GEOMORPHIC MODELING AND ROUTING IMPROVEMENTS FOR GIS-BASED WATERSHED ASSESSMENT IN ARID REGIONS by Darius James Semmens A Dissertation Submitted to the Faculty of the SCHOOL OF RENEWABLE NATURAL RESOURCES In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY WITH A MAJOR IN WATERSHED MANAGEMENT In the Graduate College THE UNIVERSITY OF ARIZONA 2004 2 The University Of Arizona g Graduate College As members of the Final Examination Committee, we certify that we have read the dissertation prepared by Darius James Semmens Geomorphic Modeling and Routing Improvements for GIS-Based Watershed entitled Assessment in Arid Regions and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy Date Waite R. Osterkamp [ ' (71 4" Date Richard H. Hawkins 151 01.1Date icente Lopes e6 3/ 510 f - Dat D. Philip Guertin 3- cot Date Dat David9> Qrodrich Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copy of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. WELat1aCzei2 Dissertation Director Wai e R. Osterkamp Date 2 6: t 9 3 STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. SIGNED: APPROVAL BY DISSERTATION DIRECTOR This thesis has been approved on the date shown below: -7)40,te,4 Waite R. Osterkamp Adjunct Professor of Watershed Management z 2 e20 Date 4 ACKNOWLEDGEMENTS This dissertation was inspired by my work at the USDA-ARS Southwest Watershed Research Center (SWRC), and drew substantially on my training in geomorphology at Northern Arizona University (NAU) and in watershed hydrology and GIS at the University of Arizona (U of A). A great number of people were instrumental to my efforts, and I am indebted to all of them. At the SWRC I benefited greatly from the experience and guidance of my supervisor, David Goodrich. Scott Miller was an invaluable source of knowledge on many fronts, and our constant exchange of ideas was invaluable during the formative stages of this research. Integrating the geomorphic and routing models into KINEROS2 could not have been accomplished without the generous assistance of Carl Unkrich, a master problem solver. Ryan Miller, Mariano Hernandez, Lainie Levick, Soren Scott, Averill Cate, Shea Burns, Ginger Paige, and Mary White all contributed significantly to a very enjoyable and productive tenure at the ARS. I came to the University of Arizona largely because of the encouragement of my advisor, Waite Osterkamp. He was instrumental in getting me to Tucson, and providing me with the funding to get me started. Although my research ultimately diverged from our original plan, Waite was unwavering in his support, and his thoughtful comments on this dissertation have improved it substantially. Phil Guertin played a central role in this research. To him I am indebted for my initial training in spatial analysis and watershed management, my position at the ARS, helping shape my ideas about the geomorphic assessment tool, and his guidance during its development and testing. Also at the U of A, Richard Hawkins, Craig Wissler, and Vicente Lopes were key instructors and patient committee members. Funding for this research came from the U.S. Environmental Protection Agency Landscape Ecology Branch for the Automated Geospatial Watershed Assessment (AGWA) project. That project was overseen by William Kepner, who has been a mentor and a friend over the course of my involvement with it. This research would not have been possible were it not for the love, support, and patience my wife, Betsy. She is the center of my life before, during, and after work. I cannot thank her enough. 5 DEDICATION To Betsy and Kira. You are the smile at the beginning and end of my everyday. 6 TABLE OF CONTENTS LIST OF ILLUSTRATIONS 8 LIST OF TABLES 13 ABSTRACT 14 1.0 INTRODUCTION 16 1.1 PROBLEM STATEMENT 1.1.1 Intermediate-Scale Watershed Runoff Models 1.1.2 Watershed Geomorphic Models 16 17 19 1.2 HYPOTHESES 1.3 APPROACH 21 21 1.3.1 Runoff Routing 1.3.2 Geomorphic Modeling 1.4 STUDY AREA 2.0 ROUTING MODEL 22 22 24 27 2.1 INTRODUCTION 2.2 SCALE-RELATED PROBLEMS IN WATERSHED MODELING 2.3 CHANNELIZED FLOW ROUTING IN WATERSHED MODELS - PREVIOUS WORK 2.3. 1 Evaluating Which Model is Applicable 2.3.2 Diffusion wave modeling with the Muskingum-Cunge scheme 2.3.3 Practical Considerations 27 28 30 34 45 50 2.4 ROUTING IN AGWA 2.5 MODEL DESCRIPTION 2.6 MVPMC4 INSTABILITY AT Low FLOW 51 52 3.0 ROUTING MODEL RESULTS 3.1 METHODOLOGY 3.2 DESIGN STORM RESULTS 3.2. 1 Komolgorov-Smirnov Comparison 3.2.2 Metric and Visual Comparisons 3.3 OBSERVED STORM RESULTS 3.4 DISCUSSION 4.0 GEOMORPHIC MODEL 4.1 CHANNEL-MORPHOLOGIC MODELS 4.2 CHANNEL NETWORK EVOLUTION (SURFICIAL PROCESS) MODELS 4.3 WATERSHED GEOMORPHIC MODELS 4.3 APPROACH 4.5 NUMERICAL MODEL 4.5.1 Channel adjustments — deposition 4.5.2 Channel adjustments — erosion 4.5.3 Erosion and Sediment Transport in K2G 4.6 GEOMORPHIC MODELING TOOL 4.6.1 Profile Smoothing 4.6.2 Results of Profile Smoothing 54 59 59 60 61 62 70 78 81 81 85 88 93 94 101 106 110 117 118 127 7 4.6.3 Mass Balance Calculations 4.7 PROBLEMS 4.8 LIMITATIONS 5.0 GEOMORPHIC MODEL RESULTS 129 131 134 137 5.1 METHODOLOGY 5.2 RESULTS 137 145 5.2.1 Graphical Output 5.2.2 Mass-Balance Calculations 5.2.3 A Perspective on Model Output 5.3 DISCUSSION 145 158 164 167 6.0 RELATIVE GEOMORPHIC CHANGE AND CHANNEL STABILITY 169 6.1 MODEL ERROR AND RELATIVE CHANGE ASSESSMENT 6.2 METHODOLOGY 6.3 RESULTS 169 171 172 6.3.1 Relative Differences for the Hydraulic-Geometry Channels 6.3.2 Relative Differences for the Observed Channel Geometries 6.3.3 Relative Differences in Runoff Transmission Loss, and Sediment Yield 6.4 DISCUSSION 172 174 176 179 7.0 SUMMARY AND CONCLUSIONS 7.1 WATERSHED RUNOFF ROUTING AND SCALE 7.1.1 Conclusions 7.1.2 Future Research 7.2 GEOMORPHIC MODELING AT THE WATERSHED SCALE 7.2.1 Conclusions 7.2.2 Future Research APPENDIX A: DESIGN AND OBSERVED EVENT HYDROGRAPHS A.1 DESIGN STORM HYDROGRAPHS A.2 OBSERVED EVENT HYDROGRAPHS APPENDIX B: DERIVATION OF CHANNEL GEOMETRY ADJUSTMENTS B.1 DEPOSITIONAL DEPTH CHANGES B.2 EROSIONAL DEPTH CHANGES B.3 BANK FAILURE RESULTING FROM EROSIONAL OVERSTEEPENING B.4 WIDTH CHANGES REFERENCES 181 181 182 182 183 184 187 190 190 190 204 204 205 207 208 209 8 LIST OF ILLUSTRATIONS Figure 1.1. Map of the Walnut Gulch Experimental Watershed showing cultural features. 25 Figure 1.2. Map showing Walnut Gulch subwatersheds. Subwatersheds used in this 26 research were WG1, WG6, WG11, and LH104 characteristic space-time a range of Graph showing hydrological processes at Figure 2.1. scales, and the superimposed spatio-temporal domains of large, small, and intermediate-scale watershed models for arid regions. 30 Figure 2.2. Graphs derived using the Ponce et al. (1978) equation for determining when the diffusion wave model yields reasonable results in comparison to the dynamic wave model. (A) Minimum wave period vs. flow depth; (B) Minimum wave period 36 vs. channel slope. Figure 2.3. Graphs illustrating the minimum wave period required for the kinematic wave model to be 95% as accurate as the diffusion wave model. Values are shown 38 for a range of (A) flow depths, and (B) channel slopes. Figure 2.4. Graph showing division of the slope/reference water depth space into four regions characterized by different wave types for a channel length of 1000 meters 39 and a Froude number of 0.4. Figure 2.5. Graphs showing river wave approximation zones obtained from the analysis 40 of the momentum equation for three values of i = 1, 8, and 20. dynamic-wave results that of kinematicand Figure 2.6. Graph showing a comparison 42 demonstrating the propagation of errors in the kinematic wave solution. Figure 2.7. Graph showing the influence of varying the time and space step during finite43 difference solutions of the kinematic wave equation. 49 Figure 2.8. Space-time discretization of Ponce and Yevjevich (1978) Figure 3.1. K2 and K2MC simulated WG6 runoff hydrographs for design storms ranging from the 5-year 30-minute event (upper left) to the 100-year 1-hour event (lower 63 right). Figure 3.2. Graph showing normalized differences in the time to peak discharge at the 65 watershed outlet between K2 and K2MC, plotted as a function of area. Figure 3.3. Graph showing normalized differences in the simulated peak discharge at the 66 watershed outlet between K2 and K2MC, plotted as a function of area. Figure 3.4. Graph showing relative error in the total discharge volume at the watershed 67 outlet between K2 and K2MC as a function of area. Figure 3.5. Graph showing the average trends in mass-balance error as a function of 69 watershed area Figure 3.6. Graph showing the best match of simulated and observed hydrographs for 72 WG1 Figure 3.7. Graph showing the best match of simulated and observed hydrographs for 72 WG6 Figure 3.8. Graph showing the best match of simulated and observed hydrographs for 74 WG11 9 Figure 3.9. Graph showing the best match of simulated and observed hydrographs for LH104 74 Figure 3.10. Graph showing average model efficiency plotted as a function of watershed area for K2 and K2MC 76 Figure 3.11. Graph showing average error (%) in the timing of flow onset at the watershed outlet plotted as a function of watershed area for K2 and K2MC 77 Figure 3.12. Graph showing average error (%) in the timing of peak flows at the watershed outlet plotted as a function of watershed area for K2 and K2MC 77 Figure 4.1. Schematic representation of erodible trapezoidal channels in CASC2D 90 Figure 4.2. Profile adjustment over 18 years in the Goodwin Creek Watershed 92 Figure 4.3. Conceptualization of a compound channel in K2G showing separate channel and overbank areas. 96 Figure 4.4. Graph showing change in sediment storage for each computational node in a reach 100 Figure 4.5. Sketch illustrating depositional width reduction for the K2G trapezoidal channels 102 Figure 4.6. Sketch illustrating depositional depth reduction for the K2G trapezoidal channels 103 Figure 4.7. Plots showing depositional adjustments at the upstream (A) and downstream (B) ends of a 210-meter reach following a 52 mm rainfall event with a peak flow of 35.3 m 3 /s (61.9 mm/hr). 105 Figure 4.8. Sketch illustrating erosional width increase for the K2G trapezoidal channels 106 Figure 4.9. Sketch illustrating erosional depth increase for the K2G trapezoidal channels 108 Figure 4.10. Plots showing erosional adjustments at the upstream (A) and downstream (B) ends of a 25-meter reach following a 52 mm rainfall event with a peak flow of 3.1 m 3/s (129.7 mm/hr). 109 Figure 4.11. Graph showing measured and simulated runoff for an event on the Catsop catchment in South Limburg, Netherlands. From Smith et al. (1999). 115 Figure 4.12. Comparison of discharge (Qt), sediment transport rate (St), and concentration (Ct) between measured rainfall simulation data (circles) and K2predicted values for three events on road plots. From Ziegler et al. (2001). 116 Figure 4.13. Graph showing the channel profile near the outlet of Walnut Gulch Subwatershed 223 before and after a K2G batch simulation of five 10-year 1-hour events and no profile smoothing between events 120 Figure 4.14. Diagrammatic representation of two-reach junction profile adjustments 121 showing variables used in derivation of the adjusted elevation, EA Figure 4.15. Diagrammatic representation of three-reach junction profile adjustments 124 showing variables used in derivation of the adjusted elevation, EA Figure 4.16. Walnut Gulch Subwatershed 223 profile near the outlet before and after a K2G batch simulation of five 10-year 1-hour events with profile smoothing between events 128 10 Figure 5.1. Maps showing the watershed discretization (A), DEM (B), SSURGO soils (C), and rain gauge configuration (D) for Walnut Gulch subwatershed 11. 139 Figure 5.2. Maps showing the four land-cover scenarios used to derive model parameters for the test simulations: (A) NALC 1973, (B) NALC 1997, (C) Partial urbanization, and (D) Total urbanization. The legend for the NALC classification scheme is shown for reference. Outlines of the discretized model elements are superimposed on the land-cover grids 142 Figure 5.4. Maps showing depth and width changes (left and right, respectively) for simulations 1964_0G_73, 1977_0G_73, and 1978_0G_73 (top to bottom, respectively). 150 Figure 5.5. Maps showing depth and width changes (left and right, respectively) for simulations 1964_HG_97, 1977_HG_97, and 1978_HG_97 (top to bottom, respectively). 151 Figure 5.6. Maps showing depth and width changes (left and right, respectively) for simulations 1964_0G_97, 1977_0G_97, and 1978_0G_97 (top to bottom, respectively). 152 Figure 5.7. Maps showing depth and width changes (left and right, respectively) for simulations 1964_HG_PU, 1977_HG_PU, and 1978_HG_PU (top to bottom, respectively). 153 Figure 5.8. Maps showing depth and width changes (left and right, respectively) for simulations 1964_0G_PU, 1977_0G_PU, and 1978_0G_PU (top to bottom, respectively). 154 Figure 5.9. Maps showing depth and width changes (left and right, respectively) for simulations 1964_HG_U, 1977_HG_U, and 1978_HG_U (top to bottom, respectively). 155 Figure 5.10. Maps showing depth and width changes (left and right, respectively) for simulations 1964_0G_U, 1977_0G_U, and 1978_0G_U (top to bottom, respectively). 156 Figure 5.11. Cumulative runoff depth (mm) per unit contributing area for the three batch simulations arranged in order of decreasing rainfall total from 1964 (top) to 1978 (bottom). Pictured results are for the hydraulic geometry simulations, but the spatial patterns for the observed geometry simulations were virtually identical 157 Figure 5.12. Bar graph showing the model (red) and geometric (blue) net mass-balance error (%) for 1964 batch simulations. 161 Figure 5.13. Bar graph showing the relative magnitude of channel erosion (kg) and the mass equivalent of the simulated geometric adjustment for the 1964 batch simulations. 162 Figure 5.14. Bar graph showing the model (red) and geometric (blue) net mass-balance error (%) for 1977 batch simulations. 163 Figure 5.15. Bar graph showing the relative magnitude of channel erosion (kg) and the mass equivalent of the simulated geometric adjustment for the 1977 batch simulations. 164 Figure 6.1. Maps showing the difference (meters) in simulated average depth change (left) and average width change (right) between the partial urbanization and NALC 11 1997 land cover scenarios for the 1964 (top) and 1977 (bottom) batch simulations based on the HG channels. Shades of red (positive values) indicate that geomorphic adjustments yielded relatively larger depths/widths for the PU scenario 173 Figure 6.2. Maps showing the difference (meters) in simulated average depth change (left) and average width change (right) between the partial urbanization and NALC 1997 land cover scenarios for the 1964 (top) and 1977 (bottom) batch simulations based on the OG channels. Shades of red (positive values) indicate that geomorphic adjustments yielded relatively larger depths/widths for the PU scenario 175 Figure 6.3. Maps showing the relative differences in cumulative runoff (mm, top), transmission loss (m 3 /km, middle), and sediment yield (kg, bottom) for the 1964 simulations. Results for the HG and OG channels are pictured on the left (A-C), and right (D-F), respectively. 177 Figure 6.4. Maps showing the relative differences in cumulative runoff (mm, top), transmission loss (m 3 /km, middle), and sediment yield (kg, bottom) for the 1977 simulations. Results for the HG and OG channels are pictured on the left (A-C), and 178 right (D-F), respectively. Figure A.1. K2 and K2MC simulated LH104 hydrographs for design storms ranging from the 5-year 30-minute event (upper left) to the 100-year 1-hour event (lower right). Plots are discharge in cubic meters per second as a function of time in 192 minutes. Figure A.2. K2 and K2MC simulated WG11 hydrographs for design storms ranging from the 5-year 30-minute event (upper left) to the 100-year 1-hour event (lower right). Plots are discharge in cubic meters per second as a function of time in minutes. 193 Figure A.3. K2 and K2MC simulated WG6 runoff hydrographs for design storms ranging from the 5-year 30-minute event (upper left) to the 100-year 1-hour event (lower right). Plots are discharge in cubic meters per second as a function of time in 194 minutes. Figure A.4. K2 and K2MC simulated WG1 hydrographs for design storms ranging from the 5-year 30-minute event (upper left) to the 100-year 1-hour event (lower right). Plots are discharge in cubic meters per second as a function of time in minutes 195 Figure A.5. Observed and simulated hydrographs for the 8/1/78 event on LH104. 196 Figure A.6. Observed and simulated hydrographs for the 7/28/81 event on LH104 196 Figure A.7. Observed and simulated hydrographs for the 9/10/83 event on LH104 196 Figure A.8. Observed and simulated hydrographs for the 8/10/86 event on LH104 197 Figure A.9. Observed and simulated hydrographs for the 8/29/86 event on LH104 197 Figure A.10. Observed and simulated hydrographs for the 7/27/76 event on WG11 197 Figure A.11. Observed and simulated hydrographs for the 8/1/78 event on WG11 198 Figure A.12. Observed and simulated hydrographs for the 8/4/80 event on WG11 198 Figure A.13. Observed and simulated hydrographs for the 8/29/86 event on WG11 198 Figure A.14. Observed and simulated hydrographs for the 8/1/90 event on WG11 199 Figure A.15. Observed and simulated hydrographs for the 7/27/76 event on WG6 199 Figure A.16. Observed and simulated hydrographs for the 8/1/78 event on WG6 199 Figure A.17. Observed and simulated hydrographs for the 8/4/80 event on WG6 200 12 Figure A.18. Observed and simulated hydrographs for the 8/29/86 event on WG6. 200 200 Figure A.19. Observed and simulated hydrographs for the 8/1/90 event on WG6 201 Figure A.20. Observed and simulated hydrographs for the 7/27/76 event on WG1 for the 8/1/78 event on WG1 201 simulated hydrographs Figure A.21. Observed and 201 Figure A.22. Observed and simulated hydrographs for the 8/4/80 event on WG1 202 Figure A.23. Observed and simulated hydrographs for the 7/30/81 event on WG1 202 Figure A.24. Observed and simulated hydrographs for the 8/17/86 event on WG1 202 Figure A.25. Observed and simulated hydrographs for the 8/2986 event on WG1 203 Figure A-26. Observed and simulated hydrographs for the 8/1/90 event on WG1. Figure B.1. Sketch illustrating the depositional depth reduction calculations for K2G 205 trapezoidal channels Figure B.2. Sketch illustrating the erosional depth increase calculations for K2G 206 trapezoidal channels Figure B.3. Sketch illustrating the bank failure calculations for K2G trapezoidal channels 207 Figure B.4. Sketch illustrating depositional width reduction calculations for K2G 208 trapezoidal channels 1.3 LIST OF TABLES Table 3.1. Results of the Komolgorov-Smirnov tests indicating whether the null hypothesis was accepted or rejected. Results are shown for the six design-storm events ranging from 5-year 30-minute (5 _ 30) to 100-year 1-hour (100_1) for each subwatershed. 62 Table 3.2. Difference between the timing of K2 and K2MC flow onset (mins) 64 Table 3.3. Mass balance error (%) computed from K2 and K2MC output files 68 Table 4.1. Sample output for one channel element from the geom.out file written during 118 a K2G simulation Table 4.2. Results of the volume-balance calculations for one 10-year 1-hour event with (PS) and without (NS) profile smoothing, and five 10-year, 1-hour events with profile smoothing. Insignificant digits are carried to prevent rounding error. 131 Batch simulations, and their associated inputs, used to evaluate the Table 5.1. 139 geomorphic model. Table 5.2. Characterization of the precipitation records for the summer monsoon on 140 WG11 during 1964, 1977, and 1978 Table 5.3. Composition of land cover classes on WG11 for the four land-cover scenarios 143 used during model testing. Table 5.4. Hydrologic parameters used by AGWA during land cover and soils 143 parameterization Table 5.5. Sediment grain-size distributions as percentages of the dry sample weight for 145 the channel and overbank elements. Table 5.6. Sediment mass balance results for the entire channel network from each simulation. Error is presented in percent, and everything else has the units of kilograms. 160 Table 5.7. Simulated cumulative outflow and sediment yield for the observed land-cover 165 scenarios Table 5.8. Summary of sediment accumulation on Walnut Gulch from Nichols and 166 Renard (2003). The italicized values are from a pond within WG11. 14 ABSTRACT Watershed models have two significant shortcomings that limit their application to management problems in arid and semi-arid regions. The first is that the performance of event-based hydrologic models for ephemeral stream networks declines significantly as watershed size increases. The second is that no single model is capable of simulating runoff, erosion, and geomorphic response in the channel network for multiple consecutive events. A diffusion-wave routing subroutine was developed for the Kinematic Runoff and Erosion Model (K_INEROS2) using a four-point iterative solution to the modified variable-parameter Muskingum-Cunge (MVPMC4) technique. It was tested against kinematic-wave routing at scales ranging from 0.05 to 150 km 2 on the Walnut Gulch Experimental Watershed in southeastern Arizona. Analyses demonstrated that MVPMC4 routing significantly improves simulated outflow hydrographs for small to moderate events on watersheds that are 95 km 2 and larger. A geomorphic model was developed by modifying KINEROS2 to compute width, depth and slope adjustments from computed changes in sediment storage at each time step. Width and depth adjustments are determined by minimizing total stream power for each reach. A GIS-based interface was developed for model parameterization, coordinating multiple-event batch simulations, tracking cumulative geomorphic change, computing the sediment mass balance, visualizing results, and comparing results from different simulations. 15 Simulated geomorphic adjustments are particularly sensitive to the number and magnitude of events in the rainfall record. Widespread erosion was predicted during the wettest sequence of rainfall events, mixed erosion and deposition during intermediate sequence, and predominantly deposition during the driest sequence. Simulation results from before and after urbanization in part of the watershed were compared for wet and intermediate rainfall records. Differences in computed geomorphic change in the unaffected part of the watershed were approximately zero. Erosion, primarily manifested as channel incision, increased within the urbanized area, and decreased downstream of it during the wet year. For the intermediate year, relative increases in deposition extended further upstream as transmission loss increased relative to the runoff volume. The overall pattern of relative geomorphic response was very similar regardless of the initial channel geometry, suggesting that the model can be used for broad-scale management and planning in the absence of detailed channel-geometry observations. 16 1.0 INTRODUCTION 1.1 Problem Statement Land, water, and biological resource managers have realized for some time that their mandates overlap substantially; the resources they manage are all parts of linked climatic, hydrologic, and geomorphic systems. These systems are defined and linked by the movement of water on the Earth's surface, and watersheds thus represent the most convenient spatial entities within which they can be described. The scientific tools available to identify, study, and remediate watershed problems, however, are often severely handicapped in that they only account for parts of the larger system, or can only be applied at very limited spatial and temporal scales. While much progress has been made towards improving watershed assessment tools in recent years, efforts have been primarily focused on humid environments. Arid and semi-arid regions, hereafter referred to collectively as arid regions, are characterized by processes operating at different spatial and temporal scales that cannot be adequately represented with assessment tools designed for humid regions. Comprising 30% of the global land surface, and housing roughly 20% of the human population, arid regions represent an important geography that has been largely overlooked in the development of landscape management and planning tools. This dissertation addresses two important limitations of arid-region watershed models that stand as obstacles to their widespread and interdisciplinary use as assessment and planning tools. The first limitation is one of scale, and the need to expand the spatial 17 limits of event-based hydrologic models by improving their performance on intermediatescale arid-region watersheds. The second limitation is the inability of watershed models to compute geomorphic change throughout a channel network in response to changing climate and land cover/use. 1.1.1 Intermediate-Scale Watershed Runoff Models A major difficulty in hydrologic modeling has been to bridge successfully the spatio-temporal transition between small watersheds with ephemeral streams and large basins with perennial flow. Numerous models are very successful on either end of this spectrum, but are less effective as changing spatial and temporal scales cause conceptual and numerical models to break down in the middle. This scale gap is generally described as occurring between about 100 and 1000 km 2 , depending on the climatic region and event magnitude (e.g. Goodrich, 1990; Goodrich et al., 1997; Syed, 1999). In humid environments, where rainfall is relatively uniform and subsurface flow buffers runoff hydrographs, large-scale watershed models operating on a daily time step can be successfully applied to smaller areas. This is not the case in arid and semi-arid environments where rainfall is highly spatially variable and runoff is generated almost exclusively by Hortonian overland flow. To model runoff in intermediate-scale arid watersheds it is thus necessary to use distributed precipitation input and a time step on the order of minutes to adequately represent the characteristic spatial and temporal variability of rainfall and runoff. This is most effectively accomplished by extending the range of scales in which small-scale (event-based) watershed models can be applied. 18 Previous studies have demonstrated that model efficiencies decrease substantially with increasing watershed size for event-based runoff models (e.g. Syed, 1999; Goodrich, 1990; Goodrich et al., 1997). In arid regions, characterized by influent channels, the runoff to rainfall ratio decreases with increasing watershed scale, thereby magnifying the effects of errors derived from the conceptual and numerical components of the model. A variety of sources of model error have been investigated, and it is widely believed that the largest source lies in the measurement of rainfall, both at a point and its distribution in space (Michaud and Sorooshian, 1994). Much recent work has been aimed at reducing rainfall error through the use of radar rainfall data (e.g. Ogden and Julien, 1993; Morin et al., 2003). Another source of error results from model conceptualization of nonchannelized surface runoff. The implementation of microtopography on upland plane elements (Smith et al., 1995), and the use of grid-based models (e.g. Ogden and Heilig, 2001) have been explored as a means of addressing this. Error resulting from the geometric complexity of watershed discretization has been investigated by numerous authors (e.g. Goodrich, 1990; Syed 1999; Miller, 2002). Errors resulting from land use/cover misclassification and the spatial resolution of soil data were explored by Miller (2002), and Bradley (2003), respectively. Whereas there is substantial literature on channel routing, the author was unable to identify any work that has specifically evaluated the merits of different routing techniques in watershed channel networks with respect to scale. The conditions under which various routing models are most suitable have been explored in detail, however, and based on this research it is clear that kinematic wave routing can accumulate error 19 when applied to large channel networks. Kinematic wave routing has been widely used in event-based watershed models because it is relatively easy to implement, and can be applied in the absence of downstream boundary conditions (e.g. USACE, 1994; Singh, 2002). The latter is particularly important for the simulation of ephemeral flows. Kinematic routing, however, cannot directly account for flood-wave diffusion, which becomes increasingly important as travel time increases and channel slopes decrease in the downstream direction. 1.1.2 Watershed Geomorphic Models The vast majority of geomorphic models are designed to facilitate the investigation of important engineering problems such as channel design and bridge scour. These models are applied on a reach basis, and require detailed channel geometry and all water and sediment inflows as input. As such, they cannot easily be used to investigate the cumulative impacts of dispersed inputs at larger scales. Land-use and climatic impacts cannot be simulated directly because the models do not contain upland routing and/or erosion subroutines. To investigate problems relating to spatially variable land management and planning it is necessary to conduct geomorphic simulations on a watershed basis. Changing conditions in one subwatershed can be negated or compounded as a result of conditions elsewhere in the watershed. Thus, accurate simulation of sediment discharge and geomorphic change requires simultaneous consideration of the upland areas and channel network throughout the watershed. Benefits of this type of approach include 20 improved estimates of sediment discharge and the ability to simulate probable geomorphic response to distributed land cover/use change. With a watershed-based geomorphic model it also becomes possible to compare the impacts of multiple change scenarios, and to identify problem areas (in the channel network and/or on the uplands) for selective remediation. Although the benefits of a watershed-based approach to geomorphic modeling have been widely recognized (e.g. Schumm et al., 1987), few watershed models possess the ability to track geomorphic change in the channel network over time, and only one of these, CASC2D (Ogden, 1998) can be applied to watersheds characterized by ephemeral flows. Over longer periods of time, however, it becomes increasingly necessary to account for changes in channel width in addition to depth (and hence slope) to account for changing sediment storage and delivery. No watershed models are available that simulate width, depth, and slope changes in arid and/or semi-arid environments. Despite the infrequent and highly variable nature of rainfall in arid regions, the rate of geomorphic change can be high. For instance, Graf (1983a) computed that approximately 6 million cubic meters of sediment were removed from the channel network of a 150 square kilometer semi-arid rangeland watershed in SE Arizona (Walnut Gulch) during a 15-year period. Similar erosional episodes were observed throughout the American Southwest during the late 1800s and early 1900s, and although no single causal factor can be identified, land use/cover change has commonly been interpreted as a significant contributor (Elliot et al., 1999). Given that the nine fastest growing cities in the United States (USCB, 2003) are in the arid and semi-arid Southwest, considerable 21 land cover/use change continues to occur in this region. In the absence of a watershed- based geomorphic model specifically designed for application in arid environments it is not possible to evaluate how cumulative landscape change impacts channel networks and the invaluable riparian corridors they support. 1.2 Hypotheses • A significant source of error in the use of kinematic routing at intermediate scales results from the inability to account directly for diffusion of the flood wave as it is routed through the channel network. • A continuous simulation event-based geomorphic model describing channel width and depth changes can simulate the anticipated geomorphic response to landscape change in semi-arid watersheds. 1.3 Approach The general approach in this research was to build on previous research by developing new components for the USDA-ARS Kinematic Runoff and Erosion Model, KINEROS2 (Smith et al., 1995), hereafter referred to as K2. Component development, testing, and evaluation varied significantly for the two objectives, however, and are described separately below. 22 1.3.1 Runoff Routing Alternative routing methods suitable for use in an intermediate-scale arid watershed model were explored, and the Muskingum-Cunge method with variable parameters, specifically the MVPMC4 methodology (Ponce and Chaganti, 1994), was selected as most suitable. A routing subroutine was developed for K2 based on this approach. The subroutine utilizes the same infiltration, rainfall, and sediment routing components of K2, and requires no additional inputs. Kinematic and MVPMC4 routing techniques were compared using a series of design (uniform) and observed (spatially variable) rainfall events. Simulated hydrographs were compared to determine if they were statistically different, and to define how they differed. Simulated and observed hydrographs were compared to evaluate model performance. All analyses were conducted for four subwatersheds in the Walnut Gulch Experimental Watershed: Lucky Hills 104 (LH104, 4.74 ha), Walnut Gulch 11 (WG11, 782 ha), Walnut Gulch 6 (WG6, 9,558 ha), and Walnut Gulch 1 (WG1, 14,664 ha). Results are presented as functions of watershed size and event magnitude. 1.3.2 Geomorphic Modeling A watershed geomorphic model was developed to compute width and depth changes on a reach basis. The geomorphic model is a modification of K2 that computes geomorphic change resulting from computed change in sediment storage following each time step. Width and depth change calculations are based on the minimum energy 23 dissipation rate theory (Yang and Song, 1979), and the theory's special case, the minimum stream power theory (e.g. Chang, 1980; Song and Yang, 1980). A GIS-based interface was developed for the geomorphic model to facilitate multiple-event simulations by tracking geomorphic change between simulations. The interface is based on the Automated Geospatial Watershed Assessment (AGWA) tool (Miller et al., 2002). The interface conducts most aspects of watershed delineation, discretization, and parameter estimation for geomorphic simulations. In addition, it provides tools for the spatial visualization of model results, and the comparison of multiple simulation outputs for change assessment. The geomorphic model was evaluated on subwatershed 11 of Walnut Gulch (WG11). Sediment mass balance was evaluated on a reach basis and for the channel network as a whole using two consecutive 10-year 1-hour design storm events. Model performance was evaluated by conducting three multiple-event (batch) simulations using all measured monsoonal events for wet (1977, 47 events), average (1964, 54 events), and dry (1978, 41 events) summers. The three batch simulations were each conducted based on parameter inputs derived from 1973 and 1997 land cover/use data sets, and a 100% urbanized artificial land-cover map. These tests were designed to evaluate the relative behavior of the model under different precipitation and land-use regimes in the absence of observed geomorphic change. Results from the geomorphic modeling are presented individually and relative to each other. The former is used as a means of evaluating whether the model produced physically realistic results in terms of the magnitude and spatial distribution of erosion in 24 the watershed. The latter is used to evaluate the model in terms of its ability to identify those reaches most susceptible to geomorphic change. 