1937; Mackin, 1948; Leopold and Maddock; 1953; Morisawa, 1968). He stated that channels that plot on the above the line half of the ternary diagram (Figure 5.3 (1)) should have a greater potential for bed load transport than those channels that plot on the below half other factors being equal. On the other hand, Leopold and Maddock (1953) stated that deep, narrow channels are associated with larger suspended sediment loads than are wide, shallow channels. At a given discharge, different rivers exhibit different values of width and depth. Those with fine-grained cohesive banks tend to be deep and narrow, and those with sandy, friable banks tend to be wide and shallow (Rhodes, 1977). Thus, those channels that plot on the upper half of first ternary diagram shown in Figure 5.3 (1) may have shapes that are best adapted to the transport of fine-grained sediments. 5.3.2. Ratio of Depth and Velocity Exponents for Competency (m = f) After Rhodes?(1977) classification , the criterion for the second division of the ternary diagram is based on the ratio of depth and velocity change rates. The inferences as to its significance are based on empirical studies and not on rigorous hydrodynamic principles. This ratio has been cited as important in considerations of sediment transport 111 by Rhodes (1977). The line of m/f = 1 is drawn on the ternary diagram as shown on the second triangle in Figure 5.3 (2). Wilcock (1971) studied the relationships between the rate of change of depth (f) and that of velocity (m), and competence. Based on considerations of a critical tractive force necessary for the initiation of particle motion, Wilcock (1971) concluded that, "Competence will tend to increase only when the rate of increase in velocity (m) equals or exceeds the rate of increase in depth (f).? Based on this assertion, Rhodes (1977) stated that those channels that are represented by points to the left side of the line m = f on Figure 5.3 (2) should experience an increase in competence with increasing discharge. Leopold, Wolman, and Miller (1964) stated ?The ratio m/f can be related to the transportation of load through the interdependence of the various channel factors?. Their finding from measurement data indicate that the higher m/f ratio, the more rapid the increase of measured sediment load with increase of discharge. The ratio of the rate of change of velocity to the rate of change of depth has been associated by Leopold and Maddock (1953) with the rate of increase of suspended load (Rhodes, 1977). They stated ?For a given width, and at a given discharge, an increase in suspended sediment requires an increase in velocity and a reduction in depth ". That is for conditions of increasing discharge, velocity must increase faster than depth, or m/f must be greater than 1, if suspended sediment load is to increase (at a constant width). If width changes with discharge, which it usually does, the relationship is complicated. The rate of increase of suspended sediment with discharge is inversely related to the rate of increase of width (Leopold and Maddock, 1953). 112 5.3.3 Ratio of Half of Depth and Velocity Exponents (m = f / 2) Hydraulic geometry values also were classified using Rhodes? (1977) third criteria which is based on hydrodynamic considerations, water flow condition usually expressed as the Froude number, F, which describes the type of flow present. The Froude number, the ratio of an inertial force to gravitational forces, is defined as F = v / (gd)1/2 . Since v = k Qm , and d = cQf, F = k Qm / (g (cQf))1/2. At the critical value F = 1, v / (gd)1/2 = 1, v = (gd)1/2 , and k Qm = g1/2(cQf)1/2. Eliminating the constants, Qm ? Qf/2 therefore, m = f/2 if the Froude number does not change with discharge. When m > f/2 (points left of the line), the Froude number increases with increasing discharge; if m< f/2 (points right of the line), F decreases with increasing discharge (Figure 5.3 (3)). According to Rhodes? (1977) explanation, if the geometry of a channel causes the velocity exponents to decrease with increasing discharge such as case v38, then that channel may be incapable of sediment transport (Simons et al., 1965), because the critical Froude number will not be attained. Transportation of large particles would require supercritical flow. Channels that have this relationship will plot at the left side of the line m = f/2. Many of the streams with such graph positions may never achieve supercritical flow because the channel dimensions would be exceeded before the necessary criteria could be met. However, it is clear that those channels that plot to the right side of the line will not reach supercritical flow on a rising stage and will therefore be unable to accomplish some types of sediment transport. 113 5.3.4 Rate of Velocity and Depth Exponents for Channel Stability and Flow Resistance (m = b + f or m = 0.50) Following Rhodes? (1977) criteria, the fourth division of the diagram is based on ratio of velocity and combined width and depth exponents (m / b + f), the line m = b + f or m = 0.50. This criterion is based on Rhodes? understanding that if velocity increases faster than the area of the channel cross-section that means the resistance must decrease rapidly with increasing discharge. This notion is based on Richards? (1973) statement "a very rapid increase in velocity with increasing discharge (m > 0.5) is associated with a rapid decrease in f'." The symbol f' represents the Darcy-Weisbach friction factor. The condition m > b + f indicates that the mean velocity is increasing faster than the cross- sectional area (channel width and channel depth) of the channel. If velocity is increasing at the same rate as cross-sectional area, then m = b + f. In order for velocity to increase faster than channel area, channel must be quite stable (Rhodes, 1977). Rhodes? assumption is that neither the bed nor banks are subject to significant erosion by the velocities attained. He also noticed that the very high rates of f are associated with stable channels. So he used the rate of increase of the width-to- depth ratio (b / f) for channel stability (decreasing w/d ratios indicating stable sections). Points plotting the left side of the m < b + f, that is m < 0.5 line represent channels that experience very rapid increase in velocity with discharge. From such rapid increases several inferences may be made regarding channel stability and flow resistance. Channels characterized by m > b + f are quite stable and experience greatly decreased resistance with increased discharge. 114 5.3.5 Ratio of Depth and Velocity Exponents for Roughness (m/f = 2/3) Following Rhodes? (1977) criteria, the fifth division was based on the Manning roughness coefficient. The b-f-m diagram is divided by the line m/f = 2/3. The Manning equation is: v = 1.486 d2/3 s1/2 / n, where s is water surface slope, n is Manning roughness coefficient (Rhodes, 1977). Slope has been defined as S ? Qz and the Manning coefficient as n ? Qy (Leopold, Wolman, and Miller, 1964). The equation can be rewritten as: kQm = 1.486 (cQf)2/3 (Q z)1/2 / Qy = 1.486 (Q 2/3 f + ? z ?y) Qm ? Q(2/3)f + (1/2) z-y, or m = (2/3)f + z / 2 ? y If m > (2/3) f (left side of the line), then z/2 ? y < 0, or s1/2/n must decrease with discharge. Likewise, if m < (2/3) f (right side of the line), then z/2 - y > 0, or s1/2/n must increase with increasing discharge. If m = (2/3) f, z/2 = y, and s1/2/n does not change with discharge. Rhodes (1977) stated that data gathered from several sources (Leopold and Maddock, 1953; Leopold, Wolman, and Miller, 1964; Langbein and Leopold, 1964; Harms and Fahnestock, 1965) indicated that slope changes little with discharge. Therefore, it is assumed that roughness is the factor that experiences the greatest change. Resistance or roughness in a hydraulic sense cannot be measured directly but it can be computed from observation of velocity, depth, and slope (Leopold et al., 1964). Direct measurements of these factors at a specific cross-section at various discharges such as Wolman?s (1961) Brandywine Creek, Pennsylvania data show that resistance decreases with increasing discharge at-a-station with the order of magnitude. 115 There are many reasons that roughness can increase or decrease in different situations such as channel size, bed material, bank material, water reference level, etc. As Rhodes (1977) pointed out roughness might decrease at-a-station for a number of reasons. Relatively small channels with large bed roughness elements (cobbles, boulders, and dunes) decrease in roughness with increasing discharge, because the effects of the bed elements are diminished as they are submerged (Wolman, 1955). Roughness may also decrease on a rising stage because the flow becomes better aligned with the channel, thus minimizing resistance due to flow distortion (Leopold, Wolman, and Miller, 1964). As shown in the equation, velocity increases with depth and slope decreases with increasing resistance to flow or bed roughness. Increased suspended sediment concentration may also result in decreased roughness (Vanoni, 1941; Leopold and Maddock, 1953). Decreased flow roughness occurs because suspended sediment increases the effective viscosity of the fluid, and the increased viscosity reduces turbulence. Roughness might increase with discharge because of the growth of bed roughness elements (dunes) or because at higher stages the channel geometry or vegetation along the banks causes greater turbulence. The Manning formula, representing a numerical value for roughness, does not explain the cause of roughness. Therefore it is difficult, in a particular instance, to be sure which elements contribute to the changes in roughness. In addition to the evaluation of the individual factor?s contribution, the evaluation of the relative contribution of each of the factors is much more difficult. 116 At present each case must be considered individually, involving a detailed analysis of many variables. The inferences concerning the relationship between roughness and hydraulic geometry are speculative. As Richards (1973) points out in discussing this problem: "Achieving a point of entry into this system for the purpose of analysis is extremely difficult, since the interaction is complete." Much work remains to be done on the relationship of hydraulic geometry and channel roughness. The inferences derived from the b-f-m diagram indicate some possible lines of inquiry. Therefore many environmental setting variables are investigated for the role of their impact on at-a-station hydraulic geometry. The next section 5.4 will identify and discuss those environmental settings and analyses and discussion will be in Chapter 6. 5.4 Classification of b-f-m Values The characteristics of the hydraulic geometry values of the ten regions are shown in Figure 5.4 and Table 5.3. 117 m Figure 5.4 b, f, and m Values with 5 Classification Criteria In Figure 5.4 the ten regions of the ternary diagram are labeled. Table 5.5 summarizes the expected relationships between increasing discharge and various fluvial parameters for each region. The labeling scheme for the regions is arbitrary. 118 Table 5.5 Ten Regions Characterized by Five Criteria Region b=f m=f m=f2 m=b+f m/f=2/3 Subbasin Identification b >f bf mf/2 mb+f m2/3 m/f<2/3 A X X X X X 11,32,34,21,5, 35 B X X X X X 6, 10, 9, 25 40, 15 C X X X X X None D X X X X X 41 E X X X X X 36 F X X X X X 7,14,37,30, 31, 38, 39, 22, 28, 26, 13, 23, 33, 19, 18 G X X X X X 17,27,43,4,1,8 H X X X X X 16,20,2,42 I X X X X X 24, 29 J X X X X X None Source: Derived by author 5.5 Environmental Settings The differences in channel systems, as well as their similarities under diverse environmental settings, pose a real challenge for study. Underlying these complexities is an assortment of interrelated environmental variables that determine the dimensions, patterns, shapes, and profiles of present-day channels. The resulting physical appearance and character of the channel is a product of an adjustment of its boundaries to current streamflow, and these streamflow characteristics can be expressed via hydraulic geometry exponents. Inversely, given current streamflow and hydraulic geometry characteristics, the channel boundaries and changes can be deduced. Since streams of similar types tend to act in similar ways (Leopold, 1994), environmental controls, which influence the hydraulic geometry exponents, and channel 119 flow characteristics are investigated for their cause and effect relationships. These relevant factors include channel width/depth ratios, patterns, shapes, and cross-section profiles. In order to investigate the relationships between channel forms and hydraulic geometry exponents, the characteristics of the channel forms and other environmental controls are classified based on commonly used criteria. Although any classification scheme risks oversimplification of very complex channel forms, researchers have found statistically discernible correlations between hydraulic geometry and environmental controls (Ferguson, 1986, Leopold, 1994). Many previous studies (Ferguson, 1986 and Knighton, 1979) indicate that there is a correlation between different channel forms and different hydraulic geometry. Leopold (1994) stated that at a given discharge, different rivers exhibit different values of width and depth. Those with fine-grained cohesive banks tend to be deep and narrow, and those with sandy, friable banks tend to be wide and shallow. Leopold (1994) asserted that a wide, dish-shaped channel would have rapid rate of increase in width with increasing discharge; a boxlike channel with straight steep sides?such as one might expect to find in cohesive materials?would have a low value for b and a high value for f. 5.5.1 Channel Pattern Channel pattern, the term used to describe how a river looks from above, has been differentiated as meandering, braided, or straight. There is no sharp distinction among these patterns. The subjectivity of these types of pattern has long been noted. 120 A meandering channel can be highly convoluted or merely sinuous, but maintains a single thread in curves having a definite geometric shape. The meandering pattern is the most common pattern seen in rivers and roughly 85-90 percent of the world rivers are classified as meandering (Leopold, 1994). Since river channels are seldom straight through a distance greater than about ten times of the channel width, the designation ?straight? may imply irregular, sinuous, or non-meandering. Straight channels are typical for short reaches along tectonic or lithological lines are sinuous but apparently random in the occurrence of bends. Even where the channel is straight, it is usual for the thalweg, line of maximum depth, to wander back and forth from near one bank to the other (Leopold et al., 1964). Opposite to the point of greatest depth, there is usually a bar or an accumulation of mud along the bank, and these bars tend to alternate from one side of the channel to the other. Braided channels are those with multiple streams separated by bars and islands, and streams with shallow channels in coarse alluvium carrying multiple threads of fast flow that subdivide and rejoin repeatedly, continually shifting their location. Their characteristics are usually unstable banks, large amount of transported sediments, slopes are steep with variations in flow. Pools and riffles are less well developed in braided than in straight, but non-meandering channels (Leopold, 1994). Braided channels are often not always associated with sandy or friable bank materials. Also, vegetation has similar effects; a change from non-braided to braided character is sometimes associated with a change from dense vegetation along the channel banks to sparse or no vegetation (Ferguson, 1986). 121 High current velocities seem to be required, and because high velocities are most readily achieved on steep slopes, it automatically follows that most braided channels are found on streams with steep longitudinal gradients. As slope increases, channels tend to become braided. Braided channels seem to be the product of a complex interaction among the flow velocity of the stream, the availability of sediment, the grain size distribution of the sediment, and flow volume irregularity. Knighton (1975) found a lower rate of change of width on straight-reach sections than in meander sections. Knighton and Rhodes (1977) stated that channel pattern has an important influence on hydraulic geometry of rivers, but the absence of adequate data has hindered definitive conclusions. However, through USGS hydrologists' field notes, data for width and depth of cross-sections in average 30 segments per each observation per gage station were collected and used to classify channels by their patterns. With these data, the ratio of lowest depth (d1) and its accompanying narrowest width (w1) and ratio of highest depth (d2) and its corresponding widest width (w2) were calculated. The morphological variables can and do change even in short distances along a river channel, due to such influences of change as geology and tributaries. Therefore, the morphological description level incorporates field measurements from selected reaches, so that the stream channel types used apply only to individual reaches of channel. Data from individual reaches are not averaged over entire basins to describe a stream system. To distinguish channel patterns, Rhodes (1977) classified channels as the ratio of the narrowest width (w1) with the associated depth (d1) to the widest width (w2) with the associated depth (d2). By adapting the Rhodes (1977)'s width and depth ratio at different stages, each channel pattern is assigned to each subbasin (Table 5.3): 122 straight channel: w1 / d1 / w2 / d2 > 1.8, meander channel: w1 / d1 / w2 / d2 ---1.8 to 1.0, and braided channel: w1 / d1 / w2 / d2 < 1.0. In the study sites, the width and depth ratios range from w1 / d1 --2.45 (m31) - 137.77 (v22), w2 / d2 -- 9.72 (m32) - 125.49 (v48), w1 / d1 / w2 / d2--- 0.101 (m311) - 2.256 (m11). These classification results are compared with topographic maps, delineated subbasin drainage, and the USGS Form 9-197 which describes each gage station. For example, the channel section of the USGS Form 9-197 for Willis Creek near Cumberland, Maryland (USGS gage station 01601500) (m10) is described as ?One channel at all stages. Channel is straight for 700 ft above gage and 2,000 ft below gage. Streambed is composed of gravel, rocks, and boulders. Right bank is high, rocky, and covered with brush. It will be overflowed only during extreme floods. Left bank is low and wooded and extends to railroad embankment. Embankment overflowed in extreme flood only.? Comparing the USGS?s gage station descriptions and calculated channel patterns by width and depth ratio, some discrepancies are revealed. For example, channel patterns for m24, v19, v25, v29, v48, and v49 are calculated as braided, but they are described as one channel at all stages on the USGS?s gage station description. In these cases, and in view of the resolution of the delineated subbasin drainage, 30 by 30 meter lattice data, and the generality of the gage station descriptions of channels, the author decided to use the calculated width and depth ratio to classify channel patterns. 123 Table 5.6 Width and Depth Ratio and Channel Pattern _______________________________________________________________ ID USGS ID Size w1/d1 w2/d2 w1/d1/w2/d2 Channel Pattern m10 1601500 247 90.32 45.51 1.985 S m11 1603000 875 83.6 37.06 2.256 S m19 1614500 494 88.92 75.8 1.173 M m21 1617800 18.9 29.55 22.5 1.313 M m24 1619500 281 24.88 32.02 0.777 B m26 1637500 66.9 50.29 67.52 0.745 B m29 1639140 31.3 33.67 39.94 0.843 B m30 1639500 102 63.83 57.12 1.117 M m31 1640965 2.14 2.45 16.79 0.146 M m311 1640970 4.01 4.09 40.47 0.101 M m32 1640980 0.38 3.24 9.72 0.333 B m34 1643000 817 84.32 160.65 0.525 B m37 1643500 62.8 77.87 37.72 2.064 S m38 1645000 101 31.54 34.13 0.924 B m43 1651000 49.4 15.15 48.26 0.314 B m45 1653600 39.5 48.01 21.93 2.189 S m47 1661050 18.5 12.01 50.8 0.236 B m48 1661500 24 28.44 39.22 0.725 B v07 1622000 379 75.28 40.82 1.844 S v12 1624800 70.1 29.4 24.64 1.193 M v13 1625000 375 36.42 45.3 0.804 B v15 1626000 127 64.47 38.4 1.679 M v17 1626850 149 78.3 35.64 2.197 S v18 1627500 212 59.38 37.79 1.571 M v19 1628060 1.94 3.6 22.78 0.158 B v22 1628500 1084 137.77 120.74 1.141 M v23 1629500 1377 100.18 120.16 0.834 B v24 1631000 1642 43.3 20.25 2.138 S v25 1632000 210 28.24 40.24 0.702 B v26 1632082 45.5 20.42 48.77 0.419 B v27 1632900 93.2 36.45 31.17 1.169 M v28 1633000 506 19.29 37.97 0.508 B v29 1634000 768 94.44 97.26 0.971 B v30 1634500 103 37.34 68.93 0.542 B v36 1638480 89.6 30.39 29.25 1.039 M v38 1643700 123 29.21 37.33 0.782 B v39 1644000 332 39.17 49.49 0.791 B v44 1646000 57.9 37.71 24.54 1.537 M v48 1653000 33.7 101.38 125.49 0.808 B v49 1654000 23.5 11.14 51.61 0.216 B v52 1656000 93.4 13.75 54.31 0.253 B v65 1658500 7.64 28.13 23.76 1.184 M v71 1660400 34.9 53.75 32.14 1.672 M _______________________________________________________________________ 124 Out of 43 channels, 7 channels were classified as straight, 14 as meandering, and 22 as braided (Table 5.6). All 7 straight channels fall in the b-f-m ternary diagram Regions F, G, and H regions, and 14 meandering channels fall in A, B, E, F, G, and I Regions. The 22 braided channels fall in A, B, F, G, and H regions. Each channel within a channel pattern type is related to the b, f, and m regions of the ternary diagram (Table 5.6). This is to better understand the channel patterns and b, f, and m relationship. 5.5.2 Channel Cross-Section Shape The shape of the cross-section of any river channel is a function of the flow, the quantity, and character of the sediment in motion through the cross-section, and the character or composition of the materials (including the vegetation) that make up the bed and banks of the channel. The position of the channel is constantly varying, but the form of the cross-section is stable, meaning more or less constant (Ferguson, 1986). Therefore, cross-sectional information is visually portrayed for the investigation of relationships between channel cross-section and hydraulic geometry and how channels adjust to changes in discharge. Ferguson (1986) classified channels by the shape of their cross-sections and noted different rates of increase of channel width with depth. The hydraulic exponents for shallow but wide channels (high value of width / depth) differ from the exponents for deep but narrow channels. Differences in channel shape cause b / f to vary from 0 for a rectangular to 1 for a triangular section. And a change in frictional characteristics alters m/f. Williams (1978) also noted that, in a deep and narrow channel, water depth increased more rapidly with increases in discharge. 125 Ferguson (1986) defined that: (1) the rectangular channel cross-section shape R typifies a rock-walled channel or one with cohesive and vegetated alluvial banks, and has constant width as depth and discharge increase so that b = 0 and f + m = 1; (2) the triangular channel cross-section, T typifies banks of non-cohesive material at a constant angle of repose and has w ? d so that b = f; (3) parabolic shape, P, which represents the most real world case where the angle of repose of sand or gravel decreases below the waterline to give approximately a parabolic shape (P in Fig 5.5) with w ? d1/2 and therefore b = f / 2; and (4) B shape of channel cross-section which represents a curved channel with cut bank and point bar, the latter convex upwards so that width increases faster than depth from medium to high discharges. R T P B Figure 5.5 Ferguson?s Four Channel Shapes Source: Ferguson (1986) Width multiplied by depth creates cross-section area. In order to categorize channels by visual inspection, channel profiles were created to classify the channels into rectangular, triangular, parabolic, and braided chanel shapes. The visual inspection was based on Ferguson?s four channel shapes, however his categorization criteria (ratio of b and f) were not used to classify the channels in this study. Using File Maker ProTM, a macro program was written to calculate cumulative width, area, and distance from the mid-point for each segment. In order to draw the channel profile, the depth values were 126 expressed as a negative value. The channel cross-section profile is drawn for each survey measurements in a given subbasin. An example of the cross-section profiles is shown in Figure 5.6. Figure 5.6 An Example of Cross-section Profiles for Wills Creek near Cumberland, Maryland (m10) Source: Created by author using ExcelTM program Based on visual inspection of the channel cross-section profiles, the 43 channel cross-sections are classified as one of the four shapes, Rectangular (R), Triangular (T), Parabolic (P), and curved channel with cut bank and point bar (B). This classification of the cross-section profile is shown in Table 5.6. Fourteen channel cross-sections were identified as Rectangular, 12 as Triangular, 7 Parabolic, and 10 curved channels with cut banks. -3.0? -2.0? -1.500 -1.000 - 0.0 -50.0? -30.0 -20.0 0.0 20.0 40.0 50.0? Channel Width Channel Depth 127 Table 5.7 Subbasin Channel Pattern and Channel Shape __________________________________________________________________ ID ID USGS Gage Size Region Pattern Shape 1 m10 1601500 247 G S B 2 m11 1603000 875 H S P 3 m19 1614500 494 F M B 4 m21 1617800 18.9 G M R 5 m24 1619500 281 A B P 6 m26 1637500 66.9 B B R 7 m29 1639140 31.3 F B T 8 m30 1639500 102 G M R 9 m31 1640965 2.14 B M R 10 m311 1640970 4.01 B M R 11 m32 1640980 0.38 A B R 12 m34 1643000 817 outside B P 13 m37 1643500 62.8 F S B 14 m38 1645000 101 F B R 15 m43 1651000 49.4 B B B 16 m45 1653600 39.5 H S B 17 m47 1661050 18.5 G B T 18 m48 1661500 24 F B R 19 v07 1622000 379 F S R 20 v12 1624800 70.1 H M B 21 v13 1625000 375 A B P 22 v15 1626000 127 F M T 23 v17 1626850 149 F S R 24 v18 1627500 212 I M T 25 v19 1628060 1.94 B B B 26 v22 1628500 1084 F M R 27 v23 1629500 1377 G B B 28 v24 1631000 1642 F S R 29 v25 1632000 210 I B T 30 v26 1632082 45.5 F B T 31 v27 1632900 93.2 F M P 32 v28 1633000 506 A B T 33 v29 1634000 768 F B R 34 v30 1634500 103 A B T 35 v36 1638480 89.6 A M T 36 v38 1643700 123 E B R 37 v39 1644000 332 F B B 38 v44 1646000 57.9 F M T 39 v48 1653000 33.7 F B P 40 v49 1654000 23.5 B B P 41 v52 1656000 93.4 D B T 42 v65 1658500 7.64 H M T 43 v71 1660400 34.9 G M B ________________________________________________________________________________ Source: Observed and visually inspected by author 128 5.5.3 Channel Asymmetry As Ferguson (1986) identified, whether the channel volume container is symmetric or asymmetric also gives impact on hydraulic geometry. Comparing the shape of the channel basin area in terms of symmetry provides another characteristic of the channel. Therefore, the asymmetry of each channel, calculated for each observation using Knighton?s (1981) index, and is treated as a predictor variables for environmental controls and hydraulic geometry relationships. The Knighton (1981) index of asymmetry was defined as: As = (Ar - Al) / A where As is the asymmetry index, Ar and Al are the cross-sectional areas to the right and the left of the channel centerline, respectively; and A = Ar + Al is the total area of the channel. If the right side occupies a larger channel area than the left side, the As will have a positive asymmetry, if the left side occupies larger channel area at that specific cross-section, then the channel will have a negative asymmetry. Areas that occupy the channel at any observation point were calculated by multiplying distance, the measurement from the previous measurement point to the observation point, by the current measurement point depth (depth of water at the observation point). Each observed point area is accumulated to calculate the right or left side of channel total area. The center point of the channel is indicated as half of the total width of the channel, distance between right and left bank. The right side of channel area is the accumulation of all the observed areas from the initial point to the mid-point of the 129 channel, and the left channel area is the accumulation of all observed areas from the center point of the channel to the opposite bank of the initial point. A File Maker ProTM macro was written to calculate; (1) the distance from the initial point to the next observation point up to the other side of bank for calculation of the mid-point of channel width; (2) the size of the area at each observed point and accumulation of these areas up to the past observed point area; and (3) the asymmetry index of channel area for each survey. The right side of the areas is accumulated up to the mid-point which is the cross-sectional areas of to the right side of the channel. The left side area of the channel was calculated by subtracting the right side banks area from the total channel area. And the asymmetry for each observation was calculated by As = (Ar - Al) / A. For example, as shown Appendix A which has all 1130 rows of surveys, an asymmetry index for Wills Creek near Cumberland, Maryland (m10) observed on June 18, 1985, is calculated as: As = (Ar - Al) / A = (27.13 - 48.93) / 76.06 = -0.287 On June 18 1985, at the cross-section at 500? above gage station, right side area of the cross-section was 28.7% smaller than the left. Two out of 29 observations show a left and right side asymmetry index of 0.0. After individual asymmetry calculations were performed for each hydrographic survey, the asymmetric indices values were averaged within each given subbasin. 130 Within each subbasin, the channel asymmetry indices are averaged and standard deviation and averaged symmetry index for each subbasin are shown Appendix A and Appendix C. The averaged asymmetry index during 18 June 1985 through 4 September 1992 is -0.131 with standard deviation of 0.147. At this point, data needed for the investigation of relationships between the hydraulic geometry and the predictor variables for the study subbasins have been identified, collected, and discussed (Appendix C). Statistical relationships are investigated for each predictor variable and hydraulic geometry exponents and findings are presented in Chapter 6. 131 CHAPTER 6 INVESTIGATION OF THE ROLES OF SCALE AND OTHER SELECTED WATERSHED VARIABLES At-a-station hydraulic geometry is a function of the interactions of many hydraulic and physical drainage system characteristics including physiography and lithology, channel pattern and shape, bed and bank materials, the sizes of channels and subbasins, the downstream locations of channels, hill slope and valley floor, and channel hierarchy in the drainage system. Leopold (1994) clearly illustrated the differences among the average slope lines of the hydraulic geometry exponents in three different sizes of drainage areas. Yet, the spatial variable, geographical scale (subbasin size) has received less research attention than its impacts on the rates of changes in channel width and depth and flow velocity (b, f, and m values) may warrant. This chapter describes and characterizes the hydraulic geometry exponents in the context of geographical scale as expressed by varying areal sizes of the study subbasins. The working hypothesis of this study is that variations in hydraulic geometry exponents, the change rates of channel width and depth, and flow velocity, are not random but that there are systematic changes in hydraulic geometry exponents as a function of subbasin size as well as subbasin and channel physical and environmental characteristics. In simple words, the b, f, and m exponent values in Amazon-size rivers are different (lower b and f and higher m values) from those of smaller rivers. My assumption is that this scale factor might explain why hydraulic geometry has different b, 132 f, and m values where all other physical variables are the same or similar regardless of subbasin size. My research attempts to discern relationships between a geographical scale factor (subbasin size) and b, f, and m via empirical correlations, and to explain roles of the possible predictor variables for b, f, and m in the scale context. Testing the overall hypothesis that b, f, and m values are functions of subbasin size will use a model: b = a0.b + a1.b * scale factor in square miles; f = a0.f + a1.f * scale factor in square miles; and m = a0.m + a1.m * scale factor in square miles where, a0.b, a0.f, and a0.m are constants and a1.b, a1.f, and a1.m are the scale factor coefficients. Theoretically, as stream discharge increases, channel width and depth, and flow velocity would increase as well, but the rates of these increases can be either positive or negative. The focus of the study is the slope term, a1.b, a1.f, and a1.m rather than the intercept (which is a positive value). The null hypotheses are H0.b: a1.b = 0, H0.f: a1.f = 0, and H0.m: a1.m = 0 and the alternative hypotheses are H1.b : a1.b ? 0, H1.f : a1.f ? 0, and H1.m : a1.m ? 0. The sensitivity of the hypothesis test to chance results will be discussed in terms of the calculated probability numbers rather than a pre-determined level of confidence such that readers can judge the statistical significance. If there is insufficient evidence against H0.b, H0.f, and H0.m in favor of H1.b, H1.f, and H1.m, I will try to further explain the outcome in terms of the physical and hydraulic characteristics, such as whether the channel bed material is non-cohesive sand vs. cohesive clay. If the hypothesis testing 133 indicates a discernable role for the scale factor, the magnitude of the scale effect will be investigated by analyzing the a1 value. This investigation also identifies the strength of relationships between each predictor variable and hydraulic geometry exponents, and which predictor variables play dominant roles for b, f, and m. For each selected predictor variable, the hydraulic geometry exponents are modeled in the context of scale such that: b = a0.b + a1.b * scale factor in square miles + a2.b * predictor variable (such as physiography, channel pattern, etc.); f = a0.f + a1.f * scale factor in square miles + a2.f * predictor variable; and m = a0.m + a1.m * scale factor in square miles + a2.m * predictor variable for each basin characteristic group, where, a0.b, a0.f, and a0.m, are constants and a1.b, a1.f, and a1.m and a2.b, a2.f, and a2.m are scale factor coefficients. This chapter will discuss influences of scale, each predictor variable, and the combination of scale and each predictor on b, f, and m separately, and conclude with investigation of the most influential variables of b, f, and m via stepwise multi-variable regression. In addition, empirical findings of f > m in sand bed and f < m in gravel bed with increasing stream discharge are investigated in a scale context. Unless otherwise noted, tables and figures are created by the author. To further examine the validity of the statistical results, two additional analyses are conducted. First of all, I examined a subset of subbasins which meet criteria of constraints, and, secondly, I analyzed all the subbasins using weighted sample sizes. Using the R2 values of the b, f, and m, each subbasin is categorized as well constrained (at least two out of three exponents have R2 of 0.9), moderately constrained (at least of 134 two exponents have R2 of 0.8), or poorly constrained subbasins, poorly constrained subbasins (personal communication with Dr. Karen Prestegaard, Oct. 2005). 6.1 Hydraulic Geometry and Scale Trainer and Watkins (1975) found that average base runoff from tributaries in the upper Potomac River Basin was approximately proportional to the drainage area, which accords with numerous studies that concluded that the size of a subbasin has a linear relationship to the volume of discharge. I have hypothesized that the size of a subbasin, a surrogate scale indicator, is a principal influence on discharge volume. Scale problems in hydrology arise because different hydrological processes are dominant at different scales. Process description or parameterizations that have been derived at the laboratory scales of the experimental plot do not necessarily hold true at the much larger scales of the subbasins. And, there is demonstrated evidence of systematic differences of b, f, and m in different sizes of subbasins. Notable examples are the average slopes of b, f, and m values on discharge lines in the hydraulic geometry of three rivers of different size: Watts Branch drainage area in Potomac River, 3.7 square miles; Seneca Creek in Potomac River, 100; and Amazon River at Obiodos drainage area of 1.9 million square miles (Leopold, 1994). For my Potomac River study, the 43 subbasin sizes range from 0.38 square miles to 1,642 square miles (Figure 6.1). Except for three subbasins, all are less than 1,000 square miles. Most of the subbasins are between 10 and 100, and from 100 to 1,000 square miles. F 135 Figure 6.1 Subbasin Size Histogram Source: Except where otherwise noted, all figures in Chapter 6 have been produced from the author?s research. The scatter plots of b, f, and m versus subbasin size are first examined via simple regression analysis using linear equations (Figure 6.2). The rate of width change (R2 = 0.154) gets smaller as the subbasin size increases, while the change rates of depth (R2 = 0.017) and velocity (R2 = 0.083) increase as the subbasin size increases. This does not mean actual width will decrease in larger subbasins. Actual width might get wider in larger subbasins, but the rate of width change decreases in the larger subbasins of the study area. 0 2 4 6 8 10 12 14 16 18 20 0 - 1 1 -10 10 - 100 100 - 1000 1000 -2000 Log (Watershed Area) in Square Miles F r e q u e n c y 136 Figure 6.2 b, f, and m Values and Subbasin Size with Linear Regression Notwithstanding the paucity of research using a non-linear model, it is of some value to empirically examine the size-exponent relationship using quadratic and logarithmic models. Regression results, shown as Table 6.1, the quadratic goodness of fit lines for b, f, and m indicate an improved coefficient of determination (R2), and the logarithmic goodness of fit lines further improves R2 from the quadratic regression. However, just looking at the correlation coefficient, the correlation is not strong. The small R2 is due to the heterogeneity of the watershed variables. The quadratic regression (Figure 6.3) shows change in the trend lines at around 1,000 square miles. This is 0 500 1000 1500 2000 Watershed Area in Mile2 - 0.5 0.0 0.5 1.0 Value m (R2 = 0.083) f (R2 = 0.017) b (R2 = 0.154) 137 consistent with Leopold?s (1994) analysis of the hydraulic geometry in three different drainage sizes.3 Figure 6.3 b, f, and m Values and Size with Quadratic Regression As shown, for subbasins of up to 1,000 square miles there is an inverse relationship between subbasin size and b. However, that relationship reverses to become proportionate when the subbasin size becomes larger than 1,000 square miles (Table 6.1). Based on these final results, when subbasin size becomes 1,000 square miles, b becomes 3 Cubic regression could show the trend lines more definitively. At the same time, there is no apparent scientific reason for the use of quadratic versus cubic regression, and going beyond quadratic and cubic regressions, more terms will be added to the regression equation, causing loss of degrees of freedom. 0 500 1000 1500 2000 -0.5 0.0 0.5 1.0 Value b (R2 = 0.2241) f (R2 = 0.0473) m (R2 = 0.1011) Watershed Area in Square Miles 138 0.0627, and when subbasin size becomes 1,500 square miles, b becomes 0.109. This trend is consistent with visual inspection of Leopold?s (1994) b slope lines in three different sizes of streams, in that small, medium, and larger size of subbasins have different rates of width changes. Even if it is based on only a few cases and polynomial relationship meant to curve, the positive proportionate relationships between the subbasin size applies to both f and m up to about 1,000 square miles. However, the relationship reverses for both exponents, when the subbasin size becomes bigger than 1,000 square miles. This trend is consistent with visual inspection of Leopold?s (1994) f and m slope lines in three different sizes of streams. 139 Table 6.1 Regression of b, f, and m with Subbasin Size b regressed on Subbasin Size (Linear, Quadratic, and Log, Log Quad) Linear Quadratic Logarithmic Log Quad R2 0.1542 0.2241 0.3105 0.3105 t (Size) -2.7239 -2.6940 -4.2968 -1.3502 t (Size*Size) 1.8851 -0.0467 Significance/ 0.0094 0.0102 0.0001 0.1845 Probability1 0.0667 0.9630 Coefficient -0.0001 -0.0004 -0.0941 -0.0910 0.0000 -0.0009 f regressed on Subbasin Size (Linear, Quadratic, and Log, Log Quad) Linear Quadratic Logarithmic Log Quad R2 0.0171 0.0473 0.1091 0.1566 t (Size) 0.8443 1.3411 2.2409 2.1651 t (Size*Size) -1.1267 -1.5011 Significance/ 0.4034 0.1875 0.0305 0.0364 Probability1 0.2666 0.1412 Coefficient 0.0000 0.0002 0.0420 0.1218 0.0000 -0.0245 m regressed on Subbasin Size (Linear, Quadratic, and Log, Log Quad) Linear Quadratic Logarithmic Log Quad R2 0.0834 0.1010 0.0857 0.1200 t (Size) 1.9317 1.4665 1.9602 -0.5308 t (Size*Size) -0.8875 1.2484 Significance/ 0.0603 0.1503 0.0570 0.5985 Probability1 0.3801 0.2191 Coefficient 0.0001 0.0002 0.0505 -0.0413 0.0000 0.0281 Note: 1 With degree of freedom 41, the critical t value at .0.01 probability is 2.704; 0.05 level, 2.021 and 0.1, 1.684. Source: All tables in Chapter 6 have been produced from the author?s research. Since the size of the study subbasins cover a wide range and the majority of subbasins are between 10 and 100 square miles, with a small numbers of subbasins under 10 and above 1, 000 square miles, it makes sense also to examine subbasin sizes 140 in log10 form. The scatter plots of the study subbasin b, f, and m values and the subbasin in log 10 size with linear and quadratic regression are shown in Figures 6.4 and 6.5, respectively. Table 6.1 incorporates the results of this version of the basic model. 1.0 10.0 100.0 1000.0 SIZE -0.5 0.0 0.5 1.0 Va lue m f b Figures 6.4 and 6.5 b, f, and m Values and Log Size with Linear and Quadratic Regressions Because the quadratic term in log10 size for each b, f, and m is not statistically significant and does not add any more statistical significance (see Table 6.1), the logarithmic linear relationships between b, f, and m with log10 subbasin size are the basis for the subsequent analyses. Because all correlations appear low and there is a paucity of research using a non-linear model, therefore it is hard to determine the expected 1.0 10.0 100.0 1000.0 SIZE -0.5 0.0 0.5 1.0 Va lue m f b 141 theoretical or empirical values of R2 of b, f, and m. Also, since there is no theoretical reasoning of selecting specific probability levels, the author opts to choose the probability level at 0.05 or 0.01. To test the hypotheses that b, f, and m change systematically with the size of the subbasin and, hence, that b, f, and m exponents are a function of subbasin scale, a model b (f and m) = a0 + a1 * scale factor in square miles for each subbasin is set. That is to say, the model now is: b = 0.389 - 0.094 * log subbasin size, f = 0.233 + 0.042 * log subbasin size, and m = 0.3843 + 0.050 * log subbasin size. Since the calculated t value for b, 4.300, is greater than 3.307, the null hypothesis (a1 = 0) is rejected with more than 0.999 probabilities and the alternative hypothesis (a1 ? 