1.4 Study Area The study area for this research is the Walnut Gulch Experimental Watershed maintained by the USDA-ARS Southwest Watershed Research Center. Walnut Gulch is approximately 120 kilometers southeast of Tucson, and contains the town of Tombstone, Arizona (figure 1.1). The watershed is representative of mixed grass-brush land (Renard et al., 1993), and lies in the transition between the Sonoran and Chihuahuan deserts. The average elevation of the watershed is -1500 m, and the topography consists of gently rolling hills incised by alluvial channels of moderate slope. Hillslopes vary from 2 to 65%, and channel slopes are predominantly between 1 and 3%. Surface soil textures (0-5 cm) include gravelly and sandy barns containing an average of 30% rock and little organic matter (Renard et al., 1993; Kustas and Goodrich, 1994). Walnut Gulch receives 250 to 500 mm of rainfall annually, most occurring during two relatively rainy seasons: winter and summer. Winter rains are predominantly from relatively uniform frontal events of low intensity and long duration. Summer rains account for approximately 2/3 of the annual total, and are most commonly associated with highly non-uniform convective thunderstorms of relatively high intensity and short duration. Rainfall is recorded in the Walnut Gulch Watershed through a network of 96 recording rain gauges, or approximately one gauge for every 1.5 km2. 25 # ARS Headquarters Roads Improved dirt A/ Paved / State highway • Unimproved dirt ,A,„/ Utility access Land Ownership Bureau of Land Management Private State Trust A/ 2 0 2 4 6 Kilometers Figure 1.1. Map of the Walnut Gulch Experimental Watershed showing cultural features. Walnut Gulch is divided into 12 primary subwatersheds varying in size from 7.85 to 148 km 2 (figure 1.2). Each of the subwatersheds is equipped with a pre-calibrated critical-depth flume for runoff measurement. Twelve smaller subwatersheds, ranging in size from 0.4 to 89 ha, are equipped with flumes or detention ponds and weirs for runoff measurement. The present study concentrates on four subwatersheds that were selected to cover the range of available sizes. These include: Walnut Gulch 1 (WG1, 148 ktn 2 ), Walnut Gulch 6 (WG6, 93.6 km 2 ), WG11 (7.85 km2 ), and Lucky Hills 104 (LH104. 0.047 km2). 26 Watershed Configuration nested subwatersheds and measunna [... Primary Watershed Watershed 1 (WG1) 2 3 4 5 6 (WG6) 7 a 148 112 942 229 221 93.6 136 14.8 239 15.8 7.85 23.7 9 10 11 (WG11) 15 A 2 0 $ Stock Pond Runoff Measuring Device Recording Rain Gauge sInactive er Active Unit Area Subwatershed Gaged Pond Subwatershed I 1 Ungaged Pond Subwatershed Primary Subwatershed 2 4 Kilometers Map Created 9-21-01 by Scott Miller Figure 1.2. Map showing Walnut Gulch subwatersheds. Subwatersheds used in this research were WG1, WG6, WG11, and LH104. 27 2.0 ROUTING MODEL 2.1 Introduction The Automated Geospatial Watershed Assessment (AGWA) tool (Miller et al., 2002) is the most comprehensive and widely used GIS-based watershed assessment tool available specifically for arid and semi-arid environments. A significant problem with the two hydrologic models currently incorporated into AGWA, however, is that neither is appropriate for simulating runoff/erosion in watersheds between about 100 and 1000 km 2 . The Soil Water Assessment Tool, SWAT (Arnold et al., ) is designed to simulate large-, or basin- scale watersheds larger than approximately 1000 km 2 . In contrast, the Kinematic Runoff and Erosion Model (K2) performs best when applied to watersheds less than about 5 km 2 , but can do reasonably well for larger events on watersheds up to about 100 km 2 (Goodrich, 1990; Goodrich et al., 1997; Syed, 1999). Whereas this multiscale approach to watershed assessment is one of AGWA's greatest strengths, the inability of either model to simulate runoff and erosion in intermediate-scale watersheds represents a significant obstacle to the practical application of the tool. A potential source of error in applying K2 at intermediate scales is theoretically the inability of the kinematic wave routing to account directly for diffusion of the flood wave as it is routed through the channel network. Alternative routing methods suitable for use in an arid-region watershed model were explored, and the Muskingum-Cunge method with variable parameters, specifically the MVPMC4 methodology (Ponce and 28 Chaganti, 1994), was adopted and tested against K2 and observed outflow hydrographs at multiple scales. 2.2 Scale-Related Problems in Watershed Modeling Watershed models are commonly associated with a scale at which they are most successful at reproducing observed water and sediment discharges. Small-watershed models concentrate on describing hillslope processes, but their predictions tend to break down as watershed area increases and runoff hydrographs become dominated by channel processes. Conversely, large-watershed or basin-scale models may employ more robust channel routing components, but often do not adequately describe spatially variable runoff generation on the hillslopes, particularly in arid and semi-arid environments where runoff duration is short relative to the model time step, which is usually 24 hours. An ideal watershed model would be physically based, would describe hillslope and channel processes equally well, and would be capable of simulations for long periods of time. In practice, however, the amount of input data, and the computational time required to achieve these objectives are insurmountable obstacles to implementing such a model. As a result, myriad modeling strategies have been developed, each with a specific purpose or problem. In AGWA, for instance, SWAT was selected for its ability to model large areas for long periods of time. To do this it simplifies the spatial resolution of its input data requirements and the temporal resolution at which computations are made. In contrast, KINEROS was selected for its ability to simulate event discharge in smaller areas. In this situation input data requirements for a given area are more extensive than 29 those for SWAT, but computational costs are about the same because the model is applied to smaller watersheds. Similarly, whereas the computational time step is much smaller in KlNEROS, the length of the simulation is much shorter. Reducing the duration of the simulation rather than the temporal resolution at which processes are computed thus minimizes computational cost. In addition to the size of the modeled area and the time length of the simulation, scale dependence in watershed modeling results at least in part from the different spatial and temporal scales at which watershed hydrologic processes themselves are active. In figure 2.1, for example, it can be seen that overland-flow processes operate at smaller spatial and temporal scales than do channel-flow processes. There is overlap between them, however, and it is within this spatial scale (-1 km) that K2 and similar smallwatershed models perform best (Goodrich, 1990; Woolhiser, 1996). As watershed size increases channel processes become increasingly dominant, and the accuracy of smallwatershed model predictions declines. Improving the numerical description of this transition from a hydrograph dominated by hillslope processes to one dominated by channel processes is the primary objective of the routing component of this research. 30 100 yrs Annual Raintall, Snowmelt, Evapo Unsaturated Flow 1 min 10m 100m Figure 2.1. Graph showing hydrological processes at a range of characteristic space-time scales, and the superimposed spatio-temporal domains of large, small, and intermediatescale watershed models for arid regions. Modified from Bloschl and Sivapalan (1995). 2.3 Channelized Flow Routing in Watershed Models — Previous Work The physical size of watersheds limits the complexity of channelized flow routing procedures in watershed models to relatively simplistic one-dimensional representations. One-dimensional models describe a uniform flow velocity parallel to the channel only, and cannot therefore account for localized effects such flow separation, eddies, or helical 31 flow around channel bends. Because flow cannot vary perpendicularly to the flow direction neither can sediment movement, and as a result channels are generally described as a series of reaches within which boundary conditions vary uniformly or not at all. Jean-Claude Barre de Saint-Venant originally derived the continuity and momentum equations for one-dimensional gradually varied unsteady flow in the late 19 th century. In classical form, the Saint-Venant equations can be expressed as: aA aQ r(x,t) Continuity +—= (2.1) \ u ay au au 0 – S f )-- r Momentum — + u— + g — = gS ( A at ax ax (2.2) at ax — Variables are defined as A - cross-sectional area of the flow Q – discharge r(x,t) - the rate of lateral inflow per unit length of the channel u - flow velocity g – acceleration due to gravity So –bed slope Sf friction, or water-surface slope ur/A – momentum exchange between the lateral inflow, r, and the main channel flow, u. - In hydraulic modeling, the continuity and momentum equations are coupled to form the nonlinear dynamic wave model. Solution of these equations yields results for flows that vary with local and convective accelerations, pressure gradient, and friction and bed slope. Obtaining solutions for the full dynamic wave model, however, is a complex and difficult process for unsteady flows in a network of channels. Most authors describe the full dynamic wave model as inappropriate for steep channels, initially dry 32 channels, transcritical flows, and complex channel networks owing to difficulties in obtaining numerical solutions (e.g. Ogden, 1998; Wu and Vieira, 2000; Meselhe and Holly, 1997; Singh, 1996). A variety of numerical techniques has been developed to work around these problems, however, and at least one author has claimed to have developed a full dynamic wave model for arid watersheds (El-Hames and Richards, 1998). Although no detailed information can be found about this model, the authors admit that it is so computationally intensive that it cannot run on normal desktop computers. Owing to the computational expense of solving the full dynamic wave equations, simplifications, or approximations, of the full equations have been derived for certain circumstances to provide simpler but acceptable solutions. These equations can be classified according to the terms of the momentum equation that they neglect or consider as quasi-steady dynamic wave, diffusion wave, or kinematic wave. The quasi-steady dynamic wave neglects only the local acceleration term (atyat ) from the momentum equation, which is then coupled with the continuity equation to form the nonlinear quasi-steady dynamic wave model (Yen, 1979). au ay u—+g—= ax ax f –s )-- 11 r 0 0 f A (2.3) This model accounts for downstream backwater effects and permits distortion, translation, and attenuation of the hydrograph peak. Upstream and downstream boundary conditions must be specified to obtain a unique solution. Neglecting only the local acceleration term, however, produces more error than the diffusion wave approximation 33 for which both local and convective terms are neglected, and it is not widely used as a result (Yen, 1979). For gradually varied unsteady flows, except in highly nonuniform channels, local and convective accelerations are commonly small, of the same order of magnitude, and have opposite signs (Yen, 1979). Assuming that they cancel each other, the momentum equation can be expressed in terms of flow depth as: ay ax =S -S 0 (2.4) when lateral inflows are not present. Combined with the continuity equation this forms the diffusion wave model. Inclusion of the pressure term substantially improves solution accuracy over the kinematic wave model (Yen, 1979). Flow peaks can be attenuated, and the hydrograph can be distorted and translated. A unique solution to the diffusion wave equation, however, requires that upstream and downstream boundary conditions and lateral inflows be specified. This requires a more complicated simultaneous or iterative solution than does the kinematic wave model (Beven, 1993). In kinematic wave theory flood waves are assumed to be long and flat so that the friction slope is nearly equal to the bed slope. Considering only the two slope terms of the momentum equation, the kinematic wave equation can be expressed as: S„ = S f(2.5) when lateral inflows are absent. Solution of this equation requires only the upstream boundary condition, making it the easiest to implement for channel networks where flow depths at the drainage divide are considered equal to zero. The kinematic wave equation 34 cannot predict backwater effects, dispersion, or attenuation of the flood peak. In practice, however, numerical diffusion associated with approximation errors in the numerical solution of the kinematic-wave equation can lead to some attenuation of the hydrograph, which is most pronounced for large time and space steps (Beven, 1993; Yen, 1979). Numerical dispersion, also a result of approximation errors, manifests itself as the steepening or flattening of the of the rising limb of the calculated runoff hydrograph, and is most pronounced when the Courant number (also a function of the time and space steps) is substantially different from 1 (Ponce, 1991). In general, when used for channelized flow routing the kinematic wave model predicts a steeper wave than actually occurs, and the effect of the accumulation of errors shows that the approximations made in the development of the kinematic-wave equations are not generally justified for most flow-routing applications (Miller, 1984; Hromadka and DeViies, 1988). 2.3.1 Evaluating Which Model is Applicable Channelized flow can be spatially generalized within a basin as conforming to one of the three primary approximations of the momentum equation (excluding the quasisteady dynamic wave). Flow in the steepest upstream channels is best described by the kinematic wave approximation (Beven, 1993). This is particularly convenient because the kinematic wave approximation also works well for overland flow on hillslopes, and can be used with only the upper boundary condition of zero flow across the drainage divide. Proceeding downstream, flows in channels of intermediate and low slope are well described by the diffusion wave approximation. For the lowest slopes, slowest velocities, 35 and greatest flow depths, local and convective accelerations are large enough to necessitate use of the full dynamic wave model. Evaluating which model is applicable for a given channel or watershed has been the subject of considerable research and debate. In general, however, the problem can be approached from two directions: identifying the various flow conditions characterized by kinematic, diffusive, and dynamic waves; and weighing the former against known limitations of the available solution techniques. Ponce et al. (1978) evaluated downstream transitions among the three wave types by using linear stability analysis. They found that the diffusion wave model yielded reasonable results in comparison with the full dynamic model when \ l/ 2 ( TpS0 g - 30 (2.6) \YN / where Tp is the wave period of a sinusoidal perturbation of steady uniform flow, yN is the steady, uniform flow depth, and So is the longitudinal slope of the channel bed. If this equation is satisfied, the diffusion model will accurately approximate the unsteady flow. Scenarios demonstrating the conditions under which this is true are presented in figure 2.2 for the range of conditions expected in the Walnut Gulch Experimental Watershed in southeastern Arizona. From these it can be seen that for small flow depths and steep slopes, a short wave period is required to get a 'reasonable' solution using the diffusionwave approximation. It appears reasonable that the diffusion-wave equations would be acceptable for the range of flow depths and channel slopes that are typical of small- to intermediately-sized semi-arid watersheds. 36 0.9 Slope = 1% 0.8 o 0.7 Diffusion .c Dynamic 0 2 0 4 6 12 10 8 Flow depth (m) A 1.6 Flow depth = 0.3 m E 0.6 Diffusion E 0.4 C 5 0.2 Dynamic • 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 Channel Slope Figure 2.2. Graphs derived using the Ponce et al. (1978) equation for determining when the diffusion wave model yields reasonable results in comparison to the dynamic wave model. (A) Minimum wave period vs. flow depth; (B) Minimum wave period vs. channel slope. 37 Ponce et al. (1978) also evaluated where to transition between diffusion and kinematic wave models, and found that if the kinematic wave model is to be 95% as accurate as the diffusion model after one wave propagation period then Tp . 171y N So u N (2.7) where UN is the normal flow velocity, So is the channel bed slope and y N is the flow depth. Figure 2.3 shows that the minimum wave period increases with increasing flow depth and decreasing channel slope. Unlike the transition between diffusion- and dynamic-wave equations, however, the transition between kinematic- and diffusion-wave equations will almost certainly be observed for streams of small- to intermediate-sized watersheds. Moramarco and Singh (2000) extended the work of Ponce et al. (1978) by subjecting a simplified dimensionless form of the momentum equation to quantitative linear analysis. Three dimensionless parameters were derived in terms of the Froude number and geometric characteristics of the river that permited quantification of the influence of inertia and pressure in the momentum equation for various flow conditions. Their results agree closely with those presented above; dynamic and diffusion waves occur on the lowest slopes and can be separated in terms of the Froude number, and kinematic waves are found on slopes greater than about 0.01, therefore having a wide range of application (figure 2.4). For the range of slopes (0.001 - 0.05) and flow depths (< 3) that are likely in channel networks of intermediate-sized arid and semi-arid watersheds, diffusive and kinematic waves are likely to occur. Dynamic waves may occur under certain circumstances, but should not be expected. 38 25.00 Channel slope = 0.01, Flow velocity = 0.3 m/s 20.00 o 0 ; 15.00 Kinematic a. o co > 10.00 — E Diffusion '"E 5.00 0.00 0 1 2 3 4 5 Flow depth (m) A 50.00 45.00 - Flow depth = 0.3 m, Flow velocity = 0.3 m/s a - o 40.00 35.00 'c 30.00 a) a. 25.00 20.00 E 15.00 Kinematic "E 10.00 5 5.00 Diffusion 0.00 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 Channel Slope Figure 2.3. Graphs illustrating the minimum wave period required for the kinematic wave model to be 95% as accurate as the diffusion wave model. Values are shown for a range of (A) flow depths, and (B) channel slopes. 39 Figure 2.4. Graph showing division of the slope/reference water depth space into four regions characterized by different wave types for a channel length of 1000 meters and a Froude number of 0.4. From Moramarco and Singh (2000). Moussa and Bocquillon (2000) evaluated the approximation zones of the SaintVenant equations given variable overbank flow. The terms from the momentum equation were analyzed as a function of the balance between friction and inertia using linear perturbation theory. Flood waves were expressed in terms of three non-dimensionalised variables: the Froude number, a dimensionless wave number, and the ratio, n, between the width of the flooded zone and the width of the main channel. As expected, large flood-plain widths introduced convective and inertial accelerations that restricted the domain of application of the diffusive and kinematic wave models (figure 2.5). Consideration of flood plains is necessary for the simulation of geomorphic change, strengthening the argument for having diffusive wave routing in K2. 40 Fo-F.A.ONINO1g.y.) 1000 Tv.W2A1/1= t 100 Gravity ware Full 10 Salnt-Venant system 1,ticn1c0.1 Kinematic WA 1/111 hacr2K0.1 hs/o21‘0.1 lukal<0.1 0.1 0.01 0.01 0.1 1 100 1000 Dimensionless wave period Full Salnt-Venant system 1000 Figure 2.5. Graphs showing river wave approximation zones obtained from the analysis of the momentum equation for three values of n = 1, 8, and 20. From Moussa and Bocquillon (2000). 41 The above evaluations of the flow conditions under which the approximations of the Saint-Venant equations are appropriate considers only the relative magnitude of the various terms in the momentum equation. The manner in which the momentum and continuity equations are solved, however, can alter the approximation zones significantly. Using the kinematic wave model as an example, two case studies are particularly illustrative of the impacts of solution techniques on the model results. Miller (1984) compared kinematic and dynamic wave models for a sinusoidal hydrograph in a hypothetical channel reach, and for flow conditions meeting the accuracy criteria of Woolhiser and Ligget (1967), and Ponce et al. (1978) for kinematic routing. The kinematic-wave model was solved using the method of characteristics, and the dynamic-wave model was solved using a linear implicit finite-difference technique. A hydrograph was routed downstream using both models for a distance of approximately 6.4 km to demonstrate that small errors incurred in the kinematic-wave approximation continuously accumulate as the flood wave travels downstream. The results are shown in figure 2.6, where it can be seen that the Ponce et al. (1978) criterion for 95% accuracy after one wave period (equation 2.7), appears to be correct. As the hydrograph is routed beyond one wave period, however, the error continues to accumulate until the solution becomes unreasonable. In addition to pointing out that errors associated with the kinematic wave approximation can be propagated downstream, this study highlights the importance of the solution method. The method of characteristics used to solve the kinematic wave equations for this study provides nearly exact solutions and therefore does not account for attenuation of the hydrograph as a result of diffusion and dispersion. 42 800 700 600 Kinematic wave model 500 400 300 I 200 0 5000 10,000 / 15,000 I 20,000 i 25,000 I 30,000 I 35,000 40,000 TIME, IN SECONDS Figure 2.6. Graph showing a comparison of kinematic- and dynamic-wave results that demonstrating the propagation of errors in the kinematic wave solution. From Miller (1984). In contrast to Miller (1984), Zoppou and 0-Neill (1982) successfully applied the kinematic wave model in situations for which the acceptance criteria of Ponce et al. (1978) were violated. A four-point fully implicit finite-difference scheme (Preissmann, 1961) was used to solve the dynamic, diffusion, and kinematic wave equations for a 33.2 km reach of the River Yarra, near Melbourne Australia. Two measured floods were routed through this reach, both of which should have been characterized by diffusive waves according to the Ponce et al. (1978) criteria. Despite this, however, the kinematic 43 wave model performed just as well as both the diffusion and dynamic wave models. The authors attribute this good agreement to specific characteristics of the channel reach in question, and suggest that the Ponce et al. (1978) criteria be reevaluated or abandoned for natural channels. The analysis is flawed, however, in that the solution technique is responsible for the good performance of the kinematic wave model in a situation that should not be characterized by kinematic waves. Numerous authors have demonstrated that finite difference methods introduce significant numerical diffusion and dispersion into the kinematic wave solutions. Hromadka and DeVries (1988), for example, demonstrated that solving the kinematic wave equations with finite difference methods could introduce substantial errors depending on the space and time steps used in the computations (figure 2.7). INFLOW IlYtOMPIAM LO 3.5 40 Figure 2.7. Graph showing the influence of varying the time and space step during finitedifference solutions of the kinematic wave equation. From Hromadka and DeVries (1988). 44 Confusion over the difference between flow conditions characterized by kinematic waves and model conditions in which the kinematic wave equations may be applied was largely put to rest by Ponce (1991). He argued that the artificial numerical effects of using finite difference solution techniques are substantial enough to represent another model parameter that requires calibration. He went so far as to state that "...it seems pointless to try to "calibrate" a kinematic wave model by varying a physical parameter such as Manning's n in order to match calculated results and observed data. This practice amounts to curve-fitting; at best it is good conceptual modeling, but it should not be interpreted as deterministic modeling." (p. 515). Given the extensive discussion following the article in which this appeared it probably could have been worded more judiciously, but the point remains: numerical diffusion can produce correct results for the wrong reasons. Physically-based flow routing with finite difference solutions of the kinematic wave model is not possible. Instead, Ponce (1991) proposed extending kinematic wave theory to encompass the related diffusion wave theory. Three methods were suggested to solve the diffusion wave equation without introducing further grid-dependent numerical diffusion: 1. Analytic solutions, leading to Hayami's (1951) diffusion-analogy solution for flood waves. 2. Numerical schemes for parabolic equations, such as the Crank-Nicolson scheme (Crandall, 1956). 3. Extending the finite difference solution of the kinematic wave to the realm of diffusion waves by matching physical and numerical diffusivities (Cunge, 1969; 45 Dooge, 1973). When the Muskingum scheme is used in this manner to model the kinematic wave, the extended method is the well-known Muskingum-Cunge model. The third method has been explored at length by Ponce and others, and represents the most practical means of accounting for diffusive flow in ephemeral watershed models such as K2. It does not require a downstream boundary condition, yet it can match physical and numerical diffusivity to ensure that flood-wave diffusion is not simply a function of the selected time and space steps. As a result, this method is discussed in greater detail in the following section. 2.3.2 Diffusion wave modeling with the Muskingum-Cunge scheme Previously described routing models can be broadly classified as hydraulic, or process-type models; they are based on the solution of the partial differential equations of unsteady open channel flow, which are also referred to as the St. Venant, or dynamic wave equations. Muskingum-Cunge routing falls into another class of routing models called hydrologic routing models. Hydrologic routing uses a conceptual or systems approach by employing the continuity equation and an analytical or empirical relation between storage within a reach and discharge at its outlet. The distinction between hydraulic and hydrologic models is appropriate for well-defined regular channels, but for natural rivers the distinction is less obvious. The complex physical properties of natural river systems defy exact mathematical representation, and any form of modeling therefore necessarily involves a conceptual element (Weinmann and Laurenson, 1979). 46 The Muskingum method is based on the assumption of a linear relation between the inflow, I, the outflow, 0, and the reach storage, V, that can be defined as V = K[X/ + (1– X)0] (2.8) where K and X are the parameters. In the original Muskingum method these parameters were determined through calibration to measured inflow and outflow hydrographs. The formula for the Muskingum method is Q n++11 = Qin e2 Q j r; + I + e3 Q n+1 i (2.9) where the routing coefficients are defined as At — CI = + 2X (2.10) K \ — + 2(1– X) At — – 2X C2 – " J — + 2(1 (2.11) – X) \ At 2(1– X )– — c3= At (2.12) —+2k1- The Muskingum-Cunge channel routing technique (Cunge, 1969) improved Muskingum routing by computing K and X based on physical properties of the reach. As such, it can be generally described as a nonlinear coefficient method based on physical channel properties. The parameters K and X are calculated as (2.13) 47 1 X=—1 2 q S o cAx i (2.14) where zlx= reach length; c = flood wave celerity; q = unit width discharge; and So= channel bed slope. While this allowed K and X to be evaluated in the absence of measured inflow and outflow hydrographs, the parameters were constant, and could not vary with the stage and celerity of passing flows, or along the channel with changes in its geometry. The assumption of constant parameters made the solution highly dependent on the reference values used to evaluate them (Dooge, 1973). Ponce and Yevjevich (1978) expressed the routing parameters of the MuskingumCunge method in terms of two physically and numerically meaningful values that could vary in space and time: the Courant and cell Reynolds numbers. These were defined, resepectivel y, as cA t C L. =- -1 AX (2.15) q S D= cdx (2.16) ' By substituting (2.13 and 2.14) into (2.10 — 2.12) and solving for the routing coefficients in terms of C and D yields Ci = C2 = C3 = 1+C —D 1+C+D (2.17) —1+C+D 1+C+D (2.18) 1—C+D 1+C+D (2.19) 48 When lateral inflows are present a fourth routing coefficient is needed C, = 2C 1+C+D (2.20) To allow C and D to vary in time and space the values of c and q were recomputed for each time and space step by solving dQ c=— dA (2.21) q= (2.22) - B where Q= total discharge, A = flow area, and B= top width. Four methods of solving for c and q were proposed, culminating in the modified iterative four-point variable- parameter method (MVPMC4; Ponce and Chaganti, 1994). In this method the routing parameters C and D are computed for each computational cell based on the average unitwidth discharge at four points in time and space as ( qa = a n + n n +1 n n+1 "/./ 4 ,n++11) (2.23) where the space-time discretization is defined in figure 2.8. Iterative solution is required because the unit-width discharge at time n+1 and space j+1 is unknown. By computing the average celerity, ca , based on the discharge q a , the routing parameters C and D are thus based on the properties of the flow at the time and space step in which they are actually applied. This nonlinear solution method significantly enhanced the technique's applicability to real-world routing problems. 49 • At Q; • • Q in±i Ax Figure 2.8. Space-time discretization of Ponce and Yevjevich (1978). With a nonlinear solution that matches physical and numerical diffusion, the Muskingum-Cunge technique is essentially a diffusion wave model. Variable parameter Muskingum-Cunge routing has been shown to compare well against the full unsteady flow equations over a wide range of flow conditions (Ponce, 1981; Brunner, 1989). In addition, it has been demonstrated that the technique is independent of the user-specified computational interval, Ax (Ponce and Chaganti, 1994). The major limitations of the MVPMC4 technique are that it cannot account for backwater effects, and the method begins to diverge from the full unsteady flow solution when very rapidly rising hydrographs are routed through channels with slopes of less than about 0.00019 (USAGE, 1994). Neither of these conditions are prevalent, however, in small- to intermediate-scale watersheds, particularly in arid and semi-arid regions. 50 2.3.3 Practical Considerations Solving the numerical equations for overland and channel flow requires simplification of complex, irregular watershed geometry. Grid and conceptual methods are the two means of accomplishing this (Singh, 1996). Representing watershed geometry using a regular grid can describe slope irregularities on uplands. At the resolution of most hydrologic model grids, however, the small surface irregularities that influence overland flow are not captured. Spatial variation of soil and vegetation parameters is also better represented using the grid method, but again these are cellaveraged values, and commonly at a coarser resolution than the elevation grid. Conceptual methods that divide a basin into model elements, such as the cascading planes in K2, further average out spatial variability in watershed geometry and hydrologic parameters, but are computationally much more efficient and therefore more easily applied to large basins. Either of these geometric representations may be referred to as quasi distributed, and it has not been shown that one performs better than the other. Geometric simplification of the channels, for watershed models that contain a channel routing component, is universally achieved by representing them with a series of cross sections. If natural channels are represented by closely spaced variable crosssections (using breakpoint data) then irregularities in the channel geometry are more likely to produce conditions where the acceleration terms are significant. For generalized trapezoidal cross-sections that are placed further apart, however, the channel irregularities are averaged, acceleration terms will be small and the full equations will perform no better than a simplification of them. In watershed modeling it is often impractical to 51 gather detailed cross-section information, and most models use generalized trapezoidal channel geometries. This does not diminish the need, however, to simulate diffusive flood wave attenuation. 2.4 Routing in AGWA Channel routing in K2 is based on a four-point finite difference solution of the kinematic wave approximation of the one-dimensional continuity and momentum equations for open channel flow. This routing method is ideal for overland flow, and flow in short, steep channel networks, but model efficiency has been shown to decline significantly when watershed size increases beyond about 100 km 2 (e.g. Syed, 1999). Because runoff hydrographs from watersheds of this size are dominated by channel processes, it seems likely that using routing algorithms that more completely describe channel flow processes would improve model performance in larger areas. Upgrading the channel routing algorithms may not eliminate the decline in model efficiency with increasing watershed size, but it is likely that a model accounting for attenuation and translation of flow hydrographs could be applicable to larger watersheds than one that does not. Channel routing in SWAT is accomplished through pseudo channels using the variable storage routing, or Muskingum river routing methods. These are both hydrologic methods, and provide less room for improving process representation. In addition, the large time step used in SWAT (1 day) precludes a detailed representation of flood hydrographs. Daily runoff totals are routed downstream as a volume using a 52 velocity calculated from Manning's equation. This works well for large areas where discharge can be considered to be approximately uniform during each 24 hour period, but without reducing the time step at which calculations are made it would be difficult to improve hydrograph representation. The objective of improving hydrograph representation in intermediate-scale watersheds can be accomplished using either diffusion- or dynamic-wave routing, both of which can account for hydrograph attenuation and translation. Based on the previous discussion of the conditions in which each of these is appropriate, however, a diffusion wave model seems adequate for small- and medium-sized watersheds, which are broadly defined as having average sizes of 25 and 500 km 2 , respectively (Pilgrim and Cordery, 1993). Local accelerations that are accounted for only in the full dynamic-wave models are less likely to be significant at these scales, which are characterized by relatively steeper slopes and shallower flows than larger watersheds. In addition, the problems associated with solving the full dynamic wave equations in channel networks with steep, initially dry channel networks that may be characterized by transcritical flows make the dynamic wave model less practical to implement in K2. 2.5 Model Description Variable parameter Muskingum-Cunge (MVPMC4) was selected as the most suitable method of implementing diffusion-wave routing within the K2 conceptual model. By adopting the K2 conceptual model and numerical framework it was possible to retain the well-established functionality of the model, and simply replace kinematic 53 with MVPMC4 routing for the channels. In addition, by holding everything else constant it is possible to facilitate direct comparison of the two channel routing techniques at multiple scales, which is the expressed objective of this exercise. Implementation of MVPMC4 routing followed the same procedure and spacetime discretization as described in section 2.3.2. The K2 numerical framework contains an outer time loop within which flow at each computational node is computed. Reaches are divided into 20 spatial nodes at which routing calculations are excuted, regardless of reach length. Flow at each spatial node is computed by the iterative 4-point technique, for which iteration ceases when the 4-point average discharge is the same before and after the routing calculation (i.e. Aga —> 0). To improve the convergence of the iterative procedure, the discharge VI++; obtained from the 3-point method (known values) is used in the first iteration. Celerity and unit-width discharge are computed for each iteration to ensure that they are representative of the flow. The top width of the flow is calculated based on the normal depth computed by means of a separate iterative procedure retained from the kinematic routing that determines the flow depth. It was necessary to retain this calculation in the model because flow depth and wetted perimeter are used in the infiltration and sediment routing subroutines. As with kinematic routing, MVPMC4 routing is sensitive to the Courant number and errors can occur if it gets too high. MVPMC4 routing works best when the Courant number is kept as close to 1 as possible, or between 0.5 and 2 in practice. For this reason the Courant time step adjustment loop from K2 was retained, and is used to adjust the 54 time step downwards when the Courant number exceeds 2. For initially dry channels, however, it was not possible to meet the lower limit; for very small discharges the Courant number is routinely less than 0.5. This may result in a small amount of numerical dispersion for the smallest flows. 2.6 MVPMC4 Instability at Low Flow Under certain conditions, generally smaller flows, the MVPMC4 routing develops an instability problem that results in negative outflows and a wildly oscilating outflow hydrograph. This problem was originally discovered when a calibrated parameter file for Walnut Gulch was used to simulate design storm runoff hydrographs with both K2 and the modified K2 with MVPMC4 routing (K2MC). Hydrographs from both models predicted very low discharges, but those simulated by K2MC displayed wild and sometimes growing oscillations. Although volume balances remained reasonable, and the discharges were low enough to be inconsequential in the larger scheme of runoff events, the underlying problem seemed significant enough to merit correction. A single 4-element subcatchment (3 planes and a channel) was used to investigate the problem further. This parameter file was run with a series of design storms from the 5-year 30-minute to the 100-year 1-hour. Oscillations developed under all but the 100year events with MVPMC4 routing, and never when kinematic routing was used. To isolate the source of the oscillations the parameter file was simplified further; twoelement (1 upland plane and a channel) and three-element (2 lateral planes and a channel) parameter files were developed and run with the series of design storm events. 55 Oscillations were not observed for any of the events using the two- and three-element parameter files, so it was concluded that the problem somehow arose when hydrographs from upstream and lateral inflows were combined in the channel. Further investigation using the four-element model revealed that the problem developed when either precipitation decreased or channel length (and transmission losses) increased to the point where flows in the channel were very small — less than about 1 m 3 s -1 . By comparing outflow hydrographs from K2 for the same events it became obvious that under these conditions hydrographs from upstream and lateral inflows separate in time to yield two distinct peaks in the outflow hydrograph. K2MC was unable to resolve this separation and developed oscillations that increased downstream until the initial dip between the two hydrographs had passed the last spatial node of the reach. Generally, the receding limb of the hydrograph was unaffected. Owing to the number of time and space steps involved in computing the routing for a reach, it was difficult to isolate the source of the problem. Extensive use was made of the Compaq array viewer provided with Visual Fortran 6 to visualize the flow profile at each time step. By using this tool it was possible to see that the oscillations started at the leading edge of the upstream inflows, and became amplified as the flood wave moved downstream through time. These same initial oscillations show up in kinematic routing, but are smoothed through time and eventually disappear. Upstream inflows, because they are added at a single section, are significantly higher than the lateral inflows, which are averaged over the reach. As a result, flow in node 1 could be two or more orders of magnitude higher than in node 2, creating a major discontinuity in the flow profile. 56 Initial attempts to define further how and where the problem developed centered around defining the Courant and cell Reynolds numbers, and comparing them (and other derived quantities) with accuracy criteria published in the literature. Specifically, accuracy criteria of Koussis (1976), Ponce and Theurer (1982), and Szel and Gaspar (2000) were evaluated, as well as general recommendations by Ponce (1989; and SDSU, written communication, 2003). Unfortunately, the results were more confusing than diagnostic. Accuracy criteria were always met during the peak of the hydrograph, but were violated during the rising and falling limbs. Increasing the time step was originally considered (space step is fixed within a simulation) to bring up the Courant number during low flow. However, further investigation demonstrated that accuracy criteria were also violated during the larger events when oscillations were not observed. In addition, increasing the time step during the rising limb when changes in discharge are large seemed likely to introduce additional error as a result of inadequately representing the hydrograph. It was concluded that if the accuracy criteria were really the source of the numerical oscillations then the oscillations should be developing whenever the criteria were violated, which was clearly not the case. A hybrid kinematic-MVPMC4 routing model was developed in an attempt to implement kinematic routing in situations where the accuracy criteria were violated. Implementing kinematic routing when the maximum Courant number for a time step was less than 0.5 did not resolve the problem. The code was then modified to implement kinematic routing within a time step, when the Courant number at any given node dropped below 0.5, but this too failed to prevent oscillations from developing. 57 Owing to a logic error in the development of a switching mechanism between MVPMC4 and kinematic routing, it was discovered that the oscillations do not develop if kinematic routing is commenced one spatial node downstream of where the problem starts. With this logic error the routing became overwhelmingly kinematic, but the results did indicate that if kinematic routing could be implemented in the node before the discontinuity that the problem might be resolved. To accomplish this, the routing subroutine was rewritten to allow a spatial backstep in the calculations for each time step. Upon identification of the discontinuity in the MVPMC4 routing at a certain spatial node, the program backs up a node and recomputes the profile using kinematic routing. Whereas this did correct the problem it produced a hydrograph that was arguably indistinguishable from pure kinematic routing. A more acceptable methodology of preventing the numerical oscillations from developing was ultimately discovered in the VPMC routing section of HEC-HMS code provided by William Sharffenberg from the US Army Corps of Engineers' Hydrologic Engineering Center. For steep hydrographs entering flat-sloped channels, the original developers of the HMS Muskingum-Cunge routing algorithm observed the same oscillations, and developed an adjustment to the C1 routing coefficient based on extensive numerical experiments where the MC method was compared to the St. Venant solution. Their solution involves modification of the routing coefficients when C1 is computed to be less than 0 by the following: C2 = C 2 + 0.5C1(2.24) C3 = C3 + 0.5C1(2.25) 58 Following these adjustments to the C2 and C3 parameters CI is set to O. This adjustment to the routing coefficients was found to be the best way to conserve mass and yield routed hydrographs close to the St. Venant solution. That work was summarized for inclusion in an official Corps of Engineers engineering manual. Later it was converted to an appendix before being dropped entirely. It was never published. When this simple fix was made it removed the numerical oscillations in the hydrograph. It has the effect, however, of predicting slightly increased peak flows and delaying the onset of the hydrograph, but the time to peak is approximately the same as for kinematic routing. This effect is most pronounced for the smallest flows, which require the most consistent modification of the routing coefficients, and decreases with increasing discharge. Modifying the routing coefficients does not result in significant volume balance errors, and the net effect on outflow hydrographs is negligible. 59 3.0 ROUTING MODEL RESULTS 3.1 Methodology Modified variable parameter Muskingum-Cunge routing was compared with the original kinematic routing of K2, hereafter referred to as the K2MC and K2 models, respectively. Comparisons were based on a set of parameter files originally developed by Syed (1999) for the Walnut Gulch subwatersheds 1, 6, 11, and Lucky Hills 104 (see description in Chapter 1). These parameter files were selected primarily because they had previously been used to investigate scale issues. Syed (1999) calibrated the WG11 watershed with marginal success, producing efficiencies of 0.46 and -1.1 for runoff volume and peak flow, respectively, when applied to the validation data. Calibration for WG1 was considerably less successful; efficiencies of 0 and -13.4 were reported. The calibrations were carried out through the use of the K2 parameter multipliers that are applied uniformly over the entire watershed, which did not modify the original values in the parameter file. The WG6 and LH104 parameter files were not calibrated. It was not deemed necessary to use the parameter multipliers of Syed (1999) for WG1 and WG11, or calibrate the two remaining parameter files. The major objective of the routing model comparison is to evaluate relative differences between the two methods as a function of scale, for which calibration is not required. Indeed, if each parameter file were calibrated separately for the two models, differences in computed hydrographs could easily result from differences in the effectiveness of the calibration. 60 The comparison first involved running the parameter files with both K2 and K2MC for a set of design storms ranging from the 5-year 30-minute to the 100-year 1hour events. This comparison was designed to establish if there were significant differences between the simulated hydrographs as a function of scale. The second comparison involved running both models for a series of rainfall events for which observed runoff data were available. This comparison was designed to evaluate relative performance of the routing techniques as a function of scale. 3.2 Design Storm Results The four subwatersheds (WG1, WG6, WG11, and LH104) were subjected to six design storm events: 5-year 30-minute, 5-year 1-hour, 10-year 30-minute, 10-year 1- hour, 100-year 30-minute, and 100-year 1-hour. Design events represent uniform rainfall over the entire watershed, so the impact of spatially-variable rainfall is removed. The design storm events were derived using the AGWA design-storm generator based on return-period depths observed at Walnut Gulch. AGWA reduces the depths based on an area-reduction factor developed by Osborn et al. (1980). The design storms used in this comparison, however, were not generated individually for each subwatershed. Instead, the design events were generated for a watershed with an area of approximately 50 hectares, and the same events were used for all comparisons. A variety of tests and metrics was used to evaluate differences (or lack thereof) between the two simulated hydrographs for each event. The Komolgorov-Smimov (K-S) test was used to determine if the hydrographs could be considered statistically different at 61 the 5% level of significance. The hydrographs were also compared in terms of runoff volume, peak flow, hydrograph shape, and the timing of the onset of the hydrograph and peak discharge. 3.2.1 Komolgorov-Smirnov Comparison The K-S test is a nonparametric test (i.e. independent of sample distribution) that is not affected by the magnitude of the values being compared. It is sensitive to deviations in the tails of the distribution where frequencies are low, but the upper tail of a cumulative plot of discharges represents the highest discharges, which also is important from a hydrologic perspective. The Komolgorov-Smirnov comparison was conducted using a two-tailed test and a 5% level of significance. Simulated discharges from the K2 and K2MC outflow hydrographs were first converted to cumulative form, normalized by the sample size. The K-S statistic is the maximum difference between these two, and is compared to critical values for a specific level of significance. The null hypothesis for a two-tailed KS test states that both samples have the same continuous distribution. Results of the K-S testing show that runoff hydrographs simulated by kinematic and MVPMC4 routing are statistically indistinguishable in most cases. Significant differences between the outflow hydrographs were observed for the smallest events in the largest watersheds (table 3.1). This is in keeping with the hypothesis that directly accounting for flood-wave diffusion will improve hydrograph representation as watershed size increases, although these results indicate only that outflow hydrographs 62 become statistically different as watershed size increases. The impact of event magnitude can be explained by referring to figure 2-5. As discharge increases the nondimensionalised wave period increases, which pushes the system from the realm of diffusion waves into that of kinematic waves. Table 3.1. Results of the Komolgorov-Smirnov tests indicating whether the null hypothesis was accepted or rejected. Results are shown for the six design-storm events ranging from 5-year 30-minute (5_30) to 100-year 1-hour (100_1) for each subwatershed. 10_1 100_30 100_1 10_30 5_1 5_30 Watershed Area (ha) LH104 4.7 Accept Accept Accept Accept Accept Accept WG11 782 Accept Accept Accept Accept Accept Accept WG6 9558 Reject Reject Accept Accept Accept Accept WG1 14664 Reject Reject Accept Accept Accept Accept 3.2.2 Metric and Visual Comparisons Visual inspection of the simulated hydrographs for the design storm events shows that the greatest differences occur for the smallest events on the largest watersheds. Hydrographs for LH104 are virtually indistinguishable, whereas those for WG1 look significantly different. Simulated hydrographs for WG6, intermediate between these two endpoints, shows a clear transition from very similar hydrographs for the largest events to substantially different hydrographs for the smallest events (figure 3.1). Simulated hydrographs for LH104, WG11, and WG1 are presented in Appendix A. 63 0.9 08 0.7 0 05 04 0.3 02 01 40 250 - 200 -.-- K2 K2MC 150 50 Figure 3.1. K2 and K2MC simulated WG6 runoff hydrographs for design storms ranging from the 5-year 30-minute event (upper left) to the 100-year 1-hour event (lower right). Plots are discharge in cubic meters per second as a function of time in minutes. 64 The timing of the onset and peak discharges, runoff volume, peak discharge, and mass-balance error were computed for each design-event simulation. Where appropriate, the normalized difference was computed using the following (3.1) (K2MC_value — K2_value) / K2_value where 'value' refers to the metric in question. Differences in the timing of the onset of flow are presented in table 3.2 as just the difference in onset time computed using the numerator of equation 2.26. As the table shows, the onset of flow is basically identical between the two models, with the maximum difference at any scale or event magnitude being 1 minute. There is no apparent pattern to the differences, and one would not be expected. Diffusion effects increase with travel time and distance, but the velocity of the wave front is primarily controlled by channel slope and roughness. Table 3.2. Difference between the timing of K2 and K2MC flow onset (mins). Watershed Area (ha) 5_30 5_1 10_30 10_1 100_30 100_1 LH104 4.7 0 0 -1 0 0 0 WG11 782.0 0 0 0 0 -1 0 WG6 9557.6 0 -1 0 -1 0 -1 WG1 14664.3 -1 0 0 0 0 0 The timing of peak discharge was substantially more variable than the onset of flow (figure 3.2). Normalized differences in the timing of peak flow almost universally increased with scale, and the change was most pronounced for the smallest events. With only one exception, the peak flow occurred at the same time or earlier in K2MC. The exception was the 100-year 30-minute event on WG1, for which it appears that a very 65 sharp and asymmetric peak in the kinematic routing was smoothed in the MVPMC4 routing, potentially due to flood-wave diffusion, which caused the peak to shift forward in time. 20 10 -10 — it— 5_30 5_1 10_30 10_1 — 3K— 100 30 100_1 50 so -70 0 2000 4000 6000 8000 10000 12000 14000 Area (ha) Figure 3.2. Graph showing normalized differences in the time to peak discharge at the watershed outlet between K2 and K2MC, plotted as a function of area. Negative values indicate that K2MC predicts an earlier peak discharge than K2, and visa versa. Normalized differences in the peak discharge were also quite sensitive to watershed size and event magnitude (figure 3.3). For most events K2MC predicted slightly lower peak flows than K2. As watershed size increased, however, K2MC became more likely to predict a larger peak flow. As with the other metrics, differences were most pronounced for the smallest events, and there was very little difference for the largest events. Visual inspection of the hydrographs reveals that the smaller peak flows do not appear to be the result of flood-wave attenuation due to diffusion. It is extremely 66 difficult, however, to resolve the cumulative impacts of diffusion in a distributed channel network, particularly when the amount of numerical diffusion in K2 varies with the grid spacing in each reach. 100 - so 5_30 NE-- 5_1 3.(-- 10_1 10_30 so —NE— 100_30 —EN— 100_1 40 20 o -20 0 2000 4000 6000 8000 10000 12000 14000 Area (ha) Figure 3.3. Graph showing normalized differences in the simulated peak discharge at the watershed outlet between K2 and K2MC, plotted as a function of area. Negative values indicate that K2MC predicts a smaller peak discharge than K2, and visa versa. Values in excess of 100% were reset to 100%. Differences in the runoff volume (water yield) mirror those associated with peak discharge (figure 3.4). In most instances water yield is slightly lower in K2MC, with the exception of values computed for the smallest events in the largest watersheds. Again, differences increase with increasing watershed size and decreasing event magnitude. 67 100 -•-- 5_30 80 5_1 10_30 --34-- 1 0_1 --*-- 100_30 60 -4---100_1 40 t Z i 20 - _ ----, -. -20 2000 0 4000 6000 8000 10000 12000 14000 Watershed Area (ha) Figure 3.4. Graph showing relative error in the total discharge volume at the watershed outlet between K2 and K2MC as a function of area. Negative values indicate that K2MC predicts less total runoff than K2, and visa versa. The parallel trends in peak discharge and runoff volume suggest a problem with the MVPMC4 routing technique. Ponce and Chaganti (1994) describe a small but persistent loss of mass associated with the variable parameter methods. The loss of mass is smallest for the MVPMC4 method, but is present none-the-less. This effect can be seen in the mass-balance error presented in table 3.3 for each simulation. In most cases mass-balance error is greater in the K2MC simulations. Error generally decreases with increasing watershed size, but there is no discernable trend associated with event magnitude (figure 3.5). For the K2 simulations, error increases with increasing watershed size and decreasing event magnitude. Differences are thus most pronounced for the smallest watersheds and largest events, which also represent those simulations for 68 which K2MC predicts slightly lower peak discharge and water yield. For smaller events and larger watersheds the difference in mass balance error decreases, and ultimately changes sign as error becomes greater in K2 (figure 3.5, and italicized values in table 3.3). The loss of mass observed by Ponce and Chaganti (1994) was evaluated for four different flows routed through a reach 800 kilometers in length with no lateral inflows. They speculated that this would represent the worst-case scenario since most practical routing applications would not consider such a long reach without intervening lateral inflows, which tend to mask the accuracy of the computation. Actual losses were shown to increase with increases in the ratio of peak inflow to baseflow, from 0.44% for a ratio of 4 to 2.43% for a ratio of 20. Table 3.3. Mass balance error (%) computed from K2 and K2MC output files. Events for which K2MC had less error are italicized. Positive values indicate that inflows exceeded outflows and storage (loss of mass). 100_1 100_30 10_1 5_1 10_30 Model 5_30 WS -0.10 -0.21 -0.40 -0.08 LH104 -0.22 -0.16 K2 2.88 3.17 2.44 2.56 2.17 LH104 K2MC 2.12 0.20 0.53 0.88 0.42 WG11 0.49 K2 1.39 1.67 2.35 2.48 1.88 2.25 1.87 WG11 K2MC 0.33 0.06 1.08 0.39 0.68 WG6 K2 1.75 1.35 1.09 0.82 1.52 1.66 WG6 K2MC 0.80 0.12 0.63 0.30 1.46 2.38 WG1 K2 1.57 0.78 0.64 0.64 1.35 WG1 1.19 K2MC 1.75 69 2.5 4 2 1.5 0.5 o -0.5 0 2000 4000 6000 8000 10000 12000 14000 Area (ha) Figure 3.5. Graph showing the average trends in mass-balance error (%) as a function of watershed area. In the present application the ratio of peak inflow to baseflow would seem to be infinite because the channels are initially dry. However, Ponce and Chaganti (1994) only considered upstream inflows relative to what was already in the channel. In K2MC lateral inflows add some water to the channel before the upstream flow arrives, and definitely before the peak. The addition of all upstream inflows at a point, however, guarantees that they will always be considerably greater than the lateral inflows, which are distributed along the length of the channel. This makes the inflow-to-baseflow ratio very large, and is a likely source of mass-balance error. Interestingly, the conditions producing the greatest loss of mass are the same as those described in section 2.5.1 that were the source of the numerical oscillations observed in smaller events. The solution to this problem was to modify the routing 70 coefficients in situations when the problem developed. Although the modifications were "found to be the best way to conserve mass and still produce routed hydrographs close to the St. Venant solution" (Scharffenberg, 2003, personal communication), their specific effects on mass conservation have not been documented. In addition, the modifications are utilized most during smaller events, for which the hydrograph differences are most pronounced. As such, these modifications may be contributing to the relatively large K2MC mass balance error in the smaller watersheds. The reduction in K2MC mass balance error with increasing watershed size (figure 3.5) may occur because the ratio of inflow discharge to that already in the channel decreases in the downstream direction as lateral inflows are afforded more time to fill the channel before upstream flows arrive. Loss of mass would be most pronounced for small watersheds with only one channel, but would tend to be averaged with increasing watershed size and greater complexity of the channel network. 3.3 Observed Storm Results The observed-event simulations were conducted to provide a comparison between kinematic and MVPMC4 routing in terms of their performance relative to measured flows as a function of scale. Simulated and observed hydrographs are compared in terms of shape and the timing of onset and peak flows (Appendix A). The timing comparisons were possible through the establishment of event time for the observed hydrographs by setting the time of flow onset equal to the time difference (in minutes) between the first recorded rainfall and the first recorded runoff. It should be noted that timing errors of 71 approximately 5 to 10 minutes are not uncommon in the observed data. In the absence of calibrated models, the following analysis emphasizes relative performance of the models rather than the actual performance. For WG1 seven runoff events were selected for the routing comparison. Peak observed discharges ranged from 1.2 to --29 m 3 s -i , for rainfall depths ranging from 7.85 to 17.81 mm over the watershed. For comparison, the 5-year 30-minute design storm event used in section 3.2 was 16.77 mm. In general, simulated runoff was substantially lower than the observed runoff, and showed a great deal less temporal variability than the observed runoff. The best, although very poor, results for WG1 are shown in figure 3.6, which shows that only the highest peaks were represented in the simulated hydrographs. In keeping with the design storm results, which demonstrated greater runoff in K2MC for small events, simulated runoff was higher in K2MC for all of the WG1 events. Potentially as a result of this, K2MC was more likely to simulate multiple peaks in the runoff hydrograph. For WG6 five events were simulated, with peak observed discharges ranging from 15 to 65 M 3 S -1 for total rainfall depths ranging from 10.81 to 23.09 mm over WG6. For reference, the 5-year 1-hour event had a rainfall depth of 26.65 mm. In general, simulated hydrographs were again smaller than the observed hydrographs, but the difference between simulated and observed values was much smaller than for WG1 (figure 3.7). Again, K2MC predicted discharges that were consistently higher than those from K2. Observed hydrographs showed significantly less temporal variation (i.e. fewer distinct peaks), and simulated hydrographs from both models were much more successful 72 30 25 Observed —K2 K2 MC 5 0 0 200 400 600 800 1000 1200 Time (min) Figure 3.6. Graph showing the best match of simulated and observed hydrographs for WG1 (14664 ha). 30 25 --•—• K2 K2MC Observed 5 I. 0 50 100 150 200 250 300 350 Time (min) Figure 3.7. Graph showing the best match of simulated and observed hydrographs for WG6 (9558 ha). 73 at reproducing it. Simulated hydrographs from K2MC, however, were more likely to capture distinct peaks in the observed hydrograph. For WG11, five events were simulated with observed peak discharges ranging from 13 to 25 m 3 s -1 , for total rainfall depths ranging from 23.12 to 31.91 mm over WG11. For reference, the 10-year 30-minute design event had a rainfall depth of 28.64 mm. Simulated hydrographs for WG11 consistently predicted greater than observed runoff, and results from both models were very comparable in terms of peak discharge (figure 3.8). K2 and K2MC were equally successful at reproducing the shape of the observed hydrograph. For LH104, five events were simulated, with observed peak discharges ranging from 0.06 to 0.4 M 3 S 1 for total rainfall depths ranging from 12.86 to 37.57 mm over LH104. For reference, the 10-year 1-hour design event had a rainfall depth of 33.31 mm. Simulated runoff for LH104 consistently overpredicted observed values, and by a much more substantial margin than for WG11 (figure 3.9). Observed hydrograph shapes were best represented at this scale, and all significant peaks in the observed hydrographs were represented by the models. K2 and K2MC predicted almost identical hydrographs for all of the events. 74 25 fi-451 K2 20 K2fVC Observed 5 0 ottmm • 70 110 90 130 150 Time (min) 170 190 Figure 3.8. Graph showing the best match of simulated and observed hydrographs for WGII (782 ha). 0.3 0.25 00 K2 K2MC Observed 0.2 1 015 a 0.1 0.05 o 90 110 150 130 170 190 Time (min) Figure 3.9. Graph showing the best match of simulated and observed hydrographs for LH 1 04 (4.7 ha). 75 Simulated and observed hydrograph shapes were compared using the NashSutcliffe statistic, or model efficiency (ME) calculation, which is essentially a measure of goodness of fit between simulated and observed discharges. The Nash-Sutcliffe (1970) statistic is calculated as ME = 1 (3.2) where 9. = observation at time i = simulated value at time i = mean of all observed values A model efficiency of 1 indicates that the modeled hydrograph is in perfect agreement with the observed hydrograph. If the value is negative, then the mean value of observed discharge would be a better predictor of observed runoff than the predicted discharge. Results of the uncalibrated K2 and K2MC model efficiency calculations show that model efficiency increases with increasing watershed size for both models (figure 3.10). Without model calibration, however, the absolute values for ME are less important than the average difference between model efficiencies at each scale. As can be seen in figure 3.10, K2MC model efficiencies are greater (or less negative) at all scales, but the magnitude of the difference shows no consistent relationship to scale. 76 0.5 0 —a-- K2 —a— K2MC 3 -3.5 0 2000 4000 6000 8000 10000 12000 14000 Area (ha) Figure 3.10. Graph showing average model efficiency plotted as a function of watershed area for K2 and K2MC. Error in the timing of the onset of flow consistently, and almost linearly, increased with watershed area for K2. K2MC, however, showed a less consistent relationship to scale (figure 3.11). Despite variation at the intermediate scales, error at the smallest scale was approximately the same as at the largest scale; just the sign is different. In addition, the difference in onset error increases from almost 0 for the smallest watershed to a maximum of approximately 60% for the largest watershed. Average error in the timing of the peak discharge increased for both models as a function of scale (figure 3.12). Increases, however, were significantly more pronounced for K2, which had a maximum error of 184% for WG1 versus a K2MC error of 24%. The difference between K2 and K2MC peak error also increased with increasing watershed size, from almost 0 for LH104 to 160% for WG1. 77 80 70 60 50 j 40 al "Zil cn oc 30 20 10 0 -10 -20 0 2000 4000 6000 8000 10000 12000 14000 Area (ha) Figure 3.11. Graph showing average error (%) in the timing of flow onset at the watershed outlet plotted as a function of watershed area for K2 and K2MC. 100 90 —.— 80 K2 —s— K2MC 70 60 50 40 30 20 10 0 -10 , 0 2000 4000 6000 8000 10000 12000 14000 Area (ha) Figure 3.12. Graph showing average error (%) in the timing of peak flows at the watershed outlet plotted as a function of watershed area for K2 and K2MC. Values in excess of 100% were reduced to 100%. 