0) accepted. That is, b is a function of geographical scale (Table 6.1). In that the calculated t statistic for f is 2.241, therefore, the null hypothesis is rejected with more than 0.950 probabilities, and it is declared that f is a function of the scale factor at 0.950. The calculated t statistic for m is 1.960. Because the calculated t value, 1.960 < 2.021, the null hypothesis (a1 = 0) cannot be rejected, and it is declared that a1 ? 0, meaning m values do not change systematically with the size of subbasin and m is not function of a scale factor (subbasin size in square miles) at 0.95 probability. In summary, the relationships between scale and b, f, and m are not strong, but the results of this study support the hypothesis that there are systematic changes in hydraulic 142 1geometry exponents except for m as a function of subbasin size at the 0.05 level of the statistical significance. Using the log of subbasin size, the R2 improved in both b and f. However because of the scarcity of research on geographic scale and at-a-station hydraulic geometry, it is hard to judge the universal goodness of the particular R2 as shown by this research. Nonetheless, recognizing the high probability of the relationship being as stated, and considering the heterogeneous nature of the subbasins in terms of physiography, lithology, channel pattern and shape, bed and bank material, the statistically discernible relationships between scale and b and f are notable (but not m). 6.2 Going Beyond Size: Selected Predictor Variables Ridenour and Giardino (1991) stated that Ferguson (1986) showed that the cross- sectional shape of a channel determines the rate of increase of stream width with depth, whereas the laws of flow resistance determine the rate of increase of mean velocity with depth. In turn, factors that affect channel geometry include vegetation (Hadley, 1961; Zimmerman et al., 1967), bank cohesiveness and sediment size (Schumm, 1960; Wolman and Brush, 1961; Knighton, 1974; Maizels, 1988), changes in suspended load (Leopold and Maddock, 1953; Wolman, 1955; Thornes, 1970), seasonal variation (Thornes, 1970), channel sinuosity (Knighton, 1975), channel roughness or flow resistance (Knighton, 1975), riffle spacing (Harvey, 1975), the processes of scour and fill (Foley, 1978; Andrews, 1979), and stream size and order (Thornes, 1970; Miller and Onesti, 1977). Hence, all of these stream parameters are candidates as covariates of the hydraulic geometry exponents. The selected predictor variables, in addition to subbasin size, that are determinative in influencing the physical and environmental characteristics of the 143 subbasins and channels are, as described in some detail in the previous chapters: physiography, lithology, landuse, channel pattern, channel shape, bed and bank materials, asymmetry of channel, topographic indices, and subbasin slope. These correlates will be examined in four ways: (1) by visually comparing b, f, and m with predictor variable categories; (2) by determining the importance of each predictor variable category by t- tests for the difference of mean values for one category vs. all others; (3) by analyses using underlying basic principles such as rank order of physiography-based erodability; and (4) by investigating the predictor variables and b, f, and m values in the context of scale. The b, f, and m are analyzed with interaction terms and by portraying residuals from size regression by predictor variable categories. The tests for difference of mean values of b, f, and m between each categorical predictor variable (e.g., Blue Ridge vs. non-Blue Ridge or Piedmont vs. non-Piedmont provinces) are created using dummy variables. Dummy variables are specified as 1, if they are the designated physiography, or 0 if they are any other physiography type. For instance, all subbasins within the Piedmont physiography are specified as 1 and all other subbasins are each coded as 0. In general, the working hypotheses are that there is no difference in the mean values of b, f, and m within one province category vs all other combined, and that each predictor variable influences the b, f, and m exponents systematically as subbasin size changes. To do this, it is necessary to classify the study subbasins according to the characteristics of each predictor variable. For example, to investigate the role of physiography on b, f, and m, study subbasins are grouped as to whether they fall in the Appalachian Plateau, Valley and Ridge, Great Valley, Blue Ridge, Piedmont, or within 144 Coastal Plain physiographic provinces. The b, f, and m values are described and correlated with each physiographic grouping of subbasins. The hypotheses models are set as: the mean values of b, f, and m of a specific group are the same as the mean values of the rest of the study area groups. The other hypotheses models are set as b (f , m) = a0.b(a0.f, a0.m) + a1.b(a1.f, a1.m) * subbasin size in square miles + a2.b(a2.f, a2.m) * predictor variable, where a0.b(a0.f, a0.m) are constants, a1.b(a1.f, a1.m) are the coefficients for scale factor, and a2.b(a2.f, a2.m) are the coefficients for the influences of an individual predictor variable. The null hypotheses are modeled H0..b : a2.b = 0, H0.f : a2 f = 0, H0.m : a2 m= 0,and the alternative hypotheses are H1.b : a2 ? 0; H1.f : a2 ? 0; and H1.m : a2 ? 0. The b, f, and m are analyzed with an interaction term because, in any form of multiple regression, interaction between predictor variables is expected. Where a lower order term, such as scale, is significant and a higher order term like physiography is not significant, the higher order term is removed from the model. Where the higher order term, the interaction term, is significant, then the lower order term is kept even if the lower term is not significant. In this case the hypotheses models are set as b (f , m) = a0.b(a0.f, a0.m) + a1.b(a1.f, a1.m) * subbasin size in square miles + a2.b(a2.f, a2.m) * predictor variable + a3.b(a3.f, a3.m) * interaction between scale and predictor variable, where a0.b(a0.f, a0.m) are constants, a1.b(a1.f, a1.m) are the coefficients for scale factor, and a2.b(a2.f, a2.m) are the coefficients for the influences of an individual predictor variable , a3.b, a3.f, a3.m are coefficients for individual predictor variable influence. The null hypotheses are H0.b : a3 = 0, H0.b : a3 = 0, and H0.b : a3 = 0, and the alternative hypotheses are H1.b : a3.b ? 0, H1.f : a3.f ? 0, and H1.m : a3.m ? 0. 145 6.3 The Role of Physiography The 43 study subbasins extend across six distinct and well known physiographic provinces (Figure 6.6). Two of the study subbasins fall in the Appalachian Plateau province, two in the Valley and Ridge Province; 7 in the Great Valley; 15 in the Blue Ridge; 12 in the Piedmont; and 5 in the Coastal Plain province4. Figure 6.6 Physiographical Provinces of the Study Area Source: US Geological Survey Water Resources Investigations Report 96-4034 4 As mentioned earlier chapter, the physiography of each subbasin is identified by overlaying subbasin plots on the physiography plots. The study area physiography file in Arc/Info ? file is acquired from the USGS Geographic Information Scientist and the delineation of each subbasin was delineated from a DEM using ESRI?s Arc/Info ?, ERDAS?s IMAGINE?, and a couple of commonly used Arc/Info ? Macro Language routines. 146 N 15 2 2 12 7 5 Mean b 0.244 0.125 0.273 0.210 0.139 0.195 Mean f 0.320 0.450 0.293 0.322 0.270 0.310 Mean m 0.440 0.430 0.434 0.473 0.592 0.504 ________________________________________________________________________ Figure 6.7 b, f, and m Average Values by Physiographic Province 6.3.1 Some Statistical Differences in Hydraulic Geometry Parameters Among Physiographic Provinces The influence of four of the six physiographic provinces to b, f and m values is tested using a t-test for difference of mean values. Two provinces are excluded for the t- test for difference of mean values because these two provinces have a sample size of only two subbasins each. 0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 Blue Ridge Appalachian Plateau Valley & Ridge Piedmont Great Valley Coastal Plain b f m 147 Using a 0.05 and 0.01 tests, only the Great Valley province is statistically discernable. Comparing subbasins in the Great Valley and non-Great Valley, the mean difference of m are greater than the t-critical value at the 0.05 level (Table 6.2). The null hypothesis (there are no differences in influencing b value and m between Great Valley and non-Great Valley) is rejected for the 0.05 level. Actually, the mean difference of m value in this province is significant at the 0.01 level. Table 6.2 Differences of Mean b, f, and m within Each Province ________________________________________________________________________ Number of Subbasins Mean b Mean b Difference Mean f Mean f Difference Mean m Mean m Difference Blue Ridge 15 0.2440 0.3146 0.4425 Non-Blue Ridge 28 0.1878 0.0562 0.3139 0.0007 0.5020 -0.0595 t-test 1.3196 0.0219 -1.3729 Piedmont 12 0.2096 0.3222 0.4725 Non-Piedmont 31 0.2066 0.0031 0.3111 0.0111 0.4846 -0.0121 t-test 0.0671 0.3210 -0.2572 Great Valley 7 0.1389 0.2699 0.5918 Non-Great Valley 36 0.2207 -0.0818 0.3228 -0.0529 0.4598 0.1319 t-test 1.4971 -1.2768 2.4744 Coastal Plain 5 0.1950 0.3105 0.5040 Non-Coastal Plain 38 0.2090 -0.0140 0.3147 -0.0042 0.4783 0.0257 t-test -0.2163 -0.0860 0.3914 Note: Provinces with less than 5 subbasins (Appalachian Plateau and Valley and Ridge) are excluded for t- test for difference of mean values. The critical value for t with 41 degrees of freedom at 0.01, 2.704; 0.05 level, 2.021; and 0.1 level, 1.684. Statistically significant at the 0.05 level and less are indicated in Bold. Just considering subbasin physiography, everything else being equal, the easiest erosion of bank and bed is expected in the Coastal Plain Province followed by the Great Valley, Piedmont, Valley and Ridge, Appalachian Plateau, and Blue Ridge Provinces (communication with USGS Geologist, Scott Southworth, Nov. 2001). With 1 148 representing the easiest to be eroded in channel width and depth or greater flow velocity change rate, Table 6.9 shows the comparative erodability of the different physiography provinces. Table 6.3 Physiography-Based Mean b, f, and m Values and Bank and Bed Erodability ___________________________________________________________________________ N Bank/Bed Erodability for b and f 1 b f Expected Velocity Change2 m Blue Ridge 15 6 0.244 0.320 1 0.440 Appalachian Plateau 2 5 0.125 0.450 2 0.430 Valley and Ridge 2 4 0.273 0.293 3 0.434 Piedmont 12 3 0.210 0.322 4 0.473 Great Valley 7 2 0.139 0.270 5 0.592 Coastal Plain 5 1 0.195 0.310 6 0.500 Mean 0.198 0.328 0.478 Standard Deviation 0.053 0.058 0.057 ____________________________________________________________________ 1 Bank/bed erodability index is a ranking of all physiographic provinces with respect to each other (communication with USGS geologist, Scott Southworth, Nov.2001). The individual physiographic province is ranked from ?easiest? to ?hardest? order and they do not constitute magnitude. 2 The ranking of expected velocity change ranking order is just the inverse of the b and f ranking order based on the equation of continuity. The expectation is that subbasins in the Coastal Plain Province would have the highest b and f values, and subbasins in the Blue Ridge the lowest b and f values. At the same time, expectation of m would be a reversed order of b and f. The rationale is that if b and f values become bigger because of rapid widening and deepening of the channel, the eroded bank and bed material would result in higher flow resistance. Therefore, flow velocity would be hampered and m becomes smaller in value. As shown in Table 6.3, the mean values of b and f do not correlate closely with the bank and bed erodability indicator which suggests that other physical and environmental controlling factors play more significant roles. The observed orders of the m values are the closest to the 149 expected ranking (R2 = 0.7714) compared to the orders of the observed b (R2 = 0.2571) or f (R2 = 0.4857) values. The m values somewhat follow the trend of the bank and bed erodabilities, that is, the Coastal Plain and Great Valley subbasins have higher m values reflecting faster stream flow changes with increasing stream discharge. The high m values in high bank and bed erodability provinces indicate that the potential erodability of bank and bed (width and depth) might not have been realized; or that the rate of change in flow velocity was much greater than that of channel width or depth erosion. 6.3.2 Particular Provinces The Appalachian Plateau Province, characterized by narrow valleys and steep, rugged ridges, is designated as resistant to erosion, and the low mean b value reflects that fact. Two study area subbasins are in this province. Since one subbasin is nested within the other, their b, f, and m values, especially b, are, not unexpectedly, very close. However, the very different f values are explainable by the differences in the slopes of the two subbasins (0.557 vs 0.235). In the Valley and Ridge Province, also with only two subbasins, the physiography results from intensely folded and faulted rocks. The b value (0.273) of this province is much higher than the average for all of the study subbasins. The subbasin (v25) with a higher f value has the steeper slope. For this subbasin, while the f value is much higher, the m value is only half of the overall subbasin average. Even though these two subbasins are in the same province, they display quite different b, f, and m values, especially for m. This suggests that other factor(s) influence b, f, and m values more than 150 the physiography. Also, considering just physiography, the erodability of channels in this province is low (4th out of 6), but the observed b and f values do not reflect the expected erodability, which amplifies the insignificant role of physiography to b, f, and m. For the Great Valley Province (seven study subbasins), which is largely underlain by carbonate bedrock with numerous sinkholes and caverns, the b, f, and m values (Figure 6.8 and Table 6.4) are highly similar to each other (similar standard deviation -Table 6.3). Even the observed b values in this province are the second lowest in contrast with the probable erodability shown in Table 6.3. The differences in mean values of b and m between the Great Valley and all other provinces are discernable. Therefore, subbasins within the Great Valley Province can be expected to have similar b, f, and m values, indicating that physiography is definitely a predictor to b, f, and m. The f values in this province are the lowest among the six and the slope of the subbasins in this province is least. Therefore, like subbasins in the Appalachian Plateau and Valley and Ridge provinces, it can be said that there is a relationship between slope and f values. As expected, since the mean b value is the lowest, the mean m value within this province is the highest of all the provinces. 151 Figure 6.8 Mean Differences of b, f, and m in Great Valley and Non-Great Valley and Blue Ridge and Non-Blue Ridge Provinces Mean Differences of b, f, and m in Great Valley and Non-Great Valley Provinces -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 1 Non-Great Valley Great Valley b, f, m b f m b Trendline f Trendline m Trendline Mean Differences of b, f, and m in Blue Ridge and Non-Blue Ridge Provinces -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 1 Non-Blue Ridge Blue Ridge b, f, m b f m b Trend f Trend m Trend 152 The Blue Ridge Province, characterized by steep relief and narrow channels, is represented by fifteen study subbasins. The expected erodability of the Blue Ridge Province is the lowest of all the provinces. However, the mean b and f values for subbasins in this province are higher than the physiography-based mean, and the m values are subsequently lower than the overall average. The differences of b and m (Table 6.2) between the Blue Ridge and the other provinces are high but statistically insignificant at 0.05 level and higher (Figure 6.8) indicating that Blue Ridge is an indiscernible predictor for b, f, and m. Twelve study subbasins are in the Piedmont Province, which is an area of gently rolling hills with low to moderate relief. The average b, f, and m values are similar to the physiography-based mean for the total 43 subbasins. The mean difference tests (Table 6.2) indicate that the differences between Piedmont and non-Piedmont subbasins in b, f, and m are the least indicating that Piedmont is not a discernible predictor for b, f, and m. Five study subbasins are in the Coastal Plain Province, which is underlain by unconsolidated sediments that form a gentle, seaward-sloping plain of low relief. The b and f are similar to the mean values of the study area. While the erodability index suggests that the expected b and f values have potential to be changed most easily in the Coastal Plain, the observed b and f values are not much different from the mean of the total subbasins. Also, the mean difference tests of b, f, and m between Coastal vs. non- Coastal provinces conclude that there is no statistically significant differences between provinces. This suggests that there are other stronger predictor variable(s) than physiography that influence b and f values. The m value is slightly higher in this gentle 153 sloping region which is consistent with findings by other researchers such as Bathurst (1985, 1993). 6.3.3 Physiography in a Scale Context To estimate the influence of the physiography on b, f, and m in a scale context, the three exponents are modeled as b (f , m) = a0.b (a0.f, a0.m) + a1.b ( a1.f, a1.m) * log10(size) + a2.b (a2.f, a2.m) * physiography variable. The interaction between scale and physiography is investigated by adding an interaction term to the model for b (f , m) = a0.b (a0.f, a0.m) + a1.b ( a1.f, a1.m) * log10(size) + a2.b (a2.f, a2.m) * physiography variable + a3.b ( a3.f, a3.m) * log10(size) * physiography variable. The null hypotheses (a2.b = 0, a2.f = 0 , a2.m = 0, and a3.f = 0, a3.m = 0) are that there are no discernible influence of physiography nor interaction between subbasin size and physiographic characteristics for the b, f, and m values. b Exponents Since the computed value of F for a1.b, 16.878 (Table 6.4), exceeds the critical value, I reject the null hypothesis for a1, and conclude that scale influences the b value at the 0.05 level. However, there is insufficient evidence (F for a2 is 0.908) to reject the null hypothesis for a2.b; it can be concluded that a discernible difference does not exist in the expected response of the contribution of physiography to the b value. Figure 6.8 supports the conclusion that physiography (as measured) does not explain the observed b values in the scale context. The expected visual pattern for the hypothesis is that the b values of each physiographic province would be shown by a set of four parallel lines (two provinces with only two subbasins are excluded) sloping downwards. However, because 154 the correlations for each province separately are weak, there is no clear evidence of parallelism. By adding the interaction term, R2 increased from 0.39 to 0.42 but the statistical significances of scale and physiography both decreased such that it is concluded that adding an interaction term does not help to account for b values as shown in Table 6.4. Therefore, it can be said that physiography, as a whole, is not a strong predictor variable influencing the b values. Table 6.4 b vs. Log Size, Physiography, and Interaction ________________________________________________________________ Dep Var: b N:43 R2: 0.3110 df F-ratio P Log Size 1 18.512 0.0000 ------------------------------------------------------------------------------------------------- Dep Var: b N:43 R2: 0.38770 df F-ratio P Log Size 1 16.877 0.0002 Physiography 5 0.9080 0.4867 ------------------------------------------------------------------------------------------------ Dep Var: b N:43 R2: 0.41897 df F-ratio P Log Size 1 0.0428 0.8357 Physiography 5 0.5168 0.7615 Physiography * Log Size 31 0.3337 0.8887 155 Log (Watershed Area in Square Miles) Blue Ridge Piedmont Great Valley Coastal Plain Overall Intercept 0.4688 0.3511 0.2915 0.1274 0.3889 Coefficient -0.1177 -0.0855 -0.0626 0.0453 -0.0944 R2 0.4743 0.2476 0.0810 0.0094 0.3111 ______________________________________________________________________ Figure 6.9 b and Subbasin Log Size in Physiography Note: Two provinces (Great Valley and Appalachian Plateau Provinces) with only two subbasins each are not graphed. f Exponents It is expected that if there is a strong relationship between f and each physiographic province, then the slope of each province will run parallel with the overall f value slope between f and scale. But results, Figure 6.10 and Table 6.5, clearly show that physiography in the scale context does not explain f values at the 0.05 and less level. -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 Blue Ridge Coastal Plain Great Valley Piedmont Trendline (All) Trendline (BR) Trendline (CP) Trendline (GV) Trendline (Pied) b 156 Unlike the case of b, the influence of physiography on f is indiscernible in the Great Valley Province (Table 6.2). As discussed in section 6.2, scale influences the f value (0.017 level of statistical significance), but when physiography and interaction terms are included, the physiography?s influence on f is of low significance (0.27 level) and the slope of the regression line is 0.04 (Figure 6.10). Adding an interaction term, the R2 improved from 0.25 to 0.38 but it does not help to explain the role of physiography to the f value. Blue Ridge Coastal Plain Great Valley Piedmont Overall Intercept 0.2616 0.0278 0.26660 0.18330 0.0233 Coeffici 0.0278 0.1892 0.00137 -0.0840 0.0420 R2 0.0614 0.1478 0.00006 0.49867 0.0793 _______________________________________________________________________ Figure 6.10 f and Subbasin in Log Size in Physiography 0 0.2 0.4 0.6 0.8 1 1.2 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 Log (Watershed Area) in Square Miles f, Blue Ridge Piedmont Great Valley Coastal Plain Trendline (BR) Trendline (Pied) Trendline (GV) Trendline (CP) Trendline (All) 157 Table 6.5 f vs. Log Size, Physiography, and Interaction ________________________________________________________________ Dep Var: f N:43 R2: 0.1090 df F-ratio P Log Size 1 5.0210 0.0310 ------------------------------------------------------------------------------------------------- Dep Var: f N:43 R2: 0.2504 df F-ratio P Log Size 1 6.2983 0.01672 Physiography 5 1.3570 0.26335 ----------------------------------------------------------------------------------------------- Dep Var: f N:43 R2: 0.3843 df F-ratio P Log Size 1 6.3154 0.0174 Physiography 5 1.0747 0.3934 Physiography * Logs Size 31 1.3479 0.2707 _______________________________________________________________ Subbasins v12 (0.471) and v18 (0.537) in the Blue Ridge physiography region have the highest f values among the study area subbasins. These high f value subbasins also have high minimum elevations (365 and 338 meters, respectively) indicating headwater subbasins. These subbasins are another example of the relationship of high f with higher hierarchy of channels. Consequently, the m values of these subbasins are lower than those of the other subbasins. This can be explained by the headwater nature of most of the streams in this province. Mean velocity increases downstream against the intuition that small tributaries flowing on steep slopes must be traveling faster than the low-gradient trunk rivers. These headwater subbasins consequently will have lower m values. 158 m Exponents Based on the data (Table 6.6), it can be concluded that scale influence on m is significant at the 0.0908 level; and physiography influences the m value at a significance level of 0.310 which is beyond the statistical significance level for this analysis. Based on the m value analyses data shown in Table 6.6, it is concluded that scale influences m value at the 0.0477 level of significance. However, the influence of physiography is, not as strong as scale?s influence, significant at the 0.122 level. Adding an interaction term does help the R2 (from 0.220 to 0.394) and the statistical significance improved from 0.3100 to 0.1224, as well as, that the m value and physiography improved to 0.0477 from 0.0908 level of significance. Therefore, it can be said that physiography, as a whole, is an insignificant predictor variable influencing m values with R2 about 0.400 with statistical significance of 0.05 level. Visual inspection of Figure 6.11 and Table 6.6 indicate that physiography in the scale context does explain m values by showing faint parallel lines by physiography along with the trend line of scale and m relationship. However, the statistical significance is not at the 0.05 level. 159 Blue Ridge Coastal Plain Great Valley Piedmont Overall Intercept 0.2747 0.8405 0.4417 0.4684 0.3842 Coeffici 0.0879 -0.2254 0.0616 0.0025 0.0504 R2 0.3191 0.1064 0.1253 0.0002 0.1384 Figure 6.11 m and Subbasin in Log Size in Physiography Table 6.6 m vs Log Size, Physiography, and Interaction ________________________________________________________________ Dep Var: m N:43 R2: 0.0860 df F-ratio P Log Size 1 3.843 0.0570 ------------------------------------------------------------------------------------------------- Dep Var: m N:43 R2: 0.22025 df F-ratio P Log Size 1 3.01926 0.09083 Physiography 5 1.24251 0.30983 ------------------------------------------------------------------------------------------------ Dep Var: m N:43 R2: 0.3939 df F-ratio P Log Size 1 4.2523 0.0477 Physiography 5 1.9028 0.1224 Physiography * Logs Size 31 1.7766 0.1469 0 0.2 0.4 0.6 0.8 -1 0 1 2 3 4 Log (Watershed Area) in Square Miles m Blue Ridge Coastal Plain Great Valley Piedmont Trendline (All ) Trendline (BR) Trendline (CP) Trendline (GV) Trendline (Pied) 160 6.3.4 Summary The conflation of graphical and statistical tools suggest that the b, f, and m values within each distinctive physiographic province are more similar than for the overall means of the 43 subbasins. Comparing the influence of physiography on the three exponents, m is the most apparent (statistical significance level of 0.122 vs 0.393 or 0.762 for f or b) as shown in Table 6.6. This can be also seen in Figure 6.11, from the linear trend line. The mean difference test indicates that the Great Valley physiography influences m at the 0.05 level. However, as Figure 6.9 and statistical analyses (Table 6.4) show, physiography, as a whole, does not strongly influence b, f, and m. 6.4 The Role of Lithology Physical and geologic characteristics are the primary natural factors affecting the hydrologic properties of ground water and surface water systems and, thus, are the primary factors affecting hydraulic geometry within major river basins. The geology of the Potomac River Basin is complex and diverse, ranging from relatively undisturbed, unconsolidated sediments to intensely deformed crystalline rocks (Milici, 1963; Cleaves, 1968). In recognition of this geologic diversity, the Potomac River Basin lithology has been divided into five major subunits for the purpose of this hydraulic geometry assessment (Figure 6.12). 161 Figure 6.12 Study Area Lithology Source: US Geological Survey Water Resources Investigations Report 96-4034 The five major rock types present in the study region are: siliciclastic sedimentary (shale and sandstone) ? 13 subbasins, crystalline metamorphic and igneous (gneiss, schist, and diabase dikes) ? 13 subbasins, undivided carbonate and siliciclastic ? 7 subbasins, carbonate sedimentary (limestone and dolomite) ? 5 subbasins, and unconsolidated sediments (sand, silt, and clay) ? 5 subbasins5. The comparative average 5 As mentioned in an earlier chapter, the lithology of each subbasin is identified by overlaying subbasin plots on the lithology plots. The study area lithology file in Arc/Info ? file format was acquired from the USGS Geographic Information Scientist and the delineation of each subbasin was delineated from DEM data using ESRI?s Arc/Info ?, ERDAS?s IMAGINE?, and a couple of commonly used Arc/Info ?Macro Language routines. 162 values of the b, f, and m for the various lithologies are graphed in Figure 6.13. N 13 13 7 5 5 Mean b 0.177 0.285 0.169 0.150 0.195 Mean f 0.287 0.310 0.330 0.370 0.310 Mean m 0.538 0.410 0.500 0.480 0.504 ______________________________________________________________________ Figure 6.13 Average b, f, and m Values in Five Lithology Regions 6.4.1 Each Lithology Region?s Influence The influence of lithology on b, f and m values is examined using a t-test for difference of mean values. Using a 0.1 % test, only the Crystalline and Siliciclastic regions are statistically discernable for b and m, and m, respectively6. Comparing subbasins in the Crystalline and non- Crystalline regions, the mean differences of b and 6 The critical value for t with 41 degrees of freedom at the 0.01 level is 2.704; 0.05 is 2.021, at 0.1 level 1.684. 0 0.1 0.2 0.3 0.4 0.5 0.6 Siliciclastic Sedimentary Crystalline Carbonate & Siliciclastic Undivided Carbonate Sedimentary Unconsolidated Sediments Mean b Mean f Mean m 163 m are greater than the t-critical value at the 0.05 level (Table 6.7). Therefore, the null hypothesis (there is no difference in influencing b and m values between Crystalline and non-Crystalline) is rejected for the 0.05 level and it can be concluded that there is a discernable difference in b and m values between the Crystalline and the other lithology regions. The mean difference of the m value in the Siliciclastic region is significant at the 0.1 level but not at 0.05 level. As the t-tests and Figure 6.14 indicate, the mean difference between Crystalline and non-Crystalline regions for f, and that of the Siliciclastic and non-Siliciclastic regions for f and m are indiscernible at the 0.05 level. Mean Differences of b, f, and m in The Siliciclastic and Non-Siliciclastic Regions 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 b, f, m b f m Trendline (b) Trendline (f) Trendline (m) Crystalline Non-Crystalline 164 Figure 6.14 Mean Differences of b, f, and m in the Crystalline and Non-Crystalline and Siliciclastic and Non-Siliciclastic Regions 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Non- Sililciclastic Sililciclastic b f m Trendline (b) Trendline (f) Trendline (m) 165 Table 6.7 Differences of Mean b, f, and m within Each Lithology Region Number of SubbasinsMean b Mean b DifferenceMean f Mean f DifferenceMean m Mean m Difference Crystalline 13 0.2849 0.3132 0.4051 Non- Crystalline 30 0.1738 0.1111 0.3146 -0.0014 0.5143 -0.1092 t-test 2.6688 -0.0421 -2.5556 Carbonate 5 0.1504 0.3666 0.4834 Non- Carbonate 38 0.2149 -0.0645 0.3073 0.0593 0.4810 0.0024 t-test -1.0105 1.2411 0.0369 Unconsolidated 5 0.1950 0.3105 0.5040 Non- Unconsolidated 38 0.2090 -0.014 0.3147 -0.0042 0.4783 0.0257 t-test -0.2163 -0.0860 0.3914 Siliciclastic 13 0.1774 0.2871 0.5377 Non-Siliciclastic 30 0.2204 -0.0430 0.3259 -0.0388 0.4568 0.0809 t-test -0.9640 -1.1607 1.8306 Carbonate & Siliciclastic Undivided 5 0.1686 0.3315 0.5000 000 Non-Carbonate & Siliciclastic Undivided 38 0.2149 -0.0463 0.3108 0.0207 0.4776 0.0224 t-test -0.8313 0.4914 0.3927 Note: The critical value for t with 41 degrees of freedom at the 0.01 level is 2.704; 0.05 is 2.021, at 0.1 level 1.684.Statistically significant at the 0.05 level and less are indicated in Bold. Considering subbasin lithology only, everything else being equal, the easiest erosion of bank and bed is expected in Unconsolidated Sediments followed by Carbonate, Carbonate and Siliciclastic Undivided, Crystalline, and Siliciclastic (communication with USGS Geologist, Scott Southworth, Nov. 2001). Table 6.8 illustrates the comparative erodability among the different lithologies, one being the lithology that is easiest to be 166 eroded, and five being the most resistant. As stated in the Physiography section, the expected m order would be the inverse of order of b and f. Table 6.8 Lithology-Based Mean b, f, and m Values and Bank and Bed Erodability ____________________________________________________________________ Bank/Bed Expected Erodability Velocity N for b and f b f Change m Siliciclastic 13 5 0.177 0.287 1 0.538 Crystalline 13 4 0.285 0.310 2 0.410 Carbonated & Siliciclastic Undivided 5 3 0.169 0.330 3 0.500 Carbonate 5 2 0.150 0.370 4 0.480 Unconsolidated Sediments 5 1 0.195 0.310 5 0.504 Mean 0.195 0.321 0.486 Standard Deviation 0.047 0.028 0.043 ____________________________________________________________________ Source: Erodability order by communication with USGS geologist, Scott Southworth, Nov. 2001. b, f, and m values are calculated by the author The average values of b, f, and m for each lithology type in erodability order do not closely follow the expected bank and bed erodability orders, indicating that other physical and other environmental controlling factors impact b, f, and m more significantly. However, the lowest standard deviations for b, f, and m indicate that, within the Unconsolidated Sediments region, the b, f, and m values are more uniform than they are for many of the other lithology regions. The thirteen subbasins that fall in Siliciclastic region have b values that range widely from 0.008 to 0.469. The average values for b and f are 0.177 and 0.287, which 167 are lower than the average for the entire study area (Table 6.8). The high value of the standard deviation, 0.120, indicates that Siliciclastic lithologies might not be a dominant predictor variable for b. Within the Siliciclastic region, the f values of the 13 subbasins are very similar which indicates that Siliciclastic lithology might play a role in the rate of depth change. Difference of means analysis indicates that t-test of siliciclastic region versus non-siliciclastic region is noticeable (t value of -1.1607) but the difference is statistically insignificant at 0.05 level. That the Siliciclastic area is the least erodable of channel banks and beds is reflected by its having the lowest f values among all the other lithology regions. The average b value for the thirteen subbasins fall within the Crystalline lithology region is 0.285, the highest among all the lithology regions. Seven subbasins out of thirteen, m31 (0.461), m311 (0.562), v19 (0.377), v38 (0.440), v49 (0.364), and v65 (0.352) have significantly high b values which do not follow the erodability order. The b values have a wide range of the set of thirteen subbasins, subbasin m311 has the highest b value in the study area, yet subbasin m34 has the lowest b value, -0.008. The values for b do not support the notion of erodability in this type of lithology, but the average f value (0.310) is small and supports the erodability order. For the seven subbasins in the Carbonate and Siliciclastic Undivided region, average b and f values are 0.169, which is the 2nd lowest followed by the Carbonate region, and 0.332, respectively. The standard deviations of b, f, and m are moderately high. Thus, lithology is not a predominant predictor variable for b, f, and m. Five study subbasins fall in the Carbonate region. Just considering lithology, high rates of change of channel width and depth are expected in this second most easily 168 erodable region. However, the very low (0.150) observed b values do not support the erodability order. As in the Carbonate and Siliciclastic Undivided region, the observation is made that in carbonate region b is small, but f compensates for the small b values. Considering subbasin lithology only, everything else being equal, the easiest erosion of bank and bed would occur in the Unconsolidated Sediments region. For the five subbasins in this region, both the average b and f values, 0.195 and 0.310, respectively, are below the total study area average and the standard deviation of f mean value is the lowest among the other subbasin lithology regions. Therefore, it can be said that the influence of Unconsolidated Sediments on f is obvious. This means that the values of b and f will change (increase) easily in the Unconsolidated Sediments and Carbonates lithology and b and f will change very little in the Siliciclastic region. 6.4.2 Lithology in a Scale Context The model for analyzing lithology predictor variables in a scale context is b (f, m) = a0.b (a0.f, a0.m) + a1.b (a1.f , a1.m) * log10(size) + a2.b (a2.f , a2.m)* lithology variable. With the interaction term, the model is b (f, m) = a0.b (a0.f , a0.m) + a1.b (a1.f , a1.m) * log10 (size) + a2.b (a2.f , a2.m)* lithology + a3.b (a3.f , a3.m)* log10 (size) * lithology. The null hypothesis is that the scale of the subbasins and lithology have no bearing on the b, f, and m values. b Exponents (Figure 6.15 and Table 6.9) As expected, the computed F value for a1, 11.627, exceeds the critical value, which reaffirms the notion that scale influences b values. However, the computed value of F for a2.b (lithology), 0.698, does not exceed the critical value; therefore the a2.b value 169 is not significantly different from 0. Hence, it is concluded that the expected response of the contribution of lithology to b value is almost 0, that is, there is no systematic influence of lithology on b. By adding an interaction term to the model for b, R2 increased from 0.359 to 0.437, and the statistical significance of lithology to b improved from 0.598 to 0.217. However, overall lithology in the scale context does not statistically account for b values. Table 6.9 b and Subbasin Log Size and Lithology ______________________________________________________________________________________ Dep Var: b N:43 R2: 0.3105 df F-ratio P Log Size 1 18.512 0.000 -------------------------------------------------------------------------------------------------------------------------- Dep Var: b N:43 R2: 0.3589 df F-ratio P Log Size 1 11.627 0.0016 Lithology 4 0.6984 0.5979 -------------------------------------------------------------------------------------------------------------------------- Dep Var: b N:43 R2: 0.4370 df F-ratio P Log Size 1 0.2603 0.6133 Lithology 4 1.5291 0.2165 Lithology * Log Size 4 1.1438 0.3532 170 Siliciclastic Crystalline Cryst & Silic Und. Carbonate Uncons Overall Intercept 0.2962 0.5215 0.1961 0.1864 0.1274 0.3889 Slope -0.0554 -0.1588 -0.0117 -0.0156 0.0453 -0.0944 R2 0.1974 0.5158 0.0013 0.0109 0.0094 0.3110 ______________________________________________________________________ Figure 6.15 b and Log Size in all Lithology Types f Exponent (Figure 6.16 and Table 6.10) Because the computed value of F for a1.f, 5.384, exceeds the critical value, 4.08, I reject the null hypothesis for a1, and conclude that scale influences f value. However, the computed value of F for a2.f, 0.7847, does not exceed the critical value, 2.60, which indicates that there is insufficient evidence to reject the null hypothesis for a2.f. Hence, it is concluded that a real difference between a2 and 0 does not exist and the expected response of the contribution of lithology to f value is indiscernible. -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 -1 -0.5 0 0.5 1 1.5 2 2.5 3 Log (Watershed Area in Square Miles) b All Siliciclastic Crystalline Crystal & Silici Undivided Carbonate Unconsolidated Trendline (All) Trendline (Siliciclastic) Trendline (Crystalline) Trendline (Crystal & Silici Undivided) Trendline (Carbonate) Trendline (Unconsolidated) 171 Adding the interaction term to the model for f explains more of the variation (R2 improved from 0.179 to 0.312). However, the model still does not make the lithology statistically significant for f even at the 0.05 level. As Figure 6.16 indicates, lithology as a whole does not explain the f variance well. The overall R2 is 0.109. The coefficient of f and of lithology regions are a mixture of negative and positive directions. Table 6.10 f and Subbasin Log Size and Lithology _________________________________________________________________ Dep Var: f N:43 R2: 0.109 df F-ratio P Log Size 1 5.021 0.0310 -------------------------------------------------------------------------------------------------- Dep Var: f N:43 R2: 0.17878 df F-ratio P Log Size 1 5.3841 0.0259 Lithology 4 0.7847 0.5425 -------------------------------------------------------------------------------------------------- Dep Var: f N:43 R2: 0.31214 df F-ratio P Log Size 1 2.7611 0.1061 Lithology 4 1.5995 0.1977 Lithology * Logs Size 4 1.5995 0.1977 __________________________________________________________________ 172 Siliciclastic Crystalline Cryst & Silic Und. Carbonate Uncons Overall Intercept 0.2025 0.2124 -0.3550 0.5085 0.0279 0.2333 Slope 0.0395 0.0677 0.2931 -0.0613 0.1892 0.0420 R2 0.1617 0.3525 0.4759 0.0751 0.1478 0.1091 ______________________________________________________________________ Figure 6.16 f and Subbasin Log Size in All Lithology m Exponent (Figure 6.17 and Table 6.11) Scale does not influence the m value at the 0.05 level of statistical significance, and the computed values of F for a1.m (scale) and a2.m (lithology) do not exceed the critical values at the 0.05 level, and there is insufficient evidence to reject the null hypothesis. Hence, it is concluded that neither scale nor lithology influences m value at the 0.05 level. However, when the interaction term is added to the multiple regression model for m, the influence of lithology on m becomes statistically significant at the 0.086 level. And, the variation is explained better, that is, R2 improves from 0.188 to 0.316 (Table 6.11). 0 0.1 0.2 0.3 0.4 0.5 0.6 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 f All Siliciclastic Crystalline Crystal & Silici Undivided Carbonate Unconsolidated Trendline (All) Trendline (Siliciclastic) Trendline (Crystalline) Trendline (Crystal & Silici Undiv) Trendline (Carbonate) Trendline (Unconsolidated) Log (Watershed Area in Square Miles) 173 Table 6.11 m and Subbasin Log Size and Lithology _________________________________________________________________ Dep Var: m N:43 R2: 0.0857 df F-ratio P Log Size 1 3.843 0.0570 ------------------------------------------------------------------------------------------------- Dep Var: m N:43 R2: 0.18840 df F-ratio P Log Size 1 1.5215 0.2252 Lithology 4 1.1707 0.3395 ------------------------------------------------------------------------------------------------ Dep Var: m N:43 R2: 0.31588 df F-ratio P Log Size 1 0.5684 0.4563 Lithology 4 2.2425 0.0857 Lithology * Logs Size 4 1.5373 0.2142 As Figure 6.17 indicates, lithology as a whole does not explain the m variance well. The overall R2 is 0.0857. The two steep slope lines of Crystalline and Siliciclastic Undivided and Unconsolidated lithology regions are in opposite directions to the rest of lithology region slopes and this fact might negate the overall slope of the lithology in m value. 174 Siliciclastic Crystalline Cryst & Silic Und. Carbonate Unconsol Overall Intercept 0.5136 0.2622 1.1608 0.3057 0.8405 0.3842 Slope 0.0112 0.0959 -0.2821 0.0767 -0.2254 0.0504 R2 0.0070 0.3054 0.3464 0.1350 0.1064 0.0857 ______________________________________________________________________ Figure 6.17 m and Subbasin Log Size in All Lithology 6.4.3 Summary Comparing subbasins, there are discernible differences in b and m values in the Crystalline vs. non-Crystalline at the 0.05 level, and m at the 0.1 level in the Siliciclastic vs. non-Siliciclastic regions. And, even though the scale influences b and f values at the 0.05 level, there is insufficient evidence as to lithology?s influence in a scale context on b and f values. Adding interaction term between scale and lithology to the multiple regression, the influence of lithology becomes statistically significant at 0.086 level, but it is still below the level of the analysis for this study. The mean b, f, and m values for each lithology do not follow the expected bank and bed erodability orders, indicating that 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -1 0 1 2 3 4 Log (Watershed Area in Square Miles) m All Siliciclastic Crystalline Crystal & Silici Undivided Carbonate Unconsolidated Trendline (All) Trendline (Siliciclastic) Trendline (Crystalline) Trendline (Crystal & Silici Undiv) Trendline (Carbonate) Trendline (Unconsolidated) 175 other physical controlling factors impact b, f, and m more significantly. Statistical analyses show lithology, as a whole, does not account for b and f values at the statistically significant level. 