78 3.4 Discussion A comparison of the K2 and K2MC models was conducted to evaluate their general behavior and performance as a function of scale. Analysis of the design storm simulations demonstrated that K2 and K2MC could only be statistically distinguished for the smallest events on the largest watersheds (4 of the 24 simulations). Although it was the 5-year return period events that were statistically different for WG6 and WG1, this was an underestimation. No area-reduction was used when the design storms were generated, so the rainfall depth associated with the 5-year return period events on WG6 and WG1 (the two largest watersheds) is comparable in magnitude to the 10-year events when area reduction is accounted for. A comparison of the peak discharge, its timing, and runoff volume showed the greatest differences between K2 and K2MC under the same circumstances. Most notable, however, was that K2MC consistently predicted earlier and larger peaks, and greater runoff volumes than did K2. An assessment of model mass-balance error showed opposite trends in model performance as a function of scale. K2MC average mass balance error was greatest for the smallest watersheds; it consistently decreased with increasing watershed area. K2 showed the opposite trend, with mass balance error increasing above that reported by K2MC for the largest watershed (WG1). Analysis of the observed events demonstrated that K2MC simulated hydrographs are marginally superior to those from K2 at approximating the shape of observed hydrographs at all scales. K2 demonstrated consistent increases in the error of its 79 hydrograph onset timing with increasing scale. In contrast, K2MC showed no consistent relationship between onset timing error and watershed scale. Differences between K2 and K2MC onset timing error, however, increased with scale. Peak discharge timing was most poorly simulated by both models at the largest scale. The relative difference between K2 and K2MC time to peak error, however, increases with watershed size, with K2MC yielding the largest improvement for WG1. The overestimation of observed runoff for small watersheds and underestimation for large watersheds by both K2 and K2MC demonstrates that the models do not describe runoff processes equally well at all scales. This problem is analogous to the decrease in runoff associated with watershed geometric simplification and information entropy loss (e.g. Miller, 2002). A contributing (channel) source area (CSA) of 0.5% of the watershed area was used during the discretization of all watersheds. The actual CSA thus varied from 0.02 ha for LH104 to 75 ha for WG1. Miller (2002) demonstrated that runoff volume decreased with increasing CSA due to significant information entropy loss among model input parameters. Strong correlations were found between simulated runoff decreases and information entropy loss associated with Manning's roughness, saturated hydraulic conductivity, and slope as model element size increased. Not considered in the analysis of Miller (2002) was the representation of specific process relationships at different scales. At the smallest scales representation of hillslopes as planar features with uniform overland flow seems reasonable. As element size increases, however, concentrated (rill) flow becomes more important, and the assumption of uniform planar flow is less suitable (e.g. Willgoose and Kuczera, 1995). 80 This observation was the primary motivation behind the development of a microtopography feature in K2 to permit flow concentration in rills, but this feature has not been tested against measured data. Flow concentrated in rills prior to its entering the channel system reduces upland infiltration, and results in more water being conveyed to the channel network and ultimately the watershed outlet. The combined effect of lower average slopes for larger model elements, and the assumption of uniform planar flow over those slopes thus is likely to be the main cause of runoff underestimation with increasing watershed scale. These complementary analyses demonstrate that: (1) MVPMC4 routing yields significantly different results from those of kinematic routing for the vast majority of all events on watersheds greater than about 100 km 2 . (2) The difference between hydrographs simulated with MVPMC4 and kinematic routing makes the MVPMC4 routing more successful at replicating observed hydrographs for watersheds larger than about 100 km2. 81 4.0 GEOMORPHIC MODEL 4.1 Channel-Morphologic Models A major objective of river and watershed managers is to simulate the morphologic response of channels to disturbance. Knowledge of how a channel will adjust its width, depth, and slope to changing water and sediment supply is crucial to the protection of water, biological, and cultural resources. Countless models have been developed with this purpose in mind, and many have been successful at predicting morphologic change under specific circumstances. A problem common to most of these, however, has been simulating changes in channel width. Despite years of research on the physics of channel width adjustment, no physically-based models of channel-width adjustment can account for overbank flow, vegetation impacts, or channel constriction. In addition, river width adjustment models are subject to the same types of limitations as are watershed models. Complex models describing physical processes in detail have been the most successful in simulating certain types of river width adjustment, but are so data intensive that they can only be applied to isolated channel reaches. Models applicable to larger areas necessarily simplify a system conceptually to reduce the data requirements and computational cost, but model output can only be as good as the conceptual framework from which it is derived. Generally, this means that river width adjustment is computed on a reach basis, and that it cannot account for unidirectional width adjustment (i.e. on the outside of meander bends). 82 Complex, multi-dimensional flow models cannot be realistically applied to watersheds, so the opportunities for simulating river width adjustment are limited. Simplified bank stability algorithms (e.g. Vieira and Wu, 2000) can be applied with onedimensional hydraulic flow models such as K2. These are designed to compute bank stability in terms of a factor of safety which, when exceeded, results in bank failure and channel widening. Bank failure algorithms are usually combined with an empirical method derived by Arulanandan et al. (1980) for computing bank toe erosion in onedimensional flows. Bank toe erosion thus provides a means of oversteepening banks to cause failure, and a method of removing failed bank material deposited at the bank toe. The bank stability approach requires initial and simulated pore water pressure in the banks, making it useful in continuous simulation models such as CCI-1E1D (Vieira and Wu, 2000), which track soil moisture between events. The only other method for simulating channel morphologic change in onedimensional routing models is the use of extremal hypotheses in an equilibrium approach. Extremal hypotheses offer a third equation (in addition to the momentum and continuity equations) that can be used to complete the set of equations needed to solve for channel width, depth, and slope. In applying an extremal hypothesis it is assumed that a channel achieves stability or equilibrium when a specified function of some combination of the hydraulic variables (water discharge, sediment discharge, sediment size, and channel width, depth, and slope) has an extremum (maximum or minimum). A variety of extremal hypotheses has been proposed in the literature, including: minimum stream power (Chang, 1980), minimum unit stream power (Yang and Song, 1979), maximum 83 friction factor (Davies and Sutherland, 1980), and maximum sediment transport rate (White et al., 1982). Of these, the minimum stream power (MSP) theory, a special case of the minimum rate of energy dissipation theory (Yang et al., 1981), has received the most attention. It has been applied in two widely used river simulation models: the Generalized Stream Tube model for Alluvial River Simulation model, GSTARS (e.g. Yang and Simôes, 1998); and the FLUVIAL-12 Mathematical Model for Alluvial Channels (e.g. Chang, 1982). The MSP hypothesis is stated as (Chang, 1980, p. 1445): For an alluvial channel, the necessary and sufficient condition of equilibrium occurs when the stream power per unit channel length yQS is a minimum subject to given constraints. Hence, an alluvial channel with water discharge Q, and sediment (discharge), Qs, as independent variables, tends to establish its width (B), depth (D) and slope (S) such that yQS is a minimum. Since Q is a given parameter, minimum yQS also means minimum channel slope. The quantity yQS has the dimension of energy per unit time per unit length, in which y is the specific weight of the fluid (water). The equilibrium methods offer predictions of the magnitude, rather than the rate of width adjustment, which is the principal conceptual objection to channel morphologic models based on extremal hypotheses (ASCE, 1998). When applied in GSTARS, however, channel geometry is adjusted at each time step to produce the lowest possible stream power given the potential geometries that could result from the computed change in sediment storage. The rate of width/depth adjustment is thus governed by the rate of erosion/deposition in the channel during a given time step. Limits on the amount of lateral change are also imposed. Width change in GSTARS is limited by the computed change in sediment storage in stream tubes bordering the channel banks (Yang and Simoes, 1998). Change in 84 sediment storage computed away from the banks can be used only for vertical change. The quasi-two dimensional approach to flow routing thus provides a mechanism for limiting erosion and deposition on the channel banks. By accounting for channel curvature and associated transverse energy gradients, Fluvial-12 simulates variable erosion/deposition along a cross-section despite its onedimensional model framework. As with GSTARS, this approach limits width change. Width adjustment as a result of bank erosion is also limited by an empirical bank erodibility factor that controls the rate of bank erosion. A major criticism of extremal hypotheses in general is that they lead to conclusions that are incompatible with observations when combined with conventional sediment transport and flow resistance equations (Griffiths, 1984). Specifically, for wide, straight, unconstrained alluvial reaches in equilibrium it can be shown that the combination of hypotheses and equations leads to the result that the Einstein sediment discharge, 41), and the Shields entrainment function, 1/w, are nearly constant. According to Griffiths (1984), the results of flume and field observations do not agree with this conclusion, but it is not clear what effect this has on computed equilibrium geometries. The American Society of Civil Engineers' (ASCE, 1998) task committee on hydraulics, bank mechanics, and modeling of river width adjustment did not raise this issue in its review of river width adjustment models based on extremal hypotheses. Despite the aforementioned criticism of the MSP technique, numerous case studies demonstrating its utility have been reported in the literature. Chang (1982) demonstrated that the concept of minimum stream power could be successfully used as a 85 physical principle governing the morphology of alluvial streams when applied within a water and sediment routing model. Both channel profile and width changes were very closely predicted during a 35-hour flood in 1978 that caused significant geomorphic change in the simulated reach. Song et al. (1995) were equally successful at simulating channel degradation and widening in the earthen spillway of the Lake Mescalero Dam and Dike, New Mexico, during a flood in 1984. By conducting the same simulation with and without stream power minimization Song et al. (1995) demonstrated that accounting for lateral sediment transport improved estimates of total sediment yield. 4.2 Channel Network Evolution (Surficial Process) Models Channel network evolution models represent another major type of geomorphic model. Unlike the morphologic models that concentrate on describing channel properties, the evolution models are designed to simulate evolution of the larger channel network and surrounding uplands on geologic time scales. This type of model is aimed at evaluating landscape evolution in response to changing climatic, geologic, and biophysical conditions. Distributed, physically-based models of drainage basin evolution have been steadily improving since the late 1980s when Roth et al. (1989) simulated the erosional development of drainage patterns. In their treatment, overland flow over a flat, tilted surface is modeled two-dimensionally according to the normal flow hypothesis, which perfectly balances the vector of flow resistance with the vector of gravitational force on the surface. Deterministic equations of mass and momentum conservation are coupled 86 with random fluctuations of sediment entrainment and transportation to provide an active source of channelization (Roth et al., 1989). This approach produces topographic lows that connect in the downstream direction and have a similar appearance to the early stages of drainage basin formation. Willgoose et al. (1991a,b) developed a numerical model capable of simulating the long-term evolution of drainage basins. Their treatment of sediment transport is similar to that of Roth et al. (1989), but a channelization equation is used to govern the development and extension of channels. This equation is adapted from one developed by Meinhardt (1982) to differentiate leaf-vein formation, and is based on the phenomenology of channel-head extension. The channelization equation provides a physically based mechanism for channel initiation and growth, but is not based on the controlling transport physics at the channel head. Howard (1994) developed a purely erosional treatment of overland flow that eliminates assumptions based on analogies to physically different problems. His model is again based on flow and sediment transport equations similar to those used by Roth et al. (1989), but it employs a much different method of transition between channelized and unchannelized surfaces. A cell in the simulation matrix is converted from nonalluvial to alluvial if the actual bedload rate exceeds the potential rate. This mechanism works because the model assumes that the system is detachment-limited, and therefore each cell is capable of transporting all of the sediment supplied to it. Howard (1994) justifies this assumption for simulating fluvial erosion of natural slopes and headwater channels because they are dominantly detachment-limited. 87 Although the temporal resolution of channel evolution models is inadequate to determine relatively short-term impacts on water quality and riparian condition, they possess important features that can benefit channel morphologic models when applied on a watershed scale. In particular, channel evolution models have necessarily paid significant attention to channel extension (headward erosion) and channel slope. Channel extension through headward erosion can be an important geomorphic process on relatively short time scales. Extensive gullying on Walnut Gulch, for instance, continues following a 15-year erosional episode beginning in about 1930 (Graf, 1983a). Whereas the vast majority of sediment generated during the erosional episode was derived from channels with large contributing areas, the effects of the change in base level that resulted continue. Unchannelized swales have transformed into valleys with small incised channels (gullies) as headcuts have migrated upstream. The cumulative impacts of sediment erosion by this process are affecting present morphologic response downstream, where sediment is now accumulating. Representation of channel extension is thus an important component of evaluating morphologic response on a watershed scale. Channel slope is a product of the delicate balance between sediment supply, transport, and deposition throughout the channel network. It is significantly influenced by sediment grain size distribution, channel roughness, geological and biological controls, contributing area, climate, and host of other factors. Regardless of the extent to which these are accurately represented in geomorphic models, their single most important feature is the ability to adjust slope dynamically in response to variable water and sediment discharges. Many watershed models proclaiming a geomorphic component 88 presume that the timescale over which channel geometry changes occur is orders of magnitude greater than the timescale of event discharges, and that morphologic change can thus be ignored during event simulations (e.g. K2). This assumption, however, breaks down as event magnitude increases and flows become capable of substantially modifying the channel morphology. 4.3 Watershed Geomorphic Models To understand how an individual stream reach responds to external stresses it is necessary to study the channel network as a whole (e.g. Schumm et al., 1987). This revelation has been the primary motivation for the development of watershed geomorphic models, which allow individual reaches to respond independently to flows within a dynamic, linked system. As described in section 4.2, many of the watershed-based geomorphic models have been developed using a surficial-processes approach for application to research questions about drainage basin evolution over long periods of time. The other approach has been to combine channel morphologic models with watershed runoff and erosion models, where the later provide input to the former. This hybrid approach carries the advantage of being able to simulate short-term changes in channel morphology that incorporate the distributed impacts of upland and upstream disturbance. The hybrid approach necessarily incorporates the use of geographic information systems (GIS) for watershed and channel network delineation and to assist with model parameterization. 89 Two examples of the hybrid approach have gained notoriety in recent years: the U.S. Army Corps of Engineers CASC2D model (Ogden and Heilig, 2001), which has been incorporated into the Watershed Modeling System (WMS) hydrologic model interface; and the One-Dimensional Channel Network Model CCRE1D (Vieira and Wu, 2002), developed at the National Center for Computational Hydroscience and Engineering. CASC2D combines a two-dimensional grid-based overland-flow routing scheme with one-dimensional grid-based channel routing based on the diffusion- or dynamicwave equations. The channel network in CASC2D is represented as a series of links and nodes, where a link is a channel segment, or reach, between two or more computational nodes, which are located a the center points of grid cells. Cross-sections, defined at links, can only be represented by trapezoids when routing sand-sized sediment. No mention is made of overbank routing, so this component of geomorphic systems is not considered. The potential to compute channel morphologic change in CASC2D is limited. Erosional and depositional changes in channel bed elevation modify channel slope, but erosion can occur only on the channel bed (figure 4.1). It does not appear that channel side slopes are modified as a result of vertical change, and there is no mechanism for bank failure or channel widening. 90 Figure 4.1. Schematic representation of erodible trapezoidal channels in CASC2D. Despite the capability of CASC2D to simulate continuously, it has not been used to simulate cumulative geomorphic change. Indeed, the model seems to have included sediment primarily for water-quality assessment, with minimal geomorphic adjustments primarily to improve the accuracy of sediment-discharge calculations. The strength of this model, however, lie in the robustness of the hydrologic model, particularly its integration of two-dimensional overland flow and one-dimensional flow routing in the channel network, and in the grid-based representation of the channel network. Twodimensional overland flow routing should improve the model's ability to represent landscape change relative to models that represent uplands as planes, although the extent to which this might be true has not been investigated. The diffusion- or dynamic-wave channelized-flow routing improves the model's ability to simulate runoff from large watersheds. The grid-based representation of the channel network yields numerous short reaches that afford the model greater flexibility to adjust channel gradient in response to localized deposition and erosion. CCHElD is perhaps the most comprehensive watershed geomorphic model available. It was developed as a detailed, continuous simulation channel-network model 91 with flow routing based on the diffusion- or dynamic-wave equations. The channel network is represented as a series of links, where a link is either a hydraulic structure or a single channel consisting of several reaches. Reaches are defined as the channel segment between two cross-sections, or computational nodes. Channels are represented by breakpoint data, where bank tops must be specified in the event of overbank flow. Variable deposition/erosion along a cross-section is facilitated by variable flow depths along the section, and through an empirical equation for computing bank toe erosion (Arulanandan et al., 1980). Bank slopes can thus increase as a result of channel degradation, or lateral erosion at the bank toe. Bank failure is computed when stability criteria are exceeded, and sediment is deposited at the bank toe. CCHElD was successfully used to predict channel profile adjustments over an 18-year period in the Goodwin Creek Watershed in Mississippi. Measured and observed profile adjustments during the 18-year period following the installation of a measuring flume are remarkably well matched (figure 4.2), with very good representation of erosion at the upstream end of the reach and deposition downstream in the backwater behind the measuring flume. 92 Measures] in he_ 1972 •Measured Ln Feb., 1992 buaal newel in Calculaboa. ha.. 1971 74 CAL, Wu et al s Fartnaia. Feb.. 1992 CAL, SEDTRA Module, Fob., 1992 - - - - C. Engelund-Hanwn. Feb., 1992 72 70 68 66 0 100)15002(100 2500 3000 Distance Upstream of Flume No, I (in) Figure 4.2. Profile adjustment over 18 years in the Goodwin Creek Watershed. Simulated profiles are shown in colors other than blue. The final observed profile is represented by orange dots. From http://hydra.cche.olemiss.edu/ccheld/. A major drawback of CCHElD is that it is a channel-network model. The watershed runoff and erosion model, AGNPS (Agricultural Non-Point Source Pollution Model), or an equivalent must be used to generate water and sediment inflows at the upstream end of each tributary in the network. Efforts to improve this cumbersome procedure are underway, but ideally there would be seamless interaction between the upland and channel network models as in CASC2D. CCHElD is designed for use in humid regions characterized by perennial flows, and requires a downstream boundary condition for the entire simulation period. As such, it is not possible to apply the model in arid-region watersheds with ephemeral flows. Another significant point is that approximately 5 years after its initial release CCHElD 3.0 is still available only for beta-testing to researchers and engineers who sign a Beta- 93 Testing Agreement with NCCHE, and only the computer server at NCCHE can be used to perform simulations. Practical application of the tool is thus difficult. 4.3 Approach To address the stated objective of developing a geomorphic model for arid-region watersheds, components of both channel morphologic and channel network evolution models were combined. Rather than adapting a channel-network model to incorporate distributed inputs from a watershed model, as was done in CCHE1D, a distributed watershed runoff and erosion model was adapted to compute channel morphologic change in the channel network. The model selected for this purpose was the USDA-ARS Kinematic Runoff and Erosion model (K2), which has been widely and successfully used in small watersheds characterized predominantly by overland flow. The numerical model, hereafter referred to as K2G, incorporated a variety of modifications for the purpose of simulating channel morphologic change: • Computation of channel slope based on channel-bed elevation at the reach endpoints to enable dynamic adjustment of slope in response to deposition/erosion • Tracking of available sediment up to a maximum erodible depth • Calculation and tracking of channel geometric change in response to changing sediment storage • Calculation of width and depth changes • Enabling bank failure if banks are oversteepened 94 The theory of total stream power minimization was adopted to compute changes in channel width in the absence of information about the lateral distribution of deposition/erosion along a cross-section. In addition, this technique requires no additional parameter information beyond that of the parent model, K2. More specifically, information about pore-water pressure and channel-bank material is not required. To facilitate the assessment of geomorphic change at the watershed scale, it was necessary to develop a GIS-based component that could work in concert with K2G as part of a larger assessment tool. This component was developed based on the Automated Geospatial Watershed Assessment tool (AGWA), which was customized to work with the geomorphic model. The GIS interface, hereafter referred to as AGWA-G, coordinates watershed and channel-network delineation, model parameterization, model simulations, spatial visualization of model results, and the comparison of multiple simulation results based on different initial conditions. The following sections describe the K2-based numerical model and the AG WA-based modeling tool. 4.5 Numerical Model The K2G geomorphic model was developed within the framework of K2. The channelized flow routing subroutine of K2 was modified to translate change in sediment storage into channel geometric change that can be tracked from one simulation (event) to the next. Because K2G channels are trapezoidal, changes in channel geometry are limited. A collection of prescribed changes, however, was developed to maximize the flexibility of geometric adjustment within the trapezoidal framework. Channels are 95 allowed to adjust by changing their width, depth, bank slopes, horizontal bank widths, and channel-bed elevations at both ends of each reach. In addition, a maximum erodible depth has been implemented below which the channel can no longer erode vertically. Slope is uniform for any given reach, and is allowed to vary as a function of elevation changes at either end of the reach. All other parameters are required as input and computed independently at either end of the reach. With the exception of slope, geometric parameters vary linearly along the reach from the values provided at the upstream end to those downstream. Bank slopes may be different on either side of the channel if data are provided by the user, but because flow is one-dimensional (i.e. no lateral variation in flow), volumetric change on either side of the reach is uniform. The original intent of this research was to incorporate the MVPMC4 diffusionwave routing described in Chapters 3 and 4 into the geomorphic model. It was not possible to do so, however, without enabling flow in compound channels. Compound channels, or channels with flood plains, are an essential component of the geomorphic model for two reasons. First, flood plains represent reservoirs of stored sediment, and the conceptual model would be incomplete if it failed to account for changes in flood-plain sediment storage. Second, when flood plains are not present the hydrologic model extends the channel banks vertically to contain all flows. This can cause erroneously high stream power when flow depth exceeds the bank tops, which results in exaggerated channel incision. Despite a concerted effort to implement compound channels in K2MC, it could not be accomplished in the available time. The original kinematic-wave routing of K2 was thus retained in K2G to permit the use of compound channels. 96 Overbank areas in K2G can be specified by the user by entering their width, side slope, and a lateral slope (figure 4.3). As in K2, overbank areas can only be represented on one side of the channel. Although not required, the use of compound channels is highly recommended when there is any potential for overbank flow. If overbank parameters are not provided K2G arbitrarily extends bank tops to contain the flow if its depth exceeds the channel depth. This can lead to inflated estimates of total stream power, and hence erroneous geomorphic adjustments. Overbank widths are adjusted in proportion to the computed channel top-width change, and overbank elevation (slope) is allowed to vary independently from channel slope. SS I Depth Width Width (overbank) Figure 4.3. Conceptualization of a compound channel in K2G showing separate channel and overbank areas. Geomorphic change is computed following each computational time step a the upstream and downstream ends of each reach. K2G computes stage, discharge, and change in sediment storage for each of 20 computational nodes in a channel reach during each time step. Given the computed volume change of sediment storage it is possible to affect only a limited amount of geometric adjustment. To determine what portion of the change in sediment storage should be applied to the bed and banks, the total stream 97 power is computed for a maximum of 100 different geometric permutations ranging from all width change to all depth change. Geometric adjustments commence by applying all of the available sediment volume towards raising or lowering the channel bed. With each successive iteration the volume applied to vertical change is decreased by 1% and the volume applied to horizontal change is increased by the same amount until the maximum permissible volume of sediment has been applied to horizontal change. For each iteration the flow depth and velocity are recomputed based on the discharge at that time step and channel geometry for that iteration. Using these values total stream power is computed for the iteration by integrating stream power along the reach: CDT j = •Wsdx (4.1) where OT represents the total stream power. This expression is discretized following Chang (1982) as: =E0.5r(asi+Qi+isi+JAxi (4.2) where N is the number of nodes/stations along the reach, dx, is the distance between stations I and I + 1, y is the specific weight of water, and S is the energy gradient. The rate of energy dissipation due to sediment movement is neglected. The energy gradient is computed using the following: S = Z, + U12 ( Z, + U '2+ 1Ax 2g 2g \ (4.3) 98 where Z is the water-surface elevation, U is the average velocity, and g is the acceleration due to gravity. The geometry that results in the minimum total stream power is adopted, and geometric parameters are updated for the next time step. Geometric change through total stream power minimization is computed only at the second and second-to-last computational nodes in the reach (nodes 2 and 19). The primary reason that nodes 2 and 19 are used instead of nodes 1 and 20 is that the energy gradients at the first and last nodes of the reach cannot be evaluated given the K2G conceptualization of the channel network as a series of connected but independent reaches. Deposition is computed for the intervals between nodes and is reported for the downstream node of each interval; change in sediment storage is not reported at node 1. Geometric change is not computed in the middle of the reach to minimize the input requirements and computational expense of applying the model to large areas. Instead, all geometric parameters are interpolated linearly between the 2 nd and 19 th nodes and to reach endpoints before each time step. As a result of the linear interpolation of geometric parameters between the reach endpoints, the computed volume of geometric change for a reach can be severely distorted, relative to the modeled net change in sediment storage, if deposition at nodes 2 and 19 is considered representative of deposition along the entire reach. Because K2G is a transport-capacity model, it contains no mechanism other than settling velocity to limit the rate of deposition when the sediment load exceeds transport capacity. When transport capacity is well below the inflowing sediment load, deposition thus predominantly 99 occurs at the upstream end of the reach, and decreases exponentially in the downstream direction. Figure 4.4 applies to a reach from Walnut Gulch Subwatershed 11 for which there was approximately uniform erosion during the rising limb of a runoff event, followed by deposition during the receding limb that was concentrated at the upstream end of the reach. As a result, using change in sediment storage at the ends of the reach to compute changes in channel geometry (that are interpolated linearly between the endpoints) causes total geometric change that severely overestimates net deposition in the reach. To ensure that the equivalent mass of geometric adjustments is equal to the net change in sediment storage for the reach it is therefore necessary to compute an average trend in the change in sediment storage along the reach. The nonlinearity is always concentrated at the upstream end of the reach, so K2G assumes that the change in sediment storage at the downstream end of the reach is representative of the downstream change and this value is used to compute downstream change in channel cross-sectional area. It is thus necessary to compute a change in cross- sectional area at the upstream end of the reach that will yield a volumetric change, VG, equal to the modeled change in sediment storage, Vm, when integrated along the reach. To accomplish this, the change in modeled sediment storage for the time step is summed for each node in the reach to obtain Vm. Because changes in geometric the geometric parameters are linear along the reach, change in channel cross-sectional area can be represented by a single value defined as 100 2.00E-01 —a— 150E-01 • Channel Deposition Overbank Deposition Ts' 1.00E-01 Je c n 5.00E-02 0 0. &&&&& a, 0.00E+00 • • -5.00E-02 -1.00E-01 4 10 7 13 16 19 Computational Node Figure 4.4. Graph showing change in sediment storage for each computational node in a reach (20 is the downstream end). Note that change in sediment storage does not vary uniformly along the reach. A Ave = Au + A D (4.4) 2 \ where Au and AD are the changes in upstream and downstream cross-sectional area. The volumetric change resulting from geometric adjustment for the reach can thus be expressed as Vm =VG = AA„ • L (4.5) where AA ve and L are the average change in channel cross-sectional area and reach length, respectively. Solving (4.4) for Au yields Au = 2. AA „, — AD(4.6) By solving (4.5) for AA„ and substituting the result into (4.6) the change in upstream cross-sectional area that will conserve mass for the reach is 101 Au =2 Vm L (4.7) The area Au is used as a volume per unit length of channel to adjust channel crosssectional geometry at the upstream end of each reach. It should be noted that K2G computes deposition as a mass (kg). This mass is converted to a volume in by assuming an average particle density of 2.65 (g/cm 3 ) and a porosity of 0.40 (40%) for the deposited material. 4.5.1 Channel adjustments — deposition During deposition channel depth and/or width can decrease in proportion to the volume of deposited sediment, resulting in the geometry that produces the minimum total stream power. Derivations of the geometric adjustments to erosion are given in Appendix B. Channel width change as a result of bank deposition proceeds by simple bank translation towards the center of the channel such that the volume of bank accretion equals the amount allotted to producing width change (figure 4.5). The volumetric change is computed such that half is added to each bank, which allows different bank side slopes to be accommodated on either side of the channel. The change in channel width is added to the overbank width to maintain the section width, which is defined as the sum of the horizontal bank widths, the channel bed width, and the overbank width. For the same reason, bank slopes and widths remain constant during channel narrowing. 