6.5 The Role of Landuse The impacts of landuse type on stream discharge behavior are well documented (Moglen and Beighley, 2000, Moglen and Berger, 1998). In the Potomac River Basin study area, the predominant landuse types, forest, agriculture, and urban areas (Figure 6.18) are investigated for their impacts on the b, f, and m. 176 Figure 6.18 Landuse Types of the Study Area Source: US Geological Survey Water Resources Investigations Report 96-4034 The study subbasins occur in three landuse types: forest (21 subbasins), agriculture (19), and urban (3) 7. The distribution of b, f, and m average values over forest, agriculture, and urban landuse types do not show clear differences among them 7As mentioned earlier chapter, the landuse of each subbasin is identified by overlaying subbasin outlines on the landuse plot. The study area landuse file in Arc/Info ? file is acquired from the USGS Geographic Information Scientist and the delineation of each subbasin was delineated from DEM using ESRI?s Arc/Info ?, ERDAS?s IMAGINE?, and a couple of commonly used Arc/Info ?Macro Language routines. 177 (Figure 6.19). The f values of the subbasins in agricultural lands are similar to each other and their standard deviation is 0.088. The subbasins in urban areas have a smaller standard deviation for the mean value, but only 3 subbasins are in this landuse type. N 21 19 3 Mean b 0.213 0.193 0.258 Mean f 0.324 0.308 0.285 Mean m 0.465 0.501 0.470 ___________________________________________________________________ Figure 6.19 Average b, f, and m Values in Landuse Types 6.5.1 Each Landuse Type?s Influence The expected correlations between watershed landuse and b, f, and m are that, agricultural areas followed by forest areas would have increased rate of changes in b and 0 0.1 0.2 0.3 0.4 0.5 0.6 Forest Agriculture Urban Mean b Mean f Mean m b, f, m 178 f. The rationale is that in agricultural areas, increased water discharge flows through plowed agricultural fields carrying more suspended and bed load material and has greater erosional impacts on channel beds and banks. On the other hand, in forest landuse areas, increased volumes of discharge occur after ground water saturation: therefore, the discharge would be more or less free of sedimentation compared to agricultural watersheds. That is, it is expected that watersheds in agricultural areas are more susceptible to erosion than those in forest areas. The expected changes in m would be the reverse because the increased bed or suspended load in stream discharge will reduce the flow velocity. The influence of landuse on b, f and m values is tested using t-test for difference of mean values. Using a 0.1 % test, none of the landuse types is statistically discernable to b, f, and m8. However, subbasins in agricultural and forest areas are statistically more significant than any others for m, but not discernible at the 0.1 level (Table 6.12). The m value slopes between agriculture vs. non-agriculture and forest vs. non-forest type of landuse are non- zero (Figure 6.20). Therefore, it is concluded that there is a discernable difference in m values between agriculture and the others and forest vs. non-forest type of landuse, however, the differences are not at the 0.05 level of statistical significance. 8 The critical value for t with 41 degrees of freedom at 0.01 level is 2.704; 0.05, 2.021, and at 0.1 level; 1.684. 179 Figure 6.20 Differences of b, f, and m in Forest vs. Non-Forest and Agriculture vs. Non- Agriculture Landuse Types Table 6.12 Differences of Mean b, f and m within each Landuse Type Number of Subbasins Mean b Mean b Difference Mean f Mean f Difference Mean m Mean m Difference Forest 21 0.2131 0.3237 0.4651 Non-Forest 22 0.2019 0.0112 0.3051 0.0186 0.4967 0.0316 t-test 0.2690 0.6007 -0.7540 Agriculture 19 0.1930 0.3082 0.5009 Non-Agriculture 24 0.2188 0.0258 0.3189 0.0107 0.4658 0.0351 t-test -0.6192 -0.3420 0.8325 Urban 3 0.2584 0.2853 0.4705 Non-Urban 40 0.2036 0.0548 0.3163 0.0310 0.4821 0.0116 t-test 0.6778 -0.5076 -0.1401 _______________________________________________________________________ Non-Agriculture 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.5 Agriculture b Linear (b) m Linear (f) Linear (m) Non-Forest 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.5 Forest f 180 6.5.2 Analyses Using Underlying Rationale Just considering subbasin landuse type only, all other predictor variables being equal, the increased rate of changes in b and f are expected in agricultural lands followed by forest areas9. The rationale is that in agriculture types of landuse, increased water discharge flows through plowed agricultural fields carrying more suspended and bed load material and has greater erosional impacts on channel beds and banks. On the other hand, in forest subbasins, increased volume of discharge occurs after ground water saturation; therefore, the discharge would be more or less free of sedimentation compared to agricultural subbasins. That is, it is expected that subbasins in agricultural areas are more susceptible to erosion than those in forest areas. The expected change in m would be the reverse order of b and m because the increased bed or suspended load in discharge will reduce the flow velocity. Table 6.13 illustrates the comparative change order of b, f, and m between forest and agriculture areas. The average values of b, f, and m for each landuse type and expected change order match, but considering only two types of landuse, it cannot be concluded definitively that b, f, and m values follow landuse. Table 6.13 Expected Change Orders and Landuse-Based Mean b, f, and m Values ____________________________________________________________________ Landuse Type* Expected b and f Expected m Change Order Change Order N Mean b Mean f Mean m Forest 19 2 0.177 0.287 1 0.538 Agriculture 13 1 0.285 0.310 2 0.410 ____________________________________________________________________ Note: Landuse type, Urban, has only 3 subbasins, so Urban is excluded in analysis. 9 The landuse type, Urban, is not included because, only three subbasins fall in this category. 181 6.5.3 b, f, and m and Landuse in a Scale Context The model for analyzing landuse predictor variables in a scale context is b (f, m) = a0.b (a0.f, a0.m) + a1.b (a1.f , a1.m) * log10(size) + a2.b (a2.f , a2.m)* landuse variable. With the interaction term, the model is b (f, m) = a0.b (a0.f , a0.m) + a1.b (a1.f , a1.m) * log10 (size) + a2.b (a2.f , a2.m)* landuse+ a3.b (a3.f , a3.m)* log10 (size) * landuse. The null hypothesis is that the scale of the subbasins and landuse have no bearing on the b, f, and m values. b Exponents As expected, the computed F value for a1.b, 16.842, exceeds the critical value, which reaffirms the notion that scale influences b values. However, the computed value of F for a2.b (landuse), 0.068, does not exceed the critical value, and the a2.b value is not significantly different from 0. Hence, it is concluded that there is no systematic influence of landuse on b. By adding an interaction term to the model for b, R2 increased from 0.313 to 0.318, and the statistical significance of landuse to b improved from 0.934 to 0.903. Figure 6.21 and Table 6.14 clearly show that landuse in the scale context does not explain b values. 182 Table 6.14 b, Subbasin Log Size, and Landuse _____________________________________________________________________ Dep Var: b N:43 R2: 0.3110 df F-ratio P Log Size 1 18.512 0.000 ------------------------------------------------------------------------------------------------- Dep Var: b N:43 R2: 0.3129 df F-ratio P Log Size 1 16.842 0.0020 Landuse 2 0.0679 0.9344 ------------------------------------------------------------------------------------------------ Dep Var: b N:43 R2: 0.3157 df F-ratio P Log Size 1 0.7900 0.3799 Landuse 2 0.1021 0.9032 Landuse * Log Size 2 0.0766 0.9264 Forest Agriculture Urban Overall Intercept 0.3786 0.4176 0.6664 0.3889 Coefficient -0.0920 -0.1058 -0.2665 -0.0944 R2 0.4191 0.1702 0.1424 0.31107 ______________________________________________________________________ Figure 6.21 b and Log Size in All Landuse Types -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 -1 0 1 2 3 4 Log (Watershed Area in Square Miles) b All Forest Agriculture Urban Trendline (All) Trendline (Forest) Trendline (Agriculture) Trendline (Urban) 183 f Exponent Because the computed value of F for a1.f, 5.33, exceeds the critical value, the null hypothesis for a1.f is rejected, and it is concluded that scale influences f values at the 0.05 level of statistical significance. However, the computed value of F for a2.f, does not exceed the critical value, which indicates that there is insufficient evidence to reject the null hypothesis for a2.f. Hence, it is concluded that a real difference between a2.f and 0 does not exist and the expected response of the contribution of landuse to f value is indiscernible. Adding the interaction term to the model, the model explains more of the variation (R2 improved from 0.131 to 0.164); however, it still does not make the landuse statistically significant for f at the 0.05 level. Table 6.15 f, Subbasin Log Size, and Landuse _________________________________________________________________ Dep Var: f N:43 R2: 0.1090 df F-ratio P Log Size 1 5.021 0.031 -------------------------------------------------------------------------------------------------- Dep Var: f N:43 R2: 0.1306 df F-ratio P Log Size 1 5.3256 0.0264 Landuse 2 0.4824 0.6210 -------------------------------------------------------------------------------------------------- Dep Var: f N:43 R2: 0.1640 df F-ratio P Log Size 1 0.0210 0.8855 Landuse 2 0.3891 0.6804 Landuse * Log Size 2 0.7395 0.4843 __________________________________________________________________ 184 As Figure 6.22 indicates, landuse as a whole does not explain the f variance well. The overall R2 is 0.109. The slopes of landuse types are a mixture of negative and positive directions. Forest Agriculture Urban Overall Intercept 0.2213 0.3167 0.2699 0.2333 Coefficient 0.0569 -0.0040 0.0101 0.0420 R2 0.2318 0.0897 0.0019 0.1091 ______________________________________________________________________ Figure 6.22 f and Subbasin Log Size in All Landuse Types m Exponent The computed values of F for a1.m (scale) and a2.m (landuse) do not exceed the critical values at 0.05 levels. However, only a1 values exceed the critical value at 0.1 level. Hence, while it is concluded that neither scale nor landuse influences the m value 0 0.1 0.2 0.3 0.4 0.5 0.6 -1 0 1 2 3 4 f All Forest Agriculture Urban Trendline (All) Trendline (Forest) Trendline (Agriculture) Trendline (Urban) Log (Watershed Area in Square Miles) 185 at 0.05 level, but it can be said that scale influences m value at 0.1 level. When the interaction term is added to the model, the role of the landuse to m becomes statistically significant at the 0.573 level, but not in a meaningful level, and the variation is explained better (R2 improves from 0.090 to 0.128). This is shown in Table 6.16. Table 6.16 m and Subbasin Log Size and Landuse _________________________________________________________________ Dep Var: m N:43 R2: 0.086 df F-ratio P Log Size 1 3.843 0.057 ------------------------------------------------------------------------------------------------- Dep Var: m N:43 R2: 0.0910 df F-ratio P Log Size 1 3.1846 0.0821 Landuse 2 0.1128 0.8936 ------------------------------------------------------------------------------------------------ Dep Var: m N:43 R2: 0.1277 df F-ratio P Log Size 1 0.4642 0.4999 Landuse 2 0.5651 0.5731 Landuse * Log Size 2 0.7787 0.4664 As Figure 6.23 indicates, landuse as a whole does not explain the m variance well. The overall R2 is 0.0857. 186 Forest Agriculture Urban Overall Intercept 0.4074 0.2566 0.0694 0.3842 Coefficient 0.0321 0.1151 0.2620 0.0505 R2 0.0510 0.1811 0.1823 0.0857 ______________________________________________________________________ Figure 6.23 m and Subbasin Log Size in All Landuse 6.5.4 Summary As expected, scale influences b and f values, but there is not sufficient evidence as to its influence on the m values. Disregarding scale, the subbasins in forest vs. non-forest and agriculture vs. non-agriculture landuse types show differences of mean in m value but the difference of means are statistically insignificant at the 0.05 level. Hence, landuse, as a whole, does not show any systematic trend in m values. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -1 0 1 2 3 4 Log (Watershed Area in Square Miles) m All Forest Agriculture Urban Trendline (All) Trendline (Forest) Trendline(Agriculture) Trendline (Urban) 187 6.6 The Role of Channel Pattern The channel pattern of a stream, whether straight, braided, or meandering, produces characteristic cross-sections. Because channel morphology is a major factor in determining the hydraulic geometry of a cross-section, it seems probable that there is a relationship between the hydraulic geometry exponents and channel pattern. Knighton and Rhodes (1977) stated that channel pattern has an important influence on the hydraulic geometry of rivers, but the absence of adequate data has hindered definitive conclusions. The variation is apparently not random but systematically related to channel pattern, straight reaches being distinguishable from both meander and braided reaches in terms of the rates of change of width. Twenty-two channels in the Potomac River Basin study area are categorized as braided, 14 as meander, and 7 as straight. The channel pattern categorization is based on the criteria described in Chapter 5. The comparative b, f, and m average values for the various channel patterns are graphed in Figure 6.24. Comparing the average values of b, f, and m for each channel pattern, there are significant differences among the three channel pattern types (Figure 6.24). The mean b value of the straight channels is 0.092, while both the meander and braided channels' b values are 0.215 and 0.239, respectively. 188 N 22 14 7 Mean b 0.239 0.215 0.092 Mean f 0.277 0.335 0.391 Mean m 0.490 0.450 0.517 ____________________________________________________________________ Figure 6.24 Average b, f, and m Values in All Channel Patterns 6.6.1 Each Channel Pattern?s Influence The influence of channel pattern on b, f, and m values is examined using a t-test for difference of mean values. Using a 0.05% test, straight channel pattern is statistically discernable for b. The straight channel?s influence on b is statistically significant at the 0.025 level (Table 6.17 and Figure 6.25). Using a 0.2% test, braided channel becomes statistically discernible for b and f, and straight channel for b. Therefore, the null hypothesis is rejected for braided for b, f, and m and straight channel for b and f at the 0.2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Braided Meander Straight b, f, m Mean b Mean f Mean m 189 level, and it is concluded that there are discernable differences in b and f in braided channels and for b and f in straight channels. -0.2 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 b f m Trend Line (b) Trend Line (f) Trend Line (m) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.2 0.4 0.6 0.8 1Non Braided Braided Non-meander Meander Non-Straight Straight Figure 6.25 Mean Differences of b, f, and m in the Braided vs. Non- Braided, Meander v Non-Meander, and Straight vs. Non-Straight Channels 190 Table 6.17 Differences of Mean b, f, and m within each Channel Pattern Type _____________________________________________________________________ Number of Subbasins Mean Mean b Difference Mean Mean f Difference Mean m Mean m Difference Braided 21 0.2392 0.2766 0.4900 Non-Braided 22 0.1741 0.0651 0.3535 0.0769 0.4721 0.0179 t-test 1.6191 -1.5589 0.1868 Meander 19 0.2150 0.3348 0.4496 Non-Meander 24 0.2037 0.0113 0.3042 0.0306 0.4966 0.0470 t-test 0.2566 0.9294 -1.0568 Straight 7 0.0922 0.3909 0.5172 Non- Straight 36 0.2298 0.1376 0.2993 0.0916 0.4743 0.0429 t-test -2.6528 1.3095 0.3250 _______________________________________________________________________ Note: Critical value at the 0.01 level is 2.704; 0.025 level, 2.423; 0.05 level, 2.021, and 0.1 level 1.684. Statistically significant at the 0.025 and less are indicated in Bold. As expected, straight channels, which are narrow and deep in nature, have the lowest b values with the highest m values, because with a low b value, f and m have to be higher in order for the summation of the b, f, and m to become unit 1. The observed mean b value in straight channels, 0.09, is not as dramatically low as Knighton?s (1975) finding, 0.03, but the observed mean b values in both meandering and braided channels are essentially in agreement with the Knighton?s (1975) finding of 0.24. Braided channels are typically broad and shallow. Characteristics are usually unstable banks and steep slopes with variations in flow. Braided channels are often, but not always, associated with sandy or friable bank materials. Therefore, higher m values are expected for braided channels than those of meander channels, and the observed mean m value are in agreement with the expected m. 191 Braided channels have the widest range of maximum and minimum values of b, f, and m and have the largest b values, as expected, because of usually unstable banks, shallow channels with high slope, among the three channel pattern types. Slope also varied with channel pattern. Braided reaches had consistently steeper slopes than undivided channels, which may account for their higher mean flow velocities. Meandering channels seem to be associated with relatively low b values, although not as low as those for straight channels. The mean m value appears to be smallest among the three channel patterns. Comparing straight, meander, and braided channels, the straight channels have the smallest range of maximum and minimum values for all b, f, and m, which indicate that straight channels have very similar b, f, and m values among themselves. And, the mean f value for braided channels has also a very small standard deviation meaning the rates of depth changes with discharge are similar within braided channels. 6.6.2 Channel Pattern in a Scale Context The role of the channel pattern on the subbasin in a scale context is b (f, m) = a0.b (a0.f, a0.m) + a1.b (a1.f , a1.m )* log10(size) + a2.b (a2.f , a2.m)* channel pattern variable. The null hypothesis is that the scale of the subbasins and channel pattern characteristics have no bearing on the b (f, m) values. b Exponents (Figure 6.26 and Table 6.18) As expected, the channel pattern and b is significant at the 0.05 level in scale context. The computed values of a1.b and a2.b 14.886 and 2.47, exceed the critical values 192 for the levels of ? = 0.05 and ? = 0.10, respectively. This statistical discernability of scale affirms the notion that scale influences b values and the role of channel pattern vis- ?-vis b values (Table 6.18). The null hypotheses for a1.b and a2.b are rejected and it is concluded that not only scale, but also channel pattern, influences b values. Specifically, the meander channels clearly show that channel pattern does explain the b value in a scale context, as revealed by an R2 of 0.758 (Figure 6.26). By adding an interaction term to the model for b, R2 increased from 0.388 to 0.493, and the statistical significance of channel pattern for b improved from 0.098 to 0.065. In general, as subbasin size increases, the b value decreases, and especially, it is noticeable in the meander channels where the slope is the steepest among the all other channel patterns. In straight channels, the role of scale appears to be minimal (Figure 6.26). 193 Table 6.18 b, Subbasin Log Size, and Channel Pattern _________________________________________________________________ Dep Var: b N:43 R2: 0.3105 df F-ratio P Log Size 1 18.512 0.000 -------------------------------------------------------------------------------------------------- Dep Var: b N:43 R2: 0.153 df F-ratio P Channel Pattern 2 3.615 0.0360 -------------------------------------------------------------------------------------------------- Dep Var: b N:43 R2: 0.3883 df F-ratio P Log Size 1 14.886 0.0004 Channel Pattern 2 2.4708 0.0977 -------------------------------------------------------------------------------------------------- Dep Var: b N:43 R2: 0.4927 df F-ratio P Log Size 1 8.2117 0.0068 Channel Pattern 2 2.9490 0.0648 Channel Pattern * Log Size 2 3.8169 0.0311 Note: Statistical significance at the 0.1 level is in Bold. 194 Braided Meander Straight Overall Intercept 0.3368 0.5093 0.1433 0.3889 Coefficient -0.0515 -0.1693 -0.0214 -0.0944 R2 0.1171 0.7584 0.0481 0.3110 ______________________________________________________________________ Figure 6.26 b and Subbasin Log Size in all Channel Pattern Types f Exponents (Figure 6.27 and Table 6.19) The computed values for a1.f, 4.0059, and a2.f, 3.9342, exceed the critical value at the ? = 0.1 and ? = 0.05 level, respectively. Hence, the null hypotheses for a1.f and a2.f, are rejected and it is concluded that geographic scale and channel pattern both influence the f value at the 0.1 and 0.05 levels, respectively. -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 Log (Watershed Area in Square Miles) b All Braided Meander Straight Trendline (All) Trendline (Braided) Trendline (Meander) Trendline (Straight) 195 The relationship between the channel pattern and f in the scale context is not strong (R2 = 0.259), but clearly the role of channel pattern on f is evidenced on R2 of 0.259 from 0.109 with just f and log size relationship. The R2 for each channel pattern and f value in a scale context is lower compared to the mean differences of Straight vs. Non-Straight and Braided vs. Non-Braided (Figure 6.25). This result can be explained by the fact that regression slope lines between channel pattern types and f values are in opposite directions among the three channel pattern types and the f values are scattered in the scale domain of braided and straight channels. Table 6.19 f, Subbasin Log Size, and Channel Pattern _________________________________________________________________ Dep Var: f N:43 R2: 0.1090 df F-ratio P Log Size 1 5.021 0.0310 -------------------------------------------------------------------------------------------------- Dep Var: f N:43 R2: 0.1830 df F-ratio P Channel Pattern 2 4.466 0.0180 -------------------------------------------------------------------------------------------------- Dep Var: f N:43 R2: 0.2587 df F-ratio P Log Size 1 4.0059 0.0523 Channel Pattern 2 3.9342 0.0278 -------------------------------------------------------------------------------------------------- Dep Var: f N:43 R2: 0.2789 df F-ratio P Log Size 1 1.0007 0.3237 Channel Pattern 2 0.8356 0.4416 Channel Pattern * Log Size 2 0.5197 0.5990 __________________________________________________________________ Note: Statistical Significance at the 0.1 level is in Bold. 196 Braided Meander Straight Overall Intercept 0.2139 0.2355 0.4253 0.2333 Coefficient 0.0331 0.0571 -0.0144 0.0420 R2 0.0943 0.1745 0.0118 0.1091 ______________________________________________________________________ Figure 6.27 f and Subbasin Log Size in all Channel Pattern Types In spite of a negative influence of straight channels on f, the overall trend is that as the subbasin size increases, the f values increase. Specifically, in meander channels the regression slope line of the f and subbasin size is the steepest (0.057). m Exponents (Figure 6.28 and Table 6.20) Since the computed a1 value, 2.938, exceeds the critical value at the ? = 0.1 level, the null hypothesis is rejected, and it is concluded that scale influences m values with a 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.5 0.5 1.5 2.5 3.5 Log (Watershed Area in Square Miles) All Braided Meander Straight Trendline (All) Trendline (Braided) Trendline (Meander) Trendline (Straight) f 197 statistical significance of 90% (Table 6.20). However, the computed value for a2.m, 0.296, does not exceed the critical value at the ? = 0.l level, which indicates that there is insufficient evidence to reject the null hypothesis. By adding the interaction term, the role of the channel pattern improves statistical significance at the 0.2 level but even with the interaction term, it is concluded that the response of the contribution of channel pattern to m value is indiscernible at 0.05 level. Table 6.20 m, Subbasin Log Size, and Channel Pattern _________________________________________________________________ Dep Var: m N:43 R2: 0.0857 df F-ratio P Log Size 1 3.843 0.0570 Dep Var: m N:43 R2: 0.032 df F-ratio P Channel Pattern 2 0.651 0.5270 -------------------------------------------------------------------------------------------------- Dep Var: m N:43 R2: 0.0994 df F-ratio P Log Size 1 2.9375 0.0945 Channel Pattern 2 0.2959 0.7455 -------------------------------------------------------------------------------------------------- Dep Var: m N:43 R2: 0.1636 df F-ratio P Log Size 1 2.1921 0.1472 Channel Pattern 2 1.6955 0.1975 Channel Pattern * Log Size 2 1.4217 0.2542 __________________________________________________________________ Note: Statistical Significance at the 0.1 level is in Bold. 198 Braided Meander Straight Overall Intercept 0.4632 0.2507 0.4332 0.3842 Coefficient 0.0141 0.1144 0.0352 0.0505 R2 0.0063 0.4596 0.0328 0.0857 ______________________________________________________________________ Figure 6.28 m and Subbasin Log Size in All Channel Pattern Types 6.6.3 Summary At the 0.1 level, there are no discernable differences in the b, f, and m values between braided and non-braided channels, and for b and f values in straight versus non- straight channels. Also, as expected, the straight channels have the smallest mean b value (0.092), but the highest m value (0.517). Braided channels have the highest mean b values, higher than meander channels. As expected, the scale influences on band f values are evident. The channel pattern?s influence on f is statistically significant at 0.05 level, respectively. However, 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 Log (Watershed Area in Square Miles) All Braided Meander Straight Trendline (All) Trendline (Braided) Trendline Meander) Trendline (Straight) m 199 the channel pattern?s influence on m is indiscernible. It is expected that channel pattern would influence width and depth change rate rather than velocity of flow change rate. Specially, the channel patterns? influence on the f value with a statistical significance level of 0.0278 is notable. 6.7 The Role of Channel Shape Considering only channel shape, it is expected, based on Rhodes? (1977) research, that rectangular channels will have the lowest b values, parabolic channels will have b = f/2, triangular channels will have b = f, and braided channels will have b > f. Differences in channel shape can vary between rectangular (b/f ? 0) and triangular (b/f ? 1). Changes in width and depth with increasing discharge may occur either by erosion of banks and beds or by progressive filling of a stable channel. Therefore, the b to f ratio gives an indication of the change in shape of the channel cross-section with changing discharge. Intrinsically, it can be assumed that the b, f, and m values for shallow wide channels would be different with increasing discharge than would the hydraulic geometry exponents for deep narrow channels. These differences in channel shape cause different friction characteristics which promote m/f differences. Therefore, it is reasonable to hypothesize that the shape of the channel cross-section causes different rates of b and f. For this research, channel shape was categorized first by visual inspection using channel depth and width profile diagrams. Channel profiles were created for each survey to classify the channels into rectangular, triangular, parabolic, and braided chanel shapes. The visual inspection was based on Ferguson?s four channel shapes, however his categorization criteria (ratio of b and f) were not used to classify the channels in this 200 study. Using File Maker ProTM, a macro program was written to calculate cumulative width, area, and distance from the mid-point for each segment. In order to draw the channel profile, the depth values were expressed as a negative value. The channel cross-section profile is drawn for each survey measurements in a given subbasin. The study subbasins are categorized as having 10 braided, 7 parabolic, 14 rectangular, and 12 triangular channels, using a channel shape classification based on criteria described in the Chapter 5. The comparative b, f, and m average values for the various channel shapes are graphed in Figure 6.29. N 10 7 14 12 b 0.190 0.174 0.199 0.251 f 0.351 0.322 0.299 0.297 m 0.462 0.514 0.504 0.452 ________________________________________________________________________ Figure 6.29 Average b, f, and m Values by Channel Shape 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Braided Parabolic Rectangular Triangular b, f, m b f m 201 6.7.1 Each Channel Shape?s Influence (Figure 6.30 and Table 6.21) The influence of channel shape on b, f and m values is examined using a t-test for difference of mean values. Comparing Braided vs. non- Braided and Triangular vs. non- Triangular channels, the mean differences of f and b, respectively, are not greater than the t-critical value at the 0.05 level. Therefore, the null hypothesis (there is no difference in influencing f values between Braided and non-Braided and b values between Triangular and non-Triangular channels) cannot be rejected for the 0.05 level, and it can be concluded that there is an indiscernable difference in f values between the Braided and the other channel shapes as well as Triangular and other channels in regard to b values. The mean difference of the b value in the Parabolic and all other channels is insignificant at the 0.1 level. Also, the differences in the mean f and m values between Rectangular and other channels as well as Triangular vs. non-Triangular channels are insignificant at the 0.1 level. 202 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.2 0.4 0.6 0.8 1 b f m Linear (b) Linear (f) Linear (m) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.2 0.4 0.6 0.8 1 b f m Linear (b) Linear (f) Linear (m) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.2 0.4 0.6 0.8 1 b f m Linear (b) Linear (f) Linear (m) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.2 0.4 0.6 0.8 1 b f m Linear (b) Linear (f) Linear (m) Non-Braided Braided Non-Parabolic Parabolic Non-Rectangular Rectangular Non-Triangular Triangular Figure 6.30 Mean Differences of b, f, and m in All Channel Shapes 203 Table 6.21 Differences of Observed Mean b, f, and m in One Type vs. All Other Channel Shapes Number of Subbasins Mean b Mean b Difference Mean f Mean f Difference Mean m Mean m Difference Braided 10 0.1902 0.3510 0.4621 Non-Braided 33 0.2126 0.0224 0.3031 0.0479 0.4871 0.0250 t-test -0.4582 1.3184 -0.5014 Parabolic 7 0.1740 0.3224 0.5136 Non-Parabolic 36 0.2139 0.0399 0.3130 0.0094 0.4750 0.0386 t-test -0.7151 0.2336 0.6786 Rectangular 14 0.1991 0.2989 0.5038 Non- Rectangular 29 0.2114 0.0123 0.3216 0.0227 0.4704 0.0334 t-test -0.2767 -0.6849 0.7467 Triangular 12 0.2508 0.2967 0.4521 Non- Triangular 31 0.1906 0.0602 0.3209 0.0242 0.4926 0.0405 t-test 1.3321 -0.7011 -0.8672 _____________________________________________________________________ Note: Critical value at the 0.01 level is 2.704; 0.025 level, 2.423; 0.05 level, 2.021, and 0.1 level 1.684. None of the t-test result is statistically significant. 204 Table 6.22 Differences of Expected and Observed Mean b, f, and m within Each Channel Shape Type Number of Subbasins Expected Change Rate (b, f) Observed Mean b Observed Mean f Observed Mean m Braided 10 b > f 0.1902 0.3510 0.4621 Triangular 12 b = f 0.2508 0.2967 0.4521 Parabolic 7 b = f/2 0.1740 0.3224 0.5136 Rectangular 14 b = lowest 0.1991 0.2989 0.5038 ________________________________________________________________________ Source: Expected b and f change rates are from Rhodes (1977). The observed mean b, f, and m values are derived by the author. The observed values and the comparative values of each b and f for channel shape type do not closely follow the expected order of b value changes. The relationship between channel shape and b and f indicates that other physical and other environmental controlling factors impact b and f more significantly. The biggest discrepancy is the mean b value of the braided channels. It is expected to have the highest mean b value, but the observed b value is the second lowest which causes b < f rather than b > f. Triangular channels have the highest mean b value which match the expected relationship (b = f) between b and f. Also, the parabolic channels match the expected relationship (b = f/2). Contrary to expectation, parabolic channels have the lowest b value and rectangular and triangular channel shapes have higher b values than braided channels. The reason that subbasins with rectangular channels have a higher b average value is that subbasin m311 which has the highest b value (0.562) in the study area, as well as subbasin m31 (0.461), are classified as having rectangular channels. Excluding these 205 subbasins, rectangular subbasins have an average b value of 0.146 which is the lowest b value among all channel shapes. However, as shown in Figure 6.31 and Figure 6.32, these subbasin channels are clearly rectangular in shape and the widths of the channels change from 0.7 ? 18.6 and 1.5 ? 31.5 feet in m31 and m311, respectively, and still maintain rectangular shapes. Therefore, m311 and m31 cannot be discarded as rectangular shape channel and it can be concluded that the b is influenced by factors other than by channel shape. Figure 6.31 Channel Shape of Subbasin m31 Source: Selected 1985 ? 1994 USGS Hydrographic Survey Data Comparing all channel shapes, braided channels have b, f, and m values that are very similar to each other. This indicates that b, f, and m values can be estimated without a large variance in braided channel shaped subbasins. -1.6 -1.000 -0.800 -0.600 -0.400 -0.200 0.000 -15 -5 0 5 15 Channel Width Channel Depth Survey Points Depth and width units are in feet 206 Figure 6.32 Channel Shape of Subbasin m311 Source: Selected 1985 ? 1991 USGS Hydrographic Survey Data 6.7.2 Channel Shape in Scale Context As discussed in 6.1, examination of b, f, and m exponents with subbasin size indicated that there is a systematic change in b, f, and m values with subbasin size. The role of the channel shape in a scale context, b (f and m) is examined by setting the model b (f, m) = a0.b (a0.f, a0.m) + a1.b (a1.f, a1.m) * scale + a2.b (a2.f, a2.m) * channel shape + a3.b (a3.f, a3.m) * channel shape * scale variable. b Exponents (Figure 6.33 and Table 6.23) The computed value of a1.b, 17.448, exceeds the critical values for level at ? = .0002, the null hypothesis for a1.b is rejected and it is concluded that scale influences b, -1.4 -1.0 -0.800 -0.600 -0.4 0.0 -20 -10 0.0 10 20 Channel Width Channel Depth Survey Points Depth and width units are in feet 207 but a2.b, 0.676, does not exceed critical value at the 0.1 level. Therefore, it is concluded that channel shape is an insignificant variable influencing b at the 0.1 level. Table 6.23 b and Subbasin Log Size and Channel Shape _____________________________________________________________________ Dep Var: b N:43 R2: 0.3105 df F-ratio P Log Size 1 18.512 0.000 ------------------------------------------------------------------------------------------------- Dep Var: b N:43 R2: 0.3454 df F-ratio P Log Size 1 17.4477 0.0002 Channel Shape 3 0.6763 0.5719 ------------------------------------------------------------------------------------------------ Dep Var: b N:43 R2: 0.3755 df F-ratio P Log Size 1 7.3755 0.0102 Channel Shape 3 0.2800 0.8409 Channel Shape * Log Size 3 0.5609 0.6444 Note: Statisitcally significant at the 0.1 level or less are indicated in Bold. 208 Braided Parabolic Rectangular Triangular Overall Intercept 0.3201 0.4198 0.4049 0.3061 0.3889 Coefficient -0.0666 -0.1093 -0.1153 -0.0295 -0.0944 R2 0.3039 0.3480 0.4352 0.0213 0.3110 ______________________________________________________________________ Figure 6.33 b and Subbasin Log Size in Channel Shape In general, as subbasin size increases, the b value decreases. However, in triangular shape channels, when the subbasins exceed 100 square miles in area, there is a discontinuity in the downward slope line. Even though the sample size is small (12), could it be an example of representative elementary area (REA) phenomenon, which is defined as a critical area at which implicit continuum assumptions may be used without knowledge of patterns of parameter values although some knowledge of the underlying distributions may still be necessary? The implicit notion of the REA is that at scales smaller than the REA the actual parameters of variability are important in determining -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.5 0.5 1.5 2.5 3.5 Log (Watershed Area in Square Miles) All Braided Parabolic Rectangular Triangular Trendline (All) Trendline (Braided) Trendline (Parabolic) Trendline (Rectangular) Trendline (Triangular) 209 the hydrological response and at larger scales a statistical description of spatial variability should suffice. Even though, REA in the literature has been historically much smaller (1 ? 5 km2) than 100 square miles in size, as Bloschl et al. (1995) and Beven (1995) suggested, it is conceivable that the concept is feasible for larger basins, as it seems likely that the appropriate size of an area might vary between watershed environments and processes. The REA might be at 100 square miles for the at-a-station hydraulic geometry in the study area? It needs to be further investigated, but it is out of scope in this research. f Exponent (Figure 6.33 and Table 6.24) The computed value of a1.f, 4.456, exceeds the critical values for level at ? = 0.05, but a2.f, 0.5839, does not exceed the critical value of the 0.1 level. Therefore, the null hypothesis for a1.f is rejected and it is concluded that scale influences f, but the channel shape is an insignificant variable to influence f at the 0.1 level. By adding an interaction term, R2 improved and the role of channel shape became more significant but still not significant at ? = 0.1 level. 210 Table 6.24 f and Subbasin Log Size and Channel Shape _____________________________________________________________________ Dep Var: f N:43 R2: 0.1090 df F-ratio P Log Size 1 5.021 0.0310 ------------------------------------------------------------------------------------------------- Dep Var: f N:43 R2: 0.1484 df F-ratio P Log Size 1 4.4564 0.0414 Channel Shape 3 0.5839 0.6292 ------------------------------------------------------------------------------------------------ Dep Var: f N:43 R2: 0.2198 df F-ratio P Log Size 1 2.1690 0.1498 Channel Shape 3 1.3180 0.2841 Channel Shape * Log Size 3 1.0688 0.3748 Note: Statisitcally significant at the 0.1 level or less are indicated in Bold. Braided Parabolic Rectangular Triangular Overall Intercept 0.3580 0.4198 0.1957 0.3128 0.2333 Coefficient -0.0037 -0.1093 0.0578 -0.0086 0.0420 R2 0.0016 0.2357 0.3667 0.0016 0.1091 ______________________________________________________________________ Figure 6.34 f and Subbasin Log Size in Channel Shape 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -0.5 0.5 1.5 2.5 3.5 Log (Watershed Area in Square Miles) f All Braided Parabolic Rectangular Triangular Trendline All Trendline Braided Trendline Parabolic Trendline Rectangular Trendline Triangular 211 m Exponent (Figure 6.35 and Table 6.25 ) The computed value of a1.m, 3.609, exceeds the critical values for level at ? = 0.050, the null hypothesis for a1 is rejected and it is concluded that scale influences m, but for a2.m, 0.488 does not exceed critical value at the 0.1 level. Therefore, it is concluded that channel shape is an insignificant variable to influence m. Table 6.25 m and Subbasin Log Size and Channel Shape _____________________________________________________________________ Dep Var: m N:43 R2: 0.086 df F-ratio P Log Size 1 3.843 0.057 ------------------------------------------------------------------------------------------------- Dep Var: m N:43 R2: 0.1196 df F-ratio P Log Size 1 3.6089 0.0651 Channel Shape 3 0.4882 0.6925 ------------------------------------------------------------------------------------------------ Dep Var: m N:43 R2: 0.1276 df F-ratio P Log Size 1 1.3889 0.2466 Channel Shape 3 0.1674 0.9177 Channel Shape * Log Size 3 0.1063 0.9558 212 Braided Parabolic Rectangular Triangular Overall Intercept 0.3376 0.4991 0.4056 0.3744 0.3842 Coefficient 0.0638 0.0064 0.0550 0.0415 0.0505 R2 0.3720 0.0016 0.1210 0.0175 0.0857 ______________________________________________________________________ Figure 6.35 m and Log Size in Channel Shape Type In general, as subbasin size increases, the m value increases as well (Figure 6.34). However, in subbasins larger than 100 square miles, the slope line is broken as has been shown in the b case. Since b and m are complementary and the correlation between b and m is strong, it is expected. The channel shape?s role to m is insignificant; however, the braided channels have the highest R2, 0.372, among all channel shapes. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.5 0.5 1.5 2.5 3.5 Log (Watershed Area in Square Miles) m All Braided Parabolic Rectangular Triangular Trendline All Trendline Braided Trendline Parabolic Trendline Rectangular Trendline Triangular 213 6.7.3 Summary In general, as subbasin size increases, the b value decreases, and f and m values increase. But there seems to be a break in size and b, f, and m trend lines around 100 square miles. The role of the channel shape is insignificant for b, f, and m, but the b values for rectangular and parabolic channels have higher R2, 0.435 and 0.348, respectively. The rectangular channels are the highest R2, 0.367, for the f values and, the braided channels (R2 = 0.372) for the m values. It is expected to find that channel shape does not strongly effect b, f, and m. That is, channel shape can be changed so easily and each hydrographic survey is a snapshot of the channel profile at a given time. Even though the number and duration of hydrographic surveys used in this study are high and span a long time frame (around 10 years), the hydrographic surveys are conducted without consideration of the channel?s resumption to its normal shape. Therefore, the hydrographic survey might have conducted in various stages of return to normal channel shape from extreme conditions. Therefore, channel shape seems to be a clear-cut variable in a theoretical sense, but it is an obscure, hard to deal with indicator for investigating b, f, and m, unless the surveys are very controlled with consideration of the normalcy of the channel shape. 