102 Figure 4.5. Sketch illustrating depositional width reduction for the K2G trapezoidal channels. Deposition on the banks must occur along the entire bank height to maintain a trapezoidal geometry. Bank deposition, however, is limited by the ratio of wetted bank perimeter to wetted bed perimeter. This limitation is necessary to determine the maximum portion of channel deposition computed by the model that can be deposited on the banks if the resultant geometry causes a lower total stream power. It has the added benefit of not allowing significant bank deposition until a significant portion of the banks is submerged. The assumption embodied in this limitation is that bank deposition per unit surface area, at a maximum, is equal to that in the channel bed. In reality bank deposition is limited by the bank width and slope, which would tend to result in lower deposition per unit surface area on the banks. Bank shear stresses, however, are also likely to be lower than on the channel bed (on average), which would have the opposite effect of resulting in greater deposition per unit surface area on the banks. When taken together these two factors are assumed to offset each other, and provide a reasonable maximum volume of sediment available for deposition on the banks. Bank deposition (channel narrowing) is limited by a minimum width. Channels are not allowed to narrow past 10% of the width computed as a function of contributing 103 area by means of a hydraulic-geometry relation developed by Miller (1995). This limit ensures that channel width does not reach zero, but it was never reached during the testing of the model. Depth change as a result of deposition on the channel bed proceeds by reducing the depth of the channel while keeping the bank widths constant (figure 4.6). As a result, deposition reduces bank slopes, but channel bottom and top widths remain constant. Figure 4.6. Sketch illustrating depositional depth reduction for the K2G trapezoidal channels. Deposition on the channel bed can proceed until a minimum depth of 10 cm is reached. This depth does not vary in proportion to the channel contributing area, and is based on the observation that defined channels are rarely much less than 10 cm deep. Unlike the minimum width, the minimum-depth limit serves a major purpose in the geomorphic model. Sediment available for vertical change after the minimum depth is met is used to raise the elevation of the cross-section uniformly. This provides a means of increasing channel slope, particularly for long channels, without causing model failure because channel depth has reached zero. It also is intended to reproduce conditions observed in depositional channels — channels become wide and shallow, or braided, 104 during extended periods of deposition. K2G mimics this process when used in conjunction with compound channels; the minimum depth of 10 cm allows flows to spread over a wide area while retaining a portion that is slightly deeper. This small inset channel can contain smaller flows and continue to effect deposition on the channel banks until the minimum width is reached. It also provides a focus for subsequent erosional events and channel incision. To illustrate the magnitude and manner of simulated depositional adjustments during a single large event, adjustments were plotted for the upstream and downstream ends of an individual reach (figure 4.7). The adjustments plotted in figure 4.7 resulted from a 52-mm rainfall event on WG11 with a peak flow of 35.3 m 3 /s (61.9 mm/hr) for the reach. They show non-uniform deposition for the reach, with approximately 0.04 meters of deposition upstream, and no significant change downstream. Channel width decreased by approximately 0.05 meters upstream, and increased by approximately 0.003 meters downstream. Bank slopes decreased from 1.02 to 0.98 upstream as a result of the depositional depth decrease, and did not change significantly downstream. Sediment was deposited on the flood plain at both ends of the reach, but most significantly upstream. All geomorphic adjustments represent net change; adjustments during an event are not output by K2G. 105 1439.1 Upstream Adjustment 1439 1438.9 1438.8 1438.7 Final 1438.6 1438.5 1438.4 1438.3 1438.2 1438.1 0 2 4 6 10 8 12 A 1437.1 Downstream Adjustment 1437 1436.9 1436.8 1436.7 - Final 1436.6 1436.5 1436.4 1436.3 1436.2 0 246 8 1012 Figure 4.7. Plots showing depositional adjustments at the upstream (A) and downstream (B) ends of a 210-meter reach following a 52-mm rainfall event with a peak flow of 35.3 m 3 /s (61.9 mm/hr). Elevation (y-axis) and distance (x-axis) both have the units of meters. 106 4.5.2 Channel adjustments — erosion During erosional events depth and/or width increases in proportion to the volume of eroded sediment, resulting in the geometry that produces the minimum total stream power. Derivations of the geometric adjustments to erosion are in Appendix B. Channel-width change as a result of bank erosion proceeds by simple bank translation outwards from the center of the channel such that the volume of material removed from the banks equals the amount allotted to width change (figure 4.8). The volumetric change is computed such that half of the volume is removed from each bank, which allows different bank side slopes to be accommodated on either side of the channel. The change in channel width is subtracted from the overbank width to maintain section width. For the same reason, horizontal bank widths remain constant during channel narrowing. Figure 4.8. Sketch illustrating erosional width increase for the K2G trapezoidal channels. As with deposition, erosion on the banks must occur along the entire bank height to maintain a trapezoidal geometry. Bank erosion, however, is limited by the ratio of wetted bank perimeter to wetted bed perimeter. This limitation is necessary to approximate the maximum portion of channel erosion computed by the model that can come from the banks in the absence of local scour information. It has the added benefit 107 of not allowing significant bank erosion until a significant portion of the banks are submerged. The assumption embodied in this limitation is that bank erosion per unit surface area, at a maximum, is equal to that in the channel bed. In reality bank erosion is limited by relatively higher shear strength of bank versus bed material, which results in less erosion per unit surface area on the banks. In addition, average bank shear stresses are lower than on the channel bed, which accentuates the preference for vertical as opposed to lateral erosion. During erosional events, however, the tendency is to minimize total stream power by incising the channel and reducing channel slope, so lateral erosion is relatively insignificant except during the largest events when the banks are overtopped. Depth change resulting from erosion of the channel bed proceeds by increasing the depth of the channel while keeping the bank widths constant (figure 4.9). As a result, vertical erosion increases bank slopes, but channel bottom and top widths remain constant. Allowing channel side slopes to vary in this fashion was also necessary because without it vertical depth changes cause the channel bottom width to be continually reduced until the section becomes a 'V'. This change in geometry causes problems in the numerical model, and is poorly representative of natural channel incision. If channel incision increases either or both channel side slopes beyond a certain critical slope, then the bank(s) fail, top width is increased, and bank slope(s) are reduced. The critical side slope is set to be 3.75 (-75'). If this slope is reached, the side slope is reset at that section to 1.75 (-60°), which is the side slope associated with the "best hydraulic section" described in basic hydraulics texts (e.g. Chang, 1988). The volume of 108 sediment removed from the banks is used to offset any volume that might have been removed from the banks, and the remainder is used to raise the channel bed uniformly. Figure 4.9. Sketch illustrating erosional depth increase for the K2G trapezoidal channels. Erosion of the channel bed can proceed until the maximum erodible depth (MED) is reached. The MED defaults to 5 meters, but can be adjusted manually by the user by adding it to the input parameter file. Once the MED is reached, further depth increases are prevented. To limit the amount of erosion computed in the sediment routing subroutine, a parameter (PAVE) is set to indicate that the portion of the channel perimeter on the bed is not erodible. A limited amount of erosion can thus continue to occur, and is manifested as channel widening. To illustrate the magnitude and manner of simulated erosional adjustments during a single large event, adjustments were plotted for the upstream and downstream ends of an individual reach (figure 4.10). The adjustments plotted in figure 4.10 resulted from a 52-mm rainfall event on WG11 with a peak flow of 3.1 m 3 /s (129.7 mm/hr) for the presented reach. They show approximately uniform erosion for the reach, with 0.067 109 1504 — Upstream Adjustment 1503.9 1503.8 1503.7 - 1503.6 1503.5 1503.4 2 4 6 8 10 12 A 1502.3 -Downstream Adjustment 1502.2 1502.1 1502 - 1501.9 - 1501.8 - 1501.7 0 2 4 6 8 10 12 Figure 4.10. Plots showing erosional adjustments at the upstream (A) and downstream (B) ends of a 25-meter reach following a 52 mm rainfall event with a peak flow of 3.1 m 3 /s (129.7 mm/hr). Elevation (y-axis) and distance (x-axis) both have the units of meters. 110 meters of erosion upstream, and 0.065 meters downstream. Width changes are approximately an order of magnitude smaller than the depth adjustments, and not readily apparent. Bank slopes increased from 1.29 to 1.53 upstream, and from 1.06 to 1.29 downstream as the channel incised. No sediment was eroded or deposited on the flood plain. 4.5.3 Erosion and Sediment Transport in K2G The general equation used in K2, and K2G, to describe the sediment dynamics at any point along a surface flow path is a mass-balance equation similar to that for kinematic water flow (Bennett, 1974): a(Ac at s ) awc, ax = q s (x,t) (4.8) in which = sediment concentration [L3112], Q = water discharge rate [L3 /T], A = cross sectional area of flow [L 2 ], e = net rate of erosion of the soil bed [L2/1], q s = rate of lateral sediment inflow for channels [L 3 /T/L]. For upland surfaces, e is assumed to result from two major components: soil erosion by rain splash on bare soil; and hydraulic erosion (or deposition) due to the interplay between the shearing force of water on the loose soil bed, and the tendency of soil particles to settle under the force of gravity. Thus e may be positive (increasing concentration in the water) or negative (decreasing concentration). The net-erosion rate is a sum of the splash-erosion rate, es , and hydraulic-erosion rate, eh, e = e, + eh (4.9) 111 For a more detailed description of splash erosion in K2G, refer to Woolhiser et al. (1990), from which much of this description was derived. The hydraulic-erosion rate represents the rate of exchange of sediment between the flowing water and the sediment over which it flows, and may be either positive or negative. K2G assumes that any given flow condition (velocity, depth, etc.) has a specific equilibrium concentration of sediment that can be carried if that flow continues steadily. The hydraulic-erosion rate is estimated as being linearly dependent on the difference between the equilibrium concentration and the current sediment concentration. Hydraulic erosion/deposition is thus modeled as a kinetic-transfer process by e h = c g—C )44 (4.10) in which C„, is the concentration at equilibrium transport capacity, C, = C,(x,t) is the current local sediment concentration, and cg is a transfer-rate coefficient [T -1 ]. When deposition is occurring, cg is equal to the particle-settling velocity divided by the hydraulic depth, h. For erosion conditions, cg is set by the user based on the properties of the soil or channel sediment. Transport capacity in K2G is computed using the Engelund and Hansen (1967) total-load formula: C„ = 5x104Suu g 2 dh(S —1) 2 in which S is specific gravity of the particles u is velocity [L/T], u * is shear velocity, defined as jghS , for which Sb is bed slope (4.11) 112 g is acceleration due to gravity [LIT/TI d is particle diameter [L], h is water depth.[L] This formula equates the work done by tractive forces to the potential energy gained by the grains as they move up the stream face of bed dunes. The effective tractive force is related to the total load using similarity considerations (Alonso et al., 1981). Although the Engelund and Hansen formula was developed using limited flume data, experimental work by Govers (1990) and others using shallow flows over soil has demonstrated relations that are similar to the transport capacity relation of Engelund and Hansen (1967). Alonso et al. (1981), in a broad comparison of transport formulas, found that the Engelund-Hansen formula yielded the best results for concave field-plot tests. For channel flow, Alonso et al. (1981) found that the Engelund-Hansen formula yielded acceptable results without excessive scatter, although it tended to overestimate systematically the field data and underestimate the flume data. For light-weight materials, Alonso et al. (1981) found that the Engelund-Hansen formula was one of only two formulas to replicate the data satisfactorily. For very fine soil particles, Alonso et al. (1981) again found the Engelund-Hansen formula to be one of only two that gave estimates close to the observed load. Particle settling velocity is calculated from particle size and density, assuming the particles have drag characteristics and terminal fall velocities similar to those of spheres (Fair and Geyer, 1954). This relation is 2 v= 4 g(p —1)d 3 C E, (4.12) 113 in which CD is the particle drag coefficient. The drag coefficient is a function of particle Reynolds number, 24 3 C D =—++0.34 R, (4.13) in which R„ is the particle Reynolds number, defined as = d (4.14) where V is the kinematic viscosity of water [L2/1]. Settling velocity of a particle is found by solving equations 4.12-14 for The general approach to sediment-transport simulation for channels is nearly the same as that for upland areas. The major difference in the equations is that splash erosion (e s ) is neglected in channel flow, and the term q s becomes important in representing lateral inflows. Equations 4.8 and 4.10 are equally applicable to either channel or distributed surface flow. The erosion computational scheme for any element uses the same time and space steps employed by the numerical solution of the surface-water flow equations. In that context, equations 4.8 and 4.10 are solved for Cs (x,t), starting at the first node below the upstream boundary, and from the upstream conditions for channel elements. If there is no inflow at the upper end of the channel, the transport capacity at the upper node is zero and any lateral input of sediment is subject to deposition. The upper boundary condition is then 114 Cs(0,t)= qc qs +v s WB (4.15) where WB is the channel bottom width. A(x,t) and Q(x,t) are assumed known from the surface-water solution. Performance of the K2 sediment-routing model has been evaluated in several studies, all of which found it to yield acceptable results. At the watershed scale, Smith et al. (1999) tested K2 against a dataset from the 41.2 ha Catsop catchment in South Limburg, Netherlands. Calibration of the model parameters was performed based on six events, with regard to the temporal distribution of runoff and sediment rather than single values such as total or peak rates. The authors concluded that the overall ability of the model to reproduce the measured data was relatively good considering the limited data regarding soil conditions (agricultural activity) between events. Results from one of the validation runs for this study are plotted in figure 4.11. The hydrograph and sedigraph were both well represented by K2. Although these results were obtained using a calibrated model, they demonstrate that K2 represents well the runoff and sediment erosion/transport process in small watersheds. 115 50 Event 22.01.93 — Rainratail — Simulated Runoff Rate Measured Runoff Rate • • Simulated Sediment, right scale Measured Sediment, right scale ! * ! 50 150 100 Time from Start, min. 200 0 Figure 4.11. Graph showing measured and simulated runoff for an event on the Catsop catchment in South Limburg, Netherlands. From Smith et al. (1999). K2 has also been shown to perform well on a plot scale. Ziegler et al. (2001) tested the performance of K2 on 3-m 2 unpaved road plots in Thailand using a rainfall simulator. Following parameter optimization based on five events, they compared simulated and observed (water) discharge, sediment-transport rate, and sediment concentration for five additional events with variable plot slopes and initial soil-moisture conditions. Average model efficiencies were 0.40 and 0.36 for sediment output and concentration, respectively, for the validation runs. Results from three of the validation events are presented graphically in figure 4.12. Based on these results, Ziegler et al. (2001) concluded that although K2 performed well overall, it had the most difficulty in predicting the time-dependent sediment outputs. Most notably, early flush peaks and the temporal decay in sediment output were not predicted, owing to the inability of K2 to model removal of a surface sediment layer of finite depth. 116 0 14 0 12 0 10* r. 0.08 . 0.08 0.04 - 0 n 002 - 1.4.OE ROACIS c:: froaskre0 — XAEFIOrs2 iot) 80 r O KHOO 0 riamiut KINE FK;r52 ROADS fripWrIllitit 01404062 0 C -40 s 15 25 25 45 S5 '55 lirra rrnrl Comparison of discharge (Qt), sediment transport rate (St), and Figure 4.12. concentration (Ct) between measured rainfall simulation data (circles) and K2-predicted values for three events on road plots. From Ziegler et al. (2001). The results of Ziegler et al. (2001) demonstrate that K2 is capable of adequately representing hydraulic and splash erosion processes at the plot scale, although they also suggest that the model could benefit substantially from temporally variable surfaceerodibility parameters. For the purpose of the present study, the sediment-routing component of K2 seems sufficiently robust to permit generalized comparisons of geomorphic response to landscape change. 117 4.6 Geomorphic Modeling Tool Model input parameters are developed through a customized GIS-based interface developed specifically for this research. The interface is a modified version of the Automated Geospatial Watershed Assessment tool (hereafter referred to as the AGWA Geomorphic tool, AGWA-G) that develops input parameter sets and visualizes distributed output specific to K2G. AGWA-G coordinates batch event simulations by conducting separate, sequential simulations using each precipitation input (*.pre) file in a specified directory and modifying input parameters between simulations based on computed change from the previous simulation. Channel geometric parameters are tracked between simulations by means of a simple output text file containing the minimum output data needed to develop a modified input parameter file for the next simulation. This output file is written to the directory containing the K2G model executable. A sample element output from this file is shown in table 4.1. The second column in table 4.1 contains parameter values for the upstream end of the reach, and the third column contains parameter values for the downstream end of the reach. Parameter names followed by an asterisk represent the minimum output required to pass channel modifications to subsequent simulations. The remaining parameters are used in AGWA-G to track and visualize cumulative change resulting from multiple simulations. The last three rows are overbank parameters that are only recorded for compound elements. 118 Table 4.1. Sample output for one channel element from the geom.out file written during a K2G simulation. Parameter abbreviations are defined in grey, and all units are in meters except slope, which is dimensionless. Four significant digits are carried to minimize rounding error. 194 194 Elem* (c/cmcra /0) 1347.9685 1348.9463 Elev* (c/cvarion, 0.2313 dcpth 0.2337 Dpth* 1.1164 0.8760 Cwid* (chtlitn('l 11 0111) /0/4 1.3602 /tic . n bank 1.2990 1* ( i SS 5 S2* (ri,:;/// bank si ( h , ,/ op c ) 1.3602 1.2990 0.0010 0.0008 TWA (rota/ wichh adji/Arnicia) 0.0162 0.0134 TDA (iota! cicp111 10.0000 10.0000 ED* (crodibie (/(7)///) 1.1144 Owid* (orci-b(///k 11 /(///i) 0.8744 0.0162 0.0134 chwincl (1(7)111 TCDA . , , , . inchtding (11 crb,1nk acpth SOB (siminica ovcrbarik acpmition) . 0.0000 0.0000 The reported depth and width adjustments (TWA, TDA, TCDA, and SOB) are calculated at nodes 2 and 19, and do not necessarily equal the difference between input and output widths and depths. The parameters required for subsequent simulations are for the reach endpoints, and are extrapolated linearly from where they are computed at nodes 2 and 19. It was not, however, deemed necessary to do this for the parameters not required in subsequent simulations. 4.6.1 Profile Smoothing The inherited conceptual model of KINEROS treats the channel network as a series of connected but independent reaches. Outflows from upstream contributing reaches are summed to determine inflow for a downstream reach, but that is the extent of the interaction between reaches. As a result, flow at channel junctions, or confluences, is poorly described by the model. Together with the KINEROS transport capacity relation 119 that exaggerates deposition or erosion immediately downstream of a channel junction, this can result in substantially different elevation adjustments among the reaches connecting at a junction. During a simulation the initially continuous profile thus can become discontinuous, with abrupt drops or ridges at the channel junctions. Whereas this poses no computational problems in the model, it results in computed geomorphic change that is unreasonable when considering the entire channel network. Figure 4.13, for example, shows the lower end of the channel profile in Walnut Gulch subwatershed 223 before and after a batch simulation of five 10-year 1-hour events without any external modification of the flow profile between simulations. It can be seen that elevations diverge significantly between reaches as the slope is adjusted independently for each reach in an attempt to convey the inflowing sediment. To correct this problem an algorithm was developed to redistribute sediment between the downstream ends of upstream reaches and the upstream ends of downstream reaches, thereby ensuring that channel elevations are the same for all reaches connecting at a junction. The algorithm is designed to determine the weighted average elevation that produces a volume change downstream equal to the volume change in the upstream contributing channel(s). For cases when downstream deposition has caused an abrupt increase in elevation at the junction a volume of sediment is transferred from the downstream reach to the upstream reaches such that the final elevations at the junction are equal. Conversely, if erosion downstream of the junction produces a sharp drop in the profile then a volume of sediment is transferred from the upstream reach(es) to the downstream reach such that the final elevations at the junction are equal. The algorithm 120 1339.0 Original 1338.5 . Final 1338.0 3: 1337.5 o t 1337.0 1336.5 1336.0 1335.5 1300 1400 1350 1450 1500 Distance (m) Figure 4.13. Graph showing the channel profile near the outlet of Walnut Gulch Subwatershed 223 before and after a K2G batch simulation of five 10-year 1-hour events and no profile smoothing between events. See text for discussion. is applied to each reach for which the upstream elevation may need to be adjusted. First order channels are not adjusted. Derivations for this volume adjustment are presented below for the cases where there are one and two upstream contributing reaches, respectively. 220.127.116.11 Two-Reach Junction Two-reach junctions are defined as junctions of one - upstream reach with one downstream reach (figure 4.14). This type of junction occurs where a single long reach must be divided into two or more shorter reaches to accomplish any of the following objectives: • Model output is required for a location where the channel was not automatically split, for instance at a gauging station 121 • Reach characteristics vary significantly within a longer reach, and it must be split into two or more shorter reaches with different geometric parameters to adequately describe the variability. • A long reach must be divided into two shorter reaches to enhance the ability of the model to adjust reach gradient through elevation changes at the endpoints Figure 4.14 shows the initial endpoint elevations for an upstream reach (El & E2) and a downstream reach (E3 E4) following a K2G simulation in which a lower slope in the downstream reach resulted in preferential deposition at its upstream end (E3). The objective of the profile-smoothing algorithm is to compute the weighted-average elevation, EA, that results in equal volume changes in the upstream and downstream reaches (i.e. VI = V2). This requirement preserves mass within the channel network even though volume changes in individual reaches may result in a net change in mass. Figure 4.14. Diagrammatic representation of two-reach junction profile adjustments showing variables used in derivation of the adjusted elevation, EA. Elevations are represented by symbols beginning with `E', volumes with 'V', and lengths with `L'. 122 Given upstream and downstream reach lengths of Li and L2, respectively, the volumes can be calculated as: V1= 0.5(W1)(L1)(EA — E2) (4.16) V2 = 0.5(W 2)(L2)(E3 — EA) (4.17) where WI and W2 are the average widths of the upstream and downstream reaches, respectively. The average width for any given reach is determined by computing summing the upstream and downstream widths, and dividing by two. Because erosion in the downstream reach could produce the opposite situation, the volumes can be expressed such that they are independently of sign by letting: (4.18) EC1=1EA— E21 EC2=1EA— E31 (4.19) VI = 0.5(W1)(L1)(EC1) (4.20) V2 = 0.5(W 2)(L2)(EC2) (4.21) to get By requiring the volume change upstream (V1) to be equal to the volume change downstream (V2), equations 4.20 and 4.21 can be combined as (W1)(L1)(EC1)= (W2)(L2)(EC2) If ED is defined as (4.23) ED =1E3—E21 EC1+ EC2= ED (4.22) (4.24) 123 yielding two equations, 4.22 and 4.24, and two unknowns, EC1 and EC2. To solve for EC I, 4.22 and 4.23 are combined to yield (W1)(L1) EC2 ED—Ed ED 1 EC1 (W2)(L2) EC1 EC1 (4.25) Let the ratio of widths and lengths, RWL be defined as RWL= (W 1)(L1) (W 2)(L2) (4.26) ED RWL+1 (4.27) Substituting and solving (4.25) for EC1 yields EC1= Because EC1 is an absolute value, EA= E2± EC1 (4.28) To determine the value for the adjusted elevation, the two possibilities are compared to the midpoint elevation between E2 and E3, which is computed as the algebraic average of these two elevations. The value of EA that is closer to the average elevation is selected. 18.104.22.168 Three-Reach Junction - Three-reach junctions are defined as junctions of two upstream reaches with one downstream reach. This type of junction is the most common, and channels are always split at this type of junction. Whereas the case of three upstream reaches is a possibility, and is permissible in the model, this type of junction is almost never observed in nature, and is therefore not considered in this derivation. Figure 4.15 shows the initial endpoint elevations for two upstream reaches (El and E2) and a downstream reach (E3) following a K2G simulation in which a lower slope 124 in the downstream reach caused deposition at its upstream end (E3). The objective of the profile-smoothing algorithm is to compute the weighted average elevation, EA, for which the combined volume change as a result of elevation adjustments for the upstream reaches equals the volume change as a result of elevation adjustment in the downstream reach (i.e. V1 + V2 = V3). Again, this requirement is designed to preserve mass within the larger channel network even though volume changes in individual reaches may result in a net change in mass. Figure 4.15. Diagrammatic representation of three-reach junction profile adjustments showing variables used in derivation of the adjusted elevation, EA. Elevations are represented by symbols beginning with `E', volumes with 'V', widths with 'W', and lengths with `L'. Following the same variable conventions as the two-reach junction scenario, average width, length, and volume change are referred to as W, L, and V, respectively. The two upstream reaches are denoted as reaches 1 and 2, and the downstream reach as 3. 125 As before, the volume adjustments can be calculated as V1= 0.5(W1)(L1)(EC1) (4.29) V2 = 0.5(W2)(L2)(EC2) (4.30) V3 = 0.5(W3)(L3)(EC3) (4.31) where the elevation-change variables, EC, are absolute values of the difference between the original and adjusted elevations. Given the preservation of mass requirement (VI + V2 = V3), the three volumes can be rewritten as (W1)(L1)(EC1)+ (W 2)(L2)(EC2) = (W3)(L3)(EC3) (4.32) Equation 4.32 contains three unknowns, so two additional equations are needed to solve it. From figure 4.15, EC1 and EC2 can be expressed in terms of the known original elevation differences between the three points as EC1= El — E31— EC3 = ED 1 _ 3 — EC3 EC 2 =1E2 — E31— EC3 = ED 2_3EG3 (4.33) (4.34) Substituting these into (4.32) yields (W1)(L1)(ED 1 _ 3 — EC3)+ (W 2)(L2)(ED 2 _ 3 — EC3) = (W3)(L3)(EC3) (4.35) Rearranging to get all EC3 values on one side of the equation yields 126 (W1)(L1)(ED,_ 3 )+ (W2)(L2)(ED 2 _ 3 ) = (W 1)(L1)(EC3)+ (W 2)(L2)(EC3)+ (W3)(L3)(EC3) (4.36) which can be simplified to (W1)(L1)(ED 1 _ 3 )+ (W2)(L2)(ED 2 _ 3 ) = (EC3)[(W 1)(L1) + (W 2)(L2) + (W3)(L3)] (4.37) The final solution for EC3 is thus EC3 = (W1)(L1)(ED1 _ 3 )+ (W2)(L2)(ED 2 _ 3 ) 1( 4 7 1)(L1) + (W 2)(L2) + (W 3)(L3)] (4.38) Because EC3 is an absolute value, EA = E3 EC3 (4.39) To determine the value for the adjusted elevation, the two possibilities are compared to the algebraic average of the three elevations, El, E2, and E3. The value of EA that is closer to the average elevation is selected. 22.214.171.124 Procedure — Junction-elevation adjustments are computed systematically for each junction in the watershed. The algorithm iterates through each reach in the channel network, and if the reach has at least one contributing reach then the adjustments are carried out for the upstream end of the reach. Contributing reaches are queried from the stream attribute table to get the parameters needed to compute the adjustment, and adjustments to the downstream ends of the contributing channels are made at the same time. The upstream ends of reaches with no upstream contributing reaches, and the downstream end of the outlet reach are never adjusted during profile smoothing because they are not located at junctions of 2 or more reaches. 127 In addition to adjusting elevations at the channel, or reach, junctions, several of the other geometric parameters are adjusted to reflect the change in elevation. The elevations represent channel-bed (thalweg) elevation, so reducing the channel-bed elevation is translated into an increase in channel depth and a decrease in the erodible depth by the same amount (as if the modeled infilling of the channel never happened). Likewise, the following parameters used to track event and cumulative change in AGWA-G are also modified by the same amount: channel depth adjustment (for the simulation), cumulative channel depth adjustment (for the batch), total channel depth adjustment (which includes overbank deposition/erosion for the simulation), and cumulative total channel depth adjustment. To ensure that section widths do not change (as in K2G), bank slopes are modified to reflect the new channel depth and a fixed channel side width. Overbank parameters are not modified during the channel junction elevation adjustments. 4.6.2 Results of Profile Smoothing Adjusting reach junction elevations after and between simulations produces a much more reasonable longitudinal profile than when elevations are not adjusted. Figure 4.16 shows the same section of the channel profile as figure 4.13 following a batch simulation with the elevation adjustments (profile smoothing). The final profile following this simulation is more reasonable from a geomorphic prospective, with profile irregularities largely removed and a final profile with decreasing slope in the downstream direction. 128 Profile smoothing causes significant sediment mass-balance errors in any given reach. A portion of the sediment deposited in a downstream reach is transferred to the upstream reach(es) to yield a common elevation at the junction. Whereas this seems illogical, it takes sediment that is deposited in a very short section at the upstream end of the downstream reach and redistributes a portion of it to the downstream end(s) of the upstream reach(es). Mass is conserved during the adjustments, so there is no net effect on the sediment balance for the channel network (see mass balance section below). 1339.0 Original 1338.5 -is Final 1338.0 î 1337.5 o LTI 1337.0 1336.5 1336.0 1335.5 1300 1350 1400 1450 1500 Distance (m) Figure 4.16. Walnut Gulch Subwatershed 223 profile near the outlet before and after a K2 0 batch simulation of five 10-year 1-hour events with profile smoothing between events. 