6.8 The Role of Channel Bed Material The main stem Potomac River and major tributaries generally have bedrock bottoms, with alluvial sediments in depositional areas. The study area stream-bottom materials range from bedrock to small cobbles and gravel in upstream areas, and from gravel, sand, and silt to eroded fine sediments in Coastal Plain streams. 214 From a rational point of view, the size and pattern/shape of a river channel at a particular point must represent a balance between erosive forces associated with water flow and with resistive forces associated with bed material size, bed structure, and bank cohesion. Variations in flow resistive forces produce not only different flow resistance magnitudes but also different rates of change of flow resistance at a site, therefore, different rates of changes in m. Bathurst (1993) stated that at-a-station hydraulic geometry exponents differ among the different channel types and he approximated that for sand-bed channels, m < 0.40; for gravel-bed rivers, m = 0.35 ? 0.45; for boulder-bed rivers, m = 0.45 - 0.55; and for steep pool/fall streams and pool-riffle sequence, m > 0.55. This pattern suggests an increase in the variation of flow resistance as the bed material size increases. The six major channel bed material types present in the study region are: rock, cobble, gravel, sand, silt, and clay. Twenty two subbasins have rock channels; 17 cobble; 27 gravel; 16 sand; 2 silt, and 2 clay bed channels10. The comparison of the b, f, and m average values for the various bed materials are graphed in Figure 6.36. 10 The channel bed material of each subbasin is identified by visual inspection by the author?s field trip and the USGS field notebook. In the case of multiple bed material subbasins, each type of bed material is counted as a separate subbasin, therefore the subbasin sample size increased to 86 subbasins. 215 N 22 17 27 16 2 2 Mean b 0.2014 0.2421 0.1813 0.1844 0.2234 0.1871 Mean f 0.3125 0.2899 0.3202 0.3536 0.3313 0.2495 Mean m 0.4873 0.4692 0.5032 0.4667 0.4478 0.5657 ______________________________________________________________ Figure 6.36 Average b, f, and m Values by Bed Material 6.8.1 Each Bed Material Type?s Influence The influence of bed material on b, f and m values is examined using a t-test for difference of mean values. Using a 0.05 % test, sand for f is statistically discernable; for a 0.1% test, gravel is statistically discernable for b (Table 6.26)11. Comparing subbasins with sand and non-sand channel beds, the mean difference of f is greater than the t- critical value at the 0.05 level (Table 6.26). Therefore, it can be concluded that there is a 11 The critical value for t with 84 sample size at 0.02 is 2.374, at 0.05 level 1.990, and 0.1 level 1.665, 0.2 level, 1.293; and 0.5 level, 0.678. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Rock Cobble Gravel Sand Silt Clay Mean b Mean f Mean m 216 discernable difference in f values between the sand and the other bed material channels at 0.025 level. The mean difference of the b value in the gravel beds is bigger compared with other bed material types, but not significant at the 0.05 level. Figure 6.37 Mean Differences between Sand, Gravel, and Cobble vs. Other Bed Material 0 0.1 0.3 0.4 0.6 0.7 0.8 Non-Sand Mixed Sand 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Non-Gravel Mixed Gravel 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Non-Cobble Cobble 0.2 0.5 217 Table 6.26 Differences of Mean b, f, and m within Each Bed Material Number of Subbasins Mean b Mean b Difference Mean f Mean f Difference Mean m Mean m Difference Rock 22 0.2014 0.3125 0.4873 Non-Rock 21 0.2137 0.0123 0.3160 0.0035 0.4750 0.0123 t-test -0.2977 -0.1135 0.3973 Cobble 17 0.2421 0.2899 0.4692 Non-Cobble 26 0.1847 0.0574 0.3300 0.0401 0.4891 0.0199 t-test 1.3837 -1.2830 -0.4618 Gravel 27 0.1813 0.3202 0.5032 Non-Gravel 16 0.2514 0.0701 0.3041 0.0161 0.4443 0.0589 t-test -1.6895 0.4995 1.3774 Sand 16 0.1844 0.3536 0.4667 Non-Sand 27 0.221 0.0366 0.2908 0.0628 0.4899 0.0232 t-test -0.8611 2.0425 -0.5337 Silt 2 0.2234 0.3313 0.4478 Non-Silt 41 0.2066 0.0168 0.3133 0.018 0.482 0.0342 t-test 0.1707 0.2421 -0.3425 Clay 2 0.1871 0.2495 0.5657 Non-Clay 41 0.2084 0.0021 0.3173 0.0678 0.4817 0.084 t-test -0.2169 -0.9264 0.8455 Note: Statistically significant at the 0.05 level is highlighted in underlined bold, and the 0.1 level and less in bold. Considering channel bed material only, everything else being equal, the expected order of b and f change is sand, gravel, cobble, and rock in the order of easiest to hardest to erode. It is also expected that, unlike bank material, bed material would influence f more than b. Table 6.27 illustrates the comparative erodability among the different lithologies, one being the lithology that is easiest to be changed, and five being the most 218 resistant. As stated in the Physiography section, the expected m order would be the inverse of order of b and f. Table 6.27 Bed Material Based Mean b, f, and m Values and Expected b, f, and m Changes ________________________________________________________________________ N Expected Expected Order of Order of b and f Observed Observed m Changes Observed Changes b f m ________________________________________________________________________ Rock 22 4 0.2014 0.3125 1 0.4873 Cobble 17 3 0.2421 0.2899 2 0.4692 Gravel 27 2 0.1813 0.3202 3 0.5032 Sand 16 1 0.1844 0.3536 4 0.4667 Source: Inferred from Bathurst?s (1993) finding. In general, an increase in stream discharge or channel depth results in a decrease in the bed materials? relative roughness and, thus, in the resistance resulting in higher m values. For sand-bed channels, however, the accompanying increase in stream power with increased water discharge can also produce a change in bed form, from plane bed to ripples to dunes. The mean m value of the 16 sand bed channel profiles illustrates this point by having low m values compared to the other channel bed types. The effect of decreasing channel bed relative roughness is, therefore, partly offset by the effect of increasing bedform drag, producing a relatively restrained rate of increase of flow velocity with higher discharge (Richards, 1982). The channel profile of m48 shows that the bedform is not smooth. However, the high flow submerged these riffle or sand dune bedforms, with resultant m values as high as 0.668. The other example is the 219 v38 channel profile case. Here the flow is low and the uneven bedform has caused high bedform drag which caused the mean m value to be as low as 0.1676. In gravel-bed (containing both gravel and cobble and gravel and rock) channels, drops in flow resistance are achieved by the drowning of roughness elements as discharge rises, therefore causing high m values in all gravel channels. Subbasins v52 and v19 both had low flows during the study period, and exhibit low m values, 0.212 and 0.346, respectively. On the other hand, in gravel-bed channels not much bedform development is expected, therefore, the b values are low and the flow velocity increases at a faster rate (Table 6.27). In rock-bed channels, the rate of increase in m is larger than for cobble channels because of greater contrasts between low- and high-flow conditions. At low flows, the rocks protrude through the flow, rock form drag is high, and only the low velocity zone between the rocks is subject to bedform changes. As discharge increases, the rock bed channels are expected to have the highest m values among the different bed material types because of drowning rough bed material and smoothing the roughness. As expected, the exponent m increases in value, moving up the network from sand bed channels, via gravel, cobble and boulder bed channels (Bathurst, 1993) except the gravel bed (Table 27). The gravel beds have the highest m value, 0.5032, which is higher than cobble and rock bed channels. Excluding the two lowest m value subbasins, m31 and m311, the rock channel beds would have the highest mean m value and follow the expectation of being high m value channels. The m31 and m311 subbasins have meander channel patterns which are expected to have high b and low m values. With highest b values of the study area, the stream power must have been consumed by making rapid 220 changes of channel width, resulting in low m values. Also, it can be explained that these two subbasins did not reach high flows which would drown out the ripple and ponding effects. 6.8.2 b, f, and m and Channel Bed Material in Scale Context The influencing role of the channel bed material to b, f, and m in a scale context is investigated by setting the model, b (f, m) = a0.b (a0.f, a0.m) + a1.b (a1.f, a1.m )* log10 (size) + a2.b (a2.f, a2.m)* channel bed material variable. The null hypothesis is that the geographical scale of the subbasins and channel bed material have no bearing on the b, f, and m values. b Exponents (Figure 6.38 and Table 6.28) As expected, the computed value of a1.b, -4.440, exceeds the critical values for level at ? = 0.00007, but a2.b, -1.1037, does not exceed the critical value even at the 0.1 level. Therefore it is concluded that channel bed material is an insignificant variable to influence b at the 0.1 level. Adding an interaction term to the model, R2 increases very little, but the bed material?s role on b becomes less significant. Therefore, it is concluded that the role of the channel bed materials is an insignificant influence on b values. 221 Table 6.28 b and Subbasin Log Size and Bed Material _____________________________________________________________________ Dep Var: b N:43 R2: 0.3110 df F-ratio P Log Size 1 18.512 0.000 ------------------------------------------------------------------------------------------------- Dep Var: b N:43 R2: 0.3309 df t-value P Log Size 1 -4.4400 0.0001 Bed Material 5 -1.1037 0.2763 ------------------------------------------------------------------------------------------------ Dep Var: b N:43 R2: 0.3399 df t-value P Log Size 1 -2.3322 0.0249 Bed Material 5 -1.0682 0.2920 Bed Material * Log Size 5 0.7311 0.4691 Rock Cobble Gravel Sand Silt* Clay* Overall N 22 17 27 16 2 2 86 Intercept 0.4391 0.4071 0.3274 0.3617 -14.8718 -0.6665 0.2333 Coefficient -0.4173 -0.0917 -0.0743 -0.0924 11.518 0.5553 -0.0944 R2 0.4833 0.3312 0.2015 0.2087 1.0000 1.0000 0.3111 Figure 6.38 b and Subbasin Log Size in all Channel Bed Materials Note: Since silt and clay have only 2 subbasins each, they are excluded from the analyses. 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.5 0.5 1.5 2.5 3.5 Log (Watershed Area in Square Miles) b All Rock Cobble Gravel Sand Trendline (All ) Trendline (Rock) Trendline (Cobble) Trendline (Gravel) Trendline (Sand) 222 f Exponents (Figure 6.39 and Table 6.29) The computed value of a1.f, 2.416, exceeds the critical values for ? = 0.021, but a2.f, -1.212, does not exceed the critical value at the 0.1 level. Therefore, it is concluded that channel bed material is an insignificant variable to influence f. Adding an interaction term to the model, the R2 increased but the influencing role of bed material on f worsened from a 0.233 level to a 0.592 level. Even though there is a correlation between f and sand bed channels at the 0.05 level, the overall influencing role of bed materials on f is statistically indiscernible. Table 6.29 f and Subbasin Log Size and Bed Material _____________________________________________________________________ Dep Var: f N:43 R2: 0.1090 df F-ratio P Log Size 1 5.021 0.0310 -------------------------------------------------------------------------------------------------------- Dep Var: f N:43 R2: 0.1407 df t P Log Size 1 2.4160 0.0204 Bed Material 4 -1.2118 0.2327 -------------------------------------------------------------------------------------------------------- Dep Var: f N:43 R2: 0.9595 df t P Log Size 1 1.4112 0.1661 Bed Material 4 0.5409 0.5917 Bed Material * Log Size 4 -1.0029 0.3221 223 Rock Cobble Gravel Sand Silt* Clay* Overall Intercept 0.2716 0.1560 0.2358 0.2464 -14.1390 0.6212 0.2333 Coefficient 0.0201 0.0744 0.0429 0.0559 11.3777 -0.1214 0.0420 R2 0.0374 0.3982 0.1035 0.1346 1.0000 0.2582 0.1091 _______________________________________________________________________ Figure 6.39 f and Subbasin Log Size in all Channel Bed Material Types Note: Since silt and clay have only 2 subbasins each, they are excluded from the analyses. m Exponents (Figure 6.40 and Table 6.30) The computed value of a1.m, 1.900, exceeds the critical values for level at ? = 0.0646, but a2.m, 0.0096, does not exceed the critical value at the 0.1 level. Therefore, it is concluded that the null hypothesis for a1.m is rejected and that subbasin scale influences m at the 0.1 level of significance, but the channel bed material is an insignificant variable to influence m. Adding an interactive term to the model, the R2 improved and the role of the bed material improved to 0.2586 from 0.9924. But in general, as shown in Figure 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.5 0.5 1.5 2.5 3.5 Log (Watershed Area in Square Miles) f All Rock Cobble Gravel Sand Trendline (All) Trendline (Rock) Trendline (Cobble) Trendline (Gravel) Trendline (Sand) 224 6.40, as subbasin size increases, the m values increase as well. However, the relationship is not strong; actually the role of the bed material is weaker than f or b. Table 6.30 m and Subbasin Log Size and Bed Material Types _____________________________________________________________________ Dep Var: m N:43 R2: 0.0858 df F-ratio P Log Size 1 3.843 0.0570 -------------------------------------------------------------------------------------------------------- Dep Var: m N:43 R2: 0.0857 df t P Log Size 1 1.9004 0.0646 Bed Material 4 0.0096 0.9924 -------------------------------------------------------------------------------------------------------- Dep Var: m N:43 R2: 0.1190 df t P Log Size 1 1.8126 0.0776 Bed Material 4 1.1465 0.2586 Bed Material * Log Size 4 -1.2134 0.2323 225 Rock Cobble Gravel Sand Silt* Clay* Overall Intercept 0.2938 0.4431 0.4496 0.3891 0.7352 -0.2867 0.3842 Coefficient 0.0954 0.0145 0.0272 0.0404 -0.2260 0.3058 0.0505 R2 0.3660 0.0079 0.0248 0.0344 1.0000 0.4359 0.0857 ________________________________________________________________________ Figure 6.40 m and Subbasin Log Size in all Channel Bed Material Types Note: Since silt and clay have only 2 subbasins each, they are excluded from the analyses. 6.8.3 f > m in Sand Bed with Increasing Stream Discharge As an empirical theory, it is known that, in sand bed channels, the change rate of depth is greater than that of velocity as discharge increases. To test this, the null hypothesis is set so that the change rate of depth is not greater than that of velocity as discharge increases, such that H0: f ? m < 0. Sixteen study area channels are categorized as sand bed channels, and 12 of these have 50% or more sand content in the channel bed. Difference of means analysis of f ? m values between sand beds with any amount of sand and non-sand bed indicates that sand beds are statistically insignificant at the 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.5 0.5 1.5 2.5 3.5 All Rock Cobble Gravel Sand Trendline (All) Trendline (Rock) Trendline (Cobble) Trendline (Gravel) Trendline (Sand) m Log (Watershed Area in Square Miles) 226 level. However, the mean difference analysis indicates that sand beds with 50% or more sand versus non-sand beds are statistically significant at the 0.1 level (actually significant at the 0.02 level). It is concluded that the role of sand bed with 50% or more sand content is statistically significant for the values of f - m. The higher the channel bed sand content, the rates of channel depth changes are greater than that for flow velocity (Figure 6.41). The average f - m values for the total 16 sand beds is -0.1131, and -0.0507 for sand beds with 50% or more sand content. The t-value between H0 mean, 0.000, and that of the beds with 50% or more sand, -0.05071, is statistically insignificant (0.3727 level). Therefore, H0 cannot be rejected for f ? m < 0 when the bed is composed of 50% or more with sand (Figure 6.42). Even though the empirical theory is not applicable in this study area, the study trend shows that depth increases faster is consistent (Table 6.31). Figure 6.41 f ? m Analysis in Sand Bed Channel -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 20 40 60 80 100 Sand Percentage f - m f - m Trendline (f - m) 227 Table 6.31 Differences of Mean f - m Values in Sand Beds and all Study Channel Beds _______________________________________________________________________ N Mean f ? m R2 t Probability Mean f - m Difference___ Sand 16 -0.1131 Non-Sand 27 -0.1991 0.0860 t-test 1.3894 0.230 Sand > 50% 12 -0.0507 Non-Sand > 50% 31 -0.2121 0.030 0.1614 t-test 2.5451 H0 0.0000 Sand > 50% 12 -0.0507 0.0801 0.9332 0.3727 ----------------------------------------------------------------------------------------------------------- Log Size 0.0251 0.5077 0.6227 ----------------------------------------------------------------------------------------------------------- Sand > 50% & Log Size 0.2137 Log Size 1.2368 0.2475 Sand > 50% 1.4694 0.1758 Adding the scale factor to the regression of sand > 50% bed channels and f ? m value, the R2 become 0.2137 from 0.0801 for just sand > 50% bed material and f ? m. The R2 for log size and f ? m values in sand > 50% bed is 0.0251 (Table 6.31). This indicates that combination of scale and sand > 50% substantially contributes the R2 value. In the scale context, the t test indicates that the statistical significance of the sand > 50% to f ? m value improved to 0.1758 from 0.3727. Therefore, it can be concluded that there is no discernable differences between f and m in the study area channels and scale factor is statistically insignificant but with the combination of scale and sand > 50% influence f ? m at the 0.1758 level. 228 Figure 6.42 f ? m and Subbasin Size in All Sand Bed Channels 6.8.4 Empirical Theory, f < m in Gravel Bed with Increasing Stream Discharge Another empirical theory, in gravel bed channels, that the change rate of flow velocity is greater than that of channel depth as discharge increases, is examined in a scale context. Similarly to sand beds, the null hypothesis is set so that the change rate of velocity is not greater than that of depth as discharge increases, such that H0: f ? m > 0. Twenty seven study area channels are categorized as gravel bed channels, and 21 of these have 50% or more gravel content in the channel bed. Difference of means analysis between gravel and non-gravel beds for the f ? m value indicates that the difference is statistically insignificant at the 0.1 level12. The t-test of the difference of means between gravel >50% and non-gravel beds demonstrated that 12 The calculated t value is -0.8060 and the critical value for t with 2 degrees of freedom at the 0.05 level is 2.021 and the 0.1 level is 1.684. -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 0.5 1 1.5 2 2.5 3 3.5 Log (Watershed Area in Square Miles) f - m f - m Trendline (f - m) 229 the f ? m values are much less significant. Clearly, f < m can be explained by the suppressed resistance of the water flow from increased stream discharge, which makes the velocity increases greater than the channel depth change rate. However, the hypothesis that in gravel channel, the f - m > 0, is statistically non-supportive in the study area. In general, the study area channels have higher m values than f. The average f - m value for subbasins with gravel bed is -0.1830, and -0.1767 for the subbasins where 50% or more of the bed content is gravel. The higher the channel gravel content, the rates of the velocity become greater than depth changes (Figure 6.42 and Table 6.32). The slope of the linear relationship between the content of gravel in the bed and the f ? m values is - 0.00086 (Figure 6.43 and Table 6.32). Figure 6.43 f ? m Analysis in Gravel Bed Channel All Gravel Bed -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 25 50 75 100% Percentage of Gravel Content in the Channel Bed f - m Trendline (f - m) f-m 230 Table 6.32 Differences of Mean f - m Values in Gravel and All Non-Gravel Channel Beds _______________________________________________________________________ N Mean f ? m R2 t Probability Mean f - m Difference_ Gravel 27 -0.1830 Non-Gravel 16 -0.1402 0.0282 t-test -0.6788 Gravel>50% 21 -0.1767 Non-Gravel>50% 22 -0.1579 0.0188 t-test -0.3093 H0 0.0000 Gravel>50% 21 -0.1767 0.0395 -0.8836 0.3879 ----------------------------------------------------------------------------------------------------------- Log Size 0.0264 0.7183 0.4813 ----------------------------------------------------------------------------------------------------------- Gravel & Log Size 0.0817 Log Size 0.9097 0.3750 Gravel>50% -1.0407 0.3118 Examining gravel bed channels and f ? m values in a scale context, the influence of scale to the f ? m in gravel bed channels is statistically insignificant (0.4813 level), and much less significant than gravel > 50% for f ? m values. As expected, with both gravel 50% and scale, the R2 improved and both gravel > 50% and scale?s roles became more statistically significant, but still not significant at the 0.1 level. By visual inspection (Figure 6.44), it is easy to conclude that subbasin scale factor is insignificant, especially since there is no correlation around the subbasins of about 100 square miles. Beyond that size the trend line may well differ from those of subbasins smaller than 100 square miles. 231 Figure 6.44 f ? m Relation in All Gravel Bed Channels 6.8.5 Summary Comparing difference of means values, it is found that f values are statistically discernible between sand and non-sand bed channels. The differences of b means between cobble and non-cobble channels as well as the difference of b and m means in gravel and non-gravel channels are also found to be sufficiently different. In the geographic scale context, even though there are statistically significant correlations between scale and b, f, and m, the overall influencing role of bed materials on b, f, and m is indiscernible. The role of sand bed is statistically significant for f > m and, the higher the sand content in the channel bed, the rates of the depth changes are greater and the difference of f > m gets smaller. Adding the scale factor to the regression of sand bed channels and f ? All Gravel Beds -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0 0.5 1 1.5 2 2.5 3 3.5 Log (Watershed Area in Square Miles) f-m f - m Trendline (f - m) 232 m, the sand bed material is statistically insignificant at 0.1 level, and scale became further insignificant. Investigating the empirical theory, that the change rate of depth gets greater than that of velocity as discharge increases in sand bed channels, it is found that the role of sand beds with 50% or more sand content is statistically significant, and that the rates of channel depth changes are greater and the difference of f ? m gets smaller. However, the rates of the depth changes in these sand > 50% channels are not greater than that for flow velocity within the entire study area. Investigation of gravel bed channels confirms the empirical theory that the change rates of flow velocity is greater than that of channel depth as discharge increases. Examining the gravel bed channels and f ? m values in a scale context, it is found that the influence of scale on the f ? m in gravel bed channels is statistically insignificant. The higher the channel gravel content, the differences between the rates of flow velocity and channel depth changes become greater. 6.9 The Role of Channel Bank Material The study region encompasses six major channel bank material types; nine subbasins have rock channels; 12 cobble; 3 gravel; 30 sand; 11 silt, and 4 clay bed channels13. Since the gravel and the clay bank material channels are represented by only 3 and 4 subbasins, these two bank materials are excluded from the analyses. The comparative analysis of the b, f, and m average values for the various bank materials are graphed (Figure 6.45). 13 The channel bed material of each subbasin is identified by visual inspection by the author?s field trip and the USGS?s field notebook. In case of multiple bed material subbasins, each bed material is counted as a separate subbasin, therefore the 43 subbasin sample size increased to 69 subbasins. 233 N 9 12 3 30 11 4 Mean b 0.231 0.229 0.2629 0.2104 0.1403 0.2052 Mean f 0.2831 0.2775 0.2148 0.3167 0.3547 0.3178 Mean m 0.4875 0.4583 0.5212 0.4753 0.5093 0.4779 ________________________________________________________________________ Figure 6.45 Average b, f, and m Values by Bank Material Note: Since gravel and clay have only 3 and 4 subbasins, respectively, they are excluded from the analyses. 6.9.1 Each Bank Material Type?s Influence The influence of bank material on b, f and m values is examined using a t-test for difference of mean values. Comparing subbasins with the silt and non-silt banks, the mean differences of b are the greatest among all other bank material type channels. Using a 0.1 % test, silt for b is statistically discernable, cobble for f and silt for f (Table 6.33)14, are insignificant at the 0.1 level. 14 The critical value for t with sample size of 62 at the 0.01 level is 2.488; at 0.05 level, 1.99, and 0.1 level 1.665. 0 0.1 0.2 0.3 0.4 0.5 0.6 Rock Cobble Gravel Sand Silt Clay Mean b Mean f Mean m 234 Figure 6.46 Mean Differences between Silt and Cobble vs. Other Bank Material 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Non-Cobble Cobble b f m Trendline (b) Trendline (f) Trendline (m) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Non-Silt Mixture Silt 235 Table 6.33 Differences of Mean b, f, and m within Each Bank Material Number of Subbasins Mean b Mean b Difference Mean f Mean f Difference Mean m Mean m Difference Rock 9 0.2310 0.2831 0.4875 Non- Rock 53 0.2011 0.0299 0.3224 0.0393 0.4796 0.0079 t-test 0.5889 -1.0403 0.1518 Cobble 12 0.2290 0.2775 0.4583 Non- Cobble 50 0.1990 0.0300 0.3284 0.0509 0.4812 0.0229 t-test 0.6511 -1.5018 -0.4800 Sand 30 0.2104 0.3167 0.4753 Non- Sand 32 0.2004 0.0100 0.3084 0.0083 0.4951 0.0198 t-test 0.2221 0.2436 -0.4312 Silt 11 0.1403 0.3547 0.5093 Non-Silt 51 0.2304 0.0901 0.3002 0.0545 0.4716 0.0377 t-test -1.986 1.5705 0.7841 Note: The critical value for t with 2 degrees of freedom at the 0.01 level is 2.488; at 0.05 level, 1.99, and 0.1 level 1.665. The statistical significance level is 0.1 and les are in Bold. Considering channel bank material only, everything else being equal, the expected order of b change is sand, gravel, cobble, and rock in the order of the easiest to erode to the most resistant (Table 6.34). In easily eroded bank material, a relatively wide shallow channel develops, while in cemented bank materials, the channel becomes deeper and narrower (Friedkin, 1945). In the case of flow velocity, as streams begin eroding their banks, their channel cross-sections become wide and shallow with consequent reduction of flow velocities (Friedkin, 1945). 236 Table 6.34 Bank Material Based Mean b, f, and m Values and Expected b, f, and m Changes _____________________________________________________________________ N Expected b Values Expected f Values Expected m Values Order of Order of Order of b Change f Change m Change Rock 9 4 0.2310 1 0.2831 4 0.4875 Cobble 12 3 0.2290 2 0.2775 3 0.4583 Sand 30 2 0.2104 3 0.3167 2 0.4753 Silt 11 1 0.1403 4 0.3547 1 0.5093 ______________________________________________________________________ Source: Expected order of b, f, and m changes are inferred from the Friedkin (1945) and Bathurst?s (1993) finding. The b, f, and m values are calculated by the author. Contrary to the expected order of changes in b, rock banks have the highest rate of b and the 2nd lowest rate of f changes. Also, more resistant banks result in higher rates of m changes than less resistant banks. In easily erodable bank materials, a relatively wide shallow channel develops but in the study area small rates of width and bigger rates of depth changes occur. Since each bank, typically, is not of uniform material, but rather a mixture of materials, it is understandable that the observed results are different from the expected order of b, f, and m changes. 6.9.2 b, f, and m and Channel Bank Material in Scale Context The role of the channel bank material relative to b, f, and m in a scale context is investigated by the model, b (f, m) = a0.b(a0.f , a0.m ) + a1.b(a1.f, a1.m) * log10(size) + a2.b (a2.f, a2.m) * channel bank material variable. The null hypothesis is that the scale of subbasins and the channel bank material have no bearing on b, f, and m values. 237 b Exponents (Figure 6.47 and Table 6.35) The computed t-test value of a1.b, -3.960, exceeds the critical values for level at ? = 0.0003, but, the computed t-test value for a2.b, -1.620, does not exceed the critical value at the 0.1 level (but exceeds it at 0.1132). Therefore, it is concluded that scale influences b, but channel bank material is not a significant predictor variable to influence b at the 0.1 level of statistical significance. Adding an interaction term to the model does increase the R2, but the role of bank material degraded. Therefore, it is concluded that the interaction term does not explain the influence of bank material on b. Table 6.35 b and Subbasin Log Size and Bank Material _____________________________________________________________________ Dep Var: b N:69 R2: 0.3105 df t-ratio P Log Size 1 18.512 0.0000 -------------------------------------------------------------------------------------------------------- Dep Var: b N:69 R2: 0.3529 df t-value P Log Size 1 -3.9604 0.0003 Bank Material 5 -1.6195 0.1132 -------------------------------------------------------------------------------------------------------- Dep Var: b N:69 R2: 0.3602 df t-value P Log Size 1 -2.3322 0.2148 Bank Material 5 -1.0682 0.8859 Bank Material * Log Size 5 -0.6679 0.5081 ______________________________________________________________________ 238 Rock Cobble Sand Silt Overall N 9 12 30 11 69 Intercept 0.3577 0.3347 0.4300 0.3098 0.3889 Coefficient -0.0758 -0.0485 -0.1151 -0.0813 -0.0944 R2 0.4151 0.0505 0.3093 0.3014 0.3111 __________________________________________________________________ Figure 6.47 b and Subbasin Log Size in Channel Bank Materials f Exponents (Figure 6.48 and Table 6.36) As expected, the computed t-test value of a1.f, 2.01, exceeds the critical values for ? = 0.0509, which is not a surprise since systematic changes of f and scale exist. However, the t-test value for a2.f, -1.235, does not exceed the critical value at the 0.1 level. Therefore, it is concluded that channel bank material is not a significant variable to influence f at the 0.1 level, which is statistically less significant compared to the relationship between b and channel bank materials. This is expected since bank material influences b more than f. There is a positive relationship between f and subbasin size, 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.5 0.5 1.5 2.5 3.5 All Rock Cobble Sand Silt Trendline (All) Trendline (Rock) Trendline (Cobble) Trendline (Sand) Trendline (Silt) b Log (Watershed Area in Square Miles) of freedom at the 0.01 level is 2.488; 239 and specific bank material types and f values in a scale context are positive, but the intensity is not strong, as for overall bank material types and f values. Since the influencing role of channel bank material on f is not significant, the interaction between scale and bank material is investigated by adding an interaction term to the model, f = a0.f + a1.f * log10(size) + a2.f * channel bank material + a3.f * log (size) * channel bank material. The statistical significance of the role of bank material type on f is significantly improved to the 0.0479 level. This indicates that variables channel bank material and scale clearly interact. Therefore, the interaction of the role of bank material to f values and geographic scale needs to be considered for the examination of the role of bank material on f. Table 6.36 f and Subbasin Log Size and Bank Material _____________________________________________________________________ Dep Var: f N:43 R2: 0.1091 df t-test P Log Size 1 2.241 0.0310 ------------------------------------------------------------------------------------------------------- Dep Var: f N:43 R2: 0.1418 df t-value P Log Size 1 2.0127 0.0509 Bank Material 5 -1.2351 0.2240 -------------------------------------------------------------------------------------------------------- Dep Var: f N:43 R2: 0.1982 df t-value P Log Size 1 -0.8486 0.4013 Bank Material 5 -2.0425 0.0479 Bank Material * Log Size 5 1.6551 0.1059 ______________________________________________________________________ Note: The statistical significance level is 0.1 and les are in Bold. 240 Rock Cobble Sand Silt Overall N 9 12 30 11 69 Intercept 0.1698 0.2453 0.2451 0.3513 0.2333 Slope 0.0678 0.0148 0.0375 0.0016 0.0420 R2 0.4607 0.0050 0.0779 0.0002 0.1091 __________________________________________________________________ Figure 6.48 f and Subbasin Log Size in Channel Bank Materials m Exponents (Figure 6.49 and Table 6.37) The computed t-test value of a1.m, 1.87, exceeds the critical values for ? = 0.069, but the t-test value for the channel bank material term, a2.m, 0.058, does not exceed the critical value at 0.1 level. Therefore, it is concluded that channel bank material alone is an insignificant variable influencing m. Overall, the relationship between m and bank material type in a scale context is positive. In particular, the relationships between silt banks and m (R2 of 0.5227) are much stronger than the overall relationship. 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 All Rock Cobble Sand Silt Trendline (All) Trendline (Rock) Trendline (Cobble) Trendline (Sand) Trendline (Silt) f Log (Watershed Area in Square Miles) 241 Considering the interaction of geographic scale and bank material type, the role of bank material to m becomes significantly improved from the 0.954 to 0.119 level of statistical significance. At the same time, the statistical significance of scale impact on m deteriorates from a level of 0.069 to 0.413. This indicates that there is a strong interaction between subbasin scale and bank materials that influences m values, and this fact is clearly illustrated by having a 0.069 level for the interaction term, a3.m * log (size) * channel bank material. Therefore, when considering the role of bank material on m values, the interaction of subbasin scale and bank material needs to be taken into account. In general, m values are high for all bank material types, especially silt channel banks where b values are low. This implies that, in addition to bank material, there are many more predictor variables such as channel pattern and shape that influence m values. Table 6.37 m and Subbasin Log Size and Bank Material _____________________________________________________________________ Dep Var: m N:43 R2: 0.0857 df t-test P Log Size 1 1.960 0.0570 ------------------------------------------------------------------------------------------------------- Dep Var: m N:43 R2: 0.0858 df t-value P Log Size 1 1.8689 0.0690 Bank Material 5 0.0577 0.9543 ----------------------------------------------------------------------------------------------------- Dep Var: m N:43 R2: 0.1610 df t-value P Log Size 1 -0.8281 0.4127 Bank Material 5 -1.5942 0.1190 Bank Material * Log Size 5 1.8703 0.0690 ____________________________________________________________________ 242 Rock Cobble Sand Silt Overall N 9 12 30 11 69 Intercept 0.4804 0.4212 0.3339 0.3332 0.3842 Coefficient 0.0043 0.0333 0.0742 0.0845 0.0505 R2 0.0010 0.0140 0.1247 0.5227 0.0857 __________________________________________________________________ Figure 6.49 m and Bank Materials on Subbasin Log Size 6.9.3 Summary Contrary to the expected order of changes in b, rock banks have the highest rate of b changes and the 2nd lowest rate of f changes. Since each bank, typically, is not of uniform material, but rather a mixture of materials, it is understandable that the observed results are different from the expect order of b, f, and m changes. Scale influences b, f, and m, but channel bank material is not a significant predictor variable to influence b, f, nor m at 0.1 level of statistical significance, but adding interaction between scale and channel bank materials, the interaction term became 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 All Rock Cobble Sand Silt Trendline (All) Trendline (Rock) Trendline (Cobble) Trendline (Sand) Trendline (Silt) m Log (Watershed Area in Square Miles) 243 statistically significant for m and f. Therefore, the interaction of the role of bank material to f and m values and geographic scale needs to be considered for the examination of the role of bank materials on f and m. 6.10 The Role of Channel Asymmetry (Figure 6.50 and Table 6.38) Without consideration of the shifting direction of channels such as left or right, the relationships between b, f, and m values and channel asymmetry are examined. The relationships between the b, f, and m and channel asymmetry are not strong. The R2 values for linear regression between b, f, and m and asymmetry are very low (under 3%). The slopes of the linear regression scatter plot between b and m and channel asymmetry are much steeper (almost 0.4) than that of f. The b values get bigger as channels become more asymmetrical and the m values get smaller. The f values do not have strong slope. The positive stronger relationship between b and asymmetry is expected because when the thalweg of channel moves from the center of the channel, the width of channel will inevitably change. Since stream power is consumed in widening the channel, the m must be reduced. It is expected that the overall depth does not have any net changes. Even though the b and m regression slopes are steep, the b and m values are not statistically significant. However, the rate of m convergence is so small that b and m will converge and start to diverge before f completely converges into a common absolute asymmetry value. The trend is as the absolute asymmetry index gets bigger, around 0.25, the b values will be bigger than f values. 244 Figure 6.50 b, f, and m and Channel Asymmetry 6.10.1 b, f, and m and Channel Asymmetry in a Scale Context (Figure 6.51) Introducing the scale of each subbasin to the relationship of b, f, and m with asymmetry, the R2 relationships increased to over 0.354 for b and over 0.114 and 0.113 for f and m from 0.023, 0.001, and 0.018, respectively. It can be explained by observing the influence of scale on the b, f, and m. Since the R2 between scale and b is 0.311, with a statistical significance of 0.0001level, the scale and asymmetry variables influenced b much more than asymmetry did by itself. Also, the statistical significance is improved by adding the scale variable, but the statistical significance is not at the 0.1 level in b (though the statistical significance of asymmetry?s role on b is at the 0.11 level), and in the f and m cases, the role of symmetry and scale are insignificant at that level (Table 6.38). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.05 0.1 0.15 0.2 b, f, m b f m Trendline b Trendline f Trendline m Asymmetry Index 245 Table 6.38 Statistical Significance of Asymmetry and b, f, and m Relationships _________________________________________________________________ N = 43 Asymmetry b f m Log Size R2 0.3105 0.1091 0.0857 Coefficient -0.0940 0.0420 0.0500 Log Size: t -4.3030 2.2410 1.9600 Log Size: p 0.0000 0.0310 0.0570 Asymmetry R2 0.0227 0.0010 0.0177 Coefficient 0.3950 -0.0640 -0.3580 Asymmetry: t 0.9690 -0.2060 -0.8586 Asymmetry: p 0.3380 0.8378 0.3956 Log Size & Asymmetry R2 0.3545 0.1137 0.1128 Log Size: t -4.5360 2.2545 2.0715 Log Size: p 0.0001 0.0297 0.0448 Asymmetry: t 1.6504 -0.4531 -1.1063 Asymmetry: p 0.1067 0.6529 0.2752 Log Size - Log Size & Asym R2 0.0440 0.0046 0.0271 Contribution* 0.0638 0.0052 0.0296 __________________________________________________________________ Note: Contribution is the ratio of the unexplained and the difference between R2 of b, f, and m with log size and with log size and asymmetry combined. The relationship between asymmetry and subbasin size is not strong; the independence of these two predictor variables are maintained for relationships of both predictor variables with b, f, and m. The contribution of scale is stronger for b than for f or m. The role of asymmetry for b, f, and m in a scale context is stronger on b than on f or on m. 246 Figure 6.51 Subbasin Size versus Asymmetry 6.10.2 Summary As expected, the relationship between b and asymmetry is the strongest among the hydraulic geometry exponents but, as a whole, the relationships between asymmetry and b, f, and m are weak, meaning that asymmetry is not a good predictor variable for b, f, and m. With the scale variable, the asymmetry variable and the b, f, and m exponents relationships become stronger than for asymmetry alone and the asymmetry?s role on b becomes statistically significant at the 0.11 level, not quite at the 0.1 level. Therefore, it is concluded that the asymmetry predictor variable is still statistically insignificant at the 0.1 level. 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0 0.5 1 1.5 2 2.5 3 3.5 Log (Watershed Area in Square Miles) A s y m m e t r y Asymmetry Trendline 247 6.11 The Role of Topographic Index Topography is recognized as an important factor in determining the stream flow response of drainage. Topography determines the effects of gravity on the movement of water in the drainage, and therefore it influences many aspects of geomorphic and hydrologic systems. The energy of stream flow is determined by the slope of the surface. A steeper slope results in greater energy, and as the energy of a stream increases, its ability to transport more and larger particles also increases. Therefore, steeper slopes result in a greater potential for erosion. Topography has been shown to affect the flow path that precipitation follows before it becomes stream flow (Wolock, et al., 1990). And the shape of a surface determines how water will flow across it. Therefore, characterization of the topography of the subbasins should provide a significant dimension of information about the watershed setting. Four levels of topographic indices: topographic maximum, topographic minimum, topographic mean, and topographic standard deviation, are correlated with b, f, and m in a scale context. 6.11.1 Strength of the Relationship between Topographic Index and b, f, and m As stated in Chapter 4, topographic indices characterize the ability for subbasins to contain and transport discharge and the discharge can be equated to the drainage size. The topographic index is considered as an important indicator effecting hydraulic geometry. The description of the topographic indices and the calculation of these indices have been reported in Chapter 4. The relationships between the b, f, and m and any topographic index are not strong. The R2 values (Table 6.39) for linear regression between b, f, and m and the 248 topographic index are very low (under 0.1). The R2 between b and the topographic index is stronger than that f and m (Figure 6.52), and the statistical significance of the topographic indices? role on b is much stronger than that for f and m. It is expected that because the topographic index is, in a way, a measurement of saturation capacity, and the wetness of both sides of a channel bank is gradually decreasing from the stream, but when as a flow runs over the previous bank, the side of channel banks will easily be saturated and the banks will be easily eroded and so width increases. If the channel banks are saturated then the stream power can be applied to erode the channel banks and therefore, b can be changed. The roles of a topographic index on other exponents are weak, but the role of the topographic index on b is statistically significant at the 0.005 level for the maximum and standard deviation of the topographic indices and minimum topographic index, and the b relationship is statistically significant at 0.104 level (Table 6.39). Figure 6.52 b, f, and m and Topographic Index -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 12 14 16 18 20 22 Topographic Index b, f, m b f m Trendline (b) Trendline (f) Trendline (m) 249 6.11.2 b, f, and m and Topographic Index in a Scale Context (Table 6.39) Introducing the scale of each subbasin to the relationship of b, f, and m with the topographic index improved the R2. It is expected that the relationship increases in multi- variable regression than that of single variable regression. However, the increased R2 are mainly due to the scale variable; the contributions of topographic indices to the R2 are under 2%, except the b and minimum topographic index (2.3%). The topographic variable impact is very low. Even including the scale variable, none of the topographic indices became statistically significant, at least at the 0.1 level. The 0.05 level of statistically significant relationship between topographic index maximum and topographic index standard deviation b became insignificant with scale predictor variable included. To determine the relative importance of topographic index variable in predicting b, f, and m, standard partial regression coefficients are calculated. The magnitudes of the coefficients (scale = 0.311 and topographic index maximum = 0.095) indicate that scale has approximately three times as much effect on predicted b as do the topographic index variables. The R2 difference between the multi-regression of the scale and topographic index maximum and the single regression of topographic index maximum and b is 0.0003 which is very insignificant and the difference contributes 0.0004 of the variation left unexplained. 250 Table 6.39 Statistical Significance of Topographic Indices and b, f, and m Relationships N = 43 Topo Index Min N = 43 Topo Index Max b f m b f m Log Size R2 0.3105 0.1091 0.0857 Log Size R2 0.3105 0.1091 0.0857 Coefficient -0.0940 0.0420 0.0500 Coefficient -0.0940 0.0420 0.0500 Log Size: t -4.3030 2.2410 1.9600 Log Size: t -4.3030 2.2410 1.9600 Log Size: p 0.0000 0.0310 0.0570 Log Size: p 0.0000 0.0310 0.0570 TopoIndexMin R2 0.0631 0.0274 0.0175 TopoIndexMax R2 0.0951 0.0271 0.0297 Coefficient 0.0160 -0.0080 -0.0080 Coefficient -0.0210 0.0080 0.0120 TopoIndexMin: t 1.6600 -1.0750 -0.8540 TopoIndexMax: t -2.0840 1.0690 1.1200 TopoIndexMin: p 0.1040 0.2890 0.3980 TopoIndexMax: p 0.0430 0.2910 0.2690 Log Size & TopoIndexMin R2 0.3265 0.1176 0.0902 Log Size& TopoIndexMax R2 0.3108 0.1102 0.0857 Log Size: t -3.9561 2.0223 1.7875 Log Size: t -3.5325 1.9330 1.5654 Log Size: p 0.0003 0.0499 0.0814 Log Size: p 0.0011 0.0603 0.1254 TopoIndexMin: t 0.9756 -0.6213 -0.4435 TopoIndexMax: t 0.1341 -0.2248 0.0177 TopoIndexMin: p 0.3351 0.5379 0.6598 TopoIndexMax: p 0.8940 0.8233 0.9860 R2 Difference 0.0160 0.0085 0.0045 R2 Difference 0.0003 0.0011 0.0000 Contribution 0.0232 0.0095 0.0049 Contribution 0.0004 0.0012 0.0000 N = 43 Topo Index Mean N = 43 Topo Index stdev b f m b f m Log Size R2 0.3105 0.1091 0.0857 Log Size R2 0.3105 0.1091 0.0857 Coefficient -0.0940 0.0420 0.0500 Coefficient -0.0940 0.0420 0.0500 Log Size: t -4.3030 2.2410 1.9600 Log Size: t -4.3030 2.2410 1.9600 Log Size: p 0.0000 0.0310 0.0570 Log Size: p 0.0000 0.0310 0.0570 TopoIndexmean R2 0.0010 0.0000 0.0020 TopoIndexstv R2 0.0973 0.0279 0.0329 Coefficient -0.0090 -0.0050 0.0140 Coefficient -0.2540 0.1030 0.1510 TopoIndexMean: t -0.1670 -0.1170 0.2610 TopoIndexstv: t -2.1010 1.0840 1.1810 TopoIndexMean: p 0.8680 0.9070 0.7960 TopoIndexstv: p 0.0420 0.2850 0.2440 Log Size& TopoIndexMean R2 0.3196 0.1096 0.0917 Log Size& TopoIndexstv R2 0.3215 0.1109 0.0912 Log Size: t -4.3298 2.2157 1.9909 Log Size: t -3.6363 1.9333 1.6016 Log Size: p 0.0001 0.0325 0.0534 Log Size: p 0.0008 0.0603 0.1171 TopoIndexMean: t -0.7309 0.1508 0.5130 TopoIndexstv: t -0.8052 0.2871 0.4925 TopoIndexMean: p 0.4691 0.8809 0.6108 TopoIndexstv: p 0.4255 0.7755 0.6250 R2 Difference 0.0091 0.0005 0.0060 R2 Difference 0.0100 0.0018 0.0055 Contribution 0.0132 0.0006 0.0066 Contribution 0.0160 0.0020 0.0060 ______________________________________________________________________ Note: Contribution is the ratio of the unexplained and the difference between R2 of b, f, and m with log size and with log size and asymmetry combined. Statistical Significance at the 0.1 level is in Bold. 251 6.11.3 Summary As expected, the relationship between b and a topographic index is stronger than for the f and m exponents. Topographic index maximum and topographic index standard deviation values for b are statistically significant at the 0.05 level, but including the scale variable, none of the topographic indices and scale variable impacts the b in a statistically significant way. By including the scale variable with to the topographic index, the R2 improved. However, the increased R2 is due to the role of scale; the topographic index has very little impact, i.e., is less than 3 % of the variation is left unexplained. 6.12 Which Predictor Variables are most influential on b, f, and m? In addition to investigating the roles of scale and selected predictor variables on b, f, and m, individually, the research examined which subset of predictor variables appear to be the most influential. 6.12.1 Size and which other Predictor Variables? This is investigated by using a backward elimination stepwise regressions method. This procedure allows a single regression equation from several possible combinations of predictor variables. Even though the predictor variables are selected because they are theoretically and empirically relevant, a smaller subset of these variables may provide a satisfactory model for the hydraulic geometry exponents. In selecting the best subset of predictor variables, a trade-off between obtaining the best prediction possible using a large number of predictor variables and keeping the model as parsimonious as possible is achieved. This clearer interpretation of the interactions 252 between the predictor(s) and the b, f, and m was made under constraints of 0.1 probability by maximizing R2 while minimizing the number of predictor variables. The results of the backward stepwise regressions for b, f, and m are shown as Table 6.40. 6.12.2 Multivariate Linear Regression for b Including all predictor variables, except size, the R2 for the regression equation for b, the R2 is 0.688. However, none of the watershed variable is statistically significant and the statistically significant variable is channel pattern (at the 0.127 level). With statistically significant predictor variables at the 0.561 level, b can be calculated with an R2 of 0.614. Excluding channel shape and physiography, using only the 0.5 level of statistically significant variables, the b can be calculated with an R2 of 0.504. With predictor variables which are statistically significant at the 0.1 level only, lithology, channel pattern, and slope, b can be calculated with an R2 of 0.381. Including all predictor variables in the regression equation for b, the R2 becomes 0.792. However, only asymmetry and size are statistically significant (at the 0.079 and 0.024 levels). With predictor variables with statistical significance (at 0.561 level), b can be calculated with an R2 of 0.614. Excluding lithology, landuse, and bed and bank material, using only the 0.4 level of statistically significant variables, the b can be calculated with an R2 of 0.704. Using predictor variables which are statistically significant at the 0.1 level only, channel pattern, topographic index minimum, topographic index maximum, topographic index standard deviation, and size, b can be calculated with an R2 of 0.541. 253 6.12.3 Multivariate Linear Regression for f Including all predictor variables, except for size, the R2 for the regression equation for f is 0.673. However, none of the predictor variables is statistically significant at the 0.1 level. Using predictor variables with statistical significance at the 0.59 level, f can be calculated with an R2 of 0.646. But with predictor variables statistically significant at the 0.395 level, f can be calculated with an R2 of 0.613. Excluding bank material (0.590 level of statistical significance) from the multivariate regression, the R2 becomes 0.488. Therefore, statistically insignificant bank material must have been erroneously included in the estimation of the f values. However, the bank material variable of my study is not a clean cut variable due to the mixed nature of heterogeneous materials. Using only a 0.46 level of statistically significant variables, the f can be estimated with an R2 of 0.484. After excluding lithology (0.46 level of statistical significance), the remaining variables, bed material and channel pattern, are statistically significant at 0.02 and 0.05 levels, respectively. Eliminating the lithology variable, with bed material and channel pattern variables which are statistically significant at the 0.002 and 0.006 levels, f can be calculated with an R2 of 0.448. It can be concluded that bed material and channel pattern are decidedly the predictor variables for estimation of the f values. Including all predictor variables in the regression equation for f, the R2 is 0.765. However, only size is statistically significant (at the 0.043 level). With predictor variables with statistical significance (at 0.549), f can be calculated with an R2 of 0.762. Excluding channel shape, asymmetry, and topographic index minimum, using only the 0.46 level of statistically significant variables, the f can be calculated with an R2 of 0.738. 254 With predictor variables which are statistically significant at the 0.2 level only, physiography, lithology, bed material, channel pattern, topographic index maximum, topographic index standard deviation, and size, f can be calculated with an R2 of 0.683. With predictor variables which are statistically significant at the 0.1 level only, that is, bed material, channel pattern, and size, f can be calculated with an R2 of 0.512. 6.12.4 Multivariate Linear Regression for m Including all predictor variables, except for size, the R2 for the regression equation for m, is 0.552. However, no predictor variable is statistically significant at the 0.1 level. Using up to the 0.49 level of statistically significant variables, the m can be calculated with an R2 of 0.510. With predictor variables which are statistically significant at the 0.1 level only, that is, lithology, m can be estimated with an R2 of 0.152. It can be concluded that m is hard to estimate even with all predictor variables and with a constraint of 0.1 statistical significance, only lithology is a useful predictor variable to estimate the m values. Including all predictor variables in the regression equation the R2 for m, is 0.559, but none of the predictor variables are statistically significant at the 0.1 level. With predictor variables with statistical significance of the 0.493 level, including predictor variables such as physiography, lithology, bed material, channel shape, and asymmetry, m can be estimated with an R2 of 0.510. Further, excluding channel shape, bed material, and asymmetry, using only the 0.22 level of statistically significant variables, the m can be calculated with an R2 of 0.385 level. Only the size variable, is statistically significant at the 0.1 level and the m can be estimated with R2 of 0.008. 255 Clearly, the estimation of m values with mostly static predictor variables, including the size variable, is statistically insignificant and it can be concluded that the rate of flow velocity change due to increasing discharge volumes is hard to estimate. However, the underlying principle of the hydraulic geometry, b + f + m = 1, allows m to be deduced from known b and f values. 256 Table 6.40 Stepwise Regression for b, f, and m with All Predictor Variables _______________________________________________________________________ Backward Stepwise Regression for b with All Predictor Variables b R2 Eliminated b w/size R2 Eliminated step 0 0.688 0.792 step 1 0.664 bed material 0.791 landuse step 2 0.652 landuse 0.772 bed material step 3 0.651 topo index min step 4 0.614 channel shape 0.729 bank material step 5 0.504 physiography 0.704 lithology step 6 0.450 bank material 0.669 channel shape step 7 0.424 asymmetry 0.562 physiography step 8 0.390 topo index max 0.541 asymmetry step 9 0.381 topo index stdev Remained lithology: 0.078 Remained channel pattern: 0.006 channel pattern: 0.011 topomin: 0.030 topomax: 0.004 topostdev: 0.007 logsize: 0.000 ---------------------------------------------------------------------------------------------------------------- Backward Stepwise Regression for f with All Predictor Variables Backward Stepwise Regression for m with All Predictor Variables f R2 Eliminated f w/size R2 Eliminated m R2 Eliminated m w/size R2 Eliminated step 0 0.673 0.765 0.552 0.559 step 1 0.673 topomin 0.765 topomin 0.543 landuse 0.555 chpatt step 2 0.670 landuse 0.762 landuse 0.536 chpatt 0.554 landuse step 3 0.646 chshape 0.741 chshape 0.516 bank mat 0.553 topostdev step 4 0.646 asym 0.738 asym 0.514 topostdev 0.522 bank mat step 5 0.613 bank mat 0.703 bank mat 0.512 topomax 0.517 topomax step 6 0.488 phys 0.510 topomin 0.515 topomin step 7 0.541 phys 0.459 chshape 0.510 logsize step 8 0.487 topomax 0.525 litho 0.401 bedmat 0.459 chshape step 9 0.484 topostdev 0.519 topostdev 0.385 asym 0.401 bedmat step10 0.448 litho 0.512 topomax 0.194 phys 0.385 asym step11 0.265 litho step12 0.066 phys Remained Remained Remained Remained bed mat: 0.002 bedmat: 0.002 litho: 0.089 none chpatt: 0.006 chpatt: 0.007 Added logsize: 0.038 logsize: 0.035 R2 becomes 0.108 Note: Statistical significances of the remained predictor variable are shown. 257 6.12.5 Conclusion Using multivariate regression methods to estimate b, f, and m values, a balance between the statistical significance of the considered predictor variables and higher correlation needs to be maintained. Considering all predictor variables for the estimation of b, f, and m values, regardless of the statistical significance of the predictor variables, b, f, and can be estimated with an R2 of 0.792, 0.765, and 0.552, respectively. The majority of the predictor variable values used for this study are more or less static and easily obtainable, and once the values are obtained, they can be used for various stream discharge volumes. Based on the statistical significance of predictor variables at the 0.1 level, the predictor variables needed to estimate b are channel pattern, topographic indices, and size of the subbasin, and b can be estimated with an R2 of 0.541. The predictor variables needed to estimate f are bed material, channel pattern, and size of the subbasin, and f can be estimated with an R2 of 0.512. Except the size of the subbasin, no predictor variable at the 0.1 level of statistical significance can predict m. Since b and f can be estimated with 0.1 level of statistically significant predictor variables, m can be calculated with the continuity law. 6.13 Constrained b, f, and m Values To further examine the validity of the statistical results, two additional analyses are conducted. First of all, I examined a subset of subbasins which meet a constraint criterion, and, secondly, I analyzed all the subbasins using weighted samples. 258 6.13.1 Constrained Hydraulic Geometry and Scale The continuity equation implies a constraint that the hydraulic exponents should sum to one. To estimate the hydraulic geometry values, a log-linear model (LLM) is used one-at-a-time, and the constraint is not imposed on the b, f, and m. The sum of the b, f, and m are noted in Chapter 5 (Table 5.1). Rhodes (1977) recommended a screening procedure which rejects b, f, and m whose sum is outside of the range of 0.95 ? 1.05. The summation of all study area b, f, and m values are within 1 ? 0.05. Out of the 43 study subbasins, two subbasins show the sum of the three exponents as 1.038 and 1.032, and two subbasins show 1.0214 and 1.0212. The rest of the subbasins are all within ? 0.01 from the unit. Therefore, all 43 subbasins are included in the analyses. Even though the summation constraints of b, f, and m values are within ? 0.05, the validity of the b, f, and m values need to be verified. This is to ensure that the study area subbasins are comprised of reasonably constrained b, f, and m values. Using the R2 values of the b, f, and m correlated with discharge and each channel width and depth, and flow velocity, each subbasin is categorized as well constrained (at least two out of three exponents have R2 of 0.9), moderately constrained (at least two exponents have R2 of 0.8), or poorly constrained subbasins (personal communication with Dr. Karen Prestegaard, Oct. 2005). Eleven subbasins are well constrained; 12, moderately; and 20 poorly constrained (Table 6.41). 259 Table 6.41 Distribution of Coefficient of Determination for b, f, and m R2b/R2f/R2m Frequency b f m <0.5 6 4 1 0.5 8 3 3 0.6 4 9 4 0.7 7 4 4 0.8 12 8 8 0.9 6 15 23 With 23 well and moderately constrained subbasins, selected statistical analyses such as correlations between b, f, and m and scale and stepwise regressions are conducted. This aids in determining the roles of scale on b, f, and m and in selecting the best statistical subset of predictor variables in order to obtain reliable estimates under constraints of maximizing the R2 with 0.1 statistically significant variables. The relationships between the b and m and the subbasin size (scale) in well and moderately constrained subbasins are stronger than that of all the study subbasins by 0.055 and 0.142, respectively. However, the relationship between scale and f deteriorated by 0.076, and the scale becomes statistically not a significant predictor variable for f at the 0.1 level. Even though the R2 improved for b, the role of scale on m became statistically less significant; however, the influence of scale is still within the 0.1 probability level. 260 Table 6.42 Regression of the Combined Well and Moderately Constrained Subbasin b, f, and m with Subbasin Size b f m N 23 23 23 R2 0.366 0.033 0.228 t Value -3.479 0.842 2.492 Significance/ 0.002 0.409 0.021 Probability1 Constant 0.404 0.303 0.301 Coefficient -0.104 0.020 0.081 Note: 1. With 1 degree of freedom, the critical value at the 0.01 is 2.831, 0.05 is 2.080, and at the 0.1 level is 1.721. Statistically significant values at 0.05 level are in bold. The poorly constrained subbasins highlighted in red on Figure 6.43 which are removed from the regression analyses of b, f, and m and scale are in a random distribution pattern and are not in statistical significant correlation with subbasin size, except that the majority (15 out of 20) of the removed subbasins are less than 150 square miles in size. That means investigations of b, f, and m in smaller subbasins with scale, and predictor variables roles need to be differentiated from those of the larger subbasins. 261 All f and Subbasin 0 0.1 0.2 0.3 0.4 0.5 0.6 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 Log (Watershed Area in Square Miles) f Trendline f f All b and Subbasin -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 Log (Watershed Area in Square Miles) b b Trendline b 262 Figure 6.53 b, f, and m with Subbasin in Log (Watershed Area in Square Miles) Even with only 23 subbasins, the role of scale is statistically significant at the 0.1 level for b and m. With the summation constraint, f can be estimated since b and m are known. It is hard to judge whether the role of the scale predictor variable is stronger in the well and moderately constrained subbasin than for the original 43 study subbasins. There are no big changes in the fundamental relationships between scale and b, f, and m such that within the study subbasin size (1,642 square miles), as scale gets bigger the b value gets smaller and f and m values get bigger. All m and Subbasin 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 m m rTrendline m Log (Watershed Area in Square Miles) 263 Table 6.43 Regression b, f, and m with Subbasin Size for All and the Combined Well and Moderately Constrained Subbasins Regression of Constrained and all Subbasins b, f, and m on Subbasin Size b f m b f m N1 23 23 23 43 43 43 R2 0.366 0.033 0.228 0.311 0.109 0.086 t Value -3.479 0.842 2.492 -4.303 2.241 1.960 Significance/ 0.002 0.409 0.021 0.000 0.031 0.057 Probability2 Constant 0.404 0.303 0.301 0.389 0.233 0.384 Slope -0.104 0.020 0.081 -0.094 0.042 0.050 Note: 1. N = 23 denotes total number of well and moderately constrained subbasins. N = 43 denotes total number of study subbasins. 2. With degree of freedom 41, the critical t value at 0.1 probability is 1.684. 3. With degree of freedom 21, the critical t value at 0.1 probability is 1.721. 6.13.2 Impact on the Choice of Predictor Variables for b, f, and m I examined the subset of subbasins in stepwise regression analyses using 23 subbasins. The results of the stepwise analyses on 23 subbasins compared with all subbasin cases, that were reported in 6.2 through 6.13, are examined for the pattern of best predictor variables rather than for a comparison of the magnitudes of correlations. Also stepwise regression results for the 23 subbasins are compared with results from weighted regression based on the sample size of each observed subbasins. The expectation of these comparisons is that well and moderately constrained subbasins would have a higher R2 than that of all study subbasins and the weighted regression analyses would also provide higher R2 values. Stepwise regression analyses of the well and moderately constrained subbasins are done with a forward selection method due to the lack of degrees of freedom with only 23 subbasins. The backward elimination methods are utilized for N = 43 stepwise 264 regression. Therefore, comparing the results from forward selection and backward elimination methods might not be prudent, but examining the results of best predictor variables for either forward or backward methods provides a valuable insight of the role of scale either in N = 43 or N = 23 cases (Table 6.44). Table 6.44 Stepwise Regressions for b, f, and m with all Predictor Variables N = 23 (Forward) N = 43 (Backward) N = 43 (Weighted1, Backward) Well and Moderately Constrained Subbasins All Study Subbasins All Study Subbasins b R2 0.599 0.541 0.718 Significant2 Predictor Channel Pattern Channel Pattern Physiography Variables Size Topo Min Index Bank Material Asymmetry Topo Max Index Asymmetry Size Topo Index Stdev Topo Index Stdev Size f R2 0.571 0.512 0.640 Significant2 Predictor Topo Min Index Bed Material Bed Material Variables Bed Material Channel Pattern Channel Shape Size Size Asymmetry Topo Min Index Size m R2 0.229 0.108 0.150 Significant2 Predictor Size Size Variables Note: 1. All study subbasins are weighted by their sample sizes. 2. Statistically significant at the 0.1 level. 265 In addition to the scale variable, including statistically significant predictor variables at the 0.1 level, the correlation of estimation of b, f, and m improved from 0.366 to 0.599 for b. For the estimation of b, the statistically significant predictor variables at the 0.1 level are channel pattern, size, and asymmetry and with these variables, b can be estimated with R2 of 0.599. Scale is one of the significant predictor variables for f, but the role of scale appears to be infinitesimally small. With the scale variable the estimation of f improved 0.08 more ( 0.571) in R2 value. The influence of scale on m is also notable. Without the scale variable, no statistically significant predictor variable is identified for the estimation of m. However, with the scale variable, m can be estimated with R2 of 0.228. This reaffirms that poorly constrained subbasins might not have the flow resistance related factors and removing these poorly constrained subbasins, m can be estimated. Statistically significant predictor variables for b, f, and m, identified for 43 subbasins in section, 6.12 (Table 6.40), are used for regression analyses for each b, f, and m in the well and moderately constrained subbasins. Besides the scale variable, not all the significant predictor variables for the original study subbasins become significant for the well and moderately constrained subbasins. However, bed material, channel pattern, and topographic minimum, and topographic maximum indices are common predictor variables for both situations. The rationale for the deterioration of the scale impact on f, even though only two subbasins with R2 f of 0.9 are removed from the analyses, might be due to the smaller sample size. Therefore, to keep the original study subbasins for the stepwise regression 266 analyses, weighted stepwise regression analyses are employed. The weights are given to each subbasin based on the sample size of each subbasin. Since several of the poorly constrained subbasins have the largest sample sizes among the study area subbasins, and since the summation of the b, f, and m are within ? 0.002, instead of removing these subbasins from the analyses, each subbasin is weighted based on its sample size (number of hydrographic survey and measurements) and examined by weighted stepwise regression analyses. Comparing the backward elimination stepwise regression results with the original study subbasins, many predictor variables become statistically significant; therefore, including all significant variables in the estimation of b, f, and m models, the R2 values for each b, f, and m become much higher. That is to say, b can be estimated with physiography, bank material, topographic standard deviation, asymmetry index, slope, and size with R2 of 0.722 (Table 6.44). However, the weighted stepwise regression does not consider the hydraulic and channel geometry constraints, and many of the predictor variables needed for the estimation of b, f, and m . Therefore, weighted stepwise regression might not be the best method to predict hydraulic geometry. Using a high number of well and moderately constrained subbasins will aid reliable estimation of b, f, and m with statistically significant predictor variables for each b, f, and m. 6.14 Summary The findings of the study on the roles of scale and watershed variables on the hydraulic geometry exponents are described in the next chapter. The scale and statistically significant, at least at the 0.05 level, watershed variables are discussed. In 267 addition, the influence of scale on b, f, and m, the ?Best? individual watershed predictor variable, and the most influential watershed predictor variables among the selected variables are addressed. Needed future research areas are also discussed. 268 CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS 7.1 Summary of Research Objectives and Hypotheses The main objectives of this study have been to determine whether (a) at-a-station hydraulic geometry changes significantly with watershed scale (which is manifested by the size differences of each subbasin area); and (b) the selected watershed variables of physiography, topography, lithology, geomorphology, landuse, channel pattern, channel shape, and bed and bank materials, affect the size and shape of stream channels and, thus, the at-a-station hydraulic geometry. The working hypothesis for the first objective of this study was that the hydraulic geometry exponents (b, f, and m values), the change rates of channel width and depth, and flow velocity with regard to discharge (not with time), are not random, but that there are systematic changes as a function of watershed scale. The expected changes in the at- a-station hydraulic geometry as a function of scale is that channel width does not change much, depth increases rapidly, and flow velocity change rates will also increase. The working hypothesis of the second objective is that watershed variables influence channel morphology by affecting stream discharge and sediment characteristics such as the size and the amount of sediments delivered to the channels. The expected correlations between particular watershed variables and b, f, and m were stated in Chapter 1. As a practical matter, the research analysis has been undertaken via empirical statistical correlations and analysis of variances, in order to: 269 (1) discern relationships between a geographical scale factor (subbasin size) and b, f, and m; (2) account for the roles of the selected individual watershed predictor variables for b, f, and m, including the empirical findings of f > m in sand beds and f < m in gravel beds; (3) account for the roles of the selected individual watershed predictor variables for b, f, and m including the empirical findings of f > m in sand beds and f < m in gravel beds within the scale context; (4) ascertain the significant watershed variables, without the scale variable, for b, f, and m via stepwise multi-variable regression; and (5) ascertain the significant watershed variables jointly with the scale variable, for b, f, and m via stepwise multi-variable regression . In addition to investigating the influence of scale and selected watershed variables on b, f, and m, individually, the study also investigated which subsets of watershed variables might be the best predictors. The use of a single stepwise regression equation helps to ascertain several possible combinations of predictor variables for the estimation of the b, f, and m values. The watershed variables used in this study were: physiography, lithology, landuse, channel pattern, channel shape, channel bed material, channel bank material, all of which are categorical variables, and quantitative measures of channel asymmetry and topography. To test the hypotheses that b, f, and m values are functions of subbasin size and selected predictor variables, the following models were used: b (f, m) = a0.b (a0.f, a0.m) + a1.b (a1.f, a1.m) * scale + a2.b (a2.f, a2.m) * predictor variable + a3.b (a3.f, a3.m) * predictor variable * scale; 270 where, a0.b ( a0.f, a0.m) are constants, a1.b ( a1.f, a1.m) are coefficients of the scale factor, a2.b (a2.f, a2.m) coefficients of a predictor variable, and a3.b (a3.f, a3.m) are coefficients of interaction between the scale factor and a watershed variable. Theoretically, as stream discharge increases, channel width and depth and flow velocity would increase as well, but the focus of the study is the coefficients of these models. The null hypotheses are H0.b (H0.f, H0.m): a1.b (a1.f, a1.m) = 0, H0.b (H0.f, H0.m): a2.b (a2.f, a2.m) = 0, and H0.b (H0.f, H0.m): a3.b (a3.f, a3.m) = 0. And the alternative hypotheses are H1.b (H1.f, H1.m): a1.b (a1.f, a1.m) ? 0, H1.b (H0.f, H0.m): a2.b (a2.f, a2.m) ? 0, and H1.b: a3.b (a3.f, a3.m) ? 0. Determination of the significant roles of scale and selected watershed predictor variables for b, f, m, f > m, and f < m is guided by the statistical significance level of 0.05. However, readers can also interpret the documented statistical significance of the predictor variables for b, f, m, f > m, and f < m based on their specific interests by using the probability values tabulated in Chapter 6. 7.2 Findings and Discussion: Scale Influence on b, f, and m Scale issues in hydrology arise because different hydrological processes are dominant at different scales. As Leopold and others have mentioned there is a relationship between at-a-station and downstream hydraulic geometry, implying a scale relationship, but the scale implications are not quantified by researchers except for Leopold?s demonstration of systematic differences of b, f, and m in different sizes of subbasins. Notable examples are the relationships among the average b, f, and m values with discharge in the hydraulic geometry of three rivers of different size: Watts Branch drainage area in Potomac River, 3.7 square miles; Seneca Creek in Potomac River, 100 271 square miles; and the Amazon River at Obiodos drainage area of 1.9 million square miles (Leopold, 1994). Despite Ridenour?s (1999) finding that there is no significant functional relationship between drainage area and hydraulic geometry, the findings in this study of the relationships between the subbasin size in log10 and b, f, and m, indicate that, even though the R2s are not high, there are statistical relationships between scale and b and for f. Both these relationships are statistically significant at the 0.05 level, thus supporting the hypothesis that there are systematic changes in hydraulic geometry exponents as a function of subbasin size. Unlike the relationships between b and scale (R2 = 0.311) and f and scale (R2 = 0.109), the relationship between scale and m (R2 = 0.086) is not statistically discernable at the 0.05 test level. For purposes of estimation, the hydraulic geometry in the study area can be determined by the empirical models (Table 7.1): b = 0.389 - 0.094 * scale (log (subbasin area (mi2)) with R-squared of 0.311; f = 0.233 + 0.042 * scale (log (subbasin area (mi2)) with R-squared of 0.109; and m = 0.384 + 0.050 * scale (log (subbasin area (mi2)) with R-squared of 0.086, provided caution is exercised in the case of m. Unexpectedly, the b value changes negatively, and b?s increase is high which is expected as b is higher in downstream hydraulic geometry. f increases not as much as expected, but m changes very much as expected. As Leopold (1953) suggested, the m value increases in downstream hydraulic geometry which is usually in large drainage basins. 272 Table 7.1 Regressions of All and the Well and Moderately Constrained Subbasin b, f, and m with Subbasin Size (in Logarithms) Regression of b, f, m on Scale for All and Well/Moderately Constrained Subbasins b f m N1 23 43 23 43 23 43 R2 0.366 0.311 0.033 0.109 0.228 0.086 t Value -3.479 -4.303 0.842 2.241 2.492 1.960 Significance/Probability2 0.002 0.000 0.409 0.031 0.021 0.057 Regression Constant 0.404 0.389 0.303 0.233 0.301 0.384 Regression Coefficient -0.104 -0.094 0.020 0.042 0.081 0.050 Note: 1. N = 23 denotes the total number of well and moderately constrained subbasins. N = 43 denotes the total number of study subbasins. 2. With 41 degrees of freedom, the critical t value at the 0.05 probability is 2.021. With 21 degrees of freedom, the critical t value at the 0.05 level is 2.080. Statistical significance at the 0.05 level is in Bold. A more robust analysis based on the removal of twenty subbasins led to some change in the statistical results (Table 7.1). Even though the summation of the b, f, and m values for the original 43 subbasins is within ? 0.02 from unity and the sample sizes of some of the subbasins are high, only 23 well and moderately constrained (at least two out of three exponents have R2 of 0.8 or higher) subbasins are kept. The sizes of 15 of the eliminated 20 poorly constrained subbasins are under 150 square miles. As the R2 improved for b and m, based on use of the constrained subbasins, the influence of scale on m also became statistically significant at the level of 0.05, a small but important improvement over the 0.057 probability value using all 43 subbasins. The relationships (R2) between the b and m and the subbasin size in well and moderately constrained subbasins are stronger than that of all the study subbasins by 0.055 and 0.142, respectively. The relationship (R2) between scale and f deteriorated by 0.076, and 273 the scale becomes, statistically, not a significant predictor variable for f at the 0.05 level. The improvement in the R2 and statistical significance may be due simply to the elimination of the poorly constrained subbasins. However, it is hard to judge whether the role of the scale predictor variable is stronger in the well and moderately constrained subbasins than for the original 43 study subbasins. As the coefficients of b, f, and m in both cases show (Table 7.1), there are no significant changes in the fundamental relationships between scale and b, f, and m such that, within the study subbasin size (1, 642 square miles), as scale increases the b value gets smaller and f and m values increase, with the m values changing more than those of f. Yet, in general, it needs to be recognized that the relationship between scale and b, f, and m is weak and that the relationship is limited to the watershed sizes of 0.38 to 1, 642 square miles in the Potomac River Basin. With these caveats, b, f, and m can be extrapolated with these estimating equations up to watershed sizes of a little over 10,000 square miles (log of 4.138 watershed size). Over 10,000 square miles of watershed size, the underlying model implies that the width of the channels will actually decrease as discharge increases, which is physically and geomorphologically impossible. Because of the paucity of research on the influence of geographic scale on at-a- station hydraulic geometry relationships, it is hard to judge the universal goodness of the particular R2 values as produced by this research. Nonetheless, recognizing the high probability of the relationship being as stated, and considering the heterogeneous nature of the study subbasins in terms of physiography, lithology, channel pattern and shape, bed and bank material, and topography, the statistically discernible relationships between scale and b and f are notable. 274 7.3 Findings and Discussion: What Are the ?Best? Predictor Watershed Variables for b, f, and m? The results of the statistical analysis for the relationships of b, f, and m, with each watershed variable, scale, and with interaction relationships, are shown in Table 7.2. Watershed predictor variables statistically significant at the 0.05 level are highlighted in red and the scale variable is highlighted in blue. An examination of the relationships of the seven categorical and two quantitative watershed variables and b, f, and m, individually both with and without consideration of scale, has led to four principal conclusions: (1) There is a more reliable prediction of the rate of width changes (based on statistical significance) using a single watershed variable, than for depth and flow velocity. This 275 Table 7.2 Summary of b, f, and m, Scale, and Selected Predictor Variable Relationships Note: Statistically significant watershed predictor (except for scale) variables are highlighted in red and scale variable highlighted in blue when it meets the 0.05 level of statistical significance. b vs Scale R2 0.3110 f vs Scale R 0.1090 m vs Scale R0.0860 Probability 0.0000 Probability 0.0310 Probability 0.0570 Coefficient -0.0940 Coefficient 0.0420 Coefficient 0.0500 Physio Litho Landuse Ch. Patt Ch. ShapeBed Mat Bank MatAsym b vsPred/scale R2 0.3877 0.3589 0.3129 0.3883 0.3454 0.3309 0.3529 0.0227 Scale Prob. 0.0000 0.0016 0.0020 0.0004 0.0002 0.0001 0.0003 Pred. Prob. 0.4867 0.5979 0.9344 0.0977 0.5719 0.2763 0.1132 0.3380 b vs pred/Inter R2 0.4190 0.4370 0.3157 0.4927 0.3755 0.3399 0.3602 0.3545 Scale Prob. 0.8357 0.6133 0.3799 0.0068 0.0102 0.0249 0.2148 0.0001 Pred. Prob. 0.7615 0.2165 0.9032 0.0648 0.8409 0.2920 0.8859 0.1067 Interaction. Prob. 0.8890 0.3532 0.9264 0.0311 0.6444 0.4691 0.5081 f vs Pred/scale R 0.2504 0.1788 0.1306 0.2587 0.2504 0.1407 0.1418 0.0010 Scale Prob. 0.0167 0.0259 0.0264 0.0523 0.0170 0.0204 0.0509 Pred. Prob. 0.2634 0.5425 0.6210 0.0278 0.2630 0.2327 0.2240 0.8378 f vs pred/Inter R2 0.3843 0.3121 0.1640 0.2789 0.3843 0.0960 0.1982 0.1137 Scale Prob. 0.0174 0.1061 0.8855 0.3237 0.0170 0.1661 0.4013 0.0297 Pred. Prob. 0.3934 0.1977 0.6804 0.4416 0.3930 0.5917 0.0479 0.6529 Interaction. Prob. 0.2707 0.1977 0.4843 0.5990 0.2710 0.3221 0.1059 m vs Pred/scale R 0.2203 0.1884 0.0910 0.0994 0.1196 0.0857 0.0858 0.0177 Scale Prob. 0.0908 0.2252 0.0821 0.0945 0.0651 0.0646 0.0690 Pred. Prob. 0.3098 0.3395 0.8936 0.7455 0.6925 0.9924 0.9543 0.3956 m vs pred/Inter R 0.3939 0.3159 0.1277 0.1636 0.1276 0.1190 0.1610 0.1128 Scale Prob. 0.0477 0.4563 0.4999 0.1472 0.2466 0.0776 0.4127 0.0448 Pred. Prob. 0.1224 0.0857 0.5731 0.1975 0.9177 0.2586 0.1190 0.2752 Interaction. Prob. 0.1469 0.2142 0.4664 0.2542 0.9558 0.2323 0.0690 276 b,f,m & Predictor Topomin Topomean Topomax Topostdev f-m in Sand R2 p b vs Pred R2 0.0631 0.0010 0.0951 0.0973 sand>50% vs non- sand 0.0310 Pred. Prob. 0.1040 0.8680 0.0430 0.0420 Sand>50% & Size 0.0251 b, Scale, vs Pred R2 0.3265 0.3196 0.3108 0.3215 Size 0.6227 Scale Prob. 0.0003 0.0001 0.0011 0.0008 Pred. Prob. 0.3351 0.4691 0.8940 0.4255 f-m & Sand>50% 0.0801 Sand >50% 0.3727 f-m, Size, Sand>50% 0.2137 f vs Pred R2 0.0274 0.0000 0.0271 0.0279 Size 0.2475 Pred. Prob. 0.2890 0.9070 0.2910 0.2850 Sand> 50% 0.1758 f, Scale, vs Pred R2 0.1176 0.1096 0.1102 0.1109 f-m in Gravel R2 p Scale Prob. 0.0499 0.0325 0.0603 0.0603 Gravel > 50% vs Non-Gravel -0.3093 Pred. Prob. 0.5379 0.8809 0.8233 0.7755 f-m & Size 0.0264 Size 0.4813 m vs Pred R2 0.0175 0.0020 0.0297 0.0329 f-m & Gravel>50% 0.0395 Pred. Prob. 0.3980 0.7960 0.2690 0.2440 Gravel> 50% 0.3879 m, Scale, vs Pred R2 0.0902 0.0917 0.0857 0.0912 f-m & Gravel>50% 0.0817 Scale Prob. 0.0814 0.0534 0.1254 0.1171 Size 0.3750 Pred. Prob. 0.6598 0.6108 0.9860 0.6250 Gravel>50% 0.3118 Note: Each b, f, and m is individually correlated with individual watershed predictor variable by itself, with scale, and with scale and a predictor variable interaction. Statistically significant watershed predictor (except for scale) variables are highlighted in red and scale variable highlighted in blue when it meets the 0.05 level of statistical significance. finding is expected, in a hydrogeomorphological sense, for more watershed variables influence directly the width and depth of the channel. That is, width changes are directly influenced by erodability of the bank and channel pattern. Depth is determined by the watershed variables and the stream power, more so than it would be for width. However, flow velocity is influenced by a combination of bed material and friction of the flow from bank and bed. The friction at the bed comes from the combination of bed material and 277 level of flow (depth). The rate of changes in flow velocity is inherently complex, and the role of individual watershed variables for m is not obvious. Even if m can be determined from the continuity of b + f + m = 1, there is a weakness because the less reliable f has to be included in the equation. Therefore, m also is not so reliably predicted. f is less reliable because depth is inherently unstable, based on the bank and bed materials. For an easily erodable bank and bed channel, based on the stream power, sediments from the eroded bank and bed materials can be deposited on the bed, which, in turn, alters the depth of the channel. (2) There are more individually influential watershed variables for b than for f, and more than for m, based on the R-squared values for single-variable regressions (Table 7.2). The most important variables, statistically speaking, are channel pattern, two measures of topography, and possibly also bed material. Considered individually, channel pattern, topographic maximum index, and topographic index standard deviation are significant predictor variables for b, but only channel pattern is for f, and there are none for m. This finding reflects the role of several geomorphic agents. However, for f, the role of an individual watershed variable is quite obscured by interactions among the process agents. Worse yet, m is a reflection of b and f which comprise channel geometry and flow dynamics within that channel geometry. (3) Prediction is improved by concentrating on subbasins with a particular group characteristic within a selected watershed variable, for example, just straight channels within the channel pattern variables. This finding clearly indicates that reducing the 278 heterogeneity of the subbasins by grouping them into a homogeneous unit will improve the predictability of b, f, and m. Generally speaking, this implies that different models should be developed for subbasins classified into different composite characteristics. To cite just one example, in the case of straight channel subbasins b values are more similar (low standard deviations) compared with the b values of the rest of the subbasins which are mixture of meandering and braided channel patterns. The difference of means test indicates that the b values for straight channel are at the 0.025 level of statistical significance. This finding confirms that fluvial characteristics of straight channels have almost vertical channel banks, seen in A low variability in b; therefore, channel flow could not erode the banks easily, which results in a small b value. (4) The results of this study suggest that the particular watershed characteristics of greater importance are the impact of sand versus non-sand bed material on f, and silt versus non- silt bank material on b. Also, the distinction between Great Valley versus non-Great Valley provinces in physiography, and crystalline versus non-crystalline lithology regions is important for the estimation of m, although neither of these watershed variables generally is influential for m. These findings need further investigation regarding geomorphic process content. Does this mean the flow resistance will be distinctly different in the Great Valley physiographic region, and the crystalline or siliciclastic lithology region compared other regions? These findings match with the fact that cohesive bank material channels have smaller b values and smaller variability among cohesive bank material channels, 279 therefore cohesive bank material is a good indicator for b value estimation. Also, this study demonstrated that channels of non-cohesive bed material (sand) versus other types of bed material channel would have distinctly different responses for f. These findings confirm Knighton?s (1998) conclusion that b is largely a function of channel geometry and, therefore, boundary composition, while the rates of change of depth (f) and velocity (m) are dependent partly on cross-sectional form and partly on transport- and resistance- related factors which tend to be more variable. In what follows only the statistically significant predictor variables by itself and with a scale context with the specific watershed predictor variables: channel pattern, bed material, bank material, and topographic indices, are examined in detail. The findings that all other watershed predictor variables (physiography, lithology, landuse, channel shape, bank material, and channel asymmetry), individually, by itself and with the scale context, were statistically insignificant, were discussed in detail in Chapter 6. The failure to detect significant statistical impacts for more variables is understandable given that the b, f, and m are responses to the interaction of various watershed influences on b, f, and m, and it might simply be that there are no relationships between the selected watershed predictor variables and b, f, and m. Hence, we need to recognize that a seemingly influential watershed variable by itself could turn out to be an insignificant predictor for b, f, and m by itself with and without consideration of scale. 