129 4.6.3 Mass Balance Calculations At the end of each batch simulation AGWA-G computes the mass balance for each channel, and for the entire channel network. This procedure involves (1) tracking event and cumulative sediment output from each upland element (plane) in the watershed, (2) computing change in sediment storage for each channel, and (3) computing the sediment yield at the watershed outlet. To accomplish each of these tasks it was necessary to create two additional database files for each batch simulation: one for tracking event and cumulative outputs for the planes, and one in which the change in sediment storage produced by geometric change can be calculated for each reach. These files are added to the project with the same simulation name followed by "_batchp.dbf" and "_volbal.dbr, respectively. Whereas the batchp.dbf file tracks model output for each event, the volbal.dbf file is used to compute the mass of sediment required for the calculated geometric changes in each reach. This calculation is accomplished by computing the volume of sediment produced by three types of geometric change: change in channel cross-sectional area, overbank deposition/erosion, and section-wide deposition that may have occurred if the channel filled to its minimum dimensions at any point during the simulations. These types of geometric change are computed separately as VC1, VC2, and VC3, respectively, so that the source of significant errors can be identified. The summed volumetric change is converted to a mass by assuming an average particle density of 2.65 g/cm 3 , and a porosity of 0.40 (40%). This mass is then compared to the value computed in K2G for each reach, and the relative error is written to the "Error" field. During the volume- 130 balance calculations, the post-simulation section widths at the upstream and downstream ends of each reach are computed and compared to the original values to ensure that no unintended section width change has taken place. These values can be found in the "SW1" and "SW2" fields, respectively. Following calculations for each individual reach, the sediment mass balance is computed for the entire channel network. A summary is written to the volbal.dbf table comments, which can be accessed under the "Table" menu by selecting "Properties" in the AGWA-G application. Sample results are shown below from a simulation of a single 10-year 1-hour event with profile smoothing. Mass balance for profile smoothing run: Network-Wide Mass-Balance Calculations for Batch Simulation: Modeled plane erosion: -1299850 kg Modeled channel erosion: -519410 kg Geometric channel erosion: -503080 kg Modeled sediment outflow: 1706310 kg (myMociPlancOut) (mvModChanDep) (my(lcoChanDcp) (niyMociSeclOut) K2G sediment mass balance error: -6.62 (%) Equation: (myModPlaneOut + myModSedOut + myModChanDep) * 100/ myModSedOut Geometric adjustment mass balance error: -5.66 (`)/0) Equation: (myModPlaneOut + myModSedOut + myGeoChanDep) * 100 / myModSedOut Error of geometric adjustment relative to modeled deposition: 3.14 (%) Equation: (myModChanDep - myGeoChanDep) * 100 / myModChanDep Volume-balance error was computed for simulations based on a single 10-year 1- hour event with and without profile smoothing (table 4.2). As this output shows, modeled change in network-wide sediment storage is very close to the mass required to 131 produce the computed geometric change. The difference between the two values can be largely attributed to rounding errors that propagate through the many computations involved in a simulation and the subsequent analysis. It is not clear why the difference is greater for the simulation with no profile smoothing. Although the error for geometric change in this simulation is less than the modeled error, it is likely that this could be reversed for a depositional scenario. To confirm that mass is not persistently lost during multiple consecutive simulations, the model was run with profile smoothing for five consecutive 10-year 1- hour events. Results show that the errors are almost identical to those from the singleevent run (table 4.2). Table 4.2. Results of the volume-balance calculations for one 10-year 1-hour event with (PS) and without (NS) profile smoothing, and five 10-year, 1-hour events with profile smoothing. Insignificant digits are carried to prevent rounding error. 1 10-yr 1-hr, PS 1 10-yr 1-hr, NS 5 10-yr 1-hr, PS -6499200 -1299800 -1299800 Modeled plane erosion (kg) -2706500 -519400 -519400 Modeled channel erosion (kg) -2619800 -453700 -503100 Geometric channel erosion (kg) 8632200 1706300 1706300 Modeled sediment outflow (kg) -6.64 -6.62 -6.62 K2G volume-balance error (%) -5.64 -2.77 -5.66 Geometric vol.-balance error (%) 3.21 12.64 3.14 K2G-geometric relative error (%) 4.7 Problems Most problems that occurred during the development and testing of the model have been eliminated. Most notable among those that remain, however, is the presence of very short channel reaches that can occur when two tributary channels connect to the 132 main stem almost directly opposite each other. Very short channel reaches are subject to massive shifts in the channel slope during the model run, as well as during the profile smoothing. This can result in failure of the numerical model, and interruption of a batch simulation. This problem is most pronounced when high-resolution DEMs are used during the watershed discretization. In one instance a reach length of approximately 2 meters was created when using 1-meter Light Detection and Ranging (LIDAR) data for the discretization. It was only possible to get the model to run by artificially lengthening the channel to 15 meters. Very long reaches can also cause problems in the numerical model, although not as severe. Channel slope can only vary as a result of elevation change at the reach endpoints, so very long reaches require a much greater combination of erosion and deposition to have a significant impact on the channel slope. If the amount of deposition required to increase the channel slope (and hence prevent continuing deposition) is greater than the original channel depth, then unrealistic geometric change can result because the channel will fill in to its minimum dimensions before the entire section can be raised. Although this occurs in nature, by representing the natural system as a network of longer reaches the model cannot adequately resolve where it happens. Computed reach-average geometric change will therefore be less representative of on-the-ground geomorphic change. The parameter that channel adjustments are most sensitive to is slope, particularly in relation to adjacent reaches. As a result, if reach slopes are derived from integer or low-resolution DEMs computed geomorphic change throughout the channel can be 133 driven predominantly by error in the initial reach slopes. Narrow and short channels are most likely to inherit errors from the elevation data, but all channels can be affected to some degree depending on the resolution and vertical accuracy of the DEM. Experience working with 30- and 10-meter DEMs, and 1-meter LIDAR has shown that as the resolution increases problems with the computed slopes become less pronounced. In the absence of a high-resolution DEM or field measurements of channel elevation at the endpoints of each reach, it may be necessary to subject the model to an initial series of flows to allow slopes to adjust. Unwanted width and depth changes are commonly associated with these adjustments, so the 'primer' events only should be used to update reach endpoint elevations. The desired simulation period should proceed with the original width and depth values determined by the user. Another problem is associated with the use of distributed precipitation files. For five of the approximately 150 distributed precipitation files used in the testing of K2G, massive deposition in certain reaches caused model failure. It appears that this occurs when computed flow depth drops to approximately zero at the leading edge of the primary flood wave, causing the majority of sediment at the leading edge of the flow to deposit. The problem arises in the kinematic routing subroutine, but it is possible that the geomorphic adjustments somehow interfere with the routing. Regardless, the volume of deposited sediment results in very large adjustments to the channel geometry that ultimately cause the model to fail. The problem is identified in K2G by comparing the change in sediment storage for each time step to the area of the channel. If the change in sediment storage is greater than the channel area, the model run is aborted and the 134 problem is reported. This error has not been observed with design (spatially uniform) rainfall events. 4.8 Limitations There is a variety of limitations associated with the K2G conceptual model and its practical application that have been identified during its development. Perhaps the most significant is that the model cannot be applied readily to large watersheds as was hoped. This is largely due to automated procedures for discretizing the watershed in a GIS that have not been able adequately to split long reaches into a series of shorter ones. In addition, the evaluation of representative upstream and downstream geometries for each reach is tedious and requires substantial time and effort. For finely discretized large watersheds this will remain a daunting task until such time as reliable automated procedures have been established, and high-resolution DEMs are widely available. Representing the slope of each reach with a single value defined by the bed elevation at each end of the reach is also limiting. This is necessary to allow dynamic slope adjustment, but it has the effect of reducing the resolution of the channel profile representation. As with any spatial averaging, a considerable amount of information is lost, and there is an associated cost in terms of the accuracy of the computed results. Considering short reaches helps minimize this problem, but as described above the more reaches the greater the time and expense of developing parameter inputs. Another limitation of the K2G model is the obvious problem of representing highly variable natural channels as uniformly varying trapezoids. This crude 135 simplification of channel geometry can be frustrating when attempting to define width and depth from a measured cross-section. Gross generalizations must be made to describe reach geometry in this fashion, and there is as yet no repeatable means of accomplishing the task. Finally, there are limitations inherited from the parent model, K2. Flow in braided or anastomosing channels cannot be represented. Sediment grain-size distributions on the channel bed and banks are the same and cannot vary through time, thus precluding armoring. Strongly non-uniform flows and flows in channels with very low slopes are not well represented. Model performance decreases with increasing watershed size (particularly beyond about 100 km 2 ) and decreasing magnitude of the rainfall input. Perhaps the largest source of error associated with K2 simulations is that of rainfall input, and K2G simulations based on observed rainfall will not be widely possible until AGWA-G has been updated to derive rainfall estimates from radar data. From a sediment-transport modeling perspective, another important limitation of K2G is the use of a transport-capacity relation to compute erosion and deposition. When transport capacity is less than sediment load the rate of deposition is limited only by the settling velocity of the particles. Thus if sand particles are being transported, the rate of deposition can be quite high. When transport capacity exceeds the sediment load the rate of erosion is limited by the transfer-rate coefficient, which can be high for noncohesive sediment on channel beds. In gradually varied natural systems these erosion/deposition rate limiters may represent the transfer of sediment between the flow and the channel bed adequately. When they are applied in the K2/K2G conceptual model, however, they are 136 problematic because of abrupt changes in channel geometry, and hence transport capacity, at reach junctions. Additional general assumptions of the K2G sediment-transport calculations include: 1) The fluid and sediment properties are steady and uniform 2) There is an infinite and continuous supply of sediment particle sizes represented in the bed material (transport-limited conditions), unless specifically limited by the user, or if the maximum erodible depth is reached 3) The particle-size distribution of the bed material does not vary as a result of preferential transport of smaller particles 4) There is a specific relation between hydraulic and sedimentological parameters and the rate at which the sediment load is transported 5) The sediment stored in a reach can be neglected These assumptions are all used to simplify governing equations and numerical procedures, and are frequently not well met in natural systems. 137 5.0 GEOMORPHIC MODEL RESULTS 5.1 Methodology Despite the extensive network of automated hydrologic measuring devices on the Walnut Gulch Experimental Watershed, there has been no systematic effort to document channel morphology throughout the watershed. A variety of research has investigated hillslope erosion and sediment yield, but only recently have efforts begun to monitor channel morphologic change, and insufficient data are presently available to test the performance of K2G in terms of its ability to reproduce observed morphologic change. Testing of the model thus concentrated on evaluating relative model response to variable land cover/use conditions, initial channel geometry, and climatic inputs. Test simulations were performed on Walnut Gulch subwatershed 11, which was discretized into model elements using a contributing (channel) source area, CSA, of 8.1 hectares, or 1% of the watershed area (figure 5.1). A 10-meter DEM was used during the watershed delineation and discretization procedure. A large pond in the upper central portion of the subwatershed contains all but the most extreme events, and was thus removed from the model. The AGWA internal gauge feature was used to break up longer reaches as much as possible, but problems with this functionality limited its success. The average channel length is 329 m, and the maximum and minimum channel lengths are 927 and 25 m, respectively. Elevations at the upstream and downstream ends of each reach were obtained from 1-meter LEDAR data to minimize channel-slope errors. 138 Twenty four batch simulations were conducted to test model performance for a range of land use/cover and climatic (rainfall) inputs, as well as for two initial channel geometry configurations. Table 5.1 lays out the initial conditions for each of these simulations. Parameter inputs from the land cover and soil data were derived using AGWA's default parameterization routines (Miller et al., 2002). The Soil Survey Geographic Database (SSURGO) was used to derive soil-based parameter inputs for all simulations. Simulations were conducted based on distributed, observed precipitation records from the summer monsoon seasons of three different years: 1964, 1977, and 1978. These years were selected as representative of wet, average, and dry years based on observations at the National Weather Service gauge in Tombstone, AZ. Unfortunately there is no easy means of evaluating annual totals for any given subwatershed, and because of the high spatial variability of rainfall on Walnut Gulch, the observations at Tombstone were not representative of precipitation at WG11. After running the simulations, which automatically interpolate rainfall depths across the watershed, it was possible to characterize the distribution of events for each year based on the average depth of rainfall over the watershed (Table 5.2). Based on this analysis, wet, dry, and intermediate years were represented, but the rainfall totals for the intermediate year are much closer to the wet year than the dry year. 1964 was the wettest year, and was characterized by 52 events with an average depth of 9.29 mm. The intermediate year, 1977, was characterized by 47 events with an average depth of 7.86 mm. The dry year, 1978, was characterized by 41 events with an average depth of 1.02 mm. 139 • • • Figure 5.1. Maps showing the watershed discretization (A), DEM (B), SSURGO soils (C), and rain gauge configuration (D) for Walnut Gulch subwatershed 11. Table 5.1. Batch simulations, and their associated inputs, used to evaluate the geomorphic model. Simulation Name 1964 HG 73 1964 HG 97 1964 HG PU 1964_HG_U 1964 OG 73 1964 OG 97 1964 OG PU 1964_0G_ U 1977 HG 73 1977_HG_97 1977_HG_PU 1977 HG U Land Cover Input Precipitation Input Channel Geometry NALC 1973 NALC 1997 Partially Urbanized 100% Urban NALC 1973 NALC 1997 Partially Urbanized 100% Urban NALC 1973 NALC 1997 Partially Urbanized 100% Urban 1964 Monsoon 1964 Monsoon 1964 Monsoon 1964 Monsoon 1964 Monsoon 1964 Monsoon 1964 Monsoon 1964 Monsoon 1977 Monsoon 1977 Monsoon 1977 Monsoon 1977 Monsoon Hydraulic geometry Hydraulic geometry Hydraulic geometry Hydraulic geometry Observed Observed Observed Observed Hydraulic geometry Hydraulic geometry Hydraulic geometry Hydraulic geometry 140 1977_0G_73 1977_0G_97 1977_0G_PU 1977_0G_U 1978_HG_73 1978_HG_97 1978_HG_PU 1978_HG_U 1978_0G_73 1978_0G_97 1978_0G_PU 1978_0G_U NALC 1973 NALC 1997 Partially Urbanized 100% Urban NALC 1973 NALC 1997 Partially Urbanized 100% Urban NALC 1973 NALC 1997 Partially Urbanized 100% Urban 1977 Monsoon 1977 Monsoon 1977 Monsoon 1977 Monsoon 1978 Monsoon 1978 Monsoon 1978 Monsoon 1978 Monsoon 1978 Monsoon 1978 Monsoon 1978 Monsoon 1978 Monsoon Observed Observed Observed Observed Hydraulic geometry Hydraulic geometry Hydraulic geometry Hydraulic geometry Observed Observed Observed Observed Table 5.2. Characterization of the precipitation records for the summer monsoon on WG11 during 1964, 1977, and 1978. 1978 1977 1964 Year Number of events Total Precip. (mm) Ave. event depth (mm) Max. event depth (mm) Standard deviation 52 483.3084 9.294393 51.92596 12.35652 47 369.1959 7.855232 40.39204 8.949365 41 41.92376 1.022531 1.765405 0.393512 The rainfall record may have differed substantially if rainfall intensity had been used as a selection criteria in addition to the annual total rainfall depth. This would have permitted the elimination of annual totals for which much of the precipitation was derived from long-duration events of low intensity. In addition, it would have been possible to characterize the rainfall record, perhaps more appropriately, in terms of its erosive capacity. Runoff in K2G is generated as infiltration excess, so higher rainfall intensities result in increased runoff. High-intensity rainfall has the greatest potential to affect erosion on upland surfaces because of its potential for sediment detachment through rain-splash erosion, and increased ability to transport that sediment because of increased flow depths. Increased runoff, and hence stream power, in the channels increases the transport capacity, and can result in increased erosion. Unfortunately, 141 rainfall intensity at a single gauge is not representative of intensity over a larger area, and spatial averaging of measured rainfall intensities from a network of gauges for every event in a year was not practical. For this reason, selecting the rainfall records in terms of annual total depth was the most suitable means of obtaining records with variable average and total rainfall depths. Parameter estimates for the simulations were derived from four land-cover grids: two observed, and two hypothetical. The land-cover grids (figure 5.2) included: 1) North American Landscape Characterization (NALC) project classified land cover for 1973 2) 1997 Landsat TM imagery classified according to the NALC scheme. 3) A hypothetical 100% urbanized condition 4) A hypothetical partially-urbanized condition for which the 1997 land cover was altered to convert part of the watershed to the urban class. The 1973 land cover is characterized predominantly by grassland and desert-scrub vegetation, with lesser amounts of mesquite woodlands (table 5.3). In 1997 the land cover is characterized by increased mesquite woodlands and decreased grassland, but the difference is less pronounced than would be ideal for comparative purposes. The 1973, 1997, and 100% urbanized land-cover grids are relatively homogenous, so the part-urban land cover was designed to introduce substantial heterogeneity by converting the upland area of the southernmost tributary to the urban class. The part- and all-urban grids produce significantly different parameter sets than the other two. For the urban landcover class, AGWA assumes that 40% of the upland areas are impervious, and hence it 142 reduces saturated hydraulic conductivity estimates, derived from the underlying soil, by 40%. In addition to the saturated hydraulic conductivity reductions, erosion is reduced by setting a parameter that designates the percentage, 40% in this case, of the surface covered by pavement (i.e. non-eroding). Vegetative cover, and hence interception, is reduced by the assumption that urban areas are characterized by 15% cover, relative to 20% for mesquite woodlands and 25% for grassland and desert scrub. Table 5.4 summarizes the parameter estimates used in AGWA for each land cover class. I= Forest Oak Woodland Mesquite Woodland Grassland Desertscrub ▪ Riparian Agriculture I= Urban Water Barren / Clouds ▪ 1 MI ill Figure 5.2. Maps showing the four land-cover scenarios used to derive model parameters for the test simulations: (A) NALC 1973, (B) NALC 1997, (C) Partial urbanization, and (D) Total urbanization. The legend for the NALC classification scheme is shown for reference. Outlines of the discretized model elements are superimposed on the landcover grids. 143 Table 5.3. Composition of land cover classes on WG11 for the four land-cover scenarios used during model testing. NALC 1973 NALC 1997 Part Urban 1997 All Urban Mesquite Woodlands 0.0 4.9 1.2 0.0 Grassland 54.8 53.0 34.4 0.0 Desertscrub 45.2 42.1 28.9 0.0 Urban 0.0 0.0 35.5 100.0 Table 5.4. Hydrologic parameters used by AGWA during land cover and soils parameterization. Imperv n Cover Int Class Name 0 20 1.15 0.050 3 Mesquite Woodland 2.00 0.150 0 4 Grassland 25 25 3.00 0.055 0 5 Desert Scrub 0.10 0.015 40 15 8 Urban All simulations were performed using compound channels, or channels with flood plains. Two different methods were used, however, to determine the initial channel geometries. The first method involved using hydraulic geometry relations derived for Walnut Gulch by Miller (1995), which is the default method for estimating channel geometry in AGWA. Miller (1995) derived relations for width and depth as: W = 0.0724A °3377 (5.1) D = 0.0502A" 523 (5.2) where A is the contributing area in square meters. Hydraulic-geometry channels were retained in this analysis to evaluate model behavior in the absence of observed-geometry data. Flood-plain widths estimated by this method were three times the channel width. Side slopes of the channel and overbank are set to 1.0 (rise/run), and the lateral slope of the flood plain is set to 0.01. 144 The second method involved defining channel cross-sections from a high resolution (1-meter) DEM that was developed from LIDAR data acquired in May, 2003. Cross-sections were obtained for at least three locations on every reach, and the data were imported into Excel where geometric parameters were computed. Translation of natural channel cross-sections into trapezoidal sections for the model was done manually. Placement of bank tops and toes was accomplished with guidance from the hydraulic- geometry relation to improve estimation of channel area when multiple banks were present. In general, however, the transformation was conducted by considering the most appropriate means of representing the natural section with a compound trapezoidal one. In many cases, for instance, channel areas were substantially larger than predicted by the hydraulic-geometry relation because the channel locally was incised. The maximum erodible depth of all reach endpoints was set to 5 meters for all simulations in the absence of field observations. This assumption is probably not justified for the majority of reaches on WG11, but the maximum simulated depth change during the model tests was 0.475 meters. The sediment grain-size distributions for the model runs were divided into five size classes (table 5.5). These size classes are global parameters for the model, and therefore apply to channel, overbank, and upland elements, although the percentage of sediment in each size class can vary for each element. Distributions for the upland elements are based on spatially averaged sand, silt, and clay fractions derived from the soils data. For the sand class, 25% was split into the medium sand class with a median diameter of 0.625 mm. The remaining 75% was left in the fine sand category, and the silt 145 and clay fractions were not modified. The coarse sand fraction was set to zero for all upland elements. Grain sizes for all channel and overbank elements followed the distributions laid out in table 5.5. Whereas the overbank distribution is based on the AGWA default values, the channel distribution was derived from distributions measured by Canfield (1998) on Walnut Gulch. Table 5.5. Sediment grain-size distributions as percentages of the dry sample weight for the channel and overbank elements. 3 0.625 0.15625 0.0332 0.002 Average particle size (mm) 4.1 13.7 22.7 59.5 0 Channel Overbank 0 10 40 40 10 5.2 Results 5.2.1 Graphical Output Results from the geomorphic model testing are presented in figures 5.3 to 5.10. Each figure displays results for a particular land-cover scenario and initial channel configuration (hydraulic geometry or observed). The six maps in each figure are arranged such that average depth changes are presented on the left, and average width changes on the right. It is important to note that the geomorphic model computes width and depth changes at both the upstream and downstream ends of each reach. To reduce the amount of data presented and simplify the analysis, these values were averaged. The map pairs A & B, C & D, and E & F are arranged in order of decreasing precipitation from top (1964) to bottom (1978). Color ramping of the results was conducted by means of an automated procedure in AGWA-G, and is unique to each set of 146 displayed results. The legend for each map is thus displayed to serve as a reference for the range of values in the simulation results. Negative values, shown in shades of blue, represent simulated channel width or depth decreases (deposition). Positive values, shown in shades of red, represent width or depth increases (erosion). For each map the class containing the zero value (no change) is shown in white. Figures 5.3 to 5.10 provide a glimpse of the enormously complex relations among landscape change, climate, and geomorphic response. It is not possible to interpret meaningfully the geomorphic response of any given reach to land changes in land cover ancUor precipitation. The spatial and temporal variability of rainfall and erosion upstream is too complex to lend itself to simple interpretation. Instead, only the broadest observations are made of the various results. The first and most obvious observation is that large, frequent precipitation events are more likely to cause widespread erosion than are small, infrequent events. The 1964 monsoon caused the greatest and most widespread erosion throughout the watershed, regardless of land cover. The 1978 monsoon produced the most widespread deposition, but because the events were so small the amount of deposition was negligible. The 1977 monsoon, with more uniformly intermediate events, produced more mixed deposition and erosion. Differences in geomorphic response to the initial channel geometries are significant. Here the magnitude of the response was less significant than the spatial distribution of geomorphic changes. Changes were less uniform and more specific to the individual reaches when observed geometries were used. Specifically, incised reaches 147 are likely to exhibit erosion, and wide reaches are likely to have deposition. In addition, the wide channels, where present, promoted transmission loss, which in turn resulted in lowered runoff volume (figure 5.11). Observed geometries were highly variable, and alternating wide and narrow sections were commonly observed within a single reach. The inability to split reaches at these transitions severely limits the model's representation of the system. In addition, when the geometry is adequately represented the absence of observed maximum erodible depths likely results in exaggerated change for incised sections where flow velocities, and hence transport capacity, are high. Geomorphic response to different land-cover conditions is less pronounced than to the different precipitation records or initial channel geometries. A minor mesquite invasion between 1973 and 1997 produced a negligible impact on channel morphology, and although a small increase in erosion can be observed it is most likely well within the error of the simulations. The partial-urbanization scenario produced the most notable differences in the spatial pattern of erosion/deposition, as would be expected given the significant heterogeneity in its land cover. The specific response to partial urbanization, however, was mixed. For the 1964 monsoon increased erosion was observed relative to that for the NALC 1997 land cover, but for the 1977 monsoon the opposite was true. These differences are discussed in more detail in the following chapter. The all-urban land-cover scenario produced increased erosion in some cases, increased deposition in others. The 1964 simulation based on the hydraulic geometry channels produced a dramatic and widespread increase in erosion, but these results are 148 sufficiently different from all of the others that they are questionable (see the mass balance discussion below for more details). The 1964 simulation based on the observed channels produced very little geomorphic adjustment throughout the watershed, with the exception of two short reaches in the lower portion of the watershed. For 1977, the allurban scenario produced widespread deposition for the hydraulic geometry channels, and a wide range of erosion and deposition for the observed channels. 149 slid A/ -0.088 - -0.071 slid -0.012 - 0.017 0.017 - 0.046 0.046 - 0.074 A/ 0.074 - 0.103 0.103 -0.132 NO.132 - 0.16 No Data wild N N -0.071 --0.055 A/-0.055 - -0.038 A/ -0.038 - -0.022 -0.022 - -0.006 -0006-0.011 0.011 - 0.027 No Data wild 511d slid -0.004 - 0.001 0.001 - 0.005 0.005 - 0.009 A/ 0.009 - 0.013 A/ 0.013 - 0.018 A/ 0.018 - 0.022 No Data wi ld A, ,Q‘ 0-0 0-0 0 - 0.001 0.001 - 0.001 A/0.001 - 0.001 r. N Decreasing Precipitation A/ 0.001 - (loot No Data wild slid -0003--0002 002- -0.002 A/ -0002--0.001 A/ -0 001 -0.001 -0.001 0-0 0- 0.001 At• 0.001 - 0.001 /V000 1 -0002 NO.002 - 0.003 No Data wild 4 Figure 5.3. Maps showing depth and width changes (left and right, respectively) in meters for simulations 1964 HG 73, 1977 HG 73, and 1978 HG 73 (top to bottom, respectively). 150 slid slid N-0_159 -0.123 N-0123--0088 irve -0 088 -0.052 /V -0 052 - -0.016 -0016-0019 0 019 - 0.055 0 055 - 0.091 /\/0091 -0.127 NO 127 - 0.162 0 162 - 0 196 A/ No Data 11111 wild /\./ -1.288 - -1.04 A/-1.04 - -0.792 /\,/ -0.792 - -0.544 A/ -0.544 - -0.296 / -0.296 - -0.048 -0.048 - 0.2 0.2 - 0.447 No Data r mi slid slid -0003- -0.002 -0.002 - 0 0- 0.002 0.002 - 0.004 A/ 0.004 - 0.006 0.006 - 0.008 0.008 - 0.01 No Data wild A/ 4 Al-0.017 -0.014 A/ -0.014 -0.01 - -0.007 A/ -0.007 - -0.004 Decreasing Precipitation -0.004 - -0.001 -0.001 - 0.002 0.002 - 0.005 ,. 0.005 - 0.009 A./ 0.009 - 0.012 No Data wild .Q‘ slid No Data slid No Data wl ld 1111 wild Figure 5.4. Maps showing depth and width changes (left and right, respectively) in meters for simulations 1964 OG 73, 1977 OG 73, and 1978 0G73 (top to bottom, respectively). 151 slid N -0.073 - -0.059 slid -0.012-0.017 0_017 - 0.046 0.046 - 0.075 A/ 0.075 - 0.104 0.104 - 0.133 0.133 - 0 162 Q‘ No Data /V /V wild slid -0.004 - 0.001 0.001 - 0.005 0.005 -0.01 001 - 0.014 A/ 0014- 0.019 A/ 0 019 - 0.023 No Data wild IV-0.059 - -0.045 N-0.045 - -0.031 -0.031 - -0.018 -0.018 - -0.004 N r -0 004 - 0.01 0.01 - 0 024 No Data MN wild slid / 0 - 0 0-0 0-0 0-0 - 0 NO - 0.001 N 0 .001 - 0.001 No Data wild Decreasing Precipitation N slid -0003- -0.002 -0002--0.002 -0002- -0.001 A/-0 001 - -0.001 0-0 0 - 0.001 Ai 0.001 - 0.001 0 001 -0002 A/ 0 002 - 0.002 No Data wild Figure 5.5. Maps showing depth and width changes (left and right, respectively) in meters for simulations 1964 HG 97, 1977 HG 97, and 1978 HG 97 (top to bottom, respectively). 152 slid A/ -0.107 - -0.076 A/ -0.076 - -0.045 -0.045 -0.015 -0015- 0.016 0016- 0.047 0.047 - 0.078 0.078 - 0.108 A/ 0.108 - 0.139 A/ 0.139 - 0.17 No Data wild slid A/ -0.004- -0.002 ' -0.002-0 0- 0.001 0.001 - 0.003 0.003 - 0.004 A/ 0.004 - 0.006 A/ 0.006 - 0.008 A/0.008 - 0.009 slid /\/-1.506 - -1.228 /\/-1.228 - -0.951 N-0.951 - -0.673 A/ -0.673 - -0.396 -0.396--0.118 -0.118- 0.159 0 159 - 0 437 A No Data MI wild X; 014--0.011 N -0.011 --0008 N -0.008 - -0.006 -0006--0.003 -0003 - 0 0 - 0.003 0.003 - 0005 0.005 - 0008 Decreasing Precipitation A/ 0.008 -0.011 A/0.011 -0014 slid 0 - 0.001 0.001 - 0.002 A/ 0.002 - 0.003 0.003 - 0.004 A/13.004 - 0.005 0.005 - 0.006 No Data wild A/ A/ Figure 5.6. Maps showing depth and width changes (left and right, respectively) in meters for simulations 1964 OG 97, 1977 OG 97, and 1978_0G_97 (top to bottom, respectively). 153 slid slid /V A/ A/ -0.012 -0025 0 025 - 0.062 0.062 - 0.098 0.098 - 0.135 0.135 - 0.172 0.172 - 0.208 No Data wild --0056 IV-°°82 A/ -0.066 --0051 A/ -0.051 --0 035 ,&J-0.035 - -0.019 -0.019--0004 -0.004 - 0.012 0.012 - 0.027 0.027 - 0.043 No Data wild slid slid -0.023- -0.011 -0.011-0 0- 0.012 0.012 - 0 023 0023- 0 034 0.034 - 0.046 0046- 0.057 No Data MI wild A/ A/ A/ A/ -0.001 - 0 0 - 0.002 /0.002 - 0.003 A/ 0.003 - 0.005 A/ 0.005 - 0.006 A/ 0.006 - 0.007 Q‘ No Data wild Decreasing Precipitation slid Al -0003- 0.002 -0002--0002 A/ -0002--0.001 - A/ -0.001 -0.001 -0 001 0-0 0 - 0.001 /1v 0.001 - 0.001 0.031 - 0.002 0.002 - 0.002 A/ A/ No Data wlid Figure 5.7. Maps showing depth and width changes (left and right, respectively) in meters for simulations 1964 HG PU, 1977 HG PU, and 1978 HG PU (top to bottom, respectively). 154 slid Al-0.144 -0.116 A/ -0.116- -0.087 A/-0.087 --0059 A/ -0.059 - -0.031 -0.031 --0.003 -C)003- 0.026 0.026 - 0.054 0 054 -0082 A/ 0.082 -011 No Data wild slid A/-1.194 - -0.974 A/ -0974--0754 A/ -0.754 - -0.534 -0534 - -0.313 -0.313 - -0.093 -0093-0127 0.127 - 0.348 0.348 - 0 568 A/ 0.568 - 0.788 No Data wild A/ slid A/ -0018--0013 A/ -0013--0008 -0.008 - -0.003 -0.003 - 0.002 0.002 - 0.007 Ar - 0.007 - 0.012 A/ 0.012 - 0.017 A/ 0.017- 0.022 A/ 0.022 - 0.027 No Data wild slid -0039--0028 A/ -0.028 - -0.017 -0.017--0007 -0.007 - 0 004 0.004 - 0.015 0.015 - 0.026 A/ 0026-0.036 0.036 - 0.047 A/0.047 - 0.058 No Data wild A. A/ Decreasing Precipitation 4mief .rneff slid -0.001 -0 0- 0.001 0.001 - 0.002 0.002 - 0.003 A/ 0 003 - 0.005 N 0.005 - 0.006 No Data wild Figure 5.8. Maps showing depth and width changes (left and right, respectively) in meters for simulations 1964 OG_PU, 1977 OG PU, and 1978_0G_PU (top to bottom, respectively). 155 slid /V -0.184-0.151 slid -0.032 - 0.008 0 008 - 0.048 0 048 - 0.089 "./ 0 089 - 0.129 A/ 0 129 - 0.169 A/ 0 169 - 0.21 No Data wild A/ -0.151 -0.118 A/ - -0.085 A/ -0.085 - -0.052 -0.052--0.02 -0.02- 0.013 0.013- 0 046 " No Data EN wild slid -0065- -0053 Af -0.053- 4104 Al -004- -0.028 A/ -0.028 - -0_016 -0.016 - -0.003 -0 003 - 0.009 0.009 - 0.021 A,• 0.021 - 0.034 /\/0.034 - 0 . 0 46 N 0 .046 - 0.058 No Data wild slid A/ -0.141 - -0.117 /V -0.117--0.093 A/ -0.093 - -0.068 Decreasing Precipitation A/-0.068 --0044 -0.044 - -0.02 -0.02 - 0.005 No Data wild slid Al0.003 - -0.002 -0002--0.002 -0002--0.001 -0001 --0001 -0.001 -0 0-0 0 - 0.001 0.001 - 0.002 N 0 .002 - 0.002 ,AV 0.002 - 0.003 No Data wild Figure 5.9. Maps showing depth and width changes (left and right, respectively) in meters for simulations 1964 HG U, 1977 _ HG _U, and 1978_HG U (top to bottom, respectively). 156 slid A/ -2.824 - -2.339 slid -0088-0.006 0.006 - 0 1 0.1 -0.194 A/ 0.194 - 0.288 /\/0.288 - 0.382 A/ 0.382 -0475 ,Q1‘ No Data wild N -2.339 - -1.853 //.1.853-- -1.368 A/ -1.368 - -0.882 7 -0.882 - -0.397 -0.397 - 0.089 0.089 - 0.575 A No Data Ili wild slid -0041--0.033 A/ -0033--0.026 A/ -0 026 --0.018 A/ A/ -0.018--0.011 -0.011 - -0.003 -0003-0005 0 005 - 0.012 0 012 - 0.02 0 02 - 0 028 0 028 - 0.035 No Data w1 1c1 slid A/ -1 .285 - -1.041 /\/-1.041 - -0.796 A/ -0.796 - -0.552 N -0.552 - -0.308 Decreasing Precipitation -0.308 - -0.063 -0.063 - 0.181 No Data 111111 wild slid -0.001 -0 0 - 0.001 0.001 - 0.002 0.002 - 0.003 A/ 0.003 - 0.005 A/ 0.005 - 0.006 N No Data wild Figure 5.10. Maps showing depth and width changes (left and right, respectively) in meters for simulations 1964 _ OG U 1977 OG U, and 1978 OG U (top to bottom, respectively). 157 slid 56.99 - 68.071 68.071 - 79.153 /V 79.153 - 90.234 /\J90.234 - 101.316 A/ 101.316 - 112.397 No Data wild slid 1.293 - 2.835 2.835 - 4.377 4.377 - 5.919 A/ 5.919 - 7.46 A/ 7.46 - 9.002 /V No Data MI wild slid 0-0 0-0 \/, 0 - 0 A/ 0 - 0.001 A/ 0.001 -0.001 /V No Data wil d Figure 5.11. Cumulative runoff depth (mm) per unit contributing area for the three batch simulations arranged in order of decreasing rainfall total from 1964 (top) to 1978 (bottom). Pictured results are for the hydraulic geometry simulations, but the spatial patterns for the observed geometry simulations were virtually identical. 158 5.2.2 Mass-Balance Calculations Cumulative changes in sediment storage and geometric adjustments were computed for each batch simulation to facilitate net mass-balance calculations for the channel network (see section 4.6.3). Results of these calculations are presented in table 5.6. Sediment inflows to the channel network are derived from erosion on the uplands (planes). Sediment outflow is the sediment leaving the outlet channel. The change in sediment storage is equivalent to the net amount of erosion and/or deposition in the channels, plus the amount of sediment remaining in suspension at the last time step. Mass-balance error (%) was calculated as: (Inflows - Outflows +- AStorage) / Outflows * 100 (5.3) where storage is added or subtracted depending on its sign. Computing the percent error by dividing by the outflow is inappropriate in the absence of outflow, which is frequently the case in arid regions. As such, the error estimates for 1978, for which sediment outflow was universally zero, were computed as: (Inflows — AStorage) / Inflows * 100 (5.4) Table 5.6 presents both the modeled change in sediment storage and the equivalent mass required to produce the simulated geometric adjustments. The modeled change in sediment storage is that computed in the K2G sediment routing, which was not altered from K2. Error associated with the modeled change in sediment storage is referred to as model error. The equivalent mass of sediment required to produce the simulated geometric change is computed at the end of the batch simulation by comparing the final geometry with the original geometry. As such, it incorporates geometric 159 adjustments computed in K2G, geometric adjustments during profile smoothing in AGWA-G, and cumulative rounding error from the calculations in each simulation. Error associated with the equivalent mass of geometric adjustment is referred to as geometric error. Model and geometric error consistently increased with decreasing precipitation for all land-cover scenarios (table 5.6). This effect reflects errors in the KINEROS sediment-routing calculations, for which the same trend is observed. The difference between computed and observed changes in sediment storage in KINEROS is roughly the same regardless of event magnitude and thus increases in proportion to the volume of routed sediment for small events. Errors in excess of 50% were common for the smallest events. Graphical presentation of the mass balance error presented in table 5.6 indicates significant trends in the simulation output. For the 1964 monsoon (figure 5.12) model error was highest for the partially urbanized land cover, and lowest for the NALC 1973 land cover. Geometric error was consistently lower than the model error when the initial channel geometries were defined by the hydraulic-geometry relation. When observed channel geometries are used as the initial condition, geometric adjustments were consistently larger than the modeled change in sediment storage. ▪ 160 (N 00 . • 0) CD • 9 (.0 r CO h 4) CD 0 • (.0 CO CO CO cc o 2 LC) N . C0 CC> (NI I 0 . CO 7 ((- Lc) CO , 8 0 0) G> (.0 0) N 0 r.-0 00 CO o •-• r--- 0 L•10. 0) 0 00 (f) J 0 -,Q .0 Cf) h- 0) CD N 0 IS) 0) C3) NCN CO CO .zi-00 CO00 Cr) r CO 00Cr) Cc) -46 Cr) r NI r r O ix; 0 CO d CNSN QC) CO CO c) • • ,cr) CO CD N 0 0) CN 0 co0) 0) ((s)(-1-; 9 cy.); 0- C) Cr) cc; cci 00 0 cc,' CT) (f5) CY) CT) — Cr) G) ' - CC 1.0 LU 0.) • f -- Cr)k-N C(') ' 0 0 co 0 r N E 7- o c, O r o d 0 D. N 0 CD .0 r CO l co ,..e 0 CO 0 C) CD CO co cs) t-- r CN ✓ r r c, co 0 .0 r--- Lf) L.r) vzt 0 r--- r-,4 r r r r-- c) co co n— O 0 co r--- 0 Lf) 1.‘) r CO CO C) Cr) •,-- r GO so C)0 0 0 0 0 cz, .7r0) 0 IC) ..(— •7J- •(-- Cf-•c••CY) 0 . ) CO CO 00 r CO r CO CN r 6). 0 0 0 6) 0, 0) CO •CO CC - 0 C-4 CO N r 0 hCO 0) hl CO Cr) tA = •—n 13 p 0 0 cc( 1.,_ 0 0 (NI 0 0 p 0 0 et Lf) r--- 0. 61 1---- f•-•- r 0 C.1. 7t- NI 1---- (.....4 CV on r..._ C) r--rn r.. " r CO r CO CO n c) 4-; ✓ CN 0 cy NI 6) •,-- ci- d (1.Y E 'I — — ,t- 0 r 0 0) c0 '71" r 1.f0 r .,-- CV 1f) Cr) hl in r. co r 0) 00 cr) 0) (-NI CO ..,i- 0) CO x--- r--- ,± r (C L!) 1,,t = I T T Ce) ‘— 0, 0 p 0 Le CN 0 ce C:) CV CD CO keN 0 c(r) CO 0 ("N r --- (0. Cr) ‹..-) CO m 00 CV CO ni- 0 0 C' C. ,=, L.r) CO CO C°CO r 0 r r se,1 1---- R I 0 .(3) 0 012 .0 th 2 Ui C "6"- - 0 0, 0 1"-IN-- LI) 0 0 CO CD 0) 0 op CNI r".-- 0 0) 0 CS) 'çr 0 0-) . (5) CO . 0 iCO i1-r) CO 4-NI CO 1 1 I CO p COQ) 00(0 CO CO r 0) CN 0 C) D 0 C) 0 0 c) CD r--- o c> co c> o, .7f- 0 a co a c) N"-- G c> keN CV D 1.10 ,_i_ CO D in r 0 In 0) CV (D L CO 0-) CO CO CO 0 CNI (4), ', L.0 r__ o co c) co -r- (0 r--- 0 0 CO CO 0 -..' a) 0-, kto o r--- CO 1.r) "ck r k-NI LI) cr) ,-, d- <,-) '.4- CD N h-CO r.___ . 7 1.. r '-'•' CN c•-- 0) LC) c ,-) Co .7,') CO 0 00 r ,._ 1 --- C",,1N , ( CD , - CN œ ( 0) , 0) I r 1 1 I I I 1 I I- CO CD CO CN CN CN CN 0 CD r r 0) CN co CO CO co CO 161 20 15 10 5 0 -5 -10 o Model Error • Geom Error -15 -20 -25 64hg73 64hg97 64hgpu 64hgu 64og73 64og97 64ogpu 64ogu Figure 5.12. Bar graph showing the model (red) and geometric (blue) net mass-balance error (%) for 1964 batch simulations. Comparison of the cumulative modeled change in sediment storage and the mass equivalent of geometric adjustments reveals a disappointing trend in the performance of the geomorphic model. For all 1964 batch simulations geometric adjustment exceeds the computed change in sediment storage (figure 5.13). This effect is most pronounced when the observed channel geometries are used as a starting point. Geometric mass-balance error is consistently less than the model error for simulations based on the hydraulicgeometry channels, but this trend is reversed when observed geometries are used. Channel geometric change and modeled change in sediment storage yield different results. Modeled erosion decreased for the two urban scenarios, which is not 162 unreasonable because sediment production on the uplands increased substantially despite the presence of erosion pavement. In addition, modeled erosion decreased when the observed geometries were used and the presence of wide, shallow reaches enabled more sediment to be captured and retained within the channel network. Geometric change, however, failed to reflect these same trends. 64hg73 64hg97 64hgpu 64hgu 64og73 64og97 64ogpu 64ogu 1000000 0 -1000000 -2000000 -3000000 -4000000 -5000000 • Channel Eros o Geometric Adjust -6000000 Figure 5.13. Bar graph showing the relative magnitude of channel erosion (kg) and the mass equivalent of the simulated geometric adjustment for the 1964 batch simulations. For the 1977 monsoon the 1-1G U batch simulation computed sediment yield was an order of magnitude different from the other seven simulations. This is thought to result from problem discussed in section 4.7 that produces excessive deposition in a single time step when flow depth drops precipitously prior to the arrival of the primary 163 flood wave. Results from the HG _U simulation were thus excluded from the following analysis. No consistent relation was observed between mass-balance error (model or geometric) and land-cover scenario for the 1977 simulations (figure 5.14). Results from the HG_U simulation were omitted for clarity. Geometric error was larger than model error for all but one of the simulations. The relation between model and geometric massbalance error was more irregular when observed channel geometries were used. 200 o Model Error • Geom Error 150 100 50 0 -50 77hg73 77hg97 77hgpu 77og73 77og97 77ogpu 77ogu Figure 5.14. Bar graph showing the model (red) and geometric (blue) net mass-balance error (%) for 1977 batch simulations. For all but one of the 1977 batch simulations geometric adjustment exceeds the computed change in sediment storage (figure 5.15). This effect is more pronounced for 164 the two urbanized scenarios, and most pronounced when the observed channel geometries are used as starting points. Unlike the 1964 simulations, however, differences for 1977 are more consistent, and the overall trend of increased deposition with urbanization is reasonably well represented by the geometric adjustments. 1900000 • Channel Eros. o Geometric Adjust. 1400000 900000 400000 -100000 77hg73 77hg97 77hgpu 77og73 77og97 77ogpu 77ogu Figure 5.15. Bar graph showing the relative magnitude of channel erosion (kg) and the mass equivalent of the simulated geometric adjustment for the 1977 batch simulations. 5.2.3 A Perspective on Model Output To put the simulated sediment yields into perspective, cumulative water and sediment outflows from the observed land-cover scenarios were compiled for comparison with observations on Walnut Gulch. The results, presented in table 5.7, are for the entire 165 watershed (WG11), which has a contributing area of 633 ha. The calculation of sediment yield as a volume assumed a particle density of 2.65 g/cm 3 , and a porosity of 0.3. Table 5.7. Simulated cumulative outflow and sediment yield for the observed land-cover scenarios. Simulation Runoff (mm) Sed. Yield (m 3/ha) sv64hg73 63.3 11.2 sv77hg73 1.5 0.1 sv78hg73 0 0.0 sv64hg97 63.6 11.3 sv77hg97 1.5 0.1 sv78hg97 0 0.0 Both runoff and sediment yield were highest for the 1964 monsoon, which yielded an order of magnitude more runoff, and two orders of magnitude more sediment than did the 1977 (intermediate) monsoon. The 1978 monsoon yielded no net runoff or sediment yield at the watershed outlet. Given the characterization of the rainfall records presented in table 5.2, the model output suggests that both runoff and sediment yield from a series of events increase exponentially as average event depth and the standard deviation of event depths increase. In comparison, Nichols and Renard (2003) computed sediment yield for several small watersheds on Walnut Gulch through repeat surveys of stock tanks. The results of their analysis is summarized in table 5.8. Although these values are not directly comparable to sediment yield from the much larger (633 ha) WG11, they indicate that the range of simulated sediment-yield values in K2G is reasonable. The 1964 monsoon was characterized by two consecutive large runoff events that transported most of the 166 sediment outflow of that year. Nichols and Renard (2003) reported only two single-year records, with computed sediment yields of 6.0 and 3.3 m 3 /ha for a watershed of 43.8 ha. These records, however, were from approximately average rainfall years, indicating that high sediment yields are likely to be substantially higher during wet years. Table 5.8. Summary of sediment accumulation on Walnut Gulch from Nichols and Renard (2003). The italicized values are from a pond within WG11. Contributing area (ha) Years in Record Sediment Yield (m 3/ha/yr) 0.4 11 92.2 35.2 18 2.3 84.2 34 1.7 43.8 46 2.8 The unit rate of sediment yield generally decreases as contributing area increases (e.g. Branson et al., 1981). This results from a host of factors, most notably including: increasing sediment storage within the channel network, decreasing average event rainfall depth, and increasing transmission loss with increasing watershed size. Given this, it is possible that K2G has overestimated sediment yield for 1964. The fact that only a partial year was simulated would accentuate that possibility, although most sediment discharge occurs during the monsoon season. However, the 1964 monsoon was exceptional, and it was selected for that reason. Sediment yield from the moderate 1977 monsoon was much closer to what one would expect based on the results of Nichols and Renard (2003). 167 5.3 Discussion Results from the test simulations highlight the importance of rainfall frequency and magnitude in governing geomorphic response. The influence of transmission losses is demonstrated by widespread deposition during intermediate and small events. When observed channel geometries were used, deposition was greatest in the lowermost channels for which flow reductions were most pronounced (figure 5.11). Graf (1983b) observed a similar response in the Henry Mountains of Utah, where periods of aggradation were associated with stream power reductions as a result of transmission losses in the downstream direction. The larger events observed during the 1964 monsoon produced less significant downstream decreases in runoff, and were associated with widespread channel scour. Test results indicate that the model is particularly sensitive to the initial channel geometry. This is not surprising, because the minimum stream power theory that was used to govern the channel adjustments is based on the hypothesis that a channel adjusts its geometry to minimize the total stream power. Hydraulic geometry channels fail to account for channel slope when they are used to assign width and depth values, and thus are not necessarily representative of well-adjusted sections. Observed geometries may be more representative of the reach and slope at the points where they were measured, but cannot account for inter-reach variability. Regardless of the initial geometries, variable adjustment was observed throughout the watershed, and those adjustments were different depending on the initial conditions. 168 The land-cover scenarios produced unexpected results. Urbanization did increase runoff as expected, but slightly larger increases in erosion resulted in increased deposition during the 1977 monsoon, and decreased erosion during the 1964 monsoon. Widespread channel incision was the expected result, but the reason it was not observed is likely due to the way AGWA-K2G treats urban uplands. By reducing saturated hydraulic conductivity, increased runoff and sediment entrainment are generated on the uplands. Increasing the PAVE parameter helps offset this by reducing computed erosion by the fraction it is set to, but it does not limit access of the runoff to sediment. In reality, urban areas concentrate runoff in paved channels and culverts to prevent erosion, and much of the additional runoff gets to the channels without a significant sediment load. The persistent overestimation of deposition by the geometric adjustments during the 1977 simulations is most likely a result of the profile-smoothing calculations between each simulation. Profile smoothing and mass-balance calculations were developed and debugged based on test simulations using design storms and initial channel geometries defined by the hydraulic-geometry relation. As such, differences between the model and geometric changes in sediment storage are least pronounced for the hydraulic geometry batch simulations. What the profile smoothing calculations fail to account for is significantly different bank slopes for the upstream and downstream channels at each junction. Elevation adjustments computed at each junction may therefore fail to conserve mass in the transfer of sediment between reaches. 169 6.0 RELATIVE GEOMORPHIC CHANGE AND CHANNEL STABILITY 6.1 Model Error and Relative Change Assessment Error in numerical models of watershed runoff and erosion can result from numerous sources. Most notable among these is how faithfully a model describes the real-world processes, both conceptually and numerically. All models are simplified abstractions of real-world systems, and assumptions are necessarily made to accomplish this. Although the simplifying assumptions made during model development are designed to minimize model error, they cannot eliminate it, and all models are thus characterized by a certain amount of inherent error. For K2, Syed (1999) obtained Nash-Sutcliffe model-efficiency values of 0.87 and 0.83 for runoff volume and peak discharge, respectively, during calibration runs on WG11 (perfect agreement of observed and simulated values produces an efficiency of 1). When the calibrated model was applied to the validation data set, however, efficiency values were reduced to 0.43 and -1.1. Syed (1999) observed that smaller events were difficult to model because the runoff to rainfall ratios are low, and input rainfall errors (as well as numerical errors) become a large percentage of the overall model output. In other words, the model did not represent rainfall-runoff processes equally well at all scales. Modeled upland- and channel-erosion predictions have been shown to represent poorly observations. Risse et al. (1993), for example, reported an overall Nash-Sutcliffe model efficiency of 0.58 for predictions of annual sediment yield from runoff plots using the Universal Soil Loss Equation (USLE). They noted that UST F. over-predicted soil loss 170 when erosion rates were low, and under-predicted when erosion rates were high. This same observation was made by Nearing (1998) in a broad analysis of erosion-model performance. Bravo-Espinosa et al. (2003) compared observed bedload discharge with predictions from seven common bedload equations under a variety of conditions. Inequality coefficients computed for each of these comparisons ranged from 0.2222 to 0.9999, and the vast majority were greater than 0.8 (perfect agreement of simulated and observed values produces an inequality coefficient of 0). Whereas these examples are not necessarily representative of conditions on Walnut Gulch, or error specifically associated with K2, they demonstrate that even carefully calibrated models cannot reliably predict water and sediment yields for any given event. For multiple events errors can be compounded, or reduced, depending on the magnitude and initial conditions of each event. As such, the simulated geomorphic changes presented in chapter 5 are useful in their ability to display where changes are likely to be greatest; the magnitude of the changes should be considered unreliable. Another way to derive useful information from the test simulation results is to compare the computed geomorphic change between two simulations for which only one initial condition varied. For instance, if the same simulation is repeated using two different land-cover grids to generate input parameters for the upland elements, then the difference between the two simulation results can be attributed to the different land-cover parameters. By holding everything else constant, repeatable errors associated with erosion and routing calculations are negated. Consistent over- or under-estimation of 171 erosion still can influence the magnitude of the differences, but their relative magnitudes are unlikely to be significantly affected. In other words, it is possible to identify where in the watershed the impacts of land-cover change are likely to be significant. 6.2 Methodology The partially urbanized (PU) and NALC 1997 (97) land-cover scenarios were selected for the relative-change assessment for two reasons. The non-urban component of partially-urbanized scenario is the same as the NALC 1997 land cover, so differences in geomorphic response can be attributed exclusively to the presence of the urban landcover class. In addition, results for these two scenarios were significantly different and spatially varied. Relative differences were computed for the batch simulations performed for the 1964 and 1977 land-cover scenarios, with initial channel geometries derived from observations (OG) and the hydraulic-geometry relation (HG). Insufficient runoff from the 1978 events yielded irregular, very small changes, so the 1978 results were not included in this analysis. Simulation outputs were compared using the AGWA-G differencing feature for batch geomorphic simulations. This feature compares all output parameters for each simulation, and writes the differences (as absolute values or in percent) to a new output file for visualization in the GIS. 172 6.3 Results Relative differences between the PU and 97 simulations were visualized for depth and width changes, and are presented in figures 6.1 and 6.2. As in the chapter 5 figures, maps are in order of decreasing precipitation from top to bottom, depth changes are on the left, and width changes are on the right. All differences are presented in meters, which are computed as: Absolute Difference = PU - 97 where PU represents an output from the partially-urbanized scenario, and 97 represents the corresponding output for the NALC 1997 land cover. Increased erosion or decreased deposition is shown in shades of red (positive values), and reduced erosion or increased deposition is shown in shades of blue (negative values). Another way to look at this is that positive values indicate relatively large widths and/or depths (greater width and/or depth increases, or smaller width and/or depth decreases), and negative values indicate relatively smalle widths and/or depths (greater width and/or depth decreases, or smaller width and/or depth increases) for the PU scenario. 6.3.1 Relative Differences for the Hydraulic-Geometry Channels Results for the HG channels (figure 6.1) indicate that the largest changes occurred during the wet year (1964). All channels in the urbanized tributary were characterized by increased erosion or decreased deposition (relatively large channels) for the PU scenario. Relative differences were of approximately the same magnitude for both width and depth changes, but did not always have the same sign. Downstream of the urbanized area 173 channel depth decreased, but widths increased for the PU scenario. This pattern of depth decreases and width increases is commonly associated with channel aggradation, and suggests that the model is yielding reasonable results. slid A/ -0.028 - -0.02 /V -0.02 - -0.012 slid A,. -0.012--0.003 -0.003 - 0.005 0.005 - 0 013 0.013 - 0.021 NO.021 -0.03 0.03 - 0 038 Ao• 0.038 - 0 046 No Data tv' /V A/ A/ A/ ' No Data Wild wild slid /\," -0 02 - -0.013 -0.013 - -0.006 -0.006 - 0 001 0 001 - 0.007 0 007 - 0.014 A/ 0 014 - 0.021 A/ 0.021 - 0.028 A/ 0 028 - 0.034 No Data wild -0.009 - -0.004 -0.004 - 0.001 0.001 - 0.006 0.006 - 0.011 0.011 - 0.016 0.016 - 0.021 0.021 - 0.026 slid -0.001 -0 0- 0.001 A, 0.001 - 0.003 A/ 0.003 - 0.004 A/ 0.004 - 0.005 A/ 0.005 - 0.007 No Data wild Figure 6.1. Maps showing the difference (meters) in simulated average depth change (left) and average width change (right) between the partial urbanization and NALC 1997 land cover scenarios for the 1964 (top) and 1977 (bottom) batch simulations based on the HG channels. Shades of red (positive values) indicate that geomorphic adjustments yielded relatively larger depths/widths for the PU scenario. Most channels in the non-urbanized area of the watershed showed little or no change, but there are some width differences where there should be none. Maximum 174 differences in the non-urbanized section of the watershed, however, are on the order of millimeters. For the 1977 batch simulations, relative-depth changes were mixed. The PU scenario yielded both increases and decreases in channel depth relative to the 97 scenario within the urbanized area. Channels downstream of the urbanized area were again characterized by depth decreases or no change. Width changes were very small, and only occurred in the uppermost channels within the urbanized area. No significant width or depth changes were reported for the non-urbanized area. 6.3.2 Relative Differences for the Observed Channel Geometries Relative differences between the PU and 97 scenarios for the 1964 batch simulations based on the observed channels (figure 6.2, top) are notable in that significant depth changes were not reported within the urbanized area. For each reach downstream of the urban area, however, deposition in the PU scenario reduced channel depths. Simulated widths for the PU scenario were larger or the same as those for the 97 scenario within the urbanized area and downstream of it. Relative width changes in the nonurbanized area were small, and less prevalent than for the simulations based on the hydraulic-geometry channels. For the 1997 simulations, relative width and depth changes again were mixed. Increased and decreased widths and depths can be observed both within and downstream of the urbanized area. The patterns of width and depth changes appear unrelated spatially. No changes were simulated for the non-urbanized area. 175 slid A/ -0.137 -0.111 A/ -0.111 - -0.085 /\/-0.085 - -0.059 N -0.059 - -0.032 , -0.032 - -0.006 -0.006 - 0.003 0.003 - 0.02 No Data wild slid A/-0.015 - -0.012 A/-0.012 --0009 -0.009--0005 -0.005 - -0.002 -0.002 - 0.001 0.001 - 0.005 Cv 0.005 - 0.008 A/0.008 - 0.011 A/0.011 -0014 /V0.014 - 0.018 No Data w11 slid -0045- 0.043 0.043 - 0.13 0.13 - 0.218 A/ 0.218 - 0.305 0.305 - 0.392 A/ 0.392 - 0.48 No Data wi ld N slid A/ -0.034 - -0 025 /V -OE025 -0.016 -0.016 - -0.006 -0.006 - 0.003 0.003 - 0.012 0.012 - 0.021 0.021 - 0.03 0.03 - 0.04 1V0.04 -0049 /\/ No Data Wi Figure 6.2. Maps showing the difference (meters) in simulated average depth change (left) and average width change (right) between the partial urbanization and NALC 1997 land cover scenarios for the 1964 (top) and 1977 (bottom) batch simulations based on the OG channels. Shades of red (positive values) indicate that geomorphic adjustments yielded relatively larger depths/widths for the PU scenario. 176 6.3.3 Relative Differences in Runoff Transmission Loss, and Sediment Yield To help explain the patterns of relative geomorphic response shown in figures 6.1 and 6.2, relative differences in cumulative runoff, transmission loss, and sediment yield were mapped for the 1964 and 1977 simulations (figures 6.3 and 6.4, respectively). For the 1964 simulations, differences in cumulative runoff (mm over the contributing area) declined in the downstream direction along the southern tributary for the HG channels. The same trend is observed for the OG channels, although the specific pattern varies slightly. Relative differences in transmission loss (m 3/1(m) and sediment yield (kg) increased in the downstream direction for simulations based on both the HG and OG channels. The specific pattern of differences varies between the HG and OG simulations for both variables, but their overall trends are similar. For the 1977 simulations, differences in cumulative runoff declined in the downstream direction along the southern tributary for both the HG and OG channels (figure 6.4). In addition, the specific pattern of downstream runoff declines is almost identical, regardless of the initial channel geometry. Relative differences in cumulative transmission loss and sediment yield initially increased in the downstream direction, and then decreased slightly for the 1977 simulations. This trend was observed regardless of the initial channel geometry, although specific patterns differed. 177 slid slid 0 - 5 686 5686- 11.373 / 11.373 - 17.059 17.059 - 22.745 A/ 22.745 - 28.432 28.432 - 34.118 No Data wild -0.178-5427 5.427- 11.032 / -N., 11.032 - 16.636 A/ 16.636 - 22.241 A/ 22.241 - 27.846 A/ 27.846 - 33.451 No Data wild A/ A/ 4isof slid -8446- 1573.799 1573.799 - 3156.044 3156.044 - 4738.289 A/ 4738.289 - 6320.535 6320.535 - 7902.78 7902.78 - 9485.025 A/ N A/ No Data wild slid A/ A/ 4 -6.804 - 1678.208 1678.208 - 3363.22 3363 22 - 5048.232 5048.232 - 6733.244 6733.244 - 8418.256 8418 256 - 10103.268 No Data vv11c1 slid slid -9661.55- 631152.795 631152795- 1271967.14 " 1271967 14- 1912781.485 1912781.485 - 2553595.83 2553595.83- 3194410.175 /\,/3194410.175 - 3835224.52 A/ A/ A/ No Data wild A/ N A/ -111836.51 - 533047.232 533047.232 - 1177930.973 1177930.973- 1822814.715 1822814.715- 2467698 457 2467698.457 - 3112582.198 3112582.198 - 3757465.94 No Data wild Figure 6.3. Maps showing the relative differences in cumulative runoff (mm, top), transmission loss (m 3 /km, middle), and sediment yield (kg, bottom) for the 1964 simulations. Results for the HG and OG channels are pictured on the left (A-C), and right (D-F), respectively. 178 slid slid 0- 3.154 3.154 - 6.307 6.307 - 9.461 9.461 - 12.614 12 614 - 15 768 No Data wild -0.039 - 2.588 2.588 - 5.214 5.214 - 7.841 7.84 1 - 10.468 A/ 10.468 - 13 095 13.095 - 15.722 No Data wild N N A/ A/ slid slid -12.061 - 1159.779 1159 779 - 2331.619 /, 2331 619 - 3503 459 Al 3503 459 - 4675.299 A/4675 299- 5847.139 A/ 5847.139 - 7018.979 No Data wild -11.094 - 1230.359 1230.359 - 2471.811 2471.811 -3713,264 3713.264 - 4954.717 A/ 4954.717 - 6196.169 A/ 6196.169 - 7437.622 / slid 0 - 144626.338 144626.338 - 289252.676 / 289252.676 - 433879.014 A/ 433879.014 - 578505.352 A/ 578505.352 - 723131.69 No Data wild A/ il wild slid -796.85 - 103452.793 103452.793 - 207702.437 207702.437 - 311952.08 311952.08- 416201.723 A/ 416201.723 - 520451.367 A/ 520451.367 -624701.01 No Data wild Figure 6.4. Maps showing the relative differences in cumulative runoff (mm, top), transmission loss (m 3 /km, middle), and sediment yield (kg, bottom) for the 1977 simulations. Results for the HG and OG channels are pictured on the left (A-C), and right (D-F), respectively. 179 6.4 Discussion Differences in the geomorphic response of the channel network were mapped for the PU land-cover scenario relative to the 97 land cover for simulations based on the observed channel geometries and channel geometries defined by the AGWA-G default hydraulic-geometry relation. The differences thus represent relative responses given the same precipitation input when part of the watershed is urbanized. They do not represent pre- and post-urbanization response. Results indicate that geomorphic response for any given reach depends largely on the initial channel geometry. The general pattern of channel adjustment, however, is surprisingly similar for the two initial channel geometries. Both yielded decreased depths and increased widths downstream of the urbanized area for the 1964 monsoon. The mixed pattern of depth changes for the 1977 simulations was almost identical for the observed and hydraulic-geometry channels. Relative width changes for the 1977 simulations, however, were more pronounced, and significant depth increases were not observed in the urbanized area for the 1964 simulations when the observed geometries were used. Similarities in relative channel response between simulations based on the two initial geometries can be attributed largely to similar spatial patterns of relative runoff, transmission loss, and sediment yield. Although values vary slightly in location and magnitude, the spatial pattern of these relative outputs was always the same between OG and HG simulations. In the absence of observed channel geometries, a suitable 180 hydraulic-geometry relation may be sufficient to provide some indication of how channels might respond to landscape change. Another important relation revealed by the relative assessment is that geomorphic response to landscape change is highly dependent on the magnitude of precipitation. Given two series of precipitation events that likely represent only a fraction of the broader spectrum of potential monsoon rainfall, it is apparent that the pattern of channel adjustments can vary significantly. The specific patterns presented above are likely to be influenced by limitations in the conceptual representation of erosion in urbanized areas (as described in chapter 5), but the spatial pattern of erosion and deposition within the channel network would be just as likely to vary with event- and seasonal-rainfall magnitude. The relevance of spatial characteristics of stream power to erosion, sediment transport, and sediment storage has been evaluated in several studies (e.g. Graf 1982, 1983a, 1983b, 1990; Magilligan, 1992; Lecce, 1997; Fonstad, 2003). All of these have reported nonlinear variation of stream power and associated variability in sediment transport and storage in the downstream direction. Although stream power was regrettably not reported in the output for these analyses, its variability can be recognized by the presented spatial trends in runoff and transmission loss. It can furthermore be deduced that spatial patterns of stream-power variation change with event- and seasonalrainfall magnitude, and that the simulated pattern of geomorphic adjustments is a direct result. 181 7.0 SUMMARY AND CONCLUSIONS Prior to listing the major conclusions that can be drawn from this research, it is appropriate to restate briefly the objectives. This study was designed to improve watershed assessment tools for arid and semi-arid regions. To that end, specific goals were identified: improving the upper watershed size limit to which event-based runoff models can be applied confidently; and developing a geomorphic model capable of simulating width, depth, and slope adjustments from multiple runoff events in ephemeral stream networks. 7.1 Watershed Runoff Routing and Scale A routing subroutine was developed for the Kinematic Runoff and Erosion Model (Smith et al., 1995) using an iterative four-point solution of the modified variableparameter Muskingum Cunge (MVPMC4) technique (Ponce and Chaganti, 1994). Analyses were designed to test the hypothesis that by accounting for flood-wave diffusion the MVPMC4 technique could improve model performance on intermediatesized watersheds relative to kinematic routing. All analyses were conducted for the Walnut Gulch Experimental Watershed (see section 1.4), and, as such, caution should be used in extrapolating the conclusions to other areas. Although Walnut Gulch is assumed typical of arid-region watersheds in many respects, the scale at which flood-wave diffusion effects become significant is significantly influenced by channel slopes. 182 7.1.1 Conclusions Major conclusions drawn from the comparison of kinematic and MVPMC4 routing at multiple scales included: 1. Hydrographs simulated by kinematic and MVPMC4 routing were statistically different at the 5% level of confidence for events with a 5-year return period or smaller on watersheds of 9558 ha (95.6 km 2 ) and larger. For larger events and/or smaller watersheds, the two techniques yield outflow hydrographs that are statistically indistinguishable. 2. Model mass-balance error decreases with increasing watershed area when MVPMC4 routing is used. Conversely, mass-balance error increases with increasing watershed area when kinematic routing is used. 3. Outflow hydrographs simulated with MVPMC4 routing more closely represent observed hydrographs than those simulated with kinematic routing at all scales, and performance gains increase with increasing watershed size. 4. Matched physical and numerical diffusion in the MVPMC4 routing technique improves its performance relative to kinematic routing under conditions where flood-wave diffusion is most pronounced: for smaller events in larger watersheds. 