7.3.1 Channel Pattern As expected, comparing the average values of b, f, and m for each channel pattern shows that there are significant differences among the three channel pattern types. 280 Discernable differences are in the b values in straight versus non-straight channels. It is expected that in straight channels width and depth do not change much, but that flow velocity would change more with increasing discharge. The found correlations between channel pattern and b, f, and m are that, in straight channels, b values are lower than those of depth; in meandering channels, relatively higher rates of changes in width than those of depth. Meandering channels seem to be associated with relatively low b values, although not as low as those for straight channels. The mean m value appears to be the smallest among the three channel pattern types. These findings make sense in that straight channels are typically symmetrical and meandering channels are normally asymmetric in cross-section at the bends; asymmetry provides for a relatively greater rate of increase in width, but f will probably still be greater than b. Braided channels are typically broad and shallow because of the unstable nature of their banks and the width-depth ratio commonly increases with increasing discharge. Braided channels are often, but not always, associated with sandy or unstable friable bank materials and steep slopes with variations in flow. As expected, higher m values are observed than for those of meander channels and they have the widest range of maximum and minimum values of b, f, and m. Channel pattern by itself is not a statistically significant predictor variable for b. However, the relationship between b and the interaction of scale and channel pattern is strong and the statistical significance is at the 0.05 level. Channel pattern is a statistically significant variable for f by itself as well as in a scale context. Specifically, the influence of channel patterns on the f value, with a statistical significance level of 0.0278, is notable. However, the channel pattern?s influence on m is indiscernible. It is expected 281 that channel pattern would influence, primarily, width and depth change rates rather than that of flow velocity. The statistical results supported the expectations. The straight channels have the highest m value (0.517) and the smallest mean b value (0.092) which is not as dramatically low as Knighton?s (1975) finding, 0.03, while the b values for the meander and braided channels' are 0.215 and 0.239, respectively. Comparing straight, meander, and braided channels, the straight channels have the smallest range of maximum and minimum values for all b, f, and m, which indicate that straight channels have very similar b, f, and m values among themselves. And, the mean f value for braided channels also has a very small standard deviation meaning that the rates of depth changes with discharge are similar within braided channels. The influence of channel patterns, on f values as a whole, has been clearly demonstrated. Specifically, the meander channels show that channel pattern does explain b value in a scale context, as revealed by an R2 of 0.758. By adding an interaction term to the model for b, R2 increased from 0.388 to 0.493, and the statistical significance of channel pattern for b improved to the level of 0.05. Even though the stepwise regression chose channel pattern as an influential predictor variable for b and f for the original 43 subbasins as well as for b values for the 23 well and moderately constrained subbasins, channel pattern is not an influential variable for f. This can be explained by the fact that other watershed variables, such as bed material and topographic minimum index, play a greater role for f. Therefore, channel pattern does not become an influential variable for f in well and moderately constrained subbasins. The R2 for each channel pattern and f value in a scale context is lower compared to the mean differences of straight vs. non-straight and braided vs. non- 282 braided channels. This can be explained by the fact that the regression slope lines between channel pattern types and f values are in opposite directions among the three channel pattern types, and the f values are scattered in the scale domain of braided and straight channels. 7.3.2 Bed Material In general, an increase in stream discharge or channel depth results in a decrease in the relative roughness of the bed materials, and, thus, in flow resistance, resulting in higher m values. For sand-bed channels, however, the accompanying increase in stream power with increased water discharge can also produce a change in bed form, from plane bed to ripples to dunes. The effect of decreasing channel bed relative roughness is, therefore, partly offset by the effect of increasing bedform drag, producing a relatively restrained rate of increase of flow velocity with higher discharge (Richards, 1982). The channel profile of basin m48 shows that the bedform is not smooth, however, the high flows submerged these riffles or sand dune bedforms with resultant m values as high as 0.668. The other example is the v38 channel profile case. Here, the flow is low and the uneven bedform must have caused high bedform drag which caused the mean m value to be as low as 0.168. The mean m value of the 16 sand bed channel profiles illustrates this point by having low m values compared to the other channel bed types. Based on Bathurst?s research (1993), as discharge increases, the expected correlations between bed material and b, f, and m are that the flow velocity increases as the bed material size increases such that, for sand-bed channels, m < 0.40; for gravel-bed channels, m = 0.35 to 0.45; and for boulder-bed channels, m = 0.45 to 0.55. 283 In gravel-bed (containing both gravel and cobble and gravel and rock) channels, drops in flow resistance are achieved by the drowning of roughness elements as discharge rises, therefore causing high m values in all gravel channels. Subbasins v52 and v19 both had low flows during the study period, and exhibit low m values, 0.212 and 0.346, respectively. On the other hand, in gravel-bed channels not much bedform development is expected; therefore, the f values are low and the flow velocity increases at a higher rate as the discharge increases. In the rock-bed channels, at low flows, the rock-bed channels are expected to have low m values, and as discharge increases, the rock bed channels are expected to have the highest m values among the different bed material types because of drowning of the rough bed material and, hence, smoothing of the roughness of the rocks protruded through the flow. As expected, the exponent m values increased, moving up the network from sand-bed channels (m = 0.467), via gravel, cobble (m = 0.503) and boulder-bed channels (m = 0.487), excepting gravel beds. Gravel-beds have the highest m value (0.503), higher than for cobble and rock bed channels. Since gravel-beds are defined as containing both gravel and cobble and gravel and rock, it is understandable for gravel-bed channels in the study area to have higher m values than would be expected for just those containing gravel. Excluding the two lowest m value subbasins, m31 and m311, the rock channel beds would have the highest mean m value and follow the expectation of being high m value channels. Subbasin m311 has the highest b value (0.562) in the study area followed by subbasin m31 (0.461). Having the higher b values, either f and m values would be lower to meet the continuum theory. 284 A total of 16 sand-bed channels, of which 12 contained 50% or more sand content, were investigated for the applicability of an empirical theory, f > m in sand bed with increasing stream discharge. A difference of means analysis of (f ? m) values between sand beds with any amount of sand and non-sand bed indicates that sand beds are statistically insignificant at the 0.05 level. However, the mean difference analysis indicates that the role of sand beds with 50% or more sand content is statistically significant at the 0.05 level for the values of (f ? m). The higher the channel bed sand content, the higher the rates of channel depth changes become. Even though f is smaller than m in the study area, the trend shows that depth increases more in 50% or more of sand-bed channels. This trend indicates that, as discharge increases, the sand-bed channels will have more eroded bed materials in the flow which would hamper the flow velocity; therefore, eventually f becomes higher than the m value. Adding the scale factor to the regression of bed channels, for sand > 50% and (f ? m) value, the R2 as well as the statistical significance, improved. However, the statistical significance is not at the 0.05 level. Therefore, it can be concluded that there is no discernable differences between f and m in the study area. Comparing difference of means values, it is found that f values are statistically discernible between sand and non-sand bed channels. The differences for b means between cobble and non-cobble channels as well as the difference of b and m means in gravel and non-gravel channels are also found to be sufficiently different. In the geographic scale context, even though there are statistically significant correlations between scale and b, f, and m, the overall influencing role of bed material on b, f, and m 285 is indiscernible. Even though there is a correlation between f and sand bed channels at the 0.05 level, the overall influence of bed material on f is statistically indiscernible. 7.3.3 Bank Material The expected correlations for channel bank material and b, f, and m are such that the value of the change in b will become larger in sequence from the least to the most resistant bank material: sand, gravel, cobble, and rock. In easily eroded channel banks, with the eroded bank material in the discharge, the rate of depth and flow velocity changes would be much less than the b values. So that in easily eroded banks, such as sand banks, a relatively wide, shallow channel develops, while in cemented bank materials (silt), the channel becomes deeper and narrower. As streams begin eroding their banks, their channel cross-sections become wide and shallow with consequent reductions of flow velocities. Without the scale context, the influence of bank material on b, f and m values is examined using a t-test for difference of mean values. Comparing subbasins with the silt and non-silt banks, the mean differences of b are the greatest among all the other bank material type channels. However, the difference between silt versus non-silt bank materials for b is statistically indiscernible at the 0.05 level. Contrary to the expected order of changes in b, rock banks have the highest b values and the 2nd lowest rate of f changes. The more resistant banks should result in higher rates of m changes than the less resistant banks. In easily erodable bank materials, a relatively wide shallow channel develops but, in the study area, small rates of width and bigger rates of depth changes occur. Since each bank, typically, is not of uniform 286 material, but rather a mixture of materials, it is understandable that the observed results differ from the expected order of b, f, and m changes. Without the scale context, the relationships (R2) between b and f with rock, m with silt bank material are 0.4151, 0.4607, and 0.5227, respectively. These are among the highest correlations of determination of all the other watershed variables with b, f, and m. These findings are not expected. There must have another watershed variables in rock bank channel which strongly influencing b values. With the scale context, the computed t-test value scale exceeds the critical values for b at the level of 0.0003, but, not the t-test value for the bank material variable for b. Therefore, it is concluded that channel bank material is not a significant predictor variable to influence b in the scale context. With the scale context, channel bank material is not a significant variable to influence f at the 0.05 level. Including the interaction term between scale and bank material in the model, the statistical significance of the role of bank material type on f is significantly improved and became statistically significant at the 0.05 level. This indicates that the channel bank material variable and scale interact. Therefore, the interaction of the role of bank material on f values and geographic scale should be considered in any examination of the role of bank material on f. Even though, the relationships between silt banks and m (R2 of 0.5227) are much stronger than the role of any other bank material on b, f, and m, in the scale context, channel bank material alone is an insignificant variable influencing m. 287 In general, m values are high for all bank material types, especially silt channel banks where the b values are low which is geomorphologically expected in the silt bank subbasins. Scale influences b and f, but channel bank material is not a significant predictor variable to influence b, f, nor m at 0.05 level of statistical significance, but adding an interaction term between scale and channel bank materials resulted in statistically significant for m and f. Therefore, interactions between the role of bank material on f and m values and geographic scale need to be considered when examining the role of bank materials on f and m. 7.3.4 Topographic Index Statistical results support somewhat the expected outcomes and the statistical significance of the relationships between topographic maximum and topographic index standard deviation by themselves (without the scale context) with b only at the level of 0.05. Topographic indices characterize the ability for subbasins to contain and transport discharge and, hence are considered as an important variable effecting hydraulic geometry. The expected correlations between topographic index and b, f, and m are that higher values of ln (area/tan?) indices are reflective of a wide floodplain or area having a greater potential for increased runoff due to easily saturated areas which implies that areas with a higher topographic index will have high b, and the areas where the small differences between maximum and minimum topographic indices also influence b. Since maximum topographic indices are influencing variable for b, it is understandable that 288 areas with highly different maximum and minimum ln (area/tan?) values might negate the maximum topographic index?s role on b which would results in not close relationship between b and topographic maximum index. The R2 and the statistical significance between b and topographic index, specifically in indices of topographic minimum, topographic maximum, and topographic standard deviation, are stronger than those of f and m. However, any of the topographic indices are not influencing variables for f and m. It is understandable that topographic indices are more for width changes rather than depth of channel nor for flow velocity measure. The relationships between the b, f, and m and topographic index in the scale context are not strong and the statistical insignificant at the 0.05 level. A little better relationship between b and topographic index is expected because it is, in a way, a measurement of saturation capacity. That is, the wetness of both sides of a channel bank gradually diminishes away from the stream. But, when a flow runs over the previous bank, the channel banks will be easily saturated and the banks will be easily eroded and so width increases. If the channel banks are saturated then the stream power can be applied to erode the channel banks and, therefore, b can be changed. It is interesting that the strong, statistically significant relationship between topographic indices and b became highly insignificant when correlated with the scale variable. The subbasin size and topographic maximum correlate well (R2 = 0.58) positively, but the b value and subbasin size correlate negatively. Could this be due to the interaction among topographic index, b, and scale? It needs further investigation, but it might be that high topographic index area (high area/tan?), might not necessarily mean only wide, flat floodplain area, but also areas of increased runoff due to saturation. 289 Out of the 54 combinations of relationships between watershed predictor variables and b, f, and m, individually, five relationships (Great Valley vs. Non-Great Valley physiography for m; Crystalline vs. Non-Crystalline lithology for b and m; straight vs. non-straight channel pattern for b; sand vs. non-sand bed material for f), and, with the scale context including interaction term, five relationships (channel pattern for b and f; bank material for f; topographic maximum index for b; and topographic standard deviation index for b) are shown as statistically significant. These statistically significant watershed variables, specifically, channel pattern, bed and bank materials are also geomorphologically significant variables for b, f, and m. We have to recognize that some of the geomorphologically significant watershed variable such as channel shape is not indicated as influential variable for b, f, and m, which indicates that some of the watershed predictor variables are simply not scale related. Also, watershed predictor variables interact with other watershed predictor variables so the true nature of the influencing factor is not easily identifiable. This implies that, in addition to a single watershed variable, researchers need to investigate the roles of subset of most influential variables for b, f, and m to understand the resultants of their individual roles to b, f, and m. 7.4 Findings and Discussion: Which Subset of Watershed Predictor Variables Are Most Influential on b, f, and m? Stepwise regressions were conducted for the original 43 subbasins and for the 23 well and moderately constrained subbasins (Table 7.3) as an aid to assess the most influential and statistically significant predictors for the estimation of b, f, and m values. 290 Since two different techniques of stepwise regressions were used, the comparisons of the regression results (Table 7.3) are presented for this purpose, rather than for comparing the magnitude of R2. Table 7.3 Most Significant Predictor Variables from Stepwise Regressions for b, f, and m N = 23 (Forward) N = 43 (Backward) N = 43 (Weighted1, Backward) Well and Moderately Constrained Subbasins All Study Subbasins All Study Subbasins b R2 0.599 0.541 0.718 Significant2 Predictor Channel Pattern Channel Pattern Physiography Variables Size Topo Min Index Bank Material Asymmetry Topo Max Index Asymmetry Size Topo Index Stdev Topo Index Stdev Size f R2 0.571 0.512 0.640 Significant2 Predictor Topo Min Index Bed Material Bed Material Variables Bed Material Channel Pattern Channel Shape Size Size Asymmetry Topo Min Index Size m R2 0.2288 0.108 0.150 Significant2 Predictor Size Size Variables Note: 1. All study subbasins are weighted by their sample sizes. 2. Statistically significant at the 0.05 level. The rate of changes in width, b, can be estimated with all the predictor variables with an R2 of 0.792. However, only asymmetry and size are statistically significant at the 291 0.05 levels in this model. Using predictor variables which are statistically significant at the 0.05 level only, channel pattern, topographic index minimum, topographic index maximum, topographic index standard deviation, and size, b can be estimated with an R2 of 0.541. In the 23 well and moderately constrained subbasins, using variables significant at the 0.05 level, b can be estimated via channel asymmetry and size, with an R2 of 0.599. A difference with that for the 43 subbasins indicates that, for the well and moderately constrained subbasins, more easily changeable dynamic predictors such as channel asymmetry are required. However, for all subbasins, without any consideration of the constraints, the more or less static predictors, such as physiography or lithology which changes little over long time, can be used to estimate b. That is, general trends of b can be estimated with static variables, but a refined estimate of b needs to be made with dynamic variables. The rate of change in depth, f, can be estimated with all predictor variables in the regression equation with the R2 of 0.765; however, only size is statistically significant at the 0.05 level for the 43 original subbasins. Progressively eliminating insignificant variables in each stepwise regression, only bed material, channel pattern, and size remained, and f can be calculated with these statistically significant variables with an R2 of 0.512. For 23 subbasins, with the 0.05 level of statistically significant variables, bed material, topographic minimum index, and size, f can be estimated with R2 of 0.571, which indicates that well and moderately constrained subbasins have higher correlation with influential predictor variables than that of all the subbasins. As expected, bed 292 material is an influential predictor variable for f for both the 43 and the subset of well and moderately constrained subbasins. The similarity of the selected watershed predictor variables for b, f, and m for the 43 and the 23 subbasins indicates that in addition to the more or less static predictor variables (size and channel pattern or topographic minimum index), a dynamic predictor variable (a channel sediment related predictor variable such as bed material), is needed. The rate of change in flow velocity, m, can be estimated with all predictor variables in the regression equation with an R2 of 0.559, and with the sole statistically significant predictor variable, size, m can be estimated with an R2 of only 0.108 in the 43 subbasin case. In the 23 subbasin case, m, can be estimated with all predictor variables in the regression equation with an R2 of 0.240, and with the sole statistically significant predictor variable, size, m can be estimated with an R2 of 0.229. Just as in the 43 subbasin case, only scale is identified as the statistically significant predictor variable in the 23 subbasins. As expected, in well and moderately constrained subbasins, the relationship between scale and m are higher than the case of 43 subbasins. Also m is harder exponents to be estimated compared to b and f. And, the scale variable is not statistically significant for m as it is for b and f. . Since estimating m is problematic, it is expected that, having a high confidence of b and f estimates, m can be derived. Comparing the pattern of variables for the estimation of b, f, and m between the cases of 43 and 23 subbasins b can be estimated with higher R2 values compared to those for m, and m is the hardest hydraulic geometry exponent to estimate. Clearly, any success in the estimation of m values is difficult with either only the significant predictor 293 variable, scale, or even with all predictor variables and with a constraint of the 0.05 or better statistical significance. 7.5 Conclusions in a Broader Context for Future Research In conclusion, this study has shown that the behavior of b, f, and m under higher discharges mainly results in higher m and f, with slight increases in cross-sectional area (f with negative b) in a scale context. With static predictor variables, the foundation of the b, f, and m can be estimated with the model. However, for a refined estimation of b, f, and m values, a more controlled selection of well constrained subbasins and flow resistance data should be used. More robust results are likely to be obtained by generating estimation equations recognizing composites of subbasin characteristics. Also, even though this study did not include flow frequency of the discharge data into the model, it is clear that including the flow frequency of the discharge data into the model for at-a-station hydraulic geometry, will provide more confident estimation of b, f, and m. Broeren and Singh?s (1990) work demonstrated flow frequency of the discharge data into the model which improved downstream hydraulic geometry estimation in Illinois. Since scale is proved to be an influential predictor variable for the estimation of b and f values, further investigation of the scale?s role across various sizes such as under 150 square miles, over 150 square miles to 1,000 square miles , and larger than 1,000 square miles streams needs to be considered. That is, usually smaller streams are unstable and out of phase with the steady state condition in the main stream. McConkey and Singh (1992) found and declared that the power function relations are reasonable for watersheds greater than 100 square miles but may not be reliable for smaller drainage 294 area streams. As their finding is evidenced in this study, by the fact that more than 2/3 of the poorly constrained subbasins are under 150 square miles in size, the power function relationship performed poorly for low flows. That is, because of the existence of riffle and pools in natural streams, the hydraulic geometry relations are rendered less reliable, especially at low stream flows. However, it needs to be recognized that the statistical relationship between scale and b, f, and m is weak and that the relationship is limited to the watershed sizes of 0.38 to 1, 642 square miles in the Potomac River Basin. The findings are reasonably useful up to watershed sizes of a little over 10,000 square miles (log of 4.138 watershed size). However, extrapolation is not warranted over 10,000 square miles of watershed size because, as the chosen mathematical model implies, the width of the channels will actually decrease as discharge increases. Clearly, this is physically and geomorphologically impossible. It has increasingly been realized that to achieve a proper understanding of fluvial processes via hydraulic geometry, a function of the interactions of many hydraulic and physical drainage system characteristics and fluvial process-response relationships with watershed variables, including the sizes of channels and subbasins, we need to assess geographic scale as a variable influencing drainage basin response. Furthermore, this study?s establishment of the differences of the influential predictor variables at various scales should help focus data collection efforts for the prediction of b, f, and m. This study demonstrated that the change rates of channel width, depth, and flow velocity at at-a-station hydraulic geometry are systematic changes and a function of watershed scale. When considering the heterogeneous nature of subbasins in terms of 295 their physiography, lithology, landuse, channel pattern and shape, bed and bank materials, and topographic index, the statistically significant relationships between geographic scale and b and f (but not m), are notable. In every selected predictor variable case, investigation of the correlations between b, f, and m with a single selected predictor variable in a scale context resulted in a noticeable improvement over the correlations between b, f, and m with a single selected predictor variable without the scale factor. Even though it may be useful to strive for further demonstrating general trends of the scale impacts on b, f, and m prediction in complex interactions in a channel, there are needs to generate comprehensive datasets which will help formulate and test physical- statistical theories of spatial and scale variability of subbasins for the calculation and analyses of hydraulic geometry. More controlled measurements of various predictor variable values would yield much more consistent general trend information of the role of scale on b, f, and m. Such could be the consideration of stream flow reference levels, the understanding of recovery and response rates, and the cycles of the flood/draught extremes during the hydrographic survey measurements periods of channel width, depth and flow velocity, and discharge volume. At best, the field hydrographic survey data used for this study are rated by the surveyors as 90% accurate and the locations of the survey points are sometimes roughly estimated from a reference point. Utilizing the advanced Light Detection and Range (LIDAR) technology, more accurate and precise measurements of channel width, depth, and flow velocity can be obtained. Also, the locations of the gage positions and hydraulic survey points can be obtained with much higher accuracy such that we can avoid the locations of stream gages being outside of their subbasins. The reason that 296 some of the current stream gages are located outside of their basins is the result of a combination of inaccuracies in the stream locations as determined from digital elevation data, and inaccuracies in the location of the stream gage as defined by coordinates of latitude and longitude. Future models and empirical analyses should be based more accurate DEM representations of the terrain. Since the summation of b, f, and m values is unity and their individual values are decimal points, a minute difference can make a big impact on the analyses of scale impacts on the channel width and depth and stream flow velocity. Therefore, more accurate width, depth, and flow velocity measurements as well as more suitable locations of the hydrographic observation points and channel gage stations will improve measurement of the general trends of scale impacts on b, f, and m values. In addition, accurate and higher spatial resolution of digital elevation data can eliminate many encountered problems such as spikes, depressions, and merging of cells for the delineation of subbasin boundaries and flow path determination using the digital elevation model. Despite the global availability of satellite-derived digital elevation data, a critical assessment of how their vertical and horizontal accuracies affect landforms is lacking and once the effect of satellite-derived digital elevation data errors on hydrogeographic analyses is better understood, delineation of drainage area boundaries and the sizes of drainage basins can better incorporate uncertainty analysis into catchment management decisions. Also, using Global Positioning System (GPS) could improve the positioning of the discharge, width, depth, and velocity survey points. In shallow channels, usage of LIDAR can eliminate manual hydrographic survey. 297 This research has demonstrated through inductive reasoning that there is evidence of geographic scale?s role on at-a-station hydraulic geometry. Now, the future scientific endeavor is to formulate the findings into a theory. However, more rigorous hydrological principles and a mathematics-based theory of the scale impacts on b, f, and m is needed for the current empirical equations of at-a-station hydraulic geometry; written as scaling relations, using the equations for momentum, flow resistance, and continuity of open channel flow. Since there is a relationship between at-a-station hydraulic geometry and downstream hydraulic geometry and there is a limited and localized hydrological principle-based downstream hydraulic geometry model, theory-based at-a-station hydraulic geometry models are feasible. The main limitation of the hydraulic geometry analysis is its neglect of the feedback from flow to form when discharge is competent to erode channel banks or bed. Hydraulic adjustments within a fixed channel are reversible, but there is plenty of evidence that hydraulic geometry relationships evolve over time. Models of channel response to changing flow must be based on the knowledge of the relative response time of hydraulic values under a range of flow conditions as well as the recovery time of subbasin into quasi-equillibrium. Even though the representative elementary area (REA) in the literature has been historically much smaller (1 - 5 km2) than 100 square miles in size, there are numerous cases of implied evidence of the similar concept of the REA in many hydraulic geometry variables in the study area. The REA is defined as a critical area at which implicit continuum assumptions may be used without knowledge of patterns of variable values although some knowledge of the underlying distributions may still be necessary. At 298 scales smaller than the REA, the actual patterns of variability are important in determining the hydrological response. At larger scales a statistical description of spatial variability should suffice. In accord with the suggestion of Bloschl et al. (1995) and Beven (1995) that the size of the rea might vary between watershed environments and processes, it could be that the rea would be at 100 square miles for the at-a-station hydraulic geometry in the study area? Another example is that Hogan and Church (1989) found that the relationships of at-a-station hydraulic geometry for Hangover Creek were well described by power functions, but that the plots displayed a discontinuity in the relationship at a certain discharge (1m3/s,) which can be equated to drainage area size. They found that below this threshold, increases in discharge are attributable primarily to increases in mean velocity and channel width, with little change in mean depth. They noted that above this threshold, increases in discharge are attributable primarily to increases in mean depth and velocity, with little change in channel width. The major predictor variables which can adjust to accommodate flows at a site are locally unstable in the sense that a change or perturbation to any system component will result in a new set of relationships among the components rather than a restoration of the previously existing quasi-equilibrium. This suggests that there may be no single, universally applicable method of predicting hydraulic geometry relationships and that stable configurations in natural alluvial channels cannot be expected to persist over time. However, this research indicates that subbasin size clearly influences hydraulic geometry and that some of the subbasin characteristics such as physiography, lithology, and scale are more or less static variables (static in longer temporal scale) that can be used for 299 prediction of hydraulic geometry of un-surveyed channels. In addition, the temporal scale, such as recovery time, needs to be considered in the development of hydraulic geometry models. Natural channels undergo continuous change, and the hydraulic geometry exponents describing a cross-section vary annually or during bankfull floods if bed and banks are modified. The flow variables are mutually interdependent, meaning that a change in any single parameter requires a response in one or more of the others. The difficulty in understanding rivers is when you consider that discharge and load are in continuous flux, and so all the hydraulic variables must always be adjusting. In situations where all hydraulic parameters can vary and where they do not respond to changing conditions at equal rates and intensities, multiple modes of adjustments are likely. Even though there is a notion of inherent instability of hydraulic geometry, with it?s implication that hydraulic geometry is not particularly precise and cannot be site independent, hydraulic geometry processes are determinate based on laws of hydraulics and relations of flow resistance. The wide range of frictional characteristic and channel shapes would not allow a universally-applicable set of at-a-station hydraulic geometry. However, including temporal information on the flow reference level (bankfull or low flow) into a model, the flow resistance and frictional characteristics can be better understood and, therefore the hydraulic geometry of channels can be better estimated. 300 APPENDICES USGS Hydrographic survey data over the study area such as stream discharge (volume per second), channel width, depth, and flow velocity, as well as information on cross section are included in Appendix A. Algorithms for subbasin boundary delineation and Topographic Index calculation algorithms are in Appendix B. Data which characterize each subbasin, which are collected from the hydrographic surveys, calculated and or derived are in Appendix C. 301 APPENDIX A USGS HYDROGRAPHIC FIELD NOTE/DATA FOR THE POTOMAC RIVER BASIN AREA (STREAM DISCHARGE, CHANNEL WIDTH, MEAN VELOCITY ON STREAM CROSS-SECTIONS bfile1a obs. # width area mean vel. gageht. Discharge depth w/d cum_wi dth Asym [1,] 538 6/18/85 79 76 1.27 2.27 96.4 0.96 82.12 38.25 -0.287 [2,] 540 9/12/85 61 41.2 0.69 1.68 28.5 0.68 90.32 28.00 -0.284 [3,] 542 1/17/86 73 81 1.09 2.23 87.9 1.11 65.79 35.00 -0.188 [4,] 544 4/8/86 90 145 1.74 2.86 253 1.61 55.86 43.50 0.000 [5,] 545 6/30/86 64 62.3 0.85 1.98 52.7 0.97 65.75 31.25 -0.115 [6,] 546 8/15/86 67 44.7 0.64 1.69 28.5 0.67 100.43 32.00 -0.076 [7,] 547 10/3/86 68 64.9 0.88 2.02 57.1 0.95 71.25 32.50 0.129 [8,] 549 3/19/87 86 131 1.88 2.84 246 1.52 56.46 41.50 -0.080 [9,] 550 6/10/87 78 87.4 1.19 2.3 104 1.12 69.61 38.50 -0.290 [10,] 551 7/16/87 69 63.6 0.73 1.94 46.4 0.92 74.86 33.50 -0.159 [1l,] 552 8/31/87 68 47.7 0.61 1.72 29 0.70 96.94 32.75 -0.217 [12,] 553 10/14/87 65 49.7 0.68 1.79 33.8 0.76 85.01 31.75 -0.166 [13,] 554 2/19/88 84 131 1.46 2.65 191 1.56 53.86 40.50 0.042 [14,] 555 3/30/88 90 181 1.68 2.97 304 2.01 44.75 43.25 -0.153 [15,] 557 6/28/88 70 61.9 0.85 1.99 52.6 0.88 79.16 34.00 -0.195 [16,] 558 8/10/88 68 63.7 0.46 1.72 29.5 0.94 72.59 32.50 0.000 [17,] 559 9/15/88 66 52.7 0.73 1.85 38.5 0.80 82.66 32.00 -0.227 [18,] 560 10/27/88 67 63.5 0.66 1.89 41.9 0.95 70.69 32.25 -0.097 [19,] 561 12/6/88 78 80.4 1.04 2.21 83.3 1.03 75.67 37.75 0.194 [20,] 563 7/19/89 90 178 1.92 3.05 342 1.98 45.51 43.50 -0.120 [21,] 564 8/17/89 78 94.2 1.2 2.34 113 1.21 64.59 38.00 -0.059 [22,] 565 10/3/89 87 129 1.83 2.83 236 1.48 58.67 42.00 -0.208 [23,] 566 11/7/89 77 100 1.58 2.55 158 1.30 59.29 36.50 -0.195 [24,] 568 3/13/90 84 119 1.35 2.56 161 1.42 59.29 42.00 0.244 [25,] 570 10/5/90 76 93.6 1.38 2.44 129 1.23 61.71 36.50 -0.232 [26,] 574 7/9/91 67 69.1 0.76 1.99 52.6 1.03 64.96 32.50 -0.253 [27,] 575 7/16/91 66 58.3 0.68 1.86 39.9 0.88 74.72 32.25 -0.143 [28,] 582 6/11/92 87 131 1.65 2.75 216 1.51 57.78 41.25 -0.266 [29,] 583 9/4/92 73 79.8 0.82 2.09 65.2 1.09 66.78 35.00 -0.387 Ave 75 88.99 1.11 2.24 114.37 1.15 68.09 36.22 -0.131 bfile2a [1,] 608 2/13/85 240 951 4.02 5.51 3820 3.96 60.57 117.50 -0.104 [2,] 609 4/1/85 249 1660 4.53 8.5 7520 6.67 37.35 123.50 0.048 [3,] 610 10/25/85 213 511 2.56 3.67 1310 2.40 88.78 104.50 0.029 [4,] 611 12/4/85 248 1350 4.58 7.2 6180 5.44 45.56 121.50 0.033 [5,] 615 2/3/87 228 654 2.74 3.99 1790 2.87 79.49 112.00 0.139 [6,] 620 5/10/88 235 742 3.27 4.51 2430 3.16 74.43 111.25 -0.040 [7,] 623 1/19/89 240 839 2.81 4.44 2360 3.50 68.65 117.25 0.029 [8,] 628 5/30/90 251 1700 4.64 8.49 7880 6.77 37.06 124.00 0.029 [9,] 632 10/24/90 250 1610 4.35 8.04 7010 6.44 38.82 125.00 0.061 [10,] 633 10/25/90 235 913 4.15 5.64 3790 3.89 60.49 117.50 0.142 [11,] 634 10/30/90 200 460 2.5 3.54 1150 2.30 86.96 100.00 0.086 302 [12,] 635 11/9/90 197 431 2.32 3.34 1000 2.19 90.04 96.75 0.087 [13,] 641 2/19/92 184 405 2.4 2.96 971 2.20 83.60 92.00 0.010 [14,] 642 2/26/92 223 1410 3.84 7.21 5420 6.32 35.27 111.90 0.066 [15,] 643 3/2/92 207 647 3.71 4.25 2400 3.13 66.23 103.50 0.082 [16,] 644 5/1/92 204 622 1.95 3.2 1210 3.05 66.91 96.75 0.017 [17,] 648 1/8/93 211 585 2.95 4.13 1730 2.77 76.10 98.50 0.037 [18,] 649 3/24/93 243 1880 4.54 9.95 8540 7.74 31.41 111.25 -0.010 [19,] 650 4/6/93 221 893 4.2 5.46 3750 4.04 54.69 107.50 -0.046 [20,] 651 5/6/93 212 524 2.98 3.76 1560 2.47 85.77 99.00 -0.040 [21,] 652 6/16/93 186 329 1.72 2.86 567 1.77 105.16 89.50 0.017 Ave 223 910.3 3.37 5.27 3447.05 2.19 65.40 108.60 0.032 bfile3a pg3 [1,] 524 1/3/85 121 228 1.66 2.29 378 1.88 64.21 58.00 0.107 [2,] 526 6/17/85 120 177 1.21 1.82 214 1.48 81.36 58.25 0.091 [3,] 527 7/29/85 118 185 1.24 1.92 230 1.57 75.26 56.75 0.118 [4,] 528 9/12/85 110 134 0.93 1.48 124 1.22 90.30 53.00 0.101 [5,] 529 10/22/85 116 150 1.02 1.62 153 1.29 89.71 55.75 0.098 [6,] 532 4/7/86 128 279 1.91 2.69 533 2.18 58.72 61.00 -0.183 [7,] 533 7/1/86 110 137 0.96 1.51 132 1.25 88.32 52.25 0.091 [8,] 534 7/30/86 110 132 0.86 1.43 114 1.20 91.67 53.50 0.157 [9,] 535 10/1/86 107 109 0.74 1.27 80.4 1.02 105.04 51.50 0.165 [10,] 536 11/18/86 110 127 0.84 1.43 107 1.15 95.28 53.00 0.000 [11,] 537 12/22/86 122 243 1.7 2.41 414 1.99 61.25 59.00 0.218 [12,] 538 3/16/87 123 248 1.86 2.49 461 2.02 61.00 59.50 0.191 [13,] 539 6/8/87 123 203 1.4 2.06 285 1.65 74.53 58.50 0.094 [14,] 540 9/1/87 108 114 0.74 1.29 84.6 1.06 102.32 52.50 0.140 [15,] 541 10/13/87 113 138 0.94 1.5 130 1.22 92.53 55.00 0.138 [16,] 542 11/23/87 120 194 1.32 1.97 257 1.62 74.23 57.00 0.113 [17,] 543 2/17/88 123 242 1.93 2.54 467 1.97 62.52 59.50 0.132 [18,] 545 6/27/88 117 164 1.16 1.74 191 1.40 83.47 56.00 0.110 [19,] 546 8/9/88 114 135 0.9 1.47 122 1.18 96.27 55.00 0.000 [20,] 547 9/14/88 115 142 0.97 1.54 138 1.23 93.13 56.00 0.117 [21,] 548 10/24/88 119 136 0.99 1.5 134 1.14 104.13 59.50 0.072 [22,] 549 12/5/88 121 176 0.9 1.61 158 1.45 83.19 58.50 0.144 [23,] 551 4/10/89 130 307 2.12 2.88 650 2.36 55.05 63.00 0.090 [24,] 553 8/16/89 126 220 1.42 2.13 312 1.75 72.16 63.00 -0.118 [25,] 554 10/3/89 125 215 1.38 2.12 296 1.72 72.67 62.50 0.135 [26,] 555 11/6/89 114 160 1.06 1.66 169 1.40 81.23 55.00 0.079 [27,] 557 3/12/90 122 210 1.39 2.07 292 1.72 70.88 61.00 0.000 [28,] 558 10/3/90 105 118 0.85 1.35 100 1.12 93.43 50.50 0.104 [29,] 559 11/8/90 127 247 1.67 2.39 413 1.94 65.30 61.50 0.126 [30,] 561 6/4/91 119 173 1.18 1.79 204 1.45 81.86 57.00 0.129 [31,] 562 7/15/91 114 134 0.9 1.44 121 1.18 96.99 55.50 0.147 [32,] 563 10/2/91 102 117 0.8 1.81 93.5 1.15 88.92 49.50 0.124 [33,] 564 2/4/92 117 147 1.05 1.58 154 1.26 93.12 58.50 0.145 [34,] 565 5/7/92 133 267 1.74 2.47 464 2.01 66.25 64.50 -0.110 [35,] 566 6/4/92 124 217 1.52 2.18 330 1.75 70.86 60.50 0.135 [36,] 567 6/10/92 136 307 1.86 2.77 570 2.26 60.25 66.50 -0.093 [37,] 568 9/2/92 121 150 1.03 1.59 155 1.24 97.61 58.50 0.125 [38,] 569 10/5/92 120 145 0.97 1.53 141 1.21 99.31 57.50 0.126 [39,] 571 2/17/93 130 247 1.62 2.37 400 1.90 68.42 63.00 0.126 [40,] 573 6/18/93 127 194 1.22 1.9 236 1.53 83.14 62.50 0.134 303 [41,] 574 8/2/93 122 150 0.95 1.56 142 1.23 99.23 59.50 0.130 [42,] 575 9/7/93 125 179 1.09 1.8 196 1.43 87.29 59.50 0.112 [43,] 576 10/5/93 126 186 1.15 1.87 214 1.48 85.35 60.00 -0.146 [44,] 577 10/27/93 127 168 1.01 1.72 170 1.32 96.01 121.00 0.056 [45,] 579 8/1/94 136 244 1.69 2.41 413 1.79 75.80 65.00 0.217 Ave 120 184.3 1.24 1.89 247.61 1.52 81.99 59.42 0.089 bfie4a [1,] 202 2/14/85 16 13.3 1.34 1.34 17.8 0.83 19.25 6.25 0.072 [2,] 204 6/17/85 15 8.66 0.49 0.95 4.23 0.58 25.98 6.75 -0.060 [3,] 205 7/29/85 13 7.6 0.43 0.89 3.25 0.58 22.24 5.75 0.047 [4,] 209 4/7/86 15.5 13.1 1.38 1.35 18.1 0.85 18.34 7.25 0.099 [5,] 210 7/1/86 13.1 7.64 0.6 0.96 4.62 0.58 22.46 6.15 0.001 [6,] 213 10/1/86 6.5 1.43 0.57 0.74 0.82 0.22 29.55 2.85 0.203 [7,] 214 11/18/86 11.8 4.34 0.27 0.81 1.16 0.37 32.08 5.30 0.177 [8,] 215 12/19/86 13 7.6 0.72 1.02 5.48 0.58 22.24 6.13 0.198 [9,] 216 3/16/87 14 8.95 1.09 1.15 9.74 0.64 21.90 6.63 0.194 [10,] 217 6/8/87 14 7.66 0.92 1.07 7.05 0.55 25.59 6.50 -0.029 [11,] 218 9/1/87 11.5 3.47 0.33 0.8 1.16 0.30 38.11 5.30 0.014 [12,] 220 11/23/87 9.5 2.72 0.65 0.85 1.77 0.29 33.18 4.38 -0.222 [13,] 221 2/17/88 14 7.39 0.81 1.04 5.99 0.53 26.52 6.38 -0.027 [14,] 222 3/29/88 12.5 7.35 0.6 0.98 4.44 0.59 21.26 5.63 0.078 [15,] 223 8/9/88 13 4.49 0.46 0.87 2.05 0.35 37.64 6.00 -0.078 [16,] 224 9/14/88 9.3 3.23 0.49 0.84 1.59 0.35 26.78 4.43 -0.186 [17,] 225 10/24/88 10 3.08 0.46 0.83 1.43 0.31 32.47 4.65 -0.127 [18,] 226 12/5/88 8.7 3.24 0.5 0.84 1.62 0.37 23.36 4.20 0.099 [19, ] 227 1/18/89 13 6.08 0.81 1 4.93 0.47 27.80 6.03 -0.006 [20,] 228 4/10/89 14 7.96 1.1 1.12 8.74 0.57 24.62 6.63 0.044 [21,] 230 8/16/89 13.7 6.82 0.59 0.97 4.04 0.50 27.52 6.60 -0.043 [22,] 231 10/2/89 14.5 7.76 0.79 1.05 6.14 0.54 27.09 7.25 0.074 [23,] 232 11/6/89 9.9 3.74 0.59 0.88 2.22 0.38 26.21 4.60 -0.051 [24,] 234 1/31/90 15 10.1 1.03 1.16 10.4 0.67 22.28 7.50 0.014 [25,] 235 4/23/90 13 6.72 0.73 1 4.93 0.52 25.15 6.00 0.028 [26,] 236 8/28/90 8.7 4.02 0.68 0.91 2.72 0.46 18.83 4.30 -0.020 [27,] 237 10/2/90 10 3.98 0.48 0.87 1.92 0.40 25.13 4.95 -0.099 [28,] 238 11/8/90 15 10.4 0.82 1.12 8.5 0.69 21.63 7.00 0.028 [29,] 239 4/22/91 15 12.8 1.11 1.29 14.2 0.85 17.58 7.50 0.012 [30,] 240 6/4/91 14 8 0.59 0.99 4.74 0.57 24.50 6.50 0.090 [31,] 241 7/15/91 12.5 5.12 0.45 0.88 2.29 0.41 30.52 5.75 0.029 [32,] 242 10/2/91 13 6.44 0.38 0.89 2.45 0.50 26.24 6.50 0.022 [33,] 243 2/3/92 14 8.33 0.54 0.98 4.5 0.60 23.53 7.00 0.044 [34,] 244 4/30/92 14.8 15.9 1.41 1.39 22.4 1.07 13.78 6.70 0.035 [35,] 245 6/4/92 15 12.7 1.31 1.3 16.7 0.85 17.72 7.00 0.071 [36,] 246 9/1/92 14 9.72 1.19 1.2 11.6 0.69 20.16 6.50 0.134 [37,] 247 10/5/92 15 9.87 0.99 1.15 9.76 0.66 22.80 6.88 -0.081 [38,] 248 11/18/92 13 9.57 0.55 1.03 5.28 0.74 17.66 6.50 -0.134 [39,] 249 1/6/93 15.3 15.3 1.31 1.39 20.1 1.00 15.30 7.03 -0.070 [40,] 250 2/17/93 16.5 12.1 1.34 1.3 16.2 0.73 22.50 7.50 0.075 [41,] 252 5/12/93 16 17.6 1.68 1.8 29.6 1.10 14.55 8.00 0.082 [42,] 253 6/23/93 15 13 1.14 1.27 14.8 0.87 17.31 7.00 0.042 [43,] 254 8/2/93 14.5 9.97 0.82 1.11 8.21 0.69 21.09 6.63 0.018 [44,] 255 9/7/93 13 7.17 0.47 0.94 3.39 0.55 23.57 6.00 0.107 [45,] 256 10/27/93 13 8.55 0.34 0.92 2.94 0.66 19.77 6.50 0.113 304 [46,] 257 12/7/93 15 17.5 1.34 1.47 23.5 1.17 12.86 7.00 -0.085 [47,] 258 2/1/94 15 14.2 1.09 1.31 15.5 0.95 15.85 7.50 0.064 Ave 13.2 8.44 0.80 1.06 7.98 0.61 23.46 6.20 0.008 bfile6a [1,] 469 1/3/85 69 150 1.51 2.83 227 2.17 31.74 33.00 -0.148 [2,] 471 6/17/83 67 131 1.54 2.75 202 1.96 34.27 31.75 -0.222 [3,] 472 7/29/85 63 116 1.34 2.59 156 1.84 34.22 30.25 -0.162 [4,] 473 9/12/85 60 104 1.25 2.47 130 1.73 34.62 29.50 -0.181 [5,] 474 10/22/85 67 128 1.72 2.81 220 1.91 35.07 32.25 -0.203 [6,] 476 4/7/86 75 156 2.37 3.18 370 2.08 36.06 36.00 -0.026 [7,] 477 7/1/86 58 106 1.49 2.6 158 1.83 31.74 27.75 -0.082 [8,] 478 7/30/86 56 99.2 1.22 2.44 121 1.77 31.61 26.75 -0.008 [9,] 479 10/1/86 56 91.4 1 2.32 91.7 1.63 34.31 26.50 0.027 [10,] 480 12/19/86 69 168 1.7 2.97 286 2.43 28.34 33.00 -0.003 [11,] 482 6/8/87 64 129 1.53 2.73 198 2.02 31.75 30.75 -0.071 [12,] 483 9/1/87 58 104 1.14 2.44 119 1.79 32.35 27.00 -0.074 [13,] 484 10/13/87 58 101 1.03 2.36 104 1.74 33.31 27.75 0.253 [14,] 485 2/17/88 64 145 1.61 2.83 234 2.27 28.25 30.25 -0.007 [15,] 486 3/29/88 65 129 1.43 2.67 184 1.98 32.75 31.00 -0.106 [16,] 487 5/9/88 78 190 1.91 3.15 362 2.44 32.02 37.50 -0.036 [17,l 488 8/9/88 59 109 1.09 2.42 119 1.85 31.94 27.75 0.004 [18,] 489 10/26/88 56 104 0.93 2.32 96.4 1.86 30.15 26.50 -0.090 [19,] 490 12/5/88 56 103 0.93 2.32 95.5 1.84 30.45 26.50 -0.090 [20,] 491 1/18/89 60 133 1.41 2.7 188 2.22 27.07 28.75 -0.084 [21,] 492 4/10/89 69 153 1.76 2.93 270 2.22 31.12 33.00 0.086 [22,] 493 8/16/89 59 120 1.31 2.57 157 2.03 29.01 27.75 -0.173 [23,] 494 10/2/89 69 127 1.59 2.73 202 1.84 37.49 33.50 0.274 [24,] 495 11/6/89 49 96.5 1.13 2.4 109 1.97 24.88 23.50 -0.100 [25,] 497 3/12/90 62 119 1.65 2.74 196 1.92 32.30 54.50 -0.087 [26,] 498 8/28/90 58 106 1.5 2.61 159 1.83 31.74 29.00 -0.056 [27,] 499 10/2/90 50 96 1.14 2.37 109 1.92 26.04 23.50 -0.091 [28,] 500 11/8/90 69 136 1.93 2.9 262 1.97 35.01 32.75 -0.261 [29,] 502 6/3/91 64 119 1.72 2.73 205 1.86 34.42 32.00 -0.302 [30,] 503 7/15/91 60 102 1.2 2.43 122 1.70 35.29 28.75 0.161 [31,] 504 10/1/91 54 107 1.26 2.48 135 1.98 27.25 25.50 0.102 [32,] 505 2/3/92 59 109 1.36 2.53 148 1.85 31.94 28.25 -0.144 [33,] 507 6/4/92 68 141 1.99 2.98 281 2.07 32.79 32.75 -0.293 [34,] 508 9/2/92 58 106 1.47 2.58 156 1.83 31.74 60.25 -0.183 [35,] 509 10/5/92 55 106 1.42 2.56 151 1.93 28.54 26.00 0.086 [36,] 511 2/17/93 70 154 2.12 3.09 326 2.20 31.82 33.25 0.107 [37,] 514 8/2/93 58 104 1.49 2.57 155 1.79 32.35 28.00 0.097 Ave 61.9 121.6 1.46 2.65 183.91 1.95 31.78 31.16 -0.056 bfile7a [1,] 382 8/18/88 16 5.09 0.42 1.43 2.12 0.32 50.29 8.00 -0.124 [2,] 383 10/11/88 16.3 6.56 0.54 1.48 3.51 0.40 40.50 8.15 0.113 [3,] 384 11/15/88 30 18.3 0.7 1.65 12.9 0.61 49.18 15.00 0.334 [4,] 385 1/3/89 32 21.4 1 1.75 21.3 0.67 47.85 16.00 0.265 [5,] 388 5/1/89 32.5 26.5 1.4 1.87 37 0.82 39.86 16.25 0.248 [6,] 389 6/13/89 43 34.4 1.46 1.94 50.2 0.80 53.75 21.50 0.304 [7,] 390 7/25/89 42.5 33.4 1.35 1.92 45 0.79 54.08 21.25 0.236 [8,] 391 10/4/89 36 19.6 0.68 1.65 13.3 0.54 66.12 18.00 0.211 [9,] 392 11/16/89 41.5 31.7 1.71 1.98 54.2 0.76 54.33 20.75 0.351 305 [10,] 393 1/22/90 41.5 28.9 1.5 1.91 43.4 0.70 59.59 20.70 0.380 [11,] 394 2/12/90 66 60.6 1.85 2.22 112 0.92 71.88 33.00 0.032 [12,] 395 3/27/90 63.5 46.6 1.53 2.07 71.5 0.73 86.53 31.75 0.057 [13,] 396 5/17/90 65.5 73.1 2.19 2.39 160 1.12 58.69 32.75 0.024 [14,] 397 6/21/90 63.5 28.9 1.05 1.82 30.3 0.46 139.52 31.75 0.112 [15,] 398 8/14/90 28 13.2 0.7 1.59 9.28 0.47 59.39 14.00 0.274 [16,] 400 11/14/90 62.5 34.1 1.21 1.9 41.3 0.55 114.55 31.25 0.078 [17,] 401 1/10/91 65 70 2 2.34 140 1.08 60.36 63.63 -0.997 [18,] 402 3/6/91 64.5 64.3 2.17 2.33 139 1.00 64.70 32.25 0.033 [19,] 403 4/18/91 65 52.3 1.72 2.14 89.8 0.80 80.78 32.50 0.028 [20,] 409 1/6/92 45 39.8 1.56 2.02 61.9 0.88 50.88 22.50 0.214 [21,] 410 2/20/92 47 46.3 1.87 2.13 86.7 0.99 47.71 23.50 -0.227 [22,] 411 4/1/92 66.3 65.1 2 2.29 130 0.98 67.52 33.15 -0.065 [23,] 412 5/20/92 59.1 46.2 1.15 1.97 53.1 0.78 75.60 29.55 0.233 [24,] 413 7/9/92 33.8 19.1 1.11 1.73 21.1 0.57 59.81 16.53 0.315 [25,] 415 10/7/92 24.6 13.1 1.17 1.67 15.3 0.53 46.20 12.30 -0.156 Ave 46 35.94 1.36 1.93 57.77 0.73 63.99 24.24 0.091 bfile8a [1,] 12 1/8/91 43 50.8 1.08 1.79 55 1.18 36.40 21.50 -0.017 [2,] 13 3/11/91 32.2 24.1 0.7 1.5 16.8 0.75 43.02 16.10 -0.017 [3,] 14 4/15/91 42 38.6 1.1 1.7 42.8 0.92 45.70 21.00 0.085 [4,] 23 1/3/92 33.6 28.6 0.86 1.56 24.7 0.85 39.47 16.80 -0.009 [5,] 24 2/18/92 43 46.3 1.51 1.86 69.8 1.08 39.94 21.50 0.089 [6,] 25 3/31/92 42 46.6 1.28 1.8 59.8 1.11 37.85 17.00 -0.053 [7,] 26 5/18/92 34 21.9 0.5 1.39 10.9 0.64 52.79 21.00 0.025 [8,] 27 5/21/92 29.9 17.3 0.37 1.31 6.4 0.58 51.68 14.95 0.025 [9,] 28 6/1/92 39.5 30.6 0.88 1.57 27 0.77 50.99 19.75 -0.042 [10,] 29 7/13/92 17.6 9.2 0.38 1.23 3.45 0.52 33.67 8.80 -0.149 [11,] 30 8/17/92 28.5 17.7 0.41 1.34 7.22 0.62 45.89 14.25 -0.069 [12,] 31 10/2/92 28.4 17 0.32 1.3 5.45 0.60 47.44 14.20 -0.018 [13,] 32 11/17/92 30.6 21.2 0.53 1.41 11.2 0.69 44.17 15.30 0.048 Ave 34.2 28.45 0.76 1.52 26.19 0.79 43.77 17.09 -0.008 bfile9a [1,] 354 11/17/88 67 56 0.63 1.03 35 0.84 80.16 33.50 -0.042 [2,] 355 2/16/89 70 117 1.44 1.78 168 1.67 41.88 35.00 0.030 [3,] 356 3/23/89 70 92.6 1.09 1.46 101 1.32 52.92 35.00 0.006 [4,] 357 5/3/89 70 111 1.38 1.74 153 1.59 44.14 35.00 -0.005 [5,] 358 7/27/89 72 67.7 0.84 1.21 56.8 0.94 76.57 36.00 -0.122 [6,] 359 10/3/89 70 74.1 0.99 1.3 73.1 1.06 66.13 35.00 -0.114 [7,] 360 11/20/89 70 74.6 1.02 1.32 75.9 1.07 65.68 35.00 -0.075 [8,] 362 1/31/90 70.5 144 1.92 2.25 276 2.04 34.52 35.25 -0.055 [9,] 363 2/14/90 70 88.3 1.3 1.55 115 1.26 55.49 35.00 -0.062 [10,] 364 3/26/90 62 69.8 1.4 1.44 97.4 1.13 55.07 31.00 -0.054 [11,] 365 5/15/90 73 117 1.46 1.84 171 1.60 45.55 36.50 -0.063 [12,] 367 6/29/90 61.5 53.9 0.98 1.16 52.9 0.88 70.17 30.75 -0.031 [13,] 368 8/8/90 60 51.4 1.01 1.15 52 0.86 70.04 30.00 -0.039 [14,] 369 10/4/90 60 43.2 0.75 0.98 32.4 0.72 83.33 30.00 -0.066 [15,] 372 11/9/90 62 57.5 0.97 1.18 56 0.93 66.85 31.00 -0.071 [16,] 373 1/8/91 62 92.7 1.78 1.77 165 1.50 41.47 31.00 -0.029 [17,] 374 3/11/91 63 77.1 1.03 1.52 79.3 1.22 51.48 31.50 -0.045 [18,] 375 4/15/91 63 86.7 1.55 1.63 134 1.38 45.78 31.75 -0.049 [19,] 376 6/4/91 61 51.6 0.77 1.07 39.6 0.85 72.11 30.50 -0.049 306 [20,] 377 7/9/91 54 37.4 0.63 0.9 23.5 0.69 77.97 27.00 -0.009 [21,] 378 7/25/91 42.5 28.3 0.41 0.78 11.7 0.67 63.83 21.25 -0.084 [22,] 379 8/19/91 44 29 0.36 0.76 10.4 0.66 66.76 22.00 -0.048 [23,] 380 10/1/91 59 41.2 0.68 0.96 27.9 0.70 84.49 29.50 -0.055 [24,] 381 11/12/91 59 43.5 0.77 1.01 33.6 0.74 80.02 29.50 -0.084 [25,] 382 1/3/92 62 64.7 1.23 1.32 79.6 1.04 59.41 31.00 -0.032 [26,] 383 2/18/92 72.5 98.4 1.23 1.59 121 1.36 53.42 36.25 -0.101 [27,] 384 3/31/92 62.5 98.5 1.84 1.86 181 1.58 39.66 31.25 -0.014 [28,] 385 5/18/92 60.5 55.8 1.06 1.21 59.1 0.92 65.60 30.25 -0.028 [29,] 386 7/13/92 57 38.5 0.69 0.95 26.6 0.68 84.39 28.50 -0.091 [30,] 388 9/2/92 58 33.22 0.49 0.83 16.2 0.58 100.00 29.00 -0.092 [31,] 391 1/7/93 70 90.5 1.46 1.6 132 1.29 54.14 35.00 0.020 [32,] 395 8/19/93 66 53.8 0.85 1.13 45.6 0.82 80.97 33.00 -0.067 [33,] 396 7/8/93 68 56.2 0.78 1.12 44 0.83 82.28 32.50 0.059 [34,] 398 4/25/94 73 93.3 1.53 1.65 143 1.28 57.12 36.50 0.073 [35,] 399 6/14/94 69 54.3 0.72 1.16 47.4 0.79 87.68 34.50 -0.111 [36,] 400 7/7/94 68 55.9 1.01 1.19 56.6 0.82 82.72 34.00 0.107 [37,] 401 8/16/94 66 44.2 0.76 1.02 33.6 0.67 98.55 33.00 0.163 [38,] 402 10/3/94 68 49.7 0.84 1.09 41.6 0.73 93.04 34.00 0.137 [39,] 403 12/2/94 69 66.4 1.18 1.36 78.7 0.96 71.70 34.25 0.085 [40,] 404 3/17/95 64.5 71 1.6 1.5 114 1.10 58.60 32.25 -0.033 [41,] 405 5/12/95 61 53 1.16 1.23 61.5 0.87 70.21 30.50 -0.044 [42,] 406 7/6/95 60 44.5 0.95 1.09 42.3 0.74 80.90 30.00 0.087 [43,] 407 8/28/95 53.6 28.5 0.5 0.82 14.4 0.53 100.81 26.80 0.123 Ave 63.8 66.42 1.05 1.29 78.57 1.02 67.76 31.87 -0.023 bfile10a [1,] 48 1/30/85 4.5 3.51 0.3 2.39 1.05 0.78 5.77 2.25 -0.176 [2,] 49 3/5/85 6.7 5.81 0.55 2.59 3.17 0.87 7.73 3.35 0.101 [3,] 50 4/18/85 6.5 5.16 0.34 2.46 1.75 0.79 8.19 3.25 0.041 [4,] 51 6/4/85 4 2.28 0.44 2.36 1 0.57 7.02 2.00 0.086 [5,] 54 7/16/85 4 2.23 0.28 2.19 0.62 0.56 7.17 2.00 0.121 [6,] 55 7/30/85 2.4 0.51 0.86 2.2 0.44 0.21 11.29 1.20 -0.043 [7,] 56 7/30/85 2.4 0.51 0.86 2.2 0.46 0.21 11.29 1.20 -0.104 [8,] 58 8/20/85 1.4 0.34 0.47 2.11 0.16 0.24 5.76 0.70 -0.176 [ 9,] 59 9/26/85 2 0.59 0.24 2.06 0.14 0.30 6.78 1.00 0.119 [10,] 60 10/8/85 1.3 0.45 0.24 2.13 0.11 0.35 3.76 0.65 0.078 [11,] 61 10/8/85 0.8 0.3 0.29 2.11 0.09 0.38 2.13 0.40 -0.033 [12,] 62 11/13/85 6.2 5.08 0.61 2.54 3.1 0.82 7.57 3.10 0.069 [13,] 63 11/26/85 6.6 5.33 1.07 2.67 5.7 0.81 8.17 3.33 0.156 [14,] 64 12/17/85 7.1 5.73 0.75 2.62 4.31 0.81 8.80 3.55 0.235 [15,] 65 1/8/86 3.5 2.04 0.69 2.41 1.4 0.58 6.00 1.75 -0.059 [16,] 68 4/2/86 6.4 4.43 0.55 2.49 2.44 0.69 9.25 3.20 0.135 [17,] 71 8/5/86 1.6 0.28 0.34 1.98 0.09 0.18 9.14 0.80 0.000 [18,] 72 10/7/86 0.9 0.18 0.28 1.95 0.05 0.20 4.50 0.45 0.000 [19,] 73 12/8/86 6.3 3.68 0.29 2.39 1.06 0.58 10.79 3.15 0.183 [20,] 74 12/31/86 6.5 4.81 0.53 2.54 2.56 0.74 8.78 6.40 -1.043 [21,] 75 1/13/87 6.2 4.49 0.32 2.43 1.45 0.72 8.56 3.10 0.047 [22,] 78 4/15/87 11.4 4.84 0.6 2.66 4.84 0.42 26.85 5.70 0.053 [23,] 79 6/4/87 4.5 2.25 0.64 2.4 1.45 0.50 9.00 2.25 0.111 [24,] 80 6/17/87 3.3 1.33 0.37 2.22 0.5 0.40 8.19 1.65 -0.019 [25,] 81 7/8/87 4 1.81 0.71 2.35 1.28 0.45 8.84 2.00 0.122 [26,] 82 8/20/87 0.7 0.2 0.2 1.94 0.04 0.29 2.45 0.35 -0.200 307 [27,] 83 10/7/87 2.2 0.76 0.28 2.16 0.21 0.35 6.37 1.10 0.105 [28,] 84 11/18/87 4.1 2.12 0.92 2.45 1.94 0.52 7.93 2.05 0.167 [29,] 85 1/13/88 3.5 1.56 0.69 2.46 1.08 0.45 7.85 1.75 0.095 [30,] 87 4/6/88 3.4 1.59 1.03 2.44 1.63 0.47 7.27 1.70 0.168 [31,] 134 8/26/92 4.1 1.43 0.45 1.66 0.64 0.35 11.76 2.05 0.147 [32,] 135 9/22/92 4.2 1.48 0.47 1.68 0.72 0.35 11.92 2.10 0.122 [33,] 136 10/6/92 4.1 1.32 0.33 1.62 0.44 0.32 12.73 2.05 0.159 [34,] 137 10/7/92 4.1 1.35 0.33 1.62 0.45 0.33 12.45 2.05 0.126 [35,] 142 3/24/93 18.6 20.6 2.59 2.54 53.4 1.11 16.79 9.30 0.071 [36,] 143 4/1/93 17.5 14.8 1.78 2.48 26.4 0.85 20.69 8.75 0.020 [37,] 144 4/14/93 15.5 8.14 1.02 2.16 8.35 0.53 29.51 7.48 -0.139 [38,] 149 10/15/93 4.5 1.51 0.36 1.73 0.53 0.34 13.41 2.53 -0.068 Ave 5.18 3.285 0.61 2.25 3.55 0.51 9.80 2.68 0.020 bfile11a [1,] 42 3/5/85 18 11.3 0.59 0.8 6.67 0.63 28.67 9.00 -0.190 [2,] 43 4/18/85 17.5 9.68 0.46 0.68 4.5 0.55 31.64 8.75 -0.122 [3,] 44 5/14/85 17.5 11.1 0.3 0.59 3.34 0.63 27.59 8.75 -0.147 [4,] 45 6/11/85 17.2 9.67 0.24 0.5 2.36 0.56 30.59 8.60 -0.183 [5,] 53 11/13/85 18.5 15.3 0.43 0.82 6.57 0.83 22.37 9.25 -0.201 [6,] 56 4/24/86 15 9.91 0.84 0.86 8.34 0.66 22.70 7.50 -0.150 [7,] 63 3/3/87 32 25.3 0.7 1.21 17.8 0.79 40.47 16.00 0.098 [8,] 69 11/18/87 16.5 12.9 0.34 0.7 4.42 0.78 21.10 8.25 -0.080 [9,] 70 2/24/88 16 11.5 0.5 0.72 5.75 0.72 22.26 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78.50 0.040 [5,] 567 6/25/87 156 260 1 1.83 261 1.67 93.60 78.00 0.024 [6,] 569 10/9/87 157 266 0.98 1.81 261 1.69 92.67 78.50 0.031 [7,] 570 11/23/87 158 314 1.24 2.23 388 1.99 79.50 79.00 -0.004 [8,] 572 2/26/88 163 440 1.92 3.09 844 2.70 60.38 81.50 0.023 [ 9,] 573 4/6/88 156 387 1.44 2.25 442 2.48 62.88 78.00 0.010 [10,] 574 7/12/88 155 249 0.65 1.52 162 1.61 96.49 77.50 0.015 [11,] 577 11/15/88 152 274 0.82 1.77 224 1.80 84.32 76.00 0.003 [12,] 578 1/3/89 154 288 0.94 1.87 271 1.87 82.35 77.00 -0.016 [13,] 579 2/14/89 156 329 1.2 2.22 395 2.11 73.97 78.00 -0.006 [14,] 580 5/1/89 157 341 1.14 2.22 389 2.17 72.28 89.75 -0.148 [15,] 581 6/13/89 161 397 1.46 2.65 579 2.47 65.29 80.50 0.020 [16,] 582 7/25/89 158 360 1.24 2.37 447 2.28 69.34 79.00 0.003 [17,] 583 9/8/89 155 239 0.64 1.44 154 1.54 100.52 77.50 0.011 [18,] 584 10/5/89 156 287 0.91 1.82 261 1.84 84.79 78.00 0.000 [19,] 587 2/21/90 165 362 1.67 2.68 604 2.19 75.21 82.50 0.014 [20,] 588 5/23/90 162 420 1.69 2.92 708 2.59 62.49 81.00 -0.011 [21,] 589 7/5/90 159 254 0.85 1.67 216 1.60 99.53 79.50 -0.017 [22,] 592 11/16/90 165 376 1.76 2.79 661 2.28 72.41 82.50 0.010 [23,] 593 2/5/91 163 416 1.86 3.07 772 2.55 63.87 81.50 0.019 [24,] 594 3/12/91 187 444 1.76 3.02 782 2.37 78.76 92.75 0.109 [25,] 595 4/12/91 164 368 1.6 2.71 590 2.24 73.09 82.00 0.005 [26,] 596 5/31/91 160 268 0.95 1.86 254 1.68 95.52 80.00 0.060 [27,] 597 7/11/91 196 180 0.72 1.36 129 0.92 213.42 98.00 0.011 [28,] 598 8/16/91 184 137 0.52 1.16 71.7 0.74 247.12 92.00 -0.037 [29,] 599 10/2/91 196 173 0.7 1.33 121 0.88 222.06 98.00 0.000 [30,] 600 11/15/91 196 197 0.73 1.46 144 1.01 195.01 98.00 -0.006 [31,] 601 1/8/92 197 390 1.4 2.59 545 1.98 99.51 98.50 -0.022 [32,] 602 3/5/92 200 376 1.37 2.5 516 1.88 106.38 100.00 -0.004 [33,] 603 4/7/92 199 431 1.6 2.86 688 2.17 91.88 99.50 0.018 [34,] 604 5/20/92 199 376 1.27 2.4 479 1.89 105.32 99.50 -0.003 [35,] 605 7/8/92 199 294 0.99 1.9 291 1.48 134.70 97.50 0.043 [36,] 606 8/18/92 199 280 1.07 1.91 297 1.41 141.43 100.00 0.008 [37,] 607 10/6/92 199 227 0.89 1.62 201 1.14 174.45 99.50 0.001 [38,] 608 11/19/92 199 365 1.37 2.47 501 1.83 108.50 99.50 0.002 [39,] 612 7/13/93 199 254 0.9 1.7 228 1.28 155.91 99.50 -0.009 [40,] 613 8/17/93 202 254 0.76 1.56 192 1.26 160.65 100.50 -0.102 [41,] 614 10/14/93 200 445 3.31 2.6 582 2.23 89.89 100.00 -0.060 [42,] 616 5/17/94 201 444 1.6 2.82 709 2.21 90.99 97.50 -0.086 [43,] 617 8/31/94 198 302 1.16 1.95 350 1.53 129.81 99.00 0.121 Ave 175 329.7 1.28 2.22 437.60 1.90 104.01 87.91 0.003 bfile14 a [1,] 272 6/12/85 64 61.3 0.48 1.83 29.3 0.96 66.82 31.00 -0.189 [2,] 273 7/23/85 63.5 75.5 0.74 2.01 55.9 1.19 53.41 31.75 -0.096 [3,] 274 8/21/85 60 49.8 0.27 1.66 13.5 0.83 72.29 30.00 -0.150 [4,] 276 11/14/85 62 56.2 0.4 1.76 22.2 0.91 68.40 31.00 -0.143 [5,] 277 2/20/86 67 134 1.93 2.67 258 2.00 33.50 33.50 -0.063 309 [6,] 278 3/25/86 64 65.1 0.91 1.99 59.5 1.02 62.92 32.00 0.085 [7,] 283 1/21/87 65 103 1.28 2.29 132 1.58 41.02 32.50 -0.086 [8,] 284 2/26/87 63 74.8 0.75 1.91 56.2 1.19 53.06 31.50 -0.120 [9,] 285 4/13/87 65 80.5 1.11 2.08 89.6 1.24 52.48 33.00 -0.118 [10,] 291 2/26/88 66 92.3 0.73 2 67.4 1.40 47.19 42.68 0.115 [11,] 292 4/6/88 64 87.8 0.66 1.96 58.2 1.37 46.65 32.00 -0.018 [12,] 293 4/14/88 63 91.6 0.68 2.04 67.7 1.45 43.33 31.50 -0.030 [13,] 299 3/21/89 67 119 1.28 2.32 152 1.78 37.72 33.50 -0.167 [14,] 300 5/12/89 66 135 1.33 2.45 180 2.05 32.27 33.00 -0.134 [15,] 312 11/14/90 64 62.5 0.95 1.88 59.1 0.98 65.54 32.00 0.139 [16,] 313 1/10/91 66.5 77.1 1.23 2.04 94.8 1.16 57.36 33.25 0.071 [17,] 314 3/6/91 63.5 59.9 0.89 1.86 53.5 0.94 67.32 31.50 0.173 [18,] 315 4/18/91 65 66 1.2 1.96 78.3 1.02 64.02 32.75 -0.087 [19,] 322 1/7/92 59 44.7 0.8 1.72 36 0.76 77.87 29.50 0.137 [20,] 323 2/20/92 62 52.2 0.98 1.81 51 0.84 73.64 31.00 0.114 [21,] 324 4/1/92 64 65.9 1.2 1.96 78.9 1.03 62.15 32.00 0.101 [22,] 325 5/20/92 60 47.8 0.82 1.74 39 0.80 75.31 30.00 0.090 [23,] 329 11/18/92 61 51.8 0.74 1.74 38.6 0.85 71.83 30.50 0.056 [24,] 330 1/6/93 65 84.2 1.27 2.05 107 1.30 50.18 32.50 -0.032 [25,] 332 5/25/93 66 75.3 0.85 1.89 63.8 1.14 57.85 33.00 0.041 [26,] 333 7/15/93 66 67.1 0.58 1.76 38.9 1.02 64.92 33.00 -0.002 Ave 63.9 76.17 0.93 1.98 76.17 1.18 57.66 32.31 -0.012 bfile15a [1,] 527 6/29/88 47.5 49.8 1.09 2.04 54.5 1.05 45.31 23.75 -0.075 [2,] 529 8/9/88 47 43 0.98 2 44 0.91 51.37 22.50 -0.056 [3,] 532 10/28/88 47 48.2 0.87 1.97 41.9 1.03 45.83 23.50 -0.040 [4,l 534 12/27/88 48 65.4 1.07 2.12 70.2 1.36 35.23 24.00 -0.088 [5,] 535 1/30/89 51 65 1.11 2.13 72 1.27 40.02 25.50 -0.104 [6,] 536 2/27/89 52.5 85.6 1.27 2.29 109 1.63 32.20 26.50 -0.126 [7,] 537 3/29/89 51 104 1.46 2.46 152 2.04 25.01 25.52 -0.123 [8,] 538 4/27/89 48.5 65 1.27 2.19 82.6 1.34 36.19 24.25 -0.046 [9,] 539 5/30/89 50 86.4 1.56 2.4 135 1.73 28.94 25.00 -0.056 [10,] 540 6/26/89 51 86.6 1.39 2.34 120 1.70 30.03 25.50 -0.023 [11,] 541 7/26/89 53 73.5 0.84 2.09 61.9 1.39 38.22 26.50 -0.037 [12,] 543 9/26/89 51 135 1.84 2.74 248 2.65 19.27 25.50 -0.046 [13,] 550 10/29/90 49 79.6 1.24 2.26 99 1.62 30.16 24.50 -0.129 [14,] 551 11/26/90 50 60.4 1 2.09 60.3 1.21 41.39 25.00 -0.139 [15,] 552 1/28/91 50.5 81.5 1.67 2.38 136 1.61 31.29 25.25 -0.104 [16,] 553 2/26/91 50 66.2 1.3 2.21 86.4 1.32 37.76 25.00 -0.154 [17,] 554 3/26/91 51 116 1.67 2.6 194 2.27 22.42 25.50 -0.071 [18,] 555 5/28/91 51 84.1 0.59 2.02 49.4 1.65 30.93 25.50 -0.042 [19,] 561 1/29/92 48.5 54.1 0.96 2.03 52.1 1.12 43.48 24.25 -0.187 [20,] 562 2/24/92 49.5 61.1 1.17 2.12 71.3 1.23 40.10 24.75 -0.130 [21,] 566 9/28/92 51.5 55 1.51 2.2 83.2 1.07 48.22 25.75 -0.039 [22,] 567 11/30/92 55 54 1.45 2.18 81.5 0.98 56.02 27.50 -0.059 [23,] 568 1/28/93 53 62.8 1.54 2.23 96.9 1.18 44.73 26.50 -0.092 [24,] 569 4/28/93 59 102 2.01 2.63 205 1.73 34.13 29.50 -0.056 [25,] 570 6/28/93 47.5 63.5 0.78 2.07 49.7 1.34 35.53 23.75 0.200 [26,] 576 5/31/94 45.6 60.2 1.17 2.13 70.4 1.32 34.54 22.80 0.188 [27,] 577 8/30/94 45 64.2 1.02 2.11 65.2 1.43 31.54 22.50 0.088 [28,] 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0.411 [15,] 849 3/26/90 47 33.3 1.39 1.74 46.4 0.71 66.34 23.50 0.221 [16,] 850 5/14/90 48 31.7 1.39 1.71 44 0.66 72.68 24.00 0.258 [17,] 851 6/18/90 42 22.5 1.01 1.41 22.7 0.54 78.40 21.00 0.198 [18,] 853 8/6/90 58 62.1 1.84 1.94 114 1.07 54.17 29.00 0.490 [19,] 854 10/1/90 34 26.9 0.42 1.34 11.3 0.79 42.97 17.00 0.226 [20,] 855 11/15/90 38.5 32.5 0.71 1.47 23.2 0.84 45.61 19.25 0.292 [21,] 856 1/10/91 48 44.2 1.36 1.69 60.2 0.92 52.13 24.00 0.369 [22,] 857 3/7/91 44.5 40.4 1.26 1.63 51 0.91 49.02 22.25 0.327 [23,] 858 4/3/91 42 35.7 1.16 1.57 41.5 0.85 49.41 21.00 0.327 [24,] 859 4/18/91 40.5 34.4 1.08 1.54 37.3 0.85 47.68 20.25 0.303 [25,] 860 6/3/91 36.5 25.1 0.68 1.03 17.2 0.69 53.08 18.25 -0.224 [26,] 861 6/28/91 33 22.3 0.49 1.11 10.9 0.68 48.83 16.50 0.150 [27,] 862 7/8/91 50 48.5 1.57 1.73 76 0.97 51.55 25.00 0.357 [28,] 863 8/19/91 33 19.1 0.24 0.99 4.58 0.58 57.02 16.50 0.028 [29,] 864 10/8/91 32 20.8 0.34 1.19 7.1 0.65 49.23 16.00 -0.040 [30,] 865 11/15/91 32 23.6 0.44 1.26 10.3 0.74 43.39 16.00 0.003 [31,] 866 1/8/92 41 41.2 0.66 1.43 27.3 1.00 40.80 20.50 0.236 [32,] 867 2/20/92 42 35.3 1.02 1.49 36.1 0.84 49.97 21.00 0.243 [33,] 868 3/27/92 53.5 79.6 2.68 2.13 213 1.49 35.96 26.75 0.305 [34,] 869 4/2/92 42 34.3 0.89 1.45 30.6 0.82 51.43 21.00 0.183 [35,] 870 5/21/92 39.5 30.5 0.83 1.39 25.2 0.77 51.16 19.75 0.205 [36,] 871 7/6/92 41.5 31.1 1.04 1.43 32.2 0.75 55.38 20.75 0.255 [37,] 872 8/17/92 38 23.6 0.92 1.35 21.8 0.62 61.19 19.00 0.178 [38,] 873 10/7/92 31.5 16.2 0.64 1.23 10.4 0.51 61.25 15.75 0.140 [39,] 874 11/25/92 43 30.8 1.53 1.52 47.2 0.72 60.03 21.50 -0.115 [40,] 875 1/4/93 50.5 29.2 0.92 1.4 27.6 0.58 87.34 25.25 0.302 [41,] 876 2/23/93 51.7 47 1.53 1.64 71.9 0.91 56.87 26.20 -0.136 [42,] 877 4/9/93 53.8 42.6 1.29 0.56 54.9 0.79 67.94 25.95 -0.129 [43,] 878 5/15/93 55.9 49.7 1.6 0.61 79.6 0.89 62.87 27.95 0.059 [44,] 879 6/10/93 37.5 22.9 1.22 0.77 29.2 0.61 61.41 18.75 -0.116 [45,] 880 6/29/93 32.5 16.8 0.94 0.99 15.8 0.52 62.87 14.35 0.177 [46,] 881 8/11/93 32 16.2 0.76 0.97 12.3 0.51 63.21 16.00 -0.025 [47,] 882 10/4/93 27 16.7 0.58 0.96 9.61 0.62 43.65 13.50 0.212 [48,] 883 10/18/93 26.8 16.9 0.56 0.96 9.44 0.63 42.50 13.40 0.216 [49,] 884 11/29/93 51 39.2 2.14 1.29 75 0.77 66.35 25.50 -0.072 [50,] 885 12/3/93 36 25.7 1.1 1.07 28.2 0.71 50.43 18.00 -0.221 [51,] 886 1/13/94 50.3 40.5 1.77 1.27 71.7 0.81 62.47 25.15 -0.100 311 [52,] 887 3/7/94 44.5 42 1.94 1.31 81.3 0.94 47.15 19.88 -0.034 [53,] 889 4/14/94 58 69.7 1.38 1.37 96.3 1.20 48.26 28.00 0.022 [54,] 890 5/3/94 35 27.2 1.34 1.1 36.4 0.78 45.04 17.50 -0.006 [55,] 891 6/14/94 33.5 19.8 0.99 1 19.6 0.59 56.68 15.88 0.020 [56,] 892 8/1/94 36.4 29.8 1.59 1.16 47.3 0.82 44.46 18.20 0.030 Ave 39.6 30.01 1.10 1.39 37.41 0.73 56.80 20.38 0.090 bfile17a [1,] 213 10/20/87 15.8 5.2 0.7 2.35 3.66 0.33 48.01 7.90 -0.195 [2,] 224 5/3/89 44 88.3 2.3 4.17 203 2.01 21.93 22.00 0.218 [3,] 226 6/14/89 38.5 45.9 1.65 3.14 75.9 1.19 32.29 19.25 -0.259 [4,] 227 7/27/89 36.5 32.6 1.52 2.98 49.4 0.89 40.87 18.25 -0.272 [5,] 228 10/3/89 38 58.7 2.01 3.44 118 1.54 24.60 19.00 -0.162 [6,] 229 11/15/89 28 31.9 0.9 2.79 28.7 1.14 24.58 14.00 0.055 [7,] 230 1/5/90 29 39.9 1.51 3.04 60.1 1.38 21.08 14.25 -0.014 [8,] 231 2/14/90 29 32.7 1.5 2.96 49.2 1.13 25.72 14.50 -0.102 [9,] 232 3/29/90 29 27.4 1.18 2.83 32.3 0.94 30.69 14.50 0.087 [10,] 233 5/16/90 29 32.1 1.21 2.88 39 1.11 26.20 14.50 0.126 [11,] 234 6/20/90 29 31.9 0.99 2.83 31.7 1.10 26.36 14.50 0.165 [12,] 239 3/12/91 29 27.8 0.77 2.7 21.3 0.96 30.25 14.50 -0.069 [13,] 240 4/17/91 28 33.2 1.34 2.97 44.5 1.19 23.61 14.00 -0.226 [14,] 246 1/7/92 29 28.5 1.11 2.83 31.7 0.98 29.51 14.50 0.210 [15,] 247 2/19/92 29.5 27.1 1.19 2.84 32.2 0.92 32.11 14.75 0.220 [16,] 248 4/1/92 29 32.4 1.44 2.96 47 1.12 25.96 14.50 0.192 [17,] 249 5/20/92 28 13.6 0.77 2.54 10.5 0.49 57.65 14.00 0.126 [18,] 251 8/20/92 29 18.5 1 2.66 18.5 0.64 45.46 14.50 0.250 [19,] 252 10/6/92 28 22.6 0.78 2.65 17.7 0.81 34.69 14.00 -0.060 [20,] 253 11/18/92 28.5 23 0.93 2.72 21.6 0.81 35.32 14.25 -0.034 [21,] 254 1/7/93 29.5 37.4 1.34 3.01 50.4 1.27 23.27 14.75 -0.021 [22,] 255 3/1/93 29.2 30.2 1.29 2.91 39 1.03 28.23 14.60 -0.053 [23,] 256 4/13/93 29.6 51.7 1.95 3.41 101 1.75 16.95 14.80 0.026 Ave 30.1 33.59 1.28 2.94 48.97 1.07 30.67 15.03 0.009 bfile18 a [1,] 133 6/13/85 9 1.81 1.22 0.97 2.2 0.20 44.75 4.50 -0.171 [2,] 136 10/9/85 12 4.87 1.55 1.16 7.56 0.41 29.57 6.00 -0.158 [3,] 140 4/1/86 18.9 8.8 1.84 1.32 16.2 0.47 40.59 11.70 0.018 [4,] 141 5/13/86 10.5 2.95 1.49 1.06 4.41 0.28 37.37 5.25 0.100 [5,] 142 6/23/86 7.3 1.83 0.28 0.81 0.51 0.25 29.12 3.65 -0.069 [6,] 143 6/24/86 7.2 1.97 0.23 0.8 0.45 0.27 26.31 3.60 -0.040 [7,] 144 8/12/86 7 1.99 0.29 0.82 0.57 0.28 24.62 3.50 0.149 [8,] 145 10/7/86 3.1 0.8 0.31 0.77 0.25 0.26 12.01 1.55 0.044 [9,] 146 12/9/86 15.7 7.87 1.82 1.3 14.3 0.50 31.32 7.85 -0.161 [10,] 150 6/8/87 15.5 5.98 1.13 1.15 6.78 0.39 40.18 7.75 0.059 [11,] 151 7/13/87 13 4.06 0.6 0.99 2.44 0.31 41.63 6.50 0.099 [12,] 152 8/24/87 4.2 1.28 0.22 0.76 0.28 0.30 13.78 2.10 0.031 [13,] 153 10/19/87 10 3.27 0.5 0.92 1.65 0.33 30.58 5.00 -0.162 [14,] 154 11/30/87 25.5 17.1 1.53 1.47 26.2 0.67 38.03 12.75 -0.145 [15,] 155 1/27/88 25.5 12.8 1.47 1.39 18.8 0.50 50.80 12.75 -0.395 [16,] 156 2/29/88 18 9.53 1.42 1.3 13.5 0.53 34.00 9.00 -0.013 [17,] 157 4/11/88 22.7 11.8 1.64 1.38 19.4 0.52 43.67 11.35 0.222 [18,] 158 6/1/88 12 3.71 0.59 0.96 2.19 0.31 38.81 6.00 0.024 [19,] 159 11/15/88 9.2 4.5 0.44 0.94 1.99 0.49 18.81 4.60 -0.252 312 [20,] 160 1/4/89 15 7.2 0.83 1.11 5.94 0.48 31.25 7.50 0.219 [21,] 161 2/14/89 17 9.24 1.03 1.2 9.52 0.54 31.28 8.25 0.224 [22,] 165 5/3/89 16 7.98 0.78 1.14 6.25 0.50 32.08 8.00 0.323 [23,] 166 7/26/89 15 6.96 0.67 1.06 4.68 0.46 32.33 7.50 0.241 [24,] 168 11/15/89 20.3 12.9 1.09 1.29 14 0.64 31.94 10.15 0.019 [25,] 169 1/4/90 21.5 18.9 1.16 1.44 21.9 0.88 24.46 10.75 0.042 [26,] 170 2/14/90 22 17.3 1.25 1.43 21.7 0.79 27.98 11.00 0.010 [27,] 171 3/28/90 21 12.5 1.08 1.29 13.5 0.60 35.28 10.50 0.019 [28,] 172 5/15/90 21.5 22 1.31 1.52 28.9 1.02 21.01 10.75 0.142 [29,] 173 8/7/90 18.2 9.04 0.49 1.06 4.46 0.50 36.64 9.10 -0.146 [30,] 175 11/13/90 20.5 11.8 0.59 1.16 6.95 0.58 35.61 10.25 0.169 [31,] 176 1/8/91 22 19.4 0.97 1.41 18.9 0.88 24.95 10.55 0.183 [32,] 177 3/11/91 19 13.5 0.66 1.21 8.98 0.71 26.74 9.50 0.122 [33,] 178 4/16/91 22.5 20.8 1.01 1.43 21.1 0.92 24.34 11.25 0.145 [34,] 179 7/9/91 20.5 13.3 0.77 1.22 10.3 0.65 31.60 10.25 0.215 [35,] 180 8/20/91 15.5 5.4 0.2 0.87 1.1 0.35 44.49 7.75 -0.219 [36,] 182 11/12/91 14.2 9.19 0.41 1.04 3.78 0.65 21.94 7.10 -0.183 [37,] 183 1/6/92 21.5 20.8 1.15 1.48 23.9 0.97 22.22 10.75 0.139 [38,] 184 2/18/92 21.3 16.5 0.81 1.31 13.3 0.77 27.50 10.90 0.234 [39,] 185 3/31/92 22.5 18.7 1.08 1.42 20.2 0.83 27.07 11.25 0.165 [40,] 186 5/19/92 19 11.3 0.62 1.16 7.03 0.59 31.95 9.50 -0.159 [41,] 187 8/19/92 22 19.5 1.32 1.32 13.9 0.89 24.82 11.00 0.188 [42,] 190 1/8/93 20.5 40.4 1.78 1.93 72.1 1.97 10.40 10.25 0.304 [43,] 191 3/1/93 17.7 18.7 1.24 1.45 23.1 1.06 16.75 8.85 -0.249 [44,] 192 4/13/93 18.5 21 1.44 1.54 30.2 1.14 16.30 9.25 0.204 [45,] 193 3/17/93 18.5 22.6 1.52 1.6 34.4 1.22 15.14 9.25 -0.234 [46,] 194 7/8/93 12 10.1 0.22 0.97 2.23 0.84 14.26 6.00 0.002 [47,] 195 8/16/93 14.8 8.24 0.15 0.89 1.21 0.56 26.58 7.40 -0.146 [48,] 196 10/5/93 11.1 10 0.1 0.87 1 0.90 12.32 5.59 -0.018 [49,] 197 12/2/93 17 12 0.72 1.2 8.71 0.71 24.08 8.25 -0.268 [50,] 198 1/25/94 16.9 19 1.23 1.45 23.4 1.12 15.03 8.45 -0.240 [51,] 199 3/9/94 20.7 25 1.5 1.61 37.6 1.21 17.14 10.35 -0.197 [52,] 200 4/25/94 18 16.3 1.32 1.41 21.3 0.91 19.88 9.00 0.218 [53,] 201 5/17/94 13.5 13.5 0.8 1.24 10.8 1.00 13.50 6.75 0.154 [54,] 202 6/20/94 14.7 7.97 0.35 1 2.76 0.54 27.11 7.35 -0.082 Ave 16.4 11.78 0.93 1.20 12.76 0.66 27.81 8.25 0.013 bfile19a [1,] 321 11/15/84 20.5 16.4 0.33 1.43 5.42 0.80 25.63 10.25 0.350 [2,] 322 12/17/84 23 19.3 0.52 1.53 10.1 0.84 27.41 11.50 -0.451 [3,] 323 2/7/85 26 25.3 1.23 1.79 31.1 0.97 26.72 14.00 -0.127 [4,] 325 4/23/85 21 17 0.31 1.41 5.31 0.81 25.94 10.50 0.372 [5,] 326 6/13/85 19 15.4 0.2 1.34 3.03 0.81 23.44 9.50 0.141 [6,] 328 8/26/85 23.5 20.9 0.56 1.54 11.8 0.89 26.42 11.75 -0.079 [7,] 329 10/9/85 23 18.3 0.5 1.49 9.09 0.80 28.91 11.50 -0.156 [8,] 330 12/2/85 24 48.7 2.44 2.46 119 2.03 11.83 12.00 0.021 [9,] 331 1/14/86 21 14.6 0.45 1.46 6.64 0.70 30.21 10.50 -0.304 [10,] 333 4/1/86 23 21 0.56 1.54 11.8 0.91 25.19 11.50 -0.118 [11,] 338 12/9/86 27 33.9 1.74 2.03 58.9 1.26 21.50 13.50 -0.188 [12,] 339 1/20/87 28 37.9 1.98 2.18 74.9 1.35 20.69 14.00 -0.285 [13,] 340 3/9/87 28 26.1 1.11 1.78 29.1 0.93 30.04 14.00 -0.384 [14,] 341 4/13/87 27 22 0.84 1.66 18.4 0.81 33.14 13.50 -0.364 [15,] 342 6/8/87 22 21.5 0.6 1.58 13 0.98 22.51 11.00 -0.151 313 [16,] 343 7/13/87 17.5 8.52 0.58 1.41 5.21 0.49 35.94 8.75 0.059 [17,] 344 8/24/87 17.6 10.7 0.18 1.27 1.88 0.61 28.95 8.80 0.035 [18,] 345 10/19/87 16 9 0.31 1.32 2.81 0.56 28.44 8.00 0.110 [19,] 346 11/30/87 28 31.1 1.33 1.9 41.3 1.11 25.21 14.00 -0.311 [20,] 347 1/27/88 27 21 0.98 1.7 20.6 0.78 34.71 13.50 0.060 [21,] 348 2/29/88 27 21.8 0.8 1.65 17.5 0.81 33.44 13.50 -0.311 [22,] 349 4/11/88 24.5 33.2 1.24 1.91 41.3 1.36 18.08 12.25 -0.055 [23,] 350 6/1/88 20.2 17.4 0.28 1.39 4.92 0.86 23.45 9.20 -0.098 [24,] 351 7/11/88 21 12.1 0.13 1.26 1.62 0.58 36.45 10.50 -0.186 [25,] 352 8/23/88 20 16.5 0.17 1.32 2.82 0.83 24.24 10.00 -0.163 [26,] 353 11/15/88 23 15.7 0.25 1.36 3.97 0.68 33.69 11.50 -0.170 [27,] 354 1/4/89 22 19.6 0.45 1.49 8.8 0.89 24.69 14.60 -0.105 [28,] 355 2/14/89 21.5 19.6 0.42 1.48 8.28 0.91 23.58 10.75 -0.108 [29,] 356 3/21/89 25 48.7 2.42 2.44 118 1.95 12.83 12.50 0.002 [30,] 358 5/2/89 23 15.6 0.44 1.46 6.94 0.68 33.91 11.50 -0.365 [31,] 359 7/26/89 19.5 11.3 0.44 1.4 4.99 0.58 33.65 9.75 -0.100 [32,] 360 10/2/89 27 31.3 1.66 1.98 52 1.16 23.29 13.50 -0.225 [33,] 361 11/14/89 20 14 0.81 1.55 11.4 0.70 28.57 10.00 0.115 [34,] 362 1/3/90 25.5 21.9 1.25 1.75 27.4 0.86 29.69 12.75 0.186 [35,] 363 2/13/90 22.5 17.7 0.94 1.64 16.7 0.79 28.60 11.25 0.025 [36,] 364 3/27/90 22.8 19 0.88 1.65 16.7 0.83 27.36 11.40 -0.126 [37,] 365 5/15/90 27 25.3 1.17 1.78 29.7 0.94 28.81 13.50 0.312 [38,] 366 8/7/90 20 13.4 0.34 1.38 4.51 0.67 29.85 10.00 -0.218 [39,] 371 4/16/91 25.2 20 1.28 1.75 25.5 0.79 31.75 12.60 0.114 [40,] 372 7/9/91 18 9.91 0.45 1.38 4.47 0.55 32.69 9.00 0.042 [41,] 373 10/1/91 24.5 12.8 0.38 1.39 4.82 0.52 46.89 12.25 -0.079 [42,] 374 11/12/91 22 19.7 0.24 1.39 4.72 0.90 24.57 11.00 -0.327 [43,] 375 1/6/92 26.5 23.1 1.42 1.82 32.8 0.87 30.40 13.25 0.130 [44,] 376 2/18/92 24.5 17 0.9 1.61 15.3 0.69 35.31 11.88 0.166 [45,] 377 3/31/92 25.5 23.2 1.32 1.82 30.7 0.91 28.03 12.75 0.126 [46,] 379 8/19/92 28.5 40.3 1.95 2.18 78.5 1.41 20.16 14.25 0.249 [47,] 380 10/5/92 23.5 19.1 0.84 1.62 16.1 0.81 28.91 12.00 0.396 [48,] 383 3/2/93 26.2 18.2 0.88 1.62 16.1 0.69 37.72 13.10 -0.251 [49,] 384 4/14/93 26 21.6 1.32 1.76 28.6 0.83 31.30 13.00 0.122 [50,] 385 5/17/93 23.5 21.2 0.73 1.61 15.4 0.90 26.05 11.75 0.095 [51,] 386 7/9/93 20.1 13.1 0.37 1.39 4.84 0.65 30.84 10.05 0.386 [52,] 387 8/16/93 22.5 10.7 0.33 1.35 3.59 0.48 47.31 11.25 0.342 [53,] 388 10/5/93 20.4 10.1 0.24 1.31 2.38 0.50 41.20 9.95 0.264 [54,] 389 12/2/93 22.5 17.7 0.36 1.43 6.39 0.79 28.60 11.25 0.092 [55,] 390 1/25/94 23 21.8 0.78 1.63 17.1 0.95 24.27 11.55 0.052 [56,] 391 3/9/94 24.7 29.2 1.73 1.98 50.6 1.18 20.89 12.35 0.119 [57,] 392 4/25/94 27.3 17.5 0.97 1.64 16.9 0.64 42.59 13.65 0.322 [58,] 393 8/3/94 27 17 1 1.63 17 0.63 42.88 13.50 0.256 [59,] 394 10/3/94 20.9 13.3 0.33 1.38 4.44 0.64 32.84 10.45 0.442 [60,] 395 11/22/94 24 27.7 1.32 1.85 36.5 1.15 20.79 12.00 0.034 [61,] 396 1/18/95 23.6 14.7 0.54 1.48 7.92 0.62 37.89 11.80 0.461 [62,] 397 3/15/95 28.7 21 0.96 1.68 20.2 0.73 39.22 12.73 0.281 [63,] 398 5/9/95 23.6 16.5 0.53 1.49 8.7 0.70 33.76 12.20 0.249 [64,] 399 7/10/95 23 15.4 0.41 1.43 6.28 0.67 34.35 11.53 0.188 Ave 23.4 20.38 0.82 1.61 20.82 0.85 29.35 11.74 0.014 bfile20a [1,] 504 8/7/89 94 155 1.36 2.43 211 1.65 57.01 47.00 0.029 314 [2,] 507 1/17/90 97 182 1.66 2.72 302 1.88 51.70 48.50 -0.003 [3,] 508 3/22/90 95 170 1.57 2.6 267 1.79 53.09 47.50 -0.021 [4,] 509 5/9/90 94 154 1.28 2.35 197 1.64 57.38 47.00 0.007 [5,] 510 6/19/90 95 135 1.09 2.22 147 1.42 66.85 47.50 0.031 [6,] 511 8/13/90 94 104 0.66 1.9 68.1 1.11 84.96 47.00 -0.005 [7,] 512 10/1/90 93 109 0.74 1.97 80.7 1.17 79.35 46.50 -0.037 [8,] 513 11/29/90 95 145 1.14 2.29 166 1.53 62.24 47.50 0.021 [9,] 514 1/31/91 97 195 1.99 2.88 388 2.01 48.25 48.50 0.003 [10,] 516 5/16/91 98 168 1.56 2.55 261 1.71 57.17 49.00 0.004 [11,] 517 7/9/91 95 143 1.32 2.35 188 1.51 63.11 47.50 -0.015 [12,] 518 9/3/91 91 110 0.86 2 94.7 1.21 75.28 45.50 -0.006 [13,] 519 9/18/91 91 95.6 0.78 1.91 74.1 1.05 86.62 45.50 0.035 [14,] 520 10/25/91 91 101 0.66 1.91 66.9 1.11 81.99 45.50 0.050 [15,] 521 12/10/91 98 146 1.29 2.35 189 1.49 65.78 49.00 -0.017 [16,] 522 2/3/92 94 136 1.08 2.21 147 1.45 64.97 47.00 -0.046 [17,] 523 4/14/92 98 148 1.3 2.38 194 1.51 64.89 49.00 -0.002 [18,] 524 6/3/92 98 168 1.54 2.61 259 1.71 57.17 49.00 0.011 [19,] 525 7/29/92 97 135 1.05 2.19 142 1.39 69.70 48.50 0.004 [20,] 526 10/14/92 94 104 0.71 1.92 73.5 1.11 84.96 47.00 -0.085 [21,] 527 12/8/92 96 148 1.34 2.4 198 1.54 62.27 48.00 -0.076 [22,] 528 2/8/93 96 163 1.49 2.53 242 1.70 56.54 48.00 0.023 [23,] 530 6/2/93 97 148 1.33 2.37 196 1.53 63.57 48.50 -0.013 [24,] 531 7/12/93 96 122 0.83 2.04 102 1.27 75.54 48.00 -0.031 [25,] 533 10/14/93 94 111 0.76 2.02 84 1.18 79.60 47.00 -0.074 [26,] 534 12/2/93 99 213 1.98 3.03 423 2.15 46.01 49.50 -0.002 [27,] 535 2/7/94 99 213 2.06 3.01 438 2.15 46.01 49.50 -0.012 [28,] 536 4/12/94 100 245 2.53 3.39 620 2.45 40.82 50.00 -0.010 [29,] 537 6/7/94 97 132 1.16 2.23 153 1.36 71.28 48.50 -0.050 [30,] 538 7/8/94 93 115 0.87 2.04 100 1.24 75.21 47.00 -0.038 [31,] 540 8/8/94 97 154 1.32 2.44 204 1.59 61.10 48.50 0.002 [32,] 541 10/11/94 94 115 0.75 1.99 85.6 1.22 76.83 47.00 0.035 [33,] 542 11/30/94 93 112 0.73 2.03 81.2 1.20 77.22 46.50 -0.024 Ave 95.5 145.3 1.24 2.34 195.24 1.52 65.59 47.74 -0.009 bfile21a [1,] 228 8/7/89 25 25.4 2.03 1.54 51.5 1.02 24.61 12.50 -0.299 [2,] 229 10/3/89 41 100 2.33 2.54 233 2.44 16.81 20.50 -0.071 [3,] 231 1/17/90 39 58.2 1.41 1.66 82 1.49 26.13 19.50 -0.154 [4,] 232 3/19/90 28 37.2 2.29 1.68 85.2 1.33 21.08 13.00 -0.139 [5,] 233 5/8/90 27 27.8 1.79 1.45 49.8 1.03 26.22 13.98 -0.300 [6,] 234 6/19/90 34 38.6 1.27 1.46 48.9 1.14 29.95 17.00 -0.333 [7,] 235 7/18/90 24 23.4 1.55 1.36 36.2 0.98 24.62 12.00 -0.210 [8,] 236 8/7/90 22 19.1 1.39 1.28 26.5 0.87 25.34 11.00 -0.199 [9,] 237 10/2/90 21 15 1.24 1.22 18.8 0.71 29.40 10.50 -0.240 [10,] 239 11/29/90 25 25 1.27 1.34 31.7 1.00 25.00 12.50 -0.263 [11,] 240 1/29/91 28 39 1.96 1.65 76.4 1.39 20.10 14.00 -0.222 [12,] 241 3/19/91 36 71.2 2.02 2 144 1.98 18.20 18.00 -0.239 [13,] 242 5/16/91 26 27.2 1.55 1.39 42.4 1.05 24.85 12.75 -0.162 [14,] 243 6/20/91 26 24.2 1.24 1.29 30 0.93 27.93 13.00 -0.223 [15,] 245 9/3/91 39 33.7 0.92 1.31 30.8 0.86 45.13 19.50 -0.047 [16,] 246 10/10/91 39 29.6 0.78 1.27 23.2 0.76 51.39 19.50 -0.083 [17,] 247 12/10/91 34 41.4 1.52 1.55 62.8 1.22 27.92 17.00 -0.227 [18,] 248 2/3/92 25 21.9 1.5 1.32 32.9 0.88 28.54 12.50 -0.150 315 [19,] 249 4/14/92 40 38.7 1.02 1.38 39.5 0.97 41.34 20.00 -0.124 [20,] 250 6/11/92 37 47.9 1.48 1.56 71 1.29 28.58 18.50 -0.190 [21,] 251 7/29/92 27 23.4 1.23 1.27 28.8 0.87 31.15 13.50 -0.077 [22,] 252 10/6/92 26 19.8 1.18 1.26 23.3 0.76 34.14 12.55 -0.146 [23,] 253 11/12/92 25 19.6 1.05 1.24 20.6 0.78 31.89 12.50 -0.102 [24,] 254 12/1/92 37 32.4 1.04 1.32 33.8 0.88 42.25 18.56 -0.186 [25,] 255 2/8/93 34 37.5 1.14 1.4 42.5 1.10 30.83 17.00 -0.210 [26,] 257 5/11/93 39 40.5 1.81 1.63 73.4 1.04 37.56 19.50 -0.105 [27,] 258 6/30/93 33 22.7 1.32 1.33 29.9 0.69 47.97 16.50 0.092 [28,] 259 8/31/93 29 17.2 0.94 1.21 16.3 0.59 48.90 15.00 0.025 [29,] 260 10/14/93 32 21.6 0.82 1.26 17.9 0.68 47.41 16.00 0.104 [30,] 261 11/30/93 33 23.3 1.18 1.29 27.6 0.71 46.74 16.50 -0.042 [31,] 262 2/2/94 39 45.4 1.66 1.6 75.3 1.16 33.50 19.50 -0.039 [32,] 263 4/8/94 42 71.6 1.72 1.89 123 1.70 24.64 21.00 -0.022 [33,] 264 6/10/94 40 35 0.95 1.32 33.4 0.88 45.71 20.00 0.029 [34,] 265 7/18/94 38 30.1 1 1.3 30 0.79 47.97 19.00 -0.001 [35,] 266 10/13/94 36 26.5 0.7 1.22 18.4 0.74 48.91 18.00 0.028 [36,] 267 11/28/94 32 26.4 0.8 1.25 21 0.83 38.79 16.00 -0.152 Ave 32.2 34.38 1.36 1.45 50.88 1.04 33.38 16.06 -0.13 bfile22a [,1] 544 9/25/89 98 212 2.92 5.2 620 2.16 45.30 4.00 -0.969 [,2] 545 11/15/89 78 127 1.86 4.05 236 1.63 47.91 3.00 -0.612 [,3] 546 1/16/90 93 156 2.49 4.5 388 1.68 55.44 4.25 0.584 [,4] 547 3/23/90 90 166 2.1 4.43 348 1.84 48.80 3.00 -0.193 [,5] 548 5/1/90 93 112 2.37 4.12 265 1.20 77.22 3.00 -0.967 [,6] 549 6/18/90 88 116 1.87 3.94 217 1.32 66.76 3.75 -0.320 [,7] 558 8/29/91 65 116 1.3 3.67 151 1.78 36.42 3.00 -0.948 [,8] 562 4/14/92 79 123 1.52 3.83 187 1.56 50.74 4.00 -0.568 [,9] 564 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168 5/16/91 81 110 1.18 2.93 130 1.36 59.65 40.50 -0.020 [18,] 169 6/18/91 90 98 0.71 2.7 71.4 1.09 82.65 45.00 -0.030 [19,] 170 6/20/91 80 117 0.63 2.68 73.7 1.46 54.70 40.00 0.046 [20,] 175 2/5/92 84 92 1.28 2.9 118 1.10 76.70 42.00 -0.086 [21,] 183 2/17/93 81 141 1.86 3.41 262 1.74 46.53 40.50 -0.036 [22,] 185 4/27/93 93 182 2.38 3.91 434 1.96 47.52 46.50 -0.097 [23,] 186 5/11/93 95 140 1.7 3.3 237 1.47 64.46 47.50 -0.066 [24,] 190 12/1/93 79 91.2 1.05 2.75 96 1.15 68.43 39.50 -0.062 [25,] 196 12/1/94 79 97.9 1.01 2.73 99 1.24 63.75 39.50 0.001 Ave 87.2 128.8 1.53 3.25 218.22 1.47 64.09 43.35 -0.026 bfile25a [1,] 400 8/7/89 64 88.2 1.58 2.94 139 1.38 46.44 32.00 -0.164 [2,] 402 11/15/89 66 88 2.33 3.11 205 1.33 49.50 33.00 -0.129 [3,] 406 5/1/90 65 105 1.95 3.08 205 1.62 40.24 32.50 0.118 [4,] 408 6/19/90 63 69.7 2.02 2.92 141 1.11 56.94 31.50 -0.123 [5,] 409 8/14/90 56 46 1.66 2.67 76.3 0.82 68.17 28.00 -0.155 [6,] 410 10/2/90 51 43.8 1.64 2.65 72 0.86 59.38 25.50 -0.031 [7,] 411 12/3/90 64 80.3 1.42 2.83 114 1.25 51.01 32.00 -0.061 [8,] 412 1/31/91 67 120 1.99 3.2 239 1.79 37.41 33.50 0.094 [9,] 413 3/28/91 69 126 2.74 3.44 345 1.83 37.79 34.50 -0.057 [10,] 414 5/14/91 63 76 1.9 2.9 144 1.21 52.22 31.50 0.010 [11,] 415 6/17/91 53 47 1.73 2.7 81.4 0.89 59.77 26.50 0.069 [12,] 417 8/29/91 60 63.4 1.98 2.86 125 1.06 56.78 30.00 -0.121 [13,] 419 12/10/91 64 69 2.06 2.94 142 1.08 59.36 32.00 -0.003 [14,] 421 4/17/92 63 65.4 1.99 2.86 130 1.04 60.69 31.50 -0.108 [15,] 422 6/4/92 67 94.2 1.88 3.01 177 1.41 47.65 33.50 0.020 [16,] 423 7/28/92 65 77.4 1.42 2.89 110 1.19 54.59 32.50 0.050 [17,] 425 12/8/92 68 107 1.63 2.97 175 1.57 43.21 34.00 0.173 [18,] 426 2/11/93 65 83.7 2.09 2.95 175 1.29 50.48 32.50 0.172 Ave 62.9 80.56 1.89 2.94 155.32 1.26 51.76 31.47 -0.014 bfile26 [1,] 104 8/7/90 3.15 0.39 0.17 1.03 0.07 0.12 25.44 1.58 -0.022 [2,] 105 10/16/90 8.5 5.41 0.88 1.63 4.75 0.64 13.35 4.25 -0.065 [3,] 106 11/13/90 8.2 2.61 1.05 1.35 2.74 0.32 25.76 4.10 -0.142 [4,] 107 1/15/91 13 7.42 1.3 1.8 9.71 0.57 22.78 6.50 0.147 [5,] 108 3/13/91 7.6 3.65 0.56 1.45 2.04 0.48 15.82 3.80 -0.294 [6,] 109 5/22/91 2 0.49 0.26 1.03 0.13 0.25 8.16 1.00 0.122 [7,] 110 6/25/91 10.2 4.08 0.41 1.4 1.69 0.40 25.50 5.05 0.114 [8,] 112 10/10/91 0.6 0.1 0.31 0.95 0.03 0.17 3.60 0.30 0.080 [9,] 113 12/4/91 11.5 9.5 1.29 1.87 12.3 0.83 13.92 5.75 -0.053 [10,] 114 2/5/92 4.75 1.72 0.24 1.11 0.42 0.36 13.12 2.38 -0.184 [11,] 115 4/2/92 8 4.05 0.25 1.39 2.3 0.51 15.80 3.85 0.037 317 [12,] 116 5/27/92 6.3 3.04 0.48 1.36 1.47 0.48 13.06 3.15 -0.224 [13,] 117 7/14/92 4.4 1.12 0.22 1.09 0.24 0.25 17.29 2.20 -0.036 [14,] 118 9/4/92 4.4 1.13 0.19 1.09 0.22 0.26 17.13 2.20 -0.026 [15,] 119 10/16/92 6.3 2.67 0.73 1.41 1.96 0.42 14.87 3.15 0.105 [16,] 120 11/16/92 7.2 5.04 2.29 1.71 11.6 0.70 10.29 3.60 0.048 [17,] 121 3/22/93 9.8 8.38 1.1 1.85 9.19 0.86 11.46 4.90 -0.132 [18,] 122 4/30/93 9 4.93 0.54 1.53 2.68 0.55 16.43 4.50 -0.067 [19,] 123 6/18/93 1.6 0.329 0.43 1.08 0.14 0.21 7.62 0.80 0.156 [20,] 125 9/22/93 0.9 0.13 0.22 0.95 0.03 0.14 6.23 0.45 -0.077 [21,] 127 11/1/93 2.5 0.25 0.49 1.15 0.13 0.10 25.00 0.75 -0.040 [22,] 128 12/15/93 5.6 2.15 0.52 1.34 1.12 0.38 14.59 3.05 -0.280 [23,] 129 3/15/94 9.3 6.16 0.98 1.67 6.02 0.66 14.04 4.65 0.160 [24,] 130 5/9/94 6.7 4.16 1.02 1.62 4.26 0.62 10.79 3.35 -0.227 [25,] 132 9/8/94 3.75 0.87 0.03 1.05 0.1 0.23 16.16 1.88 -0.130 [26,] 133 11/7/94 1.25 0.15 0.28 1.08 0.04 0.12 10.42 0.63 -0.092 Ave 6.02 3.074 0.62 1.35 2.90 0.41 14.95 2.99 -0.043 bfile27a [1,] 484 2/5/93 207 371 2.06 2.93 764 1.79 115.50 103.50 0.040 [2,] 486 6/2/93 210 356 1.77 2.84 631 1.70 123.88 102.50 0.000 [3,] 487 7/8/93 195 276 1.42 2.54 392 1.42 137.77 97.50 0.046 [4,] 488 8/31/93 205 272 1.07 2.55 292 1.33 154.50 102.50 0.055 [5,] 489 10/14/93 195 256 1.12 2.38 287 1.31 148.54 97.50 0.080 [6,] 490 12/14/93 217 390 1.72 2.83 671 1.80 120.74 108.50 -0.103 [7,] 491 2/7/94 216 462 2.46 3.34 1138 2.14 100.99 108.00 0.029 [8,] 493 6/7/94 210 355 1.34 2.57 474 1.69 124.23 105.00 0.133 [9,] 494 8/24/94 208 390 2.26 3.26 884 1.88 110.93 104.00 -0.015 Ave 207 347.6 1.69 2.80 614.78 1.67 126.34 103.22 0.029 bfile28a [1,] 230 8/8/89 149 222 3.04 3.24 675 1.49 100.00 74.50 -0.242 [2,] 235 6/21/90 172 228 2.73 3.19 622 1.33 129.75 86.00 -0.315 [3,] 239 11/29/90 161 224 3.08 3.27 688 1.39 115.72 80.50 -0.299 [4,] 241 5/16/91 178 283 3.19 3.47 902 1.59 111.96 88.75 -0.324 [5,] 242 7/11/91 158 240 2.8 3.23 674 1.52 104.02 79.00 0.218 [6,] 243 8/28/91 153 223 2.22 3.21 494 1.46 104.97 76.50 0.226 [7,] 245 11/26/91 127 161 2.39 2.96 385 1.27 100.18 63.50 -0.116 [8,] 246 1/30/92 177 258 2.71 3.27 700 1.46 121.43 88.50 -0.369 [9,] 248 6/4/92 206 330 3.3 3.58 1090 1.60 128.59 103.00 -0.372 [10,] 249 8/3/92 167 219 2.36 3.07 517 1.31 127.35 83.50 -0.248 [11,] 250 10/15/92 171 221 2.12 3.09 469 1.29 132.31 85.50 -0.352 [12,] 251 12/3/92 210 367 3.09 3.7 1130 1.75 120.16 105.00 -0.350 [13,] 256 7/29/93 176 234 1.72 3.07 401 1.33 132.38 88.00 -0.281 [14,] 261 6/8/94 190 252 2.55 3.28 643 1.33 143.25 95.00 -0.165 [15,] 262 7/26/94 208 314 2.37 3.43 743 1.51 137.78 104.00 0.244 [16,] 264 10/6/94 185 196 1.91 2.99 374 1.06 174.62 92.50 -0.398 [17,] 265 12/9/94 199 290 2.12 3.33 615 1.46 136.56 99.50 -0.270 Ave 176 250.7 2.57 3.26 654.24 1.42 124.77 87.84 -0.201 bfile29a [1, ] 447 2/4/85 204 1280 2.37 3.34 3030 6.27 32.51 102.00 0.043 [2,] 449 5/8/85 201 977 1.57 2.21 1530 4.86 41.35 100.50 0.068 [3,] 451 8/5/85 190 826 0.65 1.97 536 4.35 43.70 95.00 0.073 [4,] 452 9/17/85 197 798 0.53 1.54 423 4.05 48.63 98.50 -0.068 [5,] 453 10/22/85 195 840 1.07 1.9 895 4.31 45.27 97.50 0.096 318 [6,] 454 11/6/85 495 12100 7.48 28.45 90500 24.44 20.25 247.50 -0.095 [7,] 455 12/17/85 194 1000 1.64 2.3 1640 5.15 37.64 97.00 -0.071 [8,] 456 2/5/86 194 848 1.32 1.76 1120 4.37 44.38 97.00 0.083 [9,] 457 3/27/86 197 965 1.5 2.14 1450 4.90 40.22 98.50 0.064 [10,] 460 8/11/86 188 654 0.46 1.03 302 3.48 54.04 94.00 0.071 [11,] 461 10/7/86 198 712 0.43 1.15 307 3.60 55.06 99.00 -0.089 [12,] 464 2/19/87 194 936 1.45 2.05 1360 4.82 40.21 97.00 0.054 [13] 465 4/23/87 215 1650 3.31 4.9 5460 7.67 28.02 107.50 -0.033 [14,] 466 7/1/87 193 808 0.8 1.61 648 4.19 46.10 96.55 0.077 [15,] 467 8/5/87 198 806 0.56 1.7 446 4.07 48.64 99.40 -0.079 [16,] 468 10/20/87 179 740 0.81 1.52 598 4.13 43.30 89.50 0.117 [17,] 469 2/23/88 198 916 1.22 1.94 1120 4.63 42.80 99.00 -0.077 [18,] 470 4/12/88 195 996 1.49 2.26 1480 5.11 38.18 97.50 0.048 [19,] 471 6/16/88 196 790 0.87 1.37 688 4.03 48.63 98.00 -0.086 [20,] 472 7/21/88 193 682 0.55 1.05 375 3.53 54.62 96.50 0.096 [21,] 473 9/13/88 199 814 0.43 1.66 352 4.09 48.65 99.50 -0.102 [22,] 474 10/13/88 197 733 0.43 1.36 316 3.72 52.95 98.50 -0.080 [23,] 475 11/15/88 197 757 0.49 1.37 372 3.84 51.27 98.40 -0.065 [24,] 476 1/5/89 197 739 0.56 1.38 417 3.75 52.52 99.75 -0.105 [25,] 477 3/20/89 198 862 1.14 1.83 984 4.35 45.48 99.00 -0.068 [26,] 478 4/12/89 198 811 1.28 1.76 1040 4.10 48.34 95.50 -0.128 [27,] 479 6/1/89 197 887 1.08 1.82 960 4.50 43.75 98.50 -0.076 [28,] 480 7/6/89 199 1010 1.14 2.37 1150 5.08 39.21 99.50 -0.073 [29,] 481 8/23/89 202 1350 2.28 3.81 3140 6.68 30.23 101.00 -0.054 [30,] 482 10/4/89 211 1650 3.24 4.9 5340 7.82 26.98 105.50 0.021 [31,] 483 11/14/89 199 884 1.21 1.78 1070 4.44 44.80 99.50 -0.074 [32,] 484 1/3/90 207 1610 3.06 4.86 4920 7.78 26.61 105.50 -0.081 [33,] 485 3/6/90 196 968 1.45 2.07 1400 4.94 39.69 98.00 -0.074 [34,] 486 4/24/90 200 997 1.52 2.16 1520 4.99 40.12 100.00 -0.082 [35,] 487 6/7/90 204 1130 1.4 2.2 1580 5.54 36.83 102.00 -0.050 [36,] 488 8/7/90 199 818 0.95 1.52 780 4.11 48.41 99.50 -0.051 [37,] 489 10/12/90 202 1070 1.62 2.68 1730 5.30 38.13 101.00 -0.051 [38,] 490 11/14/90 199 987 1.51 2.14 1490 4.96 40.12 99.50 -0.069 [39,] 491 1/10/91 205 1280 2.23 3.31 2850 6.24 32.83 102.50 -0.053 [40,] 492 3/14/91 204 1120 1.81 2.55 2030 5.49 37.16 102.00 -0.069 [41,] 493 5/9/91 199 885 1.19 1.7 1050 4.45 44.75 99.50 -0.082 [42,] 494 7/15/91 199 860 1.08 1.65 932 4.32 46.05 99.50 -0.076 [43,] 495 8/28/91 195 793 0.65 1.52 519 4.07 47.95 97.50 -0.092 [44,] 496 9/23/91 197 791 0.48 1.63 383 4.02 49.06 98.50 -0.086 [45,] 497 10/22/91 198 761 0.47 1.43 358 3.84 51.52 99.00 -0.084 [46,] 499 12/20/91 199 755 0.73 1.5 550 3.79 52.45 99.50 -0.028 [47,] 500 1/29/92 197 830 0.94 1.62 782 4.21 46.76 98.50 -0.076 [48,] 501 3/24/92 202 1090 1.73 2.51 1890 5.40 37.43 101.00 -0.071 [49,] 502 5/12/92 205 1250 2.38 3.3 2970 6.10 33.62 102.50 -0.031 [50,] 503 7/7/92 198 859 0.99 1.97 847 4.34 45.64 99.00 -0.044 [51,] 504 8/25/92 199 790 0.55 1.67 438 3.97 50.13 99.50 -0.032 [52,] 505 10/13/92 196 811 0.85 1.67 689 4.14 47.37 98.00 -0.048 [53,] 506 11/12/92 200 803 0.87 1.58 696 4.02 49.81 100.00 -0.045 [54,] 507 1/6/93 195 1089 1.86 2.64 2030 5.58 34.92 97.50 -0.052 [55,] 508 3/31/93 221 1990 4.26 6.32 8480 9.00 24.54 110.50 -0.040 [56,] 509 4/20/93 218 1660 3.33 4.87 5530 7.61 28.63 109.00 -0.024 [57,] 510 6/22/93 203 847 1.05 1.61 890 4.17 48.65 101.50 -0.021 319 [58,] 513 7/29/93 206 896 0.58 1.65 515 4.35 47.36 103.00 -0.094 [59,] 515 12/14/93 197 918 1.27 1.89 1170 4.66 42.28 98.50 -0.060 [60,] 516 3/17/93 205 1510 3.07 4.44 4640 7.37 27.83 102.00 -0.035 [61,] 517 6/1/94 200 848 1.09 1.59 927 4.24 47.17 100.00 -0.068 [62,] 518 6/30/94 197 776 0.97 1.44 750 3.94 50.01 98.50 -0.064 [63,] 519 8/31/94 196 802 0.91 1.52 734 4.09 47.90 98.00 -0.060 [64,] 521 12/20/94 197 797 0.77 1.44 611 4.05 48.69 93.50 -0.073 Ave 204 1144 1.42 2.61 2902.03 5.15 42.63 101.81 -0.039 bfile30a [1,] 524 8/7/90 31 29 0.38 2.19 10.9 0.94 33.14 15.50 -0.006 [2,] 525 11/15/90 34 60.6 1.21 2.71 73.5 1.78 19.08 17.00 0.040 [3,] 527 2/27/91 45 70.4 0.83 2.65 58.2 1.56 28.76 22.50 -0.005 [4,] 529 7/15/91 36 42.7 1.46 2.65 62.2 1.19 30.35 18.00 0.025 [5,] 530 8/29/91 16.3 9.1 1.02 2.15 9.27 0.56 29.20 8.65 0.069 [6,] 531 9/23/91 12 5.1 0.67 2.25 3.43 0.43 28.24 6.00 0.265 [7,] 532 10/21/91 12.1 4.94 0.65 2.18 3.19 0.41 29.64 8.30 -0.600 [8,] 533 11/21/91 26 14.5 0.37 2.44 5.35 0.56 46.62 13.00 -0.024 [9,] 534 1/29/92 36 38.4 1.24 2.57 47.7 1.07 33.75 18.00 0.009 [10,] 537 7/7/92 52 67.2 0.84 2.62 56.4 1.29 40.24 26.00 -0.084 [11,] 538 8/26/92 20 9.87 0.65 2.04 6.42 0.49 40.53 10.00 -0.150 [12,] 539 9/16/92 32 25.7 1.12 2.46 28.9 0.80 39.84 16.00 -0.123 [13,] 540 10/8/92 21 12.5 0.84 2.21 10.5 0.60 35.28 10.50 -0.122 [14,] 546 8/4/93 17 5.44 0.51 1.91 2.76 0.32 53.13 8.50 0.305 [15,] 548 11/2/93 18.6 7.72 0.83 2.08 6.4 0.42 44.81 9.30 0.200 [16,] 549 12/15/93 46 57.4 1.37 2.71 78.5 1.25 36.86 23.00 -0.126 Ave 28.4 28.79 0.87 2.36 28.98 0.85 35.59 14.39 -0.020 bfile31a [1,] 8 11/7/85 61 76.3 1.82 1.99 138 1.25 48.77 30.50 0.1952 [2,] 10 1/10/86 23 25.9 0.89 1.22 22.9 1.13 20.42 11.00 -0.044 [3,] 26 3/9/88 40 23.9 0.77 1.19 18.5 0.60 66.95 20.00 -0.394 [4,] 37 6/27/89 34 23.3 1.32 1.42 30.8 0.69 49.61 17.00 0.1833 [5,] 39 9/27/89 33 27.6 1.1 1.58 30.5 0.84 39.46 16.50 0.062 [6,] 41 1/16/90 31 28.6 1.09 1.32 31.3 0.92 33.60 15.50 0.1281 [7,] 42 3/22/90 34 22.4 1.14 1.17 25.5 0.66 51.61 17.00 0.0738 [8,] 46 9/4/90 31 18.6 0.75 1.22 13.9 0.60 51.67 15.50 0.0701 [9,] 60 8/3/92 28 14.7 0.62 0.95 9.05 0.53 53.33 14.00 0.0085 [10,] 71 11/22/93 30 22.8 0.33 1.33 7.5 0.76 39.47 15.00 0.0205 [11,] 72 12/2/93 30 29.2 0.97 1.36 28.3 0.97 30.82 15.00 0.1932 [12,] 75 4/12/94 36 36.2 1.86 1.63 67.2 1.01 35.80 18.00 0.0719 [13,] 76 5/17/94 37 31.8 1.52 1.47 48.4 0.86 43.05 18.50 0.0866 [14,] 78 6/16/94 35 22.3 0.84 1.15 18.7 0.64 54.93 17.50 0.166 [15,] 79 7/25/94 33 20 0.67 1.09 13.5 0.61 54.45 16.50 0.0905 [16,] 82 10/13/94 27 16.5 0.55 1.05 9.09 0.61 44.18 13.50 0.0242 [17,] 85 12/20/94 28 15.5 0.52 0.99 8.16 0.55 50.58 14.00 0.0402 Ave 33.6 26.8 0.99 1.30 30.66 0.78 45.22 16.76 0.0574 bfile32a [1,] 315 8/8/89 38 31.5 0.83 1.43 26 0.83 45.84 19.00 0.000 [2,] 319 3/22/90 42 41.7 1.84 2.04 76.6 0.99 42.30 21.00 -0.041 [3,] 320 5/2/90 42 36.3 1.75 1.94 63.7 0.86 48.60 21.00 -0.030 [4l] 321 6/20/90 40 35.5 1.14 1.66 40.4 0.89 45.07 20.00 0.049 [5,] 322 8/9/90 38 26.8 0.84 1.41 22.4 0.71 53.88 19.00 0.038 [6,] 328 7/9/91 38 28.8 1.03 1.48 29.6 0.76 50.14 19.00 -0.036 320 [7,] 329 8/28/91 37 28.3 0.59 1.32 16.8 0.76 48.37 18.50 -0.084 [8,] 333 4/15/91 41 32 1.23 1.62 41.5 0.78 52.53 20.50 0.104 [9,] 335 8/3/91 37 39.8 0.56 1.45 22.3 1.08 34.40 18.50 -0.005 [10,] 336 10/14/92 27 20 0.71 1.4 14.2 0.74 36.45 13.50 -0.072 [11,] 337 11/23/92 39 51.8 1.06 2.05 54.9 1.33 29.36 19.50 -0.070 [12,] 338 11/30/92 40 22.6 1.3 1.62 30 0.57 70.80 20.00 0.138 [13,] 340 2/9/93 42 29.1 1.72 1.76 50.1 0.69 60.62 21.00 0.024 [14,] 342 5/19/93 42 56.6 2.67 2.54 151 1.35 31.17 21.00 0.012 [15,] 343 7/6/93 36 32.3 1.22 1.61 39.2 0.90 40.12 18.00 -0.037 [16,] 344 8/25/93 31 22.6 0.88 1.4 19.9 0.73 42.52 15.50 -0.104 [17,] 345 10/13/93 33 25.5 0.82 1.45 20.8 0.77 42.71 16.50 -0.020 [18,] 350 6/8/94 38 60.5 1.32 2.08 79.8 1.59 23.87 19.50 -0.064 [19,] 352 7/26/94 39 33.8 0.92 1.61 31 0.87 45.00 19.50 -0.045 [20,] 353 8/10/94 36 26.6 0.87 1.5 23.1 0.74 48.72 18.00 -0.017 [21,] 354 10/6/94 32 23.7 0.75 1.45 17.8 0.74 43.21 16.00 -0.041 [22,] 355 11/30/94 31 24 0.72 1.47 17.4 0.77 40.04 15.50 -0.023 Ave 37.2 33.17 1.13 1.65 40.39 0.88 44.35 18.64 -0.015 bfile33a 1,] 485 11/14/89 39 54.5 2.88 2.79 157 1.40 27.91 19.50 -0.037 [2,] 487 3/22/90 58 88.6 2.86 3.03 253 1.53 37.97 29.00 -0.514 [3,] 490 8/9/90 30 40.8 1.45 2.45 59 1.36 22.06 15.00 -0.074 [4,] 491 10/3/90 32 42 1.4 2.4 60.5 1.31 24.38 16.00 0.020 [5,] 492 11/29/90 38 49.2 3.2 2.74 157 1.29 29.35 19.00 0.077 [6,] 497 7/24/91 30 33.2 2.25 2.47 74.8 1.11 27.11 15.00 0.186 [7,] 498 8/23/91 28 34.8 1.49 2.38 51.8 1.24 22.53 14.00 0.022 [8,] 499 10/16/91 27 31.7 1.03 2.33 32.7 1.17 23.00 13.50 0.032 [9,] 500 11/26/91 27 28.7 1.35 2.4 38.7 1.06 25.40 13.50 -0.101 [10,] 501 2/5/92 34 34 2.89 2.55 98.3 1.00 34.00 17.00 -0.212 [11,] 503 6/2/92 40 57.2 3.76 2.9 215 1.43 27.97 20.00 -0.005 [12,] 504 8/4/92 33 37.8 2.66 2.56 100 1.15 28.81 16.50 0.108 [13,] 505 10/14/92 25 32.4 1.26 2.35 40.9 1.30 19.29 12.50 0.128 [14,] 506 1/30/92 39 65.3 3.2 2.91 209 1.67 23.29 19.50 -0.165 [15,] 510 6/2/93 36 56.6 3.16 2.82 178 1.57 22.90 18.00 0.003 [16,] 511 7/7/93 32 38.2 3.01 2.57 115 1.19 26.81 16.00 0.132 [17,] 512 8/25/93 28 28.3 1.91 2.41 53.9 1.01 27.70 13.50 0.131 [18,] 513 10/13/93 25 29.1 1.85 2.41 53.8 1.16 21.48 12.50 0.079 [19,] 518 6/7/93 33 57.5 2.96 2.81 170 1.74 18.94 16.50 0.022 [20,] 519 7/25/93 33 43.7 3.61 2.73 158 1.32 24.92 16.50 0.048 [21,] 520 10/6/93 26 34.3 1.8 2.43 61.9 1.32 19.71 13.00 0.006 [22,] 521 11/30/93 28 30 2.12 2.47 63.6 1.07 26.13 14.00 0.036 Ave 32.8 43.09 2.37 2.59 109.18 1.29 25.53 16.36 -0.004 bfile34a [1,] 469 3/20/85 184 362 0.73 2.24 263 1.97 93.52 92.00 -0.030 [2,] 470 5/8/85 185 385 0.99 2.48 381 2.08 88.90 92.50 0.053 [3,] 471 6/19/85 185 377 0.69 2.26 261 2.04 90.78 92.50 -0.014 [4,] 473 9/17/85 178 258 0.25 1.73 64.8 1.45 122.81 89.00 -0.013 [5,] 475 12/17/85 181 510 1.27 2.96 666 2.82 64.24 90.50 -0.046 [6,] 476 2/4/86 184 356 0.85 2.29 282 1.93 95.10 88.03 -0.056 [7,] 477 3/27/86 186 458 1.25 2.78 557 2.46 75.54 93.00 0.042 [8,] 479 6/17/86 186 320 0.5 2 158 1.72 108.11 93.00 0.036 [9,] 481 10/7/86 184 325 0.24 1.78 78.9 1.77 104.17 92.00 0.059 [10,] 482 11/18/86 183 273 0.39 1.84 106 1.49 122.67 91.50 -0.058 321 [11,] 483 2/19/87 184 402 1.04 2.5 417 2.18 84.22 92.00 -0.073 [12,] 484 7/1/87 184 320 0.54 2.03 173 1.74 105.80 92.00 -0.138 [13,] 485 8/5/87 185 265 0.36 1.8 96.5 1.43 129.15 92.50 0.083 [14,] 486 10/20/87 184 304 0.49 1.97 149 1.65 111.37 92.00 -0.041 [15,] 487 2/23/88 184 390 0.88 2.41 345 2.12 86.81 92.00 -0.020 [16,] 488 6/16/88 183 348 0.7 2.17 242 1.90 96.23 91.50 -0.039 [17,] 489 7/21/88 184 310 0.48 2 150 1.68 109.21 92.00 0.073 [18,] 490 9/13/88 184 277 0.37 1.82 102 1.51 122.22 92.00 0.080 [19,] 491 10/13/88 184 271 0.25 1.75 68.3 1.47 124.93 92.00 -0.072 [20,] 492 11/15/88 179 310 0.32 1.86 99 1.73 103.36 87.00 -0.879 [21,] 493 1/5/89 184 271 0.24 1.77 65.8 1.47 124.93 92.50 0.065 [22,] 494 3/7/89 184 354 0.79 2.28 280 1.92 95.64 92.00 0.059 [23,] 496 6/1/89 179 394 0.7 2.27 275 2.20 81.32 89.50 0.229 [24,] 497 7/6/89 180 407 0.9 2.44 365 2.26 79.61 90.00 -0.020 [25,] 499 11/14/89 184 376 0.77 2.28 290 2.04 90.04 92.00 -0.018 [26,] 500 3/6/90 186 438 0.99 2.57 435 2.35 78.99 93.00 -0.031 [27,] 501 4/23/90 182 447 1.26 2.77 563 2.46 74.10 91.00 0.006 [28,] 503 8/7/90 178 380 0.44 2.12 167 2.13 83.38 89.00 -0.057 [29,] 504 10/12/90 184 360 0.62 2.26 225 1.96 94.04 92.00 -0.013 [30,] 505 11/14/90 185 413 1 2.6 414 2.23 82.87 92.50 -0.021 [31,] 506 3/14/91 185 506 1.34 2.99 676 2.74 67.64 92.00 0.034 [32,] 508 7/15/91 182 430 0.73 2.48 315 2.36 77.03 91.00 -0.104 [33,] 509 8/28/91 184 291 0.41 2.02 118 1.58 116.34 91.00 0.126 [34,] 510 9/23/91 176 328 0.31 1.91 103 1.86 94.44 88.00 -0.091 [35,] 511 10/21/91 176 347 0.24 1.94 82.6 1.97 89.27 88.00 0.099 [36,] 512 11/21/91 185 236 0.3 1.8 70.4 1.28 145.02 92.50 -0.033 [37,] 513 1/28/92 187 305 0.63 2.11 190 1.63 114.65 91.50 -0.027 [38,] 518 5/24/93 189 464 0.96 2.96 443 2.46 76.98 90.00 0.000 [39,] 519 6/22/93 187 405 0.6 2.54 242 2.17 86.34 93.50 0.054 [40,] 520 7/27/93 183 386 0.61 2.52 234 2.11 86.76 91.50 -0.048 [41,] 521 8/25/93 184 407 0.35 2.23 141 2.21 83.18 92.00 -0.084 [42,] 523 12/14/93 180 473 1.04 2.69 492 2.63 68.50 90.00 -0.067 [43,] 525 6/1/94 193 383 0.87 2.4 332 1.98 97.26 96.50 0.039 [44,] 526 6/30/94 191 329 0.78 2.22 256 1.72 110.88 96.00 0.061 [45,] 528 10/5/94 179 368 0.4 1.97 146 2.06 87.07 89.50 -0.082 Ave 184 362.6 0.66 2.24 257.34 1.98 96.12 91.46 -0.022 bfile35a [1,] 541 9/27/90 58 48.8 0.65 1.69 31.8 0.84 68.93 15.00 -0.233 [2,] 555 1/25/91 30 24.1 0.29 1.59 6.94 0.80 37.34 26.50 -0.167 [3,] 556 8/27/91 30 24.8 0.3 1.61 7.53 0.83 36.29 27.53 0.012 [4,] 558 12/5/91 53 54.2 0.73 1.87 39.8 1.02 51.83 15.00 0.289 [5,] 559 1/30/92 45 46.8 0.44 1.94 20.7 1.04 43.27 22.50 -0.135 [6,] 560 4/8/92 53 52.2 1.02 1.88 53.2 0.98 53.81 26.50 -0.149 [7,] 561 6/4/92 41 37.6 0.91 1.74 34.3 0.92 44.71 20.50 -0.302 [8,] 562 7/30/92 43 37.8 0.96 1.76 36.3 0.88 48.92 21.50 0.338 [9,] 563 10/8/92 40 38.8 0.57 1.77 21.9 0.97 41.24 20.00 -0.272 [10,] 564 12/3/92 53 56.8 1.1 1.98 62.1 1.07 49.45 26.50 -0.214 [11,] 565 2/4/93 54 56.2 1.14 1.96 63.8 1.04 51.89 27.00 0.242 [12,] 566 4/8/93 55 81 2.53 2.62 205 1.47 37.35 27.50 -0.050 [13,] 567 5/20/93 53 54.2 1.28 2.01 69.2 1.02 51.83 26.50 -0.197 [14,] 570 9/30/93 38 27.6 0.5 1.55 13.7 0.73 52.32 19.00 -0.154 [15,] 571 10/21/93 32 29.1 0.67 1.71 19.5 0.91 35.19 16.00 0.265 322 [16,] 572 12/9/93 57 80 2 2.48 160 1.40 40.61 28.50 -0.051 [17,] 579 12/12/94 51 57.2 1.2 2.05 68.4 1.12 45.47 25.50 -0.248 Ave 46.2 47.48 0.96 1.89 53.77 1.00 46.50 23.03 -0.060 bfile36a [1,] 157 9/22/87 39 41.2 1.75 3.05 71.9 1.06 36.92 19.50 -0.066 [2,] 158 11/10/87 31 24.2 1.31 2.78 31.6 0.78 39.71 15.50 0.107 [3,] 159 12/10/87 54 51.8 1.25 3.07 64.5 0.96 56.29 27.00 0.405 [4,] 166 8/2/88 38 34.4 0.56 2.56 19.3 0.91 41.98 18.75 0.179 [5,] 167 9/21/88 40 39.2 0.6 2.64 23.4 0.98 40.82 20.00 0.173 [6,] 168 10/18/88 29 26.6 0.19 2.42 5.08 0.92 31.62 14.50 -0.141 [7,] 173 5/9/89 58 115 1.9 3.86 219 1.98 29.25 29.00 0.146 [8,] 176 7/12/89 41 38.2 0.6 2.62 22.8 0.93 44.01 20.50 0.092 [9,] 177 8/1/89 52 65.1 0.76 2.92 49.7 1.25 41.54 25.75 0.257 [10,] 178 9/27/89 45 49.6 0.36 2.64 17.7 1.10 40.83 22.50 0.117 [11,] 180 11/15/89 40 40.9 0.4 2.57 16.4 1.02 39.12 20.00 -0.166 [12,] 181 1/10/90 42 31.8 1.58 2.94 50.4 0.76 55.47 21.00 -0.178 [13,] 183 5/2/90 52 65.9 0.91 2.98 59.9 1.27 41.03 25.00 -0.080 [14,] 194 7/31/91 27 15.2 0.36 2.34 5.52 0.56 47.96 13.50 0.046 [15,] 195 8/13/91 19.5 9.44 0.25 2.23 2.37 0.48 40.28 9.75 0.006 [16,] 196 8/26/91 17 9.51 0.22 2.23 2.12 0.56 30.39 8.50 0.058 [17,] 198 12/4/91 51 43 0.96 2.86 41.4 0.84 60.49 25.50 0.315 Ave 39.7 41.24 0.82 2.75 41.36 0.96 42.22 19.78 0.075 bfile37a [1,] 231 10/26/89 73 77.7 0.64 2.75 49.7 1.06 68.58 36.50 -0.205 [2,] 245 11/28/90 76 102 0.67 2.8 68.1 1.34 56.63 38.00 0.018 [3,] 247 3/20/91 79 130 0.84 2.97 110 1.65 48.01 39.50 -0.006 [4,] 269 8/25/93 7.7 2.03 0.69 2.1 1.39 0.26 29.21 24.75 0.163 [5,] 272 12/8/93 83 162 1.22 3.5 198 1.95 42.52 41.50 0.050 [6,] 273 2/2/94 78 138 1.01 3.14 139 1.77 44.09 39.00 0.055 [7,] 274 4/6/94 84 189 1.41 3.61 267 2.25 37.33 42.00 -0.024 [8,] 275 6/2/94 78 103 0.35 2.55 36.1 1.32 59.07 39.00 -0.072 [9,] 276 7/20/94 78 101 0.26 2.48 26 1.29 60.24 39.00 0.056 [10,] 279 12/7/94 82 118 0.6 2.76 71.4 1.44 56.98 41.00 0.091 Ave 71.9 112.3 0.77 2.87 96.67 1.43 50.27 38.03 0.013 bfile38a [1,] 576 5/15/91 76 134 1.2 1.83 161 1.76 43.10 38.00 0.061 [2,] 577 7/10/91 43 47.2 0.53 1.1 25.2 1.10 39.17 21.50 0.367 [3,] 583 12/4/91 66 119 1.34 2.14 160 1.80 36.61 33.00 0.028 [4,] 585 4/8/92 85 146 1.09 1.79 159 1.72 49.49 42.50 0.103 [5,] 586 6/3/92 78 153 1.36 1.96 208 1.96 39.76 39.00 0.102 [6,] 587 7/29/92 77 167 1.64 2.15 274 2.17 35.50 38.50 0.086 [7,] 588 8/19/92 72 106 0.82 1.62 86.3 1.47 48.91 36.00 0.073 [8,] 589 10/7/92 62 92.3 0.85 1.58 78.5 1.49 41.65 31.00 0.235 [9,l 590 12/1/92 80 151 2.19 2.33 332 1.89 42.38 40.00 0.227 [10,] 591 2/2/93 84 165 1.57 2.15 258 1.96 42.76 42.00 0.052 [11,] 604 7/19/94 77 121 0.88 1.6 107 1.57 49.00 38.50 0.105 [12,] 605 9/6/94 73 96.8 0.57 1.39 55.3 1.33 55.05 36.50 0.155 [13,] 607 12/7/94 72 137 1.42 2 196 1.90 37.84 36.00 0.078 Ave 72.7 125.8 1.19 1.82 161.56 1.70 43.17 36.35 0.129 bfile39a [1,] 535 9/26/89 51 106 1.35 4.3 143 2.08 24.54 25.50 -0.030 [2,] 536 11/14/89 40 54.4 0.5 3.1 27.1 1.36 29.41 20.00 -0.119 323 [3,] 537 1/9/90 49 64.6 1.07 3.52 68.9 1.32 37.17 24.50 0.161 [4,] 538 3/6/90 40 58.2 0.71 3.26 41.1 1.46 27.49 20.00 -0.118 [5,] 542 10/9/90 35 38.5 0.3 2.86 11.4 1.10 31.82 17.50 -0.068 [6,] 543 11/27/90 41 52.7 0.51 3.09 27.1 1.29 31.90 20.50 -0.132 [7,] 544 1/23/91 41 66 0.8 3.39 52.9 1.61 25.47 21.00 -0.053 [8,] 545 3/19/91 46 75.6 1.02 3.62 76.7 1.64 27.99 23.00 -0.016 [9,] 546 5/14/91 41 50.9 0.57 3.12 28.8 1.24 33.03 20.50 -0.059 [10,] 547 7/9/91 40 45.9 0.77 3.19 35.2 1.15 34.86 20.00 -0.041 [11,] 548 7/31/91 36 24.8 0.72 2.91 17.7 0.69 52.26 18.00 0.040 [12,] 549 8/26/91 29 22.3 0.29 2.53 6.52 0.77 37.71 14.50 0.094 [13,] 552 1/28/92 37 45.5 0.62 3.11 28 1.23 30.09 18.50 -0.086 [14,] 556 10/6/92 36 32.8 0.64 3 21 0.91 39.51 18.00 0.072 [15,] 560 5/18/93 44 56.3 1.26 3.57 71 1.28 34.39 22.00 0.069 [16,] 572 12/6/94 41 62.9 0.88 3.42 55.3 1.53 26.72 20.50 -0.112 Ave 40.4 53.59 0.75 3.25 44.48 1.29 32.77 20.25 -0.025 bfile40a [1,] 482 12/18/89 54 26.7 0.43 0.56 11.6 0.49 109.21 27.50 0.123 [2,] 483 1/29/90 69 35.8 0.67 0.65 23.9 0.52 132.99 34.50 0.394 [3,] 485 3/28/90 67 35 0.48 0.6 16.9 0.52 128.26 33.50 -0.430 [4,] 486 4/26/90 58 26.1 0.67 0.61 17.6 0.45 128.89 29.00 -0.489 [5,] 494 12/26/90 72 36.8 0.61 0.88 22.6 0.51 140.87 36.00 0.050 [6,] 495 1/29/91 70 33.4 0.61 0.79 17.1 0.48 146.71 35.00 0.183 [7,] 496 2/25/91 70 31.2 0.42 0.76 13 0.45 157.05 35.00 0.226 [8,] 497 3/27/91 75 47.4 0.75 0.99 35.6 0.63 118.67 37.50 -0.030 [9,] 498 4/25/91 71 32 0.54 0.81 17.4 0.45 157.53 35.50 0.109 [10,] 499 5/28/91 67 23.6 0.38 0.69 8.99 0.35 190.21 33.50 0.226 [11,] 500 6/26/91 70 25.8 0.34 0.67 8.72 0.37 189.92 35.00 0.248 [12,] 501 7/29/91 81 80 1.68 1.49 134 1.25 64.80 38.00 0.012 [13,] 502 8/27/91 99 78.1 1.03 1.28 80.3 0.79 125.49 49.50 -0.100 [14,] 519 2/24/93 83 57.9 0.45 0.91 25.9 0.70 118.98 41.50 0.016 [15,] 520 3/29/93 96 86.9 0.77 1.2 67.3 0.91 106.05 48.00 -0.045 [16,] 521 4/28/93 93 65.3 0.4 0.99 25.3 0.70 132.45 46.50 0.049 [17,] 522 5/25/93 91 66.5 0.33 0.98 21 0.73 124.53 45.50 -0.006 [18,] 523 6/28/93 91 56 0.09 0.84 5.12 0.62 147.88 45.50 0.055 [19,] 532 4/26/94 67 31.1 0.67 0.82 20.9 0.46 144.34 33.50 0.183 [20,] 534 6/28/94 64 25 0.44 0.72 11 0.39 163.84 32.00 0.173 [21,] 535 6/28/94 42 17.4 0.44 0.7 7.71 0.41 101.38 21.00 -0.104 [22,] 536 9/28/94 62 30.8 0.52 0.78 16.1 0.50 124.81 31.00 0.282 Ave 73.3 43.13 0.58 0.67 27.64 0.58 134.31 36.55 0.051 bfile41a [1,] 430 2/27/85 28 26 0.52 1.8 13.6 0.93 30.15 13.20 0.043 [2,] 482 10/25/90 28 19.8 0.6 1.79 11.9 0.71 39.60 14.00 0.154 [3,] 484 1/23/91 30 22.6 0.59 1.83 13.4 0.75 39.82 15.00 -0.239 [4,] 485 3/19/91 28 24.6 0.88 1.97 21.7 0.88 31.87 14.00 -0.101 [5,] 488 8/5/91 7 4.4 0.35 1.44 1.54 0.63 11.14 3.50 -0.089 [6,] 499 5/18/93 31 36.5 1.07 2.28 39.1 1.18 26.33 15.50 0.018 [7,] 500 6/10/93 34 25.8 0.64 1.98 16.5 0.76 44.81 17.00 0.006 [8,] 502 8/24/93 16 5.45 0.29 1.47 1.6 0.34 46.97 8.00 -0.028 [9,] 511 10/3/94 34 22.4 0.26 1.8 5.76 0.66 51.61 17.00 -0.036 Ave 26.2 20.84 0.58 1.82 13.90 0.76 35.81 13.02 -0.030 bfile42a [1,] 404 9/19/89 17 9.64 0.84 2.37 8.11 0.57 29.98 8.50 -0.062 324 [2,] 417 7/9/91 22 7.9 1.19 2.43 9.4 0.36 61.27 11.00 0.013 [3,] 418 8/6/91 2.8 0.57 0.94 1.97 0.54 0.20 13.75 1.40 0.140 [4,] 419 8/20/91 4.5 1.4 0.35 1.98 0.49 0.31 14.46 2.25 -0.314 [5,] 435 10/19/93 10.5 2.88 0.32 2.2 0.92 0.27 38.28 5.25 0.215 [6,] 436 12/7/93 61 115 1.63 3.72 188 1.89 32.36 30.50 0.047 [7,] 438 4/5/94 65 77.8 1.44 3.3 112 1.20 54.31 32.50 0.415 [8,] 439 6/1/94 16.2 4.98 0.95 2.25 4.72 0.31 52.70 9.23 0.233 Ave 24.9 27.52 0.96 2.53 40.52 0.64 37.14 12.58 0.086 bfile43a [1,] 383 9/17/90 8.4 2.21 0.25 1.47 0.55 0.26 31.93 4.20 0.208 [2,] 384 10/26/90 12 5.28 0.47 1.7 2.46 0.44 27.27 6.00 -0.055 [3,] 388 7/22/91 1.5 0.08 0.27 1.17 0.02 0.05 28.13 0.75 0.000 [4,] 391 5/20/92 6.6 1.64 1.23 1.53 2.02 0.25 26.56 3.30 0.396 [5,] 397 6/22/93 9.2 4.54 0.22 1.47 1.16 0.49 18.64 4.60 0.277 [6,] 400 1/26/94 14 10.2 1.24 2.19 12.6 0.73 19.22 6.75 0.158 [7,] 401 3/16/94 14 8.25 1.14 1.94 9.38 0.59 23.76 7.00 0.190 [8,] 402 4/28/94 13 3.84 1.14 1.67 4.37 0.30 44.01 6.50 -0.055 [9,] 403 6/16/94 4.2 0.78 0.68 1.35 0.53 0.19 22.62 2.10 0.128 10,] 405 8/31/94 10.8 2.06 0.52 1.48 1.09 0.19 56.62 5.40 0.116 Ave 9.37 3.888 0.72 1.60 3.42 0.35 29.87 4.66 0.136 bfile44a [1,] 148 8/3/87 17 8.69 0.18 1.32 1.56 0.51 33.26 8.50 0.285 [2,] 176 1/23/91 30 32.7 0.91 1.52 29.7 1.09 27.52 15.00 0.343 [3,] 177 3/19/91 38 57.5 1.36 1.86 78 1.51 25.11 19.00 0.203 [4,] 178 5/14/91 27 26 0.47 1.28 12.2 0.96 28.04 13.50 0.290 [5,] 197 7/26/93 14.7 4.02 0.22 1.09 0.9 0.27 53.75 7.35 -0.247 [6,] 200 10/19/93 15 7.05 0.28 2.05 1.98 0.47 31.91 7.50 -0.077 [7,] 205 6/1/94 23 16.2 0.46 1 7.43 0.70 32.65 11.50 0.302 [8,] 206 7/14/94 20 11.3 0.31 0.9 3.5 0.57 35.40 10.00 0.272 [9,] 208 10/4/94 18.5 12.6 0.29 1.39 3.7 0.68 27.16 9.25 0.300 [10,] 210 12/7/94 39 57.3 0.37 1.8 21.3 1.47 26.54 19.50 0.316 Ave 24.2 23.34 0.49 1.42 16.03 0.82 32.14 12.11 0.199 325 APPENDIX B ALGORITHMS FOR SUBBASIN/DRAINAGE AREA DELINEATION Process.aml &s sitecov [response 'Name of input site coverage in UTM'] &s elevgrid [response 'Name of output filled elevation grid'] &s shedname [response 'Name of output watershed mask grid'] &s atngrid [response 'Name of output ATN grid'] &s dirgrid [response 'Name of output direction grid'] &s i = 1 &label top &s dem%i% [response 'Name of DEM grid'] &s i = %i% + 1 &if [query 'Additional DEMS'] &then &goto top &do j = 1 &to [calc %i% - 1] arc demlattice [value dem%j%] dem%j%grid &end &if not [null %dem2%] &then &do arc latticemerge commongrid &do k = 1 &to [calc %i% - 1] dem%k%grid &end end fill commongrid %elevgrid% # # %dirgrid% xxflowacc = flowaccumulation(%dirgrid%) setcell %elevgrid% setwindow %elevgrid% xxpointgrid = pointgrid(%sitecov%) seeds = snappour(xxpointgrid,xxflowacc,100) %shedname% = watershed(%dirgrid%,seeds) &r modatn %elevgrid% %dirgrid% %atngrid% kill (! commongrid xxflowacc xxpointgrid seeds !) &do k = 1 &to [calc %i% - 1] 326 kill dem%k%grid &end &return Modpro.aml &s sitecov [response 'Name of input site coverage in UTM'] &s elevgrid [response 'Name of output filled elevation grid'] &s shedname [response 'Name of output watershed mask grid'] &s atngrid [response 'Name of output ATN grid'] &s dirgrid [response 'Name of output direction grid'] xxflowacc = flowaccumulation(%dirgrid%) setcell %elevgrid% setwindow %elevgrid% xxpointgrid = pointgrid(%sitecov%) seeds = snappour(xxpointgrid,xxflowacc,100) %shedname% = watershed(%dirgrid%,seeds) &r modatn %elevgrid% %dirgrid% %atngrid% kill (! xxflowacc xxpointgrid seeds !) &return Rerun.aml &s sitecov [response 'Name of input site coverage in UTM'] &s tmpgrid [response 'Name of input temp elevation grid'] &s elevgrid [response 'Name of output filled elevation grid'] &s shedname [response 'Name of output watershed mask grid'] &s atngrid [response 'Name of output ATN grid'] &s dirgrid [response 'Name of output direction grid'] arc copy %tmpgrid% commongrid /* &s i = 1 /* &label top /* &s dem%i% [response 'Name of DEM grid'] /* &s i = %i% + 1 327 /* &if [query 'Additional DEMS'] &then &goto top /* &do j = 1 &to [calc %i% - 1] /* arc demlattice [value dem%j%] dem%j%grid /* &end /* if not [null %dem2%] &then &do /* arc latticemerge commongrid /* &do k = 1 &to [calc %i% - 1] /* dem%k%grid /* end /* end fill commongrid %elevgrid% # # %dirgrid% xxflowacc = flowaccumulation(%dirgrid%) setcell %elevgrid% setwindow %elevgrid% xxpointgrid = pointgrid(%sitecov%) seeds = snappour(xxpointgrid,xxflowacc,100) %shedname% = watershed(%dirgrid%,seeds) &r modatn %elevgrid% %dirgrid% %atngrid% kill (! commongrid xxflowacc xxpointgrid seeds !) /* &do k = 1 &to [calc %i% - 1] /* kill dem%k%grid /* &end &return 328 APPENDIX C WATERSHED DATA COLLECTED, CALCULATED AND/OR DERIVED Num Id$ Idd b f m Size 1 m10 1601500 0.1416 0.3605 0.4981 247 2 m11 1603000 0.1084 0.5349 0.3567 875 3 m19 1614500 0.0989 0.3936 0.5078 494 4 m21 1617800 0.1699 0.3841 0.4467 18.9 5 m24 1619500 0.2441 0.1891 0.5661 281 6 m26 1637500 0.3477 0.2669 0.3842 66.9 7 m29 1639140 0.225 0.2597 0.5119 31.3 8 m30 1639500 0.1121 0.3941 0.4962 102 9 m31 1640965 0.4607 0.2403 0.2895 2.14 10 m311 1640970 0.5622 0.1442 0.2976 4.01 11 m32 1640980 0.2972 0.0906 0.6336 0.38 12 m34 1643000 -0.008 0.4667 0.5736 817 13 m37 1643500 0.04 0.3581 0.6011 62.8 14 m38 1645000 0.065 0.4289 0.5113 101 15 m43 1651000 0.2741 0.2663 0.463 49.4 16 m45 1653600 0.1868 0.4474 0.3668 39.5 17 m47 1661050 0.2769 0.2784 0.4488 18.5 18 m48 1661500 0.1 0.2326 0.6684 24 19 v07 1622000 0.034 0.357 0.6072 379 20 v12 1624800 0.1302 0.4717 0.3992 70.1 21 v13 1625000 0.2286 0.2081 0.5632 375 22 v15 1626000 0.0906 0.2665 0.6411 127 23 v17 1626850 0.0507 0.3832 0.5697 149 24 v18 1627500 0.1761 0.5368 0.2872 212 25 v19 1628060 0.3773 0.2978 0.3461 1.94 26 v22 1628500 0.0573 0.3363 0.6063 1084 27 v23 1629500 0.2195 0.3008 0.4797 1377 28 v24 1631000 0.0839 0.2954 0.6208 1642 29 v25 1632000 0.339 0.416 0.2443 210 30 v26 1632082 0.1865 0.2601 0.555 45.5 31 v27 1632900 0.1437 0.2687 0.587 93.2 32 v28 1633000 0.2729 0.1612 0.5681 506 33 v29 1634000 0.0075 0.2406 0.7541 768 34 v30 1634500 0.2061 0.1703 0.6242 103 35 v36 1638480 0.2498 0.2162 0.5363 89.6 36 v38 1643700 0.4398 0.3903 0.1676 123 37 v39 1644000 0.193 0.2523 0.5559 332 38 v44 1646000 0.1671 0.3015 0.5314 57.9 39 v48 1653000 0.1374 0.3277 0.573 33.7 40 v49 1654000 0.3637 0.2618 0.3754 23.5 41 v52 1656000 0.4685 0.3194 0.2123 93.4 42 v65 1658500 0.3515 0.3744 0.2646 7.64 43 v71 1660400 0.2404 0.359 0.4032 34.9 329 physiogra$ phys1$ ordinary for b ordinary for f ordinary for m plat (100%) plat 5 5 2 plat (100%) plat 5 5 2 gv/br/vr(50/10/40) gv 2 2 5 br br 6 6 1 br br 6 6 1 br br 6 6 1 pied pied 3 3 4 pied pied 3 3 4 br br 6 6 1 br br 6 6 1 br/pied (70/30) pied 3 3 4 pied pied 3 3 4 pied Pied 3 3 4 pied Pied 3 3 4 pied/coap (80/20) coap 1 1 6 coap coap 1 1 6 coap coap 1 1 6 coap coap 1 1 6 gv/vr (60/40) gv 2 2 5 gv/br (50/50) br 6 6 1 gv/br/vr(70/20/10) gv 2 2 5 br/gv (80/20) br 6 6 1 br/gv (80/20) br 6 6 1 br br 6 6 1 br br 6 6 1 br/gv/vr (30/50/20) br 6 6 1 br/gv/vr (40/40/20) br 6 6 1 br/gv/vr (45/40/15) br 6 6 1 vr vr 4 4 3 gv gv 2 2 5 gv gv 2 2 5 vr/gv (45/55) gv 2 2 5 vr/gv (20/80) gv 2 2 5 vr vr 4 4 3 br BR 6 6 1 br BR 6 6 1 br/pied (95/5) pied 3 3 4 pied pied 3 3 4 pied/coap (70/30) coap 1 1 6 pied pied 3 3 4 pied/br (60/40) pied 3 3 4 pied pied 3 3 4 pied pied 3 3 4 330 piedmont coastal plateau greatv_val blue_ridge vr 0 0 100 0 0 0 0 0 100 0 0 0 0 0 0 50 10 40 0 0 0 0 100 0 0 0 0 0 100 0 0 0 0 0 100 0 100 0 0 0 0 0 100 0 0 0 0 0 0 0 0 0 100 0 0 0 0 0 100 0 30 0 0 0 70 0 100 0 0 0 0 0 100 0 0 0 0 0 100 0 0 0 0 0 80 20 0 0 0 0 0 100 0 0 0 0 0 100 0 0 0 0 0 100 0 0 0 0 0 0 0 60 0 40 0 0 0 50 50 0 0 0 0 70 20 10 0 0 0 20 80 0 0 0 0 20 80 0 0 0 0 0 100 0 0 0 0 0 100 0 0 0 0 50 30 20 0 0 0 40 40 20 0 0 0 40 45 15 0 0 0 0 0 100 0 0 0 100 0 0 0 0 0 100 0 0 0 0 0 55 0 45 0 0 0 80 0 20 0 0 0 0 0 100 0 0 0 0 100 0 0 0 0 0 100 0 5 0 0 0 95 0 100 0 0 0 0 0 70 30 0 0 0 0 100 0 0 0 0 0 60 0 0 0 40 0 100 0 0 0 0 0 100 0 0 0 0 0 331 Lithology$ Litho1$ ordinary for b ordinary for f sili/und (70/30) und 3 3 sili/und (70/30) und 3 3 carb/sili/crys/und (45/40/10/5) carb 2 2 carb carb 2 2 carb/und/crys (70/10/20) und 3 3 crys crys 4 4 sili/crys (60/40) sili 5 5 crys/sili (70/30) crys 4 4 crys crys 4 4 crys crys 4 4 crys/sili (60/40) sili 5 5 crys crys 4 4 crys crys 4 4 crys/sili (90/10) sili 5 5 crys/uncon (80/20) uncon 1 1 uncon uncon 1 1 uncon uncon 1 1 uncon uncon 1 1 carb/sili (50/50) carb 2 2 carb/sili (60/40) sili 5 5 carb/und/sili (60/10/30) sili 5 5 carb/und/crys (40/20/40) und 3 3 carb/und/crys (40/20/40) und 3 3 carb/und/crys (40/20/40) carb 2 2 crys crys 4 4 crys/und/carb/sili (20/10/50/20) sili 5 5 crys/und/carb/sili (10/10/50/30) sili 5 5 crys/und/carb/sili (25/5/50/20) sili 5 5 sili/und (85/15) und 3 3 carb/sili (70/30) sili 5 5 carb/sili (50/50) sili 5 5 sili/und/carb (45/10/45) carb 2 2 sili/und/carb (50/10/40) sili 5 5 sili/und (40/60) und 3 3 crys crys 4 4 crys crys 4 4 crys/sili (90/10) sili 5 5 crys crys 4 4 crys/unc (50/50) uncon 1 1 crys crys 4 4 crys/sili (70/30) sili 5 5 crys crys 4 4 crys crys 4 4 332 ordinary for m sili1 sili_und1 carbonate1 unconsol1 3 70 30 0 0 3 70 30 0 0 4 40 5 45 0 4 0 0 100 0 3 0 10 70 0 2 0 0 0 0 1 60 0 0 0 2 30 0 0 0 2 0 0 0 0 2 0 0 0 0 1 40 0 0 0 2 0 0 0 0 2 0 0 0 0 1 10 0 0 0 5 0 0 0 20 5 0 0 0 100 5 0 0 0 100 5 0 0 0 100 4 50 0 50 0 1 40 0 60 0 1 30 10 60 0 3 0 20 40 0 3 0 20 40 0 4 0 20 40 0 2 0 0 0 0 1 20 10 50 0 1 30 10 50 0 1 20 5 50 0 3 85 15 0 0 1 30 0 70 0 1 50 0 50 0 4 45 10 45 0 1 50 10 40 0 3 40 60 0 0 2 0 0 0 0 2 0 0 0 0 1 10 0 0 0 2 0 0 0 0 5 0 0 0 50 2 0 0 0 0 1 30 0 0 0 2 0 0 0 0 2 0 0 0 0 333 crystalline1 Landuse$ landuse1$ ordinary for b ordinary for f 0 f f 0 f f 10 a/f (70/30) a 0 a a 20 a/f (80/20) f 100 a a 40 a a 70 a a 100 f f 100 f f 60 f f 100 a a 100 a/f (90/10) f 90 a/f (90/10) f 80 a/u (20/80) u 0 a/f/u (20/50/30) f 0 a/f (30/70) a 0 f f 0 a/f (50/50) a 0 a/f (90/10) a 0 a a 40 a/f (50/50) a 40 a/f/u (40/50/10) f 40 a/f (40/60) f 100 f f 20 a/f (60/40) f 10 a/f (60/40) f 25 a/f (50/50) f 0 f f 0 a a 0 a/f (70/30) a 0 a/f (90/10) a 0 a/f (60/40) a 0 f f 100 a a 100 a a 90 a a 100 a/f/u (20/30/50) f 50 u u 100 f/u (10/90) u 70 a a 100 f f 100 f f 334 ordinary for m forest2 agrric2 urban2 Bed material$ 100 0 0 c/r (50/50) 100 0 0 g/s (50/50) 30 70 0 r/g (90/10) 0 100 0 r/g/si (33/33/34) 20 80 0 g 0 100 0 g/s (50/50) 0 100 0 g/c (50/50) 0 100 0 g/r (80/20) 100 0 0 c/r (50/50) 100 0 0 c/r (50/50) 100 0 0 g/c (50/50) 0 100 0 g/s (50/50) 10 90 0 g/s/r(33/33/34) 10 90 0 g/s (50/50) 0 20 80 clay/s (70/30) 50 20 30 g/s (50/50) 70 30 0 s/si (50/50) 100 0 0 g/s/clay (25/25/50) 50 50 0 r/g (50/50) 10 90 0 g/s (50/50) 0 100 0 r 50 50 0 r/c (50/50) 50 40 10 g/c (50/50) 60 40 0 r/c(50/50) 100 0 0 r/g (80/20) 40 60 0 r/c (50/50) 40 60 0 r/c (50/50) 50 50 0 r/g/s (80/10/10) 100 0 0 r/c (50/50) 0 100 0 r/c (50/50) 30 70 0 c/g (50/50) 10 90 0 r/g (50/50) 40 60 0 r/g (50/50) 100 0 0 g/c (50/50) 0 100 0 c/s (50/50) 0 100 0 s/g (50/50) 0 100 0 g 30 20 50 r/s (50/50) 0 0 100 g/s (50/50) 10 0 90 r/g (50/50) 0 100 0 c/g (50/50) 100 0 0 s 100 0 0 r/c (50/50) 335 ordinary for b ordinary for f ordinary for m cobble3 1.5 5.5 1.5 50 3.5 3.5 3.5 0 1.2 5.8 1.2 0 3 4 3 0 3 4 3 0 3.5 3.5 3.5 0 2.5 4.5 2.5 50 2.6 4.4 2.6 0 1.5 5.5 1.5 50 1.5 5.5 1.5 50 2.5 4.5 2.5 50 3.5 3.5 3.5 0 2.7 4.3 2.7 0 3.5 3.5 3.5 0 5.4 1.6 5.4 0 3.5 3.5 3.5 0 4.5 2.5 4.5 0 4.75 2.25 4.75 0 2 5 2 0 3.5 3.5 3.5 0 1 6 1 0 1.5 5.5 1.5 50 2.5 4.5 2.5 50 1.5 5.5 1.5 50 1.4 5.6 2 0 1.5 5.5 1.5 50 1.5 5.5 1.5 50 1.5 5.5 1.5 0 1.5 5.5 1.5 50 1.5 5.5 1.5 50 2.5 4.5 2.5 50 2 5 2 0 2 5 2 0 2.5 4.5 2.5 50 3 4 3 50 3.5 3.5 3.5 0 3 4 3 0 3 4.5 2.5 0 3.5 3.5 3.5 0 2 5 2 0 2.5 2.5 2.5 50 4 3 4 0 3.5 5.5 1.5 50 336 rock3 gravel3 sand3 silt3 clay3 bank material 50 0 0 0 0 co/rock (50/50) 0 50 50 0 0 san 90 10 0 0 0 silt 33 33 0 34 0 silt 0 100 0 0 0 silt 0 50 50 0 0 san 0 50 0 0 0 g/san (50/50) 20 80 0 0 0 silt 50 0 0 0 0 rock 50 0 0 0 0 g/san (50/50) 0 50 0 0 0 rock 0 50 50 0 0 silt 34 33 33 0 0 san/silt (50/50) 0 50 50 0 0 san/silt (50/50) 0 0 30 0 70 san/silt (50/50) 0 50 50 0 0 silt 0 0 50 50 0 san/silt (50/50) 0 25 25 0 50 san 50 50 0 0 0 rock/san (50/50) 0 50 50 0 0 san 100 0 0 0 0 co/san (50/50) 50 0 0 0 0 co/san (50/50) 0 50 0 0 0 san/clay (50/50) 50 0 0 0 0 co/san (50/50) 80 20 0 0 0 san 50 0 0 0 0 san/clay (50/50) 50 0 0 0 0 co/san (50/50) 80 10 10 0 0 silt 50 0 0 0 0 rock 50 0 0 0 0 san/co (50/50) 0 50 0 0 0 san/co (50/50) 50 50 0 0 0 co/clay (50/50) 50 50 0 0 0 san/g (50/50) 0 50 0 0 0 co/rock (50/50) 0 0 50 0 0 co/san (50/50) 0 50 50 0 0 san/clay (50/50) 0 100 0 0 0 san/rock (50/50) 50 0 50 0 0 san/rock (50/50) 0 50 50 0 0 san 50 50 0 0 0 san/co (50/50) 0 50 0 0 0 san/co (50/50) 0 0 100 0 0 san 50 0 0 0 0 rock/san (50/50) 337 ordinary for b ordinary for f ordinary for m cobble4 rock4 3.5 2.5 1.5 50 50 1 6 3 0 0 5 5 4 0 0 5 5 4 0 0 5 5 4 0 0 1 6 3 0 0 3.5 3.5 3.5 0 0 5 2 5 0 0 1 6 1 0 100 3.5 3.5 3.5 0 0 1 6 1 0 100 5 2 5 0 0 4.5 2.5 4.5 0 0 4.5 2.5 4.5 0 0 4.5 2.5 4.5 0 0 5 2 5 0 0 4.5 2.5 4.5 0 0 4 3 4 0 0 2.5 4.5 2.5 0 50 4 3 4 0 0 3 4 3 50 0 3 4 3 50 0 5 2 5 0 0 3 4 3 50 0 4 3 4 0 0 5 2 5 0 0 3 4 3 50 0 5 2 5 0 0 1 6 1 0 100 3 4 3 50 0 3 4 3 50 0 4 3 4 50 0 3.5 3.5 3.5 0 0 1.5 5.5 1.5 50 50 3 4 3 50 0 5 2 5 0 0 2.5 4.5 2.5 0 50 2.5 4.5 2.5 0 50 4 3 4 0 0 3 4 3 50 0 3 4 3 50 0 4 3 4 0 0 2.5 4.5 2.5 0 50 338 sand4 silt4 gravel4 s4 clay4 grav4 ch_Pattern$ 0 0 0 0 0 0 S 100 0 0 0 0 0 S 0 100 0 0 0 0 M 0 100 0 0 0 0 M 0 100 0 0 0 0 B 100 0 0 0 0 0 B 50 0 50 0 0 0 B 0 100 0 0 0 0 M 0 0 0 0 0 0 M 50 0 50 0 0 0 M 0 0 0 0 0 0 B 0 100 0 0 0 0 B 50 50 0 0 0 0 S 50 50 0 0 0 0 B 50 50 0 0 0 0 B 0 100 0 0 0 0 S 50 50 0 0 0 0 B 100 0 0 0 0 0 B 50 0 0 0 0 0 S 100 0 0 0 0 0 M 50 0 0 0 0 0 B 50 0 0 0 0 0 M 50 0 0 0 0 0 S 50 0 0 0 0 0 M 100 0 0 0 0 0 B 50 0 0 0 50 0 M 50 0 0 0 0 0 B 0 100 0 0 0 0 S 0 0 0 0 0 0 B 50 0 0 0 0 0 B 50 0 0 0 0 0 M 0 0 0 0 50 0 B 50 0 50 0 0 0 B 0 0 0 0 0 0 B 50 0 0 0 0 0 M 50 0 0 0 50 0 B 50 0 0 0 0 0 B 50 0 0 0 0 0 M 100 0 0 0 0 0 B 50 0 0 0 0 0 B 50 0 0 0 0 0 B 100 0 0 0 0 0 M 50 0 0 0 0 0 M 339 ordinary for b ordinary for f ordinary for m ch_Shape$ ordinary for b 3 3 1 B 1 3 3 1 P 3 2 2 2 B 1 2 2 2 R 4 1 1 3 P 3 1 1 3 R 4 1 1 3 T 2 2 2 2 R 4 2 2 2 R 4 2 2 2 R 4 1 1 3 R 4 1 1 3 P 3 3 3 1 B 1 1 1 3 R 4 1 1 3 B 1 3 3 1 B 1 1 1 3 T 2 1 1 3 R 4 3 3 1 R 4 2 2 2 B 1 1 1 3 P 3 2 2 2 T 2 3 3 1 R 4 2 2 2 T 2 1 1 3 B 1 2 2 2 R 4 1 1 3 B 1 3 3 1 R 4 1 1 3 T 2 1 1 3 T 2 2 2 2 P 3 1 1 3 T 2 1 1 3 R 4 1 1 3 T 2 2 2 2 T 2 1 1 3 R 4 1 1 3 B 1 2 2 2 T 2 1 1 3 P 3 1 1 3 P 3 1 1 3 T 2 2 2 2 T 2 2 2 2 B 1 340 ordinary for f ordinary for m Asymmetry abs_asym topomin 4 1 -0.131 0.131 3.165 2 3 0.032 0.032 -3.578 4 1 0.089 0.089 -3.588 1 4 0.008 0.008 2.952 2 3 -0.056 0.056 4.404 1 4 0.091 0.091 4.163 3 2 -0.008 0.008 4.317 1 4 -0.023 0.023 -3.59 1 4 0.02 0.02 3.711 1 4 -0.048 0.048 3.219 1 4 0.029 0.029 -2.393 2 3 0.002 0.002 1.475 4 1 -0.012 0.012 4.995 1 4 -0.047 0.047 3.912 4 1 0.09 0.09 3.47 4 1 0.009 0.009 3.544 3 2 0.013 0.013 3.584 1 4 0.014 0.014 3.807 1 4 -0.009 0.009 2.832 4 1 -0.13 0.13 3.507 2 3 -0.123 0.123 1.2 3 2 -0.044 0.044 2.996 1 4 -0.026 0.026 2.996 3 2 -0.014 0.014 2.974 4 1 -0.043 0.043 3.551 1 4 0.029 0.029 1.2 4 1 -0.201 0.201 1.2 1 4 -0.039 0.039 1.2 3 2 -0.02 0.02 2.567 3 2 0.057 0.057 3.667 2 3 -0.015 0.015 2.742 3 2 -0.004 0.004 2.567 1 4 -0.022 0.022 2.567 3 2 -0.06 0.06 3.387 3 2 0.075 0.075 3.584 1 4 0.013 0.013 3.501 4 1 0.125 0.125 3.368 3 2 -0.025 0.025 4.002 2 3 0.051 0.051 4.099 2 3 -0.03 0.03 4.327 3 2 0.086 0.086 3.832 3 2 0.136 0.136 4.3 4 1 0.199 0.199 4.078 341 topomax topomean topostdev topomin tmin/size 20.624 6.828 1.991 3.165 0.013 21.89 6.965 2.023 -3.578 -0.004 21.344 7.588 2.125 -3.588 -0.007 20.754 7.427 2.168 2.952 0.156 18.557 7.935 1.994 4.404 0.016 15.878 7.112 1.722 4.163 0.062 12.207 6.812 1.434 4.317 0.138 21.695 7.644 2.075 -3.59 -0.035 19.261 7.17 2.134 3.711 1.734 19.392 7.438 2.018 3.219 0.803 19.014 7.448 2.055 -2.393 -6.297 18.789 7.325 2.181 1.475 0.002 18.315 8.212 1.809 4.995 0.080 18.035 7.548 2.177 3.912 0.039 19.321 7.087 2.076 3.47 0.070 19.745 7.463 2.1 3.544 0.090 16.491 7.277 1.96 3.584 0.194 18.009 7.04 2.148 3.807 0.159 21.033 6.868 2.143 2.832 0.007 19.403 6.967 2.309 3.507 0.050 21.033 6.997 2.287 1.2 0.003 19.954 6.923 2.286 2.996 0.024 20.13 6.934 2.288 2.996 0.020 20.467 6.982 2.285 2.974 0.014 15.779 6.048 1.801 3.551 1.830 22.093 6.998 2.236 1.2 0.001 22.335 6.982 2.236 1.2 0.001 22.509 6.972 2.239 1.2 0.001 20.46 6.396 2.128 2.567 0.012 18.922 7.173 2.18 3.667 0.081 19.663 7.158 2.137 2.742 0.029 21.251 6.867 2.182 2.567 0.005 21.761 6.964 2.186 2.567 0.003 19.701 6.885 2.078 3.387 0.033 19.61 7.406 2.136 3.584 0.040 19.88 7.096 2.087 3.501 0.028 20.897 7.284 2.11 3.368 0.010 19.17 7.343 2.078 4.002 0.069 18.643 7.592 2.035 4.099 0.122 18.284 7.567 2.008 4.327 0.184 19.65 7.635 2.082 3.832 0.041 17.148 7.539 1.966 4.3 0.563 18.676 7.551 2.062 4.078 0.117 342 topomax tmax/size topomean tmean/size topostdev 20.624 0.083 6.828 0.028 1.991 21.89 0.025 6.965 0.008 2.023 21.344 0.043 7.588 0.015 2.125 20.754 1.098 7.427 0.393 2.168 18.557 0.066 7.935 0.028 1.994 15.878 0.237 7.112 0.106 1.722 12.207 0.390 6.812 0.218 1.434 21.695 0.213 7.644 0.075 2.075 19.261 9.000 7.17 3.350 2.134 19.392 4.836 7.438 1.855 2.018 19.014 50.037 7.448 19.600 2.055 18.789 0.188 7.325 0.073 2.181 18.315 0.366 8.212 0.164 1.809 18.035 0.361 7.548 0.151 2.177 19.321 0.386 7.087 0.142 2.076 19.745 0.197 7.463 0.075 2.1 16.491 0.330 7.277 0.146 1.96 18.009 0.750 7.04 0.293 2.148 21.033 0.055 6.868 0.018 2.143 19.403 0.277 6.967 0.099 2.309 21.033 0.056 6.997 0.019 2.287 19.954 0.157 6.923 0.055 2.286 20.13 0.135 6.934 0.047 2.288 20.467 0.097 6.982 0.033 2.285 15.779 8.134 6.048 3.118 1.801 22.093 0.020 6.998 0.006 2.236 22.335 0.016 6.982 0.005 2.236 22.509 0.014 6.972 0.004 2.239 20.46 0.097 6.396 0.030 2.128 18.922 0.416 7.173 0.158 2.18 19.663 0.211 7.158 0.077 2.137 21.251 0.042 6.867 0.014 2.182 21.761 0.028 6.964 0.009 2.186 19.701 0.191 6.885 0.067 2.078 19.61 0.219 7.406 0.083 2.136 19.88 0.162 7.096 0.058 2.087 20.897 0.063 7.284 0.022 2.11 19.17 0.331 7.343 0.127 2.078 18.643 0.553 7.592 0.225 2.035 18.284 0.778 7.567 0.322 2.008 19.65 0.210 7.635 0.082 2.082 17.148 2.245 7.539 0.987 1.966 18.676 0.535 7.551 0.216 2.062 343 tsddev/size w1/d1 w2/d2 w1/d1/w2/d2 w1 x1 0.008 90.32 45.51 1.985 44.75 l00.43 0.002 83.6 37.06 2.256 31.41 105.16 0.004 88.92 75.8 1.173 55.05 105.04 0.115 29.55 22.5 1.313 12.86 38.11 0.007 24.88 32.02 0.777 24.88 37.49 0.026 50.29 67.52 0.745 39.86 139.52 0.046 33.67 39.94 0.843 33.67 52.79 0.020 63.83 57.12 1.117 34.52 100.81 0.997 2.45 16.79 0.146 2.13 29.51 0.503 4.09 40.47 0.101 4.09 53.66 5.408 3.24 9.72 0.333 2.94 10.73 0.022 84.32 160.65 0.525 53.05 247.12 0.036 77.87 37.72 2.064 32.27 77.87 0.044 31.54 34.13 0.924 19.27 56.02 0.042 15.15 48.26 0.314 15.15 102.93 0.021 48.01 21.93 2.189 16.95 57.65 0.039 12.01 50.8 0.236 10.40 50.80 0.090 28.44 39.22 0.725 11.83 47.31 0.006 75.28 40.82 1.844 40.82 86.62 0.033 29.4 24.64 1.193 16.81 51.39 0.006 36.42 45.3 0.804 36.42 77.22 0.018 64.47 38.4 1.679 37.12 72.72 0.015 78.3 35.64 2.197 35.64 108.43 0.011 59.38 37.79 1.571 37.41 68.17 0.928 3.6 22.78 0.158 3.60 25.76 0.002 137.77 120.74 1.141 100.99 154.50 0.002 100.18 120.16 0.834 100.00 136.56 0.001 43.3 20.25 2.138 20.25 55.06 0.010 28.24 40.24 0.702 19.08 53.13 0.048 20.42 48.77 0.419 20.42 66.95 0.023 36.45 31.17 1.169 23.87 70.80 0.004 19.29 37.97 0.508 18.94 37.97 0.003 94.44 97.26 0.971 64.24 145.02 0.020 37.34 68.93 0.542 35.19 68.93 0.024 30.39 29.25 1.039 29.25 60.49 0.017 29.21 37.33 0.782 29.21 68.58 0.006 39.17 49.49 0.791 35.50 55.05 0.036 37.71 24.54 1.537 24.54 52.26 0.060 101.38 125.49 0.808 64.80 190.21 0.085 11.14 51.61 0.216 11.14 51.61 0.022 13.75 54.31 0.253 13.75 61.27 0.257 28.13 23.76 1.184 18.64 56.62 0.059 53.75 32.14 1.672 25.11 53.75 344 w/x Min Elev Max Elev Elev diff. Mean Elev Std Devia Perimeter(Mi) 2.24 214 911 697 566.213 146.317 108.090 3.35 189 1254 1065 693.802 266.490 285.080 1.91 118 749 631 268.562 123.116 248.340 2.96 107 197 90 152.000 14.038 35.550 1.51 97 660 563 248.755 106.152 137.230 3.50 113 582 469 315.581 109.402 59.690 1.57 124 260 136 177.034 27.966 44.040 2.92 111 346 235 201.412 42.433 77.580 13.84 324 582 258 471.690 57.098 9.200 13.12 302 582 280 484.572 48.147 12.070 3.65 261 509 248 417.406 56.376 2.940 4.66 38 602 564 198.272 114.934 204.180 2.41 73 387 314 167.840 37.308 56.520 2.91 66 273 207 144.432 32.859 65.540 6.79 6 179 173 99.839 52.002 59.840 3.40 4 86 82 61.018 15.538 49.740 4.88 13 177 164 109.784 31.943 29.770 4.00 3 48 45 30.120 6.955 32.120 2.12 332 1338 1006 629.720 236.035 143.040 3.06 365 677 312 482.991 65.622 63.160 2.12 315 1360 1045 503.772 130.505 138.460 1.96 396 1172 776 571.471 146.897 89.500 3.04 377 840 463 460.694 78.588 101.310 1.82 338 1027 689 498.621 138.874 131.270 7.16 457 882 425 668.902 104.118 7.710 1.53 299 1024 725 404.065 93.505 245.210 1.37 224 1180 956 467.043 173.052 323.530 2.72 140 1231 1091 379.287 187.577 394.920 2.78 320 1230 910 602.348 162.464 98.810 3.28 314 827 513 412.785 51.060 50.710 2.97 272 996 724 423.548 114.118 76.760 2.00 243 814 571 379.034 78.229 171.280 2.26 151 1006 855 370.394 152.408 229.560 1.96 204 953 749 421.646 157.441 84.240 2.07 78 514 436 175.617 58.769 63.830 2.35 105 673 568 234.813 103.924 80.560 1.55 74 537 463 159.970 49.864 137.900 2.13 45 159 114 107.311 19.523 50.750 2.94 11 150 139 82.024 24.808 41.540 4.63 59 150 91 106.284 15.970 36.290 4.45 63 416 354 131.296 43.182 68.780 3.04 73 129 57 104.479 10.967 18.070 2.14 53 136 83 94.998 16.194 40.800 345 perim/3.14 r size r square square root r region 34.424 17.212 247 78.662 8.869 G 90.790 45.395 875 278.662 16.693 H 79.089 39.545 494 157.325 12.543 F 11.322 5.661 18.9 6.019 2.453 G 43.704 21.852 281 89.490 9.460 A 19.010 9.505 66.9 21.306 4.616 B 14.025 7.013 31.3 9.968 3.157 F 24.707 12.354 102 32.484 5.699 G 2.930 1.465 2.14 0.682 0.826 B 3.844 1.922 4.01 1.277 1.130 B 0.936 0.468 0.38 0.121 0.348 A 65.025 32.513 817 260.191 16.130 OUT 18.000 9.000 62.8 20.000 4.472 F 20.873 10.436 101 32.166 5.671 F 19.057 9.529 49.4 15.732 3.966 B 15.841 7.920 39.5 12.580 3.547 H 9.481 4.740 18.5 5.892 2.427 G 10.229 5.115 24 7.643 2.765 F 45.554 22.777 379 120.701 10.986 F 20.115 10.057 70.1 22.325 4.725 H 44.096 22.048 375 119.427 10.928 A 28.503 14.252 127 40.446 6.360 F 32.264 16.132 149 47.452 6.889 F 41.806 20.903 212 67.516 8.217 I 2.455 1.228 1.94 0.618 0.786 B 78.092 39.046 1084 345.223 18.580 F 103.035 51.518 1377 438.535 20.941 G 125.771 62.885 1642 522.930 22.868 F 31.468 15.734 210 66.879 8.178 I 16.150 8.075 45.5 14.490 3.807 F 24.446 12.223 93.2 29.682 5.448 F 54.548 27.274 506 161.146 12.694 A 73.108 36.554 768 244.586 15.639 F 26.828 13.414 103 32.803 5.727 A 20.328 10.164 89.6 28.535 5.342 A 25.656 12.828 123 39.172 6.259 E 43.917 21.959 332 105.732 10.283 F 16.162 8.081 57.9 18.439 4.294 F 13.229 6.615 33.7 10.732 3.276 F 11.557 5.779 23.5 7.484 2.736 B 21.904 10.952 93.4 29.745 5.454 D 5.755 2.877 7.64 2.433 1.560 H 12.994 6.497 34.9 11.115 3.334 G 346 BIBLIOGRAPHY Abrahams, A., 1984, Spatial dependency of hydraulic geometry exponents in a sub-alpine stream--A comment. 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