7.1.2 Future Research A comparative analysis of routing performance for compound channels was not attempted in this research. Hydrograph attenuation resulting from overbank flows increases the importance of flood-wave diffusion, which has the effect of making 183 diffusive effects important on smaller watersheds when flood-plain spillage occurs than for an equivalent channel network without flood plains. Flood plains represent important reservoirs for sediment storage, and are thus crucial in geomorphic modeling. The geomorphic model developed in this research thus requires MVPMC4 routing for compound channels before it can be applied to larger watersheds. The variable-parameter Muskingum-Cunge technique has been implemented for compound channels in the hydrologic Drainage Network Channel Flow Routing model, DNCFR, (Garbrecht and Brunner, 1991). DNCFR, however, is a continuous simulation model for large watersheds with perennial flow. Variable-parameter Muskingum-Cunge routing has not been applied previously to channel networks in event-based ephemeral flow models. 7.2 Geomorphic Modeling at the Watershed Scale A geomorphic model, K2G, was developed based on the Kinematic Runoff and Erosion Model to compute width, depth, and slope adjustments from simulated changes in sediment storage. Width and depth adjustments are computed based on the minimum rate of energy dissipation theory (Yang and Song, 1986), and specifically the theory's special case, the minimum stream-power theory (Chang, 1980). A GIS-based interface was developed to coordinate model parameterization, multiple-event simulations, and the visualization of model output. In the absence of observed data, testing of the geomorphic model was designed to accomplish the following: 184 • Demonstrate that the model conserves mass • Confirm that the model behaves reasonably given different land use/cover conditions and precipitation inputs • Evaluate the importance of utilizing observed channel-geometry information 7.2.1 Conclusions Conclusions drawn from the individual batch simulation results presented in Chapter 5: 1. The model conserved mass reasonably well when initial channel geometry was defined by a hydraulic-geometry relation and bank slopes defaulted to 1.0. When observed geometries with variable bank slopes were used, however, the mass imbalance was significant. 2. Erosion was greatest and most widespread for the wettest precipitation record. Mixed erosion and deposition was simulated for the intermediate record, and deposition was most widespread for the driest record. This trend was expected, and strongly indicated that a sediment balance could be achieved if the model was run for a long enough period of time. 3. Simulated channel adjustments were particularly sensitive to the initial channel geometry. The pattern of geomorphic adjustments for each land-cover scenario varied significantly when observed geometries were used, and was much more uniform when the default hydraulic geometry channels were used. Use of the default hydraulic-geometry relation for model parameterization is not 185 recommended when estimates of channel adjustment are needed for a specific reach or reaches. Conclusions drawn from the differencing of simulation results presented in Chapter 6 for the partially urbanized and NALC 1997 land-cover inputs (relative change assessment): 4. The relative assessment of partial urbanization and un-urbanized scenarios did an excellent job of removing background change. Results of the scenario-output differencing show the concentration of impacts within and downstream of the urbanized area, and no significant changes in the unaffected areas. 5. Partial urbanization of the upland area resulted in increased channel incision within the urbanized area and increased deposition downstream for the wettest precipitation record (1964). Increases were less pronounced than expected due to contemporary increases in upland erosion that resulted from the lack of direct hydraulic connectivity between impervious surfaces and the channels. 6. Increased aggradation downstream of the urbanized area for the 1964 simulations was characterized by depth reductions and width increases. This pattern commonly is observed in natural systems during periods of aggradation, and provides a strong indication that the model is behaving properly. 7. Urbanization of the upland areas resulted in mixed increases in erosion and deposition within the urbanized area, and increased deposition downstream for the intermediate precipitation record (1977). The difference in relative response 186 between the 1964 and 1977 simulations indicates that any assessment of stability, or vulnerability to degradation, requires consideration of a range of return-period rainfall. 8. Spatial patterns of geomorphic adjustment to urbanization were closely linked to variability in cumulative runoff depth and transmission loss. Runoff initially increases as flows coalesce in the headwaters, and then declines as transmission losses increase in the downstream direction. Sediment yield is highest at this transition and begins to decline downstream, which closely corresponds with where increased deposition was predicted by the model. The specific location of the transition shifts upstream for simulations with small average event discharges for which transmission losses can cause significant reductions in sedimenttransport capacity. When runoff volumes are large relative to transmission losses, the locus of deposition shifts downstream. 9. The large-scale patterns of relative geomorphic response to urbanization were strikingly similar regardless of the initial channel geometry. This suggests that channel slopes and lengths are the most important parameters controlling watershed-scale geomorphic response, even though specific adjustments may vary from reach to reach. Watershed-scale assessments of geomorphic response to landscape change can thus be carried out with reasonable confidence in the absence of detailed channel-geometry information. 187 7.2.2 Future Research Much work remains to be done before K2G/AGWA-G is ready for broad distribution as an assessment tool. In particular, the profile-smoothing calculations need to be corrected to account for variable widths, depths, and bank slopes within each reach and between reaches connecting at a junction. With this correction, mass conservation should be greatly improved (or at least be as good as it is in the K2 sediment-transport calculations), particularly when observed sections are used. Rigorous testing of the model must be conducted to demonstrate that simulated geomorphic adjustments are representative of observed adjustments, and to quantify error in the simulated adjustments over a range of conditions. Data for this type of testing rarely is available on a watershed scale, but repeat LlDAR over-flights promise to be a useful source. Some problems with vertical and horizontal offset errors have occurred with surface matching for repeat LIDAR data (Crowell et al., 2003), but the errors can be largely removed. In addition, LIDAR data, as opposed to at-a-point field measurements, provides the ability to derive much more detailed information about the spatial variability of channel response. In its present form, the K2G/AGWA-G geomorphic model can only be applied to areas smaller than about 10 km 2 . For larger areas, channel lengths are too long to represent adequately channel-slope variability, and hence geomorphic response. In addition, longer reaches limit the model's ability to adjust reach slope because bedelevation adjustments become vanishingly small relative to the length of the channel. To increase the upper size limit of the geomorphic model, automated, GIS-based procedures 188 for channel network discretization need to be developed for which a maximum reach length can be specified. Ideally, the channel-network discretization would identify the most significant changes in channel slope. Another improvement to the modeling tool that would increase the scale at which it can be applied is the addition of MVPMC4 routing described above. The results presented in chapters 5 and 6 provide a tantalizing glimpse of the many questions that can be addressed with a model such as K2G that can represent relations between rainfall, transmission losses, stream power, and geomorphic response. In particular, the geomorphic model holds great promise in its ability to address the issue of short-term channel stability in response to landscape and climate change. Before this can be undertaken, however, a good deal of work needs to be conducted to ascertain model behavior under the conditions in which channel destabilization has been observed. For instance, spatial patterns of geomorphic adjustment need to be mapped over a broader range of precipitation records and over longer periods of time (years to decades). In addition, the model must be pushed to evaluate its response to major disturbances in the channel network and on the uplands, and whether it can recover its initial geometry after the disturbance is removed. In other words, it is necessary to determine if externally imposed disturbances result in the crossing of an extrinsic threshold to a new process regime, which is characterized as responsive behavior (Werritty, 1997), and whether the new regime is persistent (transitive) or short-lived (intransitive) using the terminology of Chappell (1983). 189 When and if the assessment of channel stability can be accomplished, it would represent a significant advancement in surficial-process modeling for arid regions. Used in coordination with an alternative futures scenario-generation tool developed for AGWA (Levick et al., 2003), the geomorphic modeling tool would provide the first comprehensive assessment and planning tool for arid-region watersheds and their stream networks. 190 APPENDIX A: DESIGN AND OBSERVED EVENT HYDROGRAPHS A.I Design Storm Hydro graphs Hydrographs from both K2 and K2MC are plotted side by side in figures A.1-4 for six design events in each of the four Walnut Gulch subwatersheds: LH104 (4.74 ha), WG11 (782 ha), WG6 (9558 ha), and WG1 (14664 ha). Each figure contains all six design events in the same order; from upper left to lower right these are the 5-year 30- minute, 5-year 1-hour, 10-year 30-minute, 10-year 1-hour, 100-year 30-minute, and 100year 1-hour. To facilitate visual comparison, the hydrographs are presented as discharge in cubic meters per second as a function of time in minutes. A.2 Observed Event Hydrographs Simulated hydrographs from K2 and K2MC are plotted together with observed runoff hydrographs from the Walnut Gulch event database in figures A.5-26. The timing of simulated and observed hydrographs was synchronized by adding the difference in time between the observed start of rainfall and the observed start of runoff to the observed hydrograph times. A minimum of five events was simulated for each of the four subwatersheds (LH104, WG11, WG6, and WG1). Events were selected to provide a range of peak discharge values, but very small events were ignored because of poor model performance for the smallest events. All graphs for the observed events plot discharge in cubic meters per second as a function of event time in minutes. Data points were omitted from the WG1 plots for 191 clarity; the relatively small discharge values for K2 and K2MC were very difficult to see when data points were plotted. The time step for the simulation models was always one minute. The time step for the observed hydrographs is variable; data points are collected when changes in discharge are detected (breakpoint data). 0.7 06 r 192 -.- i 0.5 04 . K2 K2MC ‘ , 03 02 \. 1 0.1 i __. ç 0 10 30 50 70 90 0.9 08 K2 • 07 K2MC 06 05 0.4 03 0.2 0.1 0 10 30 50 70 90 1.4 1.2 K2 • K2MC 0.8 0.6 0.4 0.2 0 10 30 50 70 90 Figure A.1. K2 and K2MC simulated LH104 hydrographs for design storms ranging from the 5-year 30-minute event (upper left) to the 100-year 1-hour event (lower right). Plots are discharge in cubic meters per second as a function of time in minutes. 193 6 30 5 25 • A K2 • K2MC 20 I 15 i 10 5 0 0 100 50 150 200 • 100 K2 K2MC BO Y 60 40 20 0 0 50 100 150 200 Figure A.2. K2 and K2MC simulated WG11 hydrographs for design storms ranging from the 5-year 30-minute event (upper left) to the 100-year 1-hour event (lower right). Plots are discharge in cubic meters per second as a function of time in minutes. 194 114 12 -4-- K2 • K2MC 10 8 4 2 -«- K2 K2MC 250 - t ‘ 200 150 100 0 50 100 150 200 250 300 Figure A.3. K2 and K2MC simulated WG6 runoff hydrographs for design storms ranging from the 5-year 30-minute event (upper left) to the 100-year 1-hour event (lower right). Plots are discharge in cubic meters per second as a function of time in minutes. 195 20 18 16 K2 • K2MC 14 12 10 0 100 150 200 250 300 350 0 400 50 100 150 200 250 900 350 400 4 180 160 250 140 K2 K,2 M C K2 • K2MC 200 120 100 150 100 50 20 0 50 100 150200 250 300 350 400 4 CI 50 100 150 200 250 300 350 400 4501 Figure A.4. K2 and K2MC simulated WG1 hydrographs for design storms ranging from the 5-year 30-minute event (upper left) to the 100-year 1-hour event (lower right). Plots are discharge in cubic meters per second as a function of time in minutes. 196 0.3 0.25 —4,-- K2 .4t. - Ig K2MC 0.2 Observed 0.15 I 0.1 0.05 0 simm.**90 101101PPP•Pni.c . 110 100 120 130 140 150 170 160 180 19 Figure A.5. Observed and simulated (uncalibrated) hydrographs for the 8/1/78 event on LH104 (discharge in cubic meters per second plotted as a function of time in minutes). 0.3 0.25 —4— K2 K2MC 0.2 Observed 0.15 0.1 0.05 o 50 70 110 90 130 150 170 190 230 210 25 Figure A.6. Observed and simulated (uncalibrated) hydrographs for the 7/28/81 event on LH104 (discharge in cubic meters per second plotted as a function of time in minutes). 0.8 f 0.7 0.6 .1 0.5 --•— K2 Observed ,.. \,.. J 0.4 0.3 "1".•.... 0.2 -. 0.1 0 K2 MC • * \* -41...,.”" . 4. 45 4.6 or • r 55 65 75 85 1P ,P 111411.0.4. 95 105 Figure A.7. Observed and simulated (uncalibrated) hydrographs for the 9/10/83 event on LH104 (discharge in cubic meters per second plotted as a function of time in minutes). 197 1 MC Observed 0.7 0.6 \.., 0.5 .T 0.4 0.3 lO• 1 0.2 % # 0.1 0 • 20 %.11414141'744‘'--"IRP liT 'low n vim.ord. - — 0 10 30 40 50 60 70 90 80 1.. Figure A.8. Observed and simulated (uncalibrated) hydrographs for the 8/10/86 event on LH104 (discharge in cubic meters per second plotted as a function of time in minutes). 0.35 a** •••• 0.3 K2 K2MC 0.25 Observed 0.2 0.15 0.1 0.05 90 100 110 120 130 140 150 160 170 180 19 Figure A.9. Observed and simulated (uncalibrated) hydrographs for the 8/29/86 event on LH104 (discharge in cubic meters per second plotted as a function of time in minutes). Figure A.10. Observed and simulated (uncalibrated) hydrographs for the 7/27/76 event on WG11 (discharge in cubic meters per second plotted as a function of time in minutes). 198 25 20 —10— K2 K2 MC Observed 15 10 ••••••n• • •=, 190 170 150 130 110 90 70 Figure A.11. Observed and simulated (uncalibrated) hydrographs for the 8/1/78 event on WG11 (discharge in cubic meters per second plotted as a function of time in minutes). 35 30 —4.-- K2 - K2 MC 25 Observed 20 15 10 5 4411111411"41461111"1001 01011V1411111010rmwo NIMIllemsrmdadadam. 0 70 50 110 90 130 150 190 170 Figure A.12. Observed and simulated (uncalibrated) hydrographs for the 8/4/80 event on WG11 (discharge in cubic meters per second plotted as a function of time in minutes). 40 35 —4.-- 30 a K2 K2 MC Observed 25 20 15 10 5 •nnn •nnn•••"Ilt 100 120 140 160 180 200 220 240 260 280 30 Figure A.13. Observed and simulated (uncalibrated) hydrographs for the 8/29/86 event on WG11 (discharge in cubic meters per second plotted as a function of time in minutes). 199 - K2 - K2MC Observed _ 3IFJP.PIW 11•••••nn•••_ 0 40 20 60 80 100 140 120 180 160 Figure A.14. Observed and simulated (uncalibrated) hydrographs for the 8/1/90 event on WG11 (discharge in cubic meters per second plotted as a function of time in minutes). 45 40 —4.— 35 • K2 K2 MC Observed 30 25 20 15 10 5 0 50 150 100 300 250 200 Figure A.15. Observed and simulated (uncalibrated) hydrographs for the 7/27/76 event on WG6 (discharge in cubic meters per second plotted as a function of time in minutes). 70 60 K2 K2 MC 50 Obs erved 40 30 20 10 o 1111111. 1111011. 1111mor- 111WArar Immoirrier 100 150 200 250 300 Figure A.16. Observed and simulated (uncalibrated) hydrographs for the 8/1/78 event on WG6 (discharge in cubic meters per second plotted as a function of time in minutes). 200 16 14 12 --411-- K2 --MI— K2 MC Observed 10 8 6 4 2 o•• —•••n•! —46-44aglimells—limilmint_asmor_pww_wr 150 100 350 300 250 200 Figure A.17. Observed and simulated (uncalibrated) hydrographs for the 8/4/80 event on WG6 (discharge in cubic meters per second plotted as a function of time in minutes). 25 20 K2 K2MC Observed 15 10 o 11, mummer majn V 100 250 200 150 300 350 450 400 50 Figure A.18. Observed and simulated (uncalibrated) hydrographs for the 8/29/86 event on WG6 (discharge in cubic meters per second plotted as a function of time in minutes). 30 25 —4-- K2 K2MC 20 Observed 15 10 5 0 50 100 150 200 250 Figure A.19. Observed and simulated (uncalibrated) hydrographs for the 8/1/90 event on WG6 (discharge in cubic meters per second plotted as a function of time in minutes). 201 25 K2 20 - K2IVIC Observed 15 10 0 50 100 150 200 250 300 350 400 450 Figure A.20. Observed and simulated (uncalibrated) hydrographs for the 7/27/76 event on WGI (discharge in cubic meters per second plotted as a function of time in minutes). 35 30 Observed -K2 25 - K2MC 20 15 10 o 150 200 250 300 350 400 450 Figure A.21. Observed and simulated (uncalibrated) hydrographs for the 8/1/78 event on WG1 (discharge in cubic meters per second plotted as a function of time in minutes). 1.2 Observed 1 -K2 K2 MC 0.8 0.6 0.4 0.2 0 250 300 350 400 450 500 Figure A.22. Observed and simulated (uncalibrated) hydrographs for the 8/4/80 event on WG I (discharge in cubic meters per second plotted as a function of time in minutes). 202 30 Observed 25 - 20 K2 K2IVIC 15 10 5 0 o 200 600 400 1000 800 1200 Figure A.23. Observed and simulated (uncalibrated) hydrographs for the 7/30/81 event on WG1 (discharge in cubic meters per second plotted as a function of time in minutes). 12 Observe 10 -K2 MC 8 K2 6 4 2 0 --- 0 , -r---- 100 200 300 -r- , 400 500 600 700 80 Figure A.24. Observed and simulated (uncalibrated) hydrographs for the 8/17/86 event on WGI (discharge in cubic meters per second plotted as a function of time in minutes). 20 Observed 16 K2 K2 MC 12 8 4 0 300 350 400 450 500 550 600 650 Figure A.25. Observed and simulated (uncalibrated) hydrographs for the 8/2986 event on WG1 (discharge in cubic meters per second plotted as a function of time in minutes). 203 1 Observed -K2 K2 MC 0 150 200 250 300 350 400 45 Figure A-26. Observed and simulated (uncalibrated) hydrographs for the 8/1/90 event on WGI (discharge in cubic meters per second plotted as a function of time in minutes). 204 APPENDIX B: DERIVATION OF CHANNEL GEOMETRY ADJUSTMENTS Channel-depth adjustments in K2G are derived for depositional and erosional scenarios. Channel bottom width is preserved during both adjustments, but bank side slopes are allowed to vary. Both calculations compare the original channel crosssectional area (A 0 , volume per unit length of channel) with the sum of the original area and the area change, IV B.1 Depositional Depth Changes To compute depth decreases as a result of deposition, the adjusted cross-sectional area, A a , is computed first as: = A0 — (B.1) The adjusted cross-sectional area can also be written as: A, = d a • w+ 0.5.d, • swl + 0.5. d a • sw2 (B.2) where da is the adjusted channel depth, and sw/ and sw/ are the bank widths on either side of the channel (figure B.1). Bank widths are computed from the original geometric parameters, and are only adjusted in the case of bank failure. Solving equation B.2 for the adjusted channel depth yields 205 da = A, (B.3) [w+ 0.5(swl + sw2)] for which everything on the right hand side is known. Following the computation of adjusted channel depth, the bank slopes (rise/run) are adjusted by zl = d (B.4) swl and z2 = • d (B.5) sw2 04 04 sw2 sw 1 Sketch illustrating the depositional depth reduction Figure B.1. calculations for K2G trapezoidal channels. B.2 Erosional Depth Changes To compute depth increases as a result of erosion, the adjusted cross-sectional area, A a , is computed first as: = A, + A, The adjusted cross-sectional area can also be written as: (B.6) 206 A, = d a • w + 0.5 d a • swl + 0.5 d a • sw2 (B.7) where da is the adjusted channel depth, and sw/ and sw/ are the bank widths on either side of the channel (figure B.2). Bank widths are computed from the original geometric parameters, and are only adjusted in the case of bank failure. Solving equation B.2 for the adjusted channel depth yields da = Aa [w + 0.5(swl + sw2)] (B.8) for which everything on the right hand side is known. Following the computation of adjusted channel depth, the bank slopes (rise/run) are adjusted by d zl = a swl (B.9) z 2= d ' (B.10) and sw2 swl w sw2 Figure B.2. Sketch illustrating the erosional depth increase calculations for K2G trapezoidal channels. 207 B.3 Bank Failure Resulting from Erosional Oversteepening Bank failure occurs when channel incision increases bank slope above 3.75, and bank slope is automatically reduced to 1.75. The volume of failed material per unit length of channel is computed by subtracting the channel area above the new bank from that over the original (figure B.3). Since these are two right triangles the calculation becomes Vf = 0.5. d • sw f 0.5. d • sw o(B.11) where d is the channel depth, sw o is the original channel bank width (known), and swf is defined as SW f = d 1.75 because the final bank slope of 1.75 is known. d Figure B.3. Sketch illustrating the bank failure calculations for K2G trapezoidal channels. (B.12) 208 B.4 Width Changes Depositional and erosional width changes in K2G are accomplished simply by translating the channel banks towards or away from the center of the channel, respectively. As a result, the volume per unit length of channel is computed as the area of a parallelogram, which is defined as the base length times the height (perpendicular distance between the top and bottom). To calculate this area, consider the parallelogram on its side such that the base width is equal to the width change, and the height is equal to the depth of the channel (figure B.4). The area of the parallelogram is known in this case, and is defined as one half of the volume per unit length of channel that is available to produce width changes. Width change, wc, (the base width) can thus be computed as wc -- Ap d (B.13) where Ap is the area of the parallelogram. The other half of the volume available for width change is applied to the other side of the channel. Figure B.4. Sketch illustrating depositional width reduction calculations for K2G trapezoidal channels. Width reduction is pictured here, but the variables and procedure is identical for width increases. 209 REFERENCES Arnold, J.G., J.R. Williams, R. Srinivasan, K.W. King, and R.H. Griggs, 1994, SWAT: Soil Water Assessment Tool; U. S. Department of Agriculture, Agricultural Research Service, Grassland, Soil and Water Research Laboratory, Temple, TX. Arulanandan, K., Gillogley, E., and Tully, R., 1980, Development of a quantitative method to predict critical shear stress and rate of erosion of naturally undisturbed cohesive soils. Rep. GL-80-5, U.S. Army Corps of Engineers, Waterway Experiment Station, Vicksburg, Mississippi. ASCE, 1998, River width adjustment. II: Modeling; Journal of Hydraulic Engineering, v. 124, n. 9, p. 903-917. Bennett, J.P., 1974, Concepts of mathematical modeling of sediment yield; Water Resources Research, v. 10, n. 3, p. 485-492. Beven, K., 1993, Models of channel networks: theory and predictive uncertainty; in Channel Network Hydrology, Beven, K., and Kirkby, M.J., eds., John Wiley & Sons Ltd, p. 295-310. Bloschl, G., and Sivapalan, M., 1995, Scale issues in hydrological modeling: a review; in Scale Issues in Hydrological Modelling, Kalma, J.D., and Sivapalan, M., eds., John Wiley and Sons, New York, 489 pp. Bradley, C.M., 2003, Effects of soil data resolution on modeling results using a physically based rainfall-runoff model. Unpublished Masters Thesis, University of Arizona, Department of Watershed Management. Branson, F.A., Gifford, G.F., Renard, K.G., and Hadley, R.F., 1981, Rangeland Hydrology, Range Science Series 1, Society for Range Management, Denver, CO. Bravo-Espinosa, M., Osterkamp, W.R., Lopes, V.L., 2003, Bedload transport in alluvial channels; Journal of Hydraulic Engineering, v. 129, n. 10, p. 783-795. Brunner, G.W., 1989, Muskingum-Cunge Channel Routing; Lecture Notes, Hydrologic Engineering Center, U.S. Army Corps of Engineers, Davis, CA. Canfield, H.E., 1998, Use of geomorphic indicators in parameterizing an event-based sediment yield model; unpublished PhD Dissertation, University of Arizona, 296 pp. Chang, H.H., 1979, Geometry of gravel streams; Journal of the Hydraulics Division, v. 106, n. HY9, p. 1443-1456. 210 Chang, H.H., 1982, Mathematical model for erodible channels; Journal of the Hydraulics Division, ASCE, v. 108, n. HY5, p. 678-689. Chang, H.H., 1988, Fluvial Processes in River Engineering. John Wiley and Sons, Inc. 432 pp. Chappell, J., 1983, Thresholds and lags in geomorphic changes; Australian Geographer, v. 15, p. 357-366. Crandall, S.H., 1956, Engineering Analysis: A survey of numerical procedures; McGrawHill Book Co., New York, N.Y. Crowell, K.J., Wilson, C.J., Canfield, H.E., 2003, Application of local surface matching to multi-date ALSM data for improved calculation of flood-driven sediment deposition and erosion; Eos Transactions, AGU, v. 84, n. 46, Fall Meeting Supplement, Abstract Gl1A-0242. Cunge, J.A., 1969, On the subject of a flood propagation computation method (Muskingum method); Journal of Hydraulic Research, v. 7, n. 2, p. 205-230. Davies, T.R.H., and Sutherland, A.J., 1980, Extremal hypotheses for river behavior; Water Resources Research, v. 19, n. 1, p. 141-148. Dooge, J.C.I., 1973, Linear theory of hydrologic systems; Technical Bulletin No. 1468, USDA Agricultural Research Service, Washington D.C. El-Hames, A.S., and Richards, K.S., 1998, An integrated, physically based model for arid region flash flood prediction capable of simulating dynamic transmission losses; Hydrological Processes, v. 12, p. 1219-1232. Elliott, J.G., Gellis, A.C., and Aby, S.B., 1999, Evolution of arroyos: incised channels of the southwestern United States; in Incised River Channels, Darby, S.E., and Simon, A., eds., John Wiley and Sons, New York, p. 153-185. Engelund, F., and Hansen, E., 1967, A Monograph on Sediment Transport in Alluvial Streams; Teknisk Forlag, Copenhagen, 62 pp. Fair, G.M., and J.C. Geyer. 1954. Water Supply and Wastewater Disposal. John Wiley and Sons, New York, 973 pp. Fondstad, M.A., 2003, Saptial variation in the power of mountain streams in the Sangre de Cristo Mountains, New Mexico; Geomorphology, v. 55, p. 75-96. 211 Garbrecht, J., and Brunner, G.W., 1991, A Muskingum-Cunge channel flow routing method for drainage networks; U.S. Army Corps of Engineers, Technical Paper No. 135, 30 pp. Goodrich, D.C., 1990, Geometric simplification of a distributed rainfall-runoff model over a range of basin scales; PhD Dissertation, Dept. of Hydrology and Water Resources, University of Arizona, Tucson. Goodrich, D.C., Lane, L.J., Shillito, R.M., Miller, S.N., Syed, K.H., and Woolhiser, D.A., 1997, Linearity of basin response as a function of scale in a semiarid watershed; Water Resources Research, v. 33, n. 12, p. 2951-2965. Govers, G., 1990, Empirical relationships for the transport capacity of overland flow. Erosion, Transport and Deposition Processes; Proceedings of the Jerusalem Workshop, March-April 1987;. IASH Publication no. 189, pp. 45-63. Graf, W.L., 1982, Spatial variation of fluvial processes in semi-arid lands; in Space and Time in Geomorphology, Thorne, C.E., ed., p. 193-217, London, George Allen and Unwin. Graf, W.L., 1983a, Variability of sediment removal in a semiarid watershed; Water Resources Research, v. 19, n. 3, p. 643-652. Graf, W.L., 1983b, Flood-related change in an arid region river; Earth Surface Processes and Landforms, v. 8, p. 125-139. Graf, W.L., 1990, Fluvial dynamics of Thorium-230 in the Church Rock event, Perco River, New Mexico; Annals of the Association of American Geographers, v. 80, n. 3, p. 327-342. Griffiths, G.A., 1984, Extremal hypotheses for river regime: an illusion of progress; Water Resources Research, v. 20, n. 1., p. 113-118. Hayami, S., 1951, On the propagation of flood waves; Bulletin of the Disaster Prevention Research Institute, Disaster Prevention Research Institute, Kyoto, Japan, v. 1, n. 1, p. 1-16. Howard, A D., 1994, A detachement-limited model of drainage basin evolution; Water Resources Research, v. 30, n. 7, p. 2261-2285. Hromadka, T.V., and DeVries, J.J., 1988, Kinematic wave routing and computational error; Journal of Hyraulic Engineering, v. 114, n. 2, p. 207-217. 212 Koussis, A., 1976, An approximate dynamic flood routing method; Proceedings of the International Symposium on Unsteady Flow in Open Channels, Paper Li, NewcastleUpon-Tyne, England. Kustas, W.P., Goodrich, D.C., 1994, Preface; Water Resources Research, v. 30, p. 12111225. Lecce, S.A., 1997, Nonlinear downstream changes in stream power on Wisconsin's Blue River; Annals of the Association of American Geographers, v. 87, n. 3, p. 471-486. Levick, L.R., Scott, S.N., Miller, S.N., Semmens, D.J., Guertin, D.P., and Goodrich, D.C., 2003, Application of a land cover modification tool and the Automated Geospatial Watershed Assessment (AGWA) tool at Fort Huachuca, AZ; Poster presented at the 6 th Annual Conference for the Society of Conservation GIS, July 3-5, Monterey, CA. Magilligan, F.J., 1992, Thresholds and the spatial variability of flood power during extreme floods; Geomorphology, v. 5, p. 373-390. Meinhardt, H.A., 1982, Models of Biological Pattern Formation. Academic, San Diego, CA. Meselhe, E.A., and Holly Jr., F.M., 1997, Invalidity of the Preissmann scheme for transcritical flow; Journal of Hydraulic Engineering, v. 123, n. 7, p. 652-655. Meyer, L.D., and Wischmeier, W.H., 1969, Mathematical simulation of the process of soil erosion by water; Transactions of the American Society of Agricultural Engineers, v. 12, n. 6, p. 754-762. Michaud, J., and Sorooshian, S., 1994, Effect of rainfall-sampling errors on simulations of desert flash floods; Water Resources Research, v. 30, n. 10, p. 2765-2776. Miller, J.E., 1984, Basic concepts of kinematic-wave models; U.S. Geological Survey Professional Paper 1302, 29 pp. Miller, S.N., 1995, An Analysis of Channel Morphology at Walnut Gulch Linking Field Research With GIS Applications; Unpublished Masters Thesis, University of Arizona, 167 pp. Miller, S.N., 2002, Scale Effects of Geometric Complexity, Misclassification Error and Land Cover Change in Distributed Hydrologic Modeling; Unpublished Ph.D. Dissertation, University of Arizona, 169 pp. 213 Miller, S.N., Semmens, D.J., Miller, R.C., Hernandez, H., Goodrich, D.C., Miller, W.P., Kepner, W.G., and Ebert, D., 2002, Automated Geospatial Watershed Assessment (AGWA): A GIS-based hydrologic modeling tool. Proceedings of the Second Federal Interagency Hydrologic Modeling Conference, Las Vegas, NV, 7/28 — 8/1. Moramarco, T., and Singh, V.P., 2000, A practical method for analysis of river waves and for kinematic wave routing in natural channel networks; Hydrological Processes, v. 14, p. 51-62. Morin, E., Krajewski, W.F., Goodrich, D.C., Gao, X., Sorooshian, S., 2003, Estimating rainfall intensities from weather radar data: the scale-dependency problem; Journal of Hydrometeorology, v. 4, n. 5, p. 782-797. Moussa, R., and Bocquillon, C., 2000, Approximation zones for the Saint-Venant equations for flood routing with overbank flow; Hydrology and Earth System Sciences, v. 4, n. 2., p. 251-261. Nash, J.E., and Sutcliffe, J.V., 1970, River flow forecasting through conceptual models, I. A discussion of principles; Journal of Hydrology, v. 10, p. 282-290. Nearing, M.A., 1998, Why soil erosion models over-predict small soil losses and underpredict large soil losses; Catena, v. 32, p. 133-155. Nichols, M.H., and Renard, K.G., 2003, Sediment yield from semiarid watersheds; Proceedings of the First Interagency Conference on Research in the Watersheds, Benson, AZ, Oct. 27-30, p. 161-166. Ogden, F.L, and Julien, P.Y., 1993, Runoff sensitivity to temporal and spatial rainfall variability at runoff plane and small basin scales; Water Resources Research, v. 29, p. 2589-2597. Ogden, F.L., 1998, CASC2D Version 1.18 User's Manual; University of Connecticut Press, Storrs, CT. Ogden, F.L., and Heilig A., 2001, Two-Dimensional Watershed-Scale Erosion Modeling with CASC2D, in Landscape Erosion and Evolution Modeling, R.S. Harmon and W.W. Doe III, eds., Kluwer Academic Publishers, New York, ISBN 0-306-4618-6, 535 pp. Osborn, H. B., Lane, L. J., and Myers, V. A., 1980, Rainfall/watershed relationships for Southwestern thunderstorms; Transactions of the ASAE, v. 23, n. 1, p. 82-91. Pilgrim, D.H., and Cordery, I., 1993, Flood runoff; in Handbook of Hydrology, Maidment, D.R., ed., McGraw-Hill, Inc. 214 Ponce, V.M., 1981, Development of an algorithm for the linearized diffusion method of flood routing; SDSU, Civil Engineering Series No. 81144, San Diego State University, San Diego, CA. Ponce, V.M., 1989, Engineering Hydrology, Principles and Practices; Prentice Hall, New Jersey, 627 pp. Ponce, V.M., 1991, The kinematic wave controversy; Journal of Hydraulic Engineering, v. 117, n. 4, p. 511-525 . Ponce, V.M., and Chaganti, P.V., 1994, Variable-parameter Muskingum-Cunge method revisited; Journal of Hydrology, v. 162, p. 433-439. Ponce, V.M., and Theurer, F.D., 1982, Accuracy criteria in diffusion routing; Journal of the Hydraulics Division, ASCE, v. 108, n. HY6, p. 747-757. Ponce, V.M., and Yevjevich, V., 1978, Muskingum-Cunge method with variable parameters; Journal of the Hydraulics Division, ASCE, v. 104, n. 12, p. 1663-1667. Ponce, V.M., Li, R.N., and Simons, D.B., 1978, Applicability of kinematic and diffusion models; Proceedings of the American Society of Civil Engineers, Journal of the Hydraulics Division, v. 104, n. HY3, p. 353-360. Preissmann, A., 1961, Propagation of translatory waves in channels and rivers; Proceedings of the First French Association for Computation, Grenoble, France, p. 433-442. Renard, K.G., Lane, L.J., Simanton, J.R., Emmerich, W.E., Stone, J.J., Weltz, M.A., Goodrich, D.C., Yakowitz, D.S., 1993, Agricultural impacts in an arid environment: Walnut Gulch studies; Hydrological Science and Technology, v. 9, n. 1-4, p. 145-190. Risse, L.M., Nearing, M.A., Nicks, A.D., Laflen, J.M., 1993, Error assessment in the Universal Soil Loss Equation; Soil Science Society of America Journal, v. 57, n. 3, p. 824-833. Roth, G., Siccardi, F., and Rosso, R., 1989, Hydrodynamic description of the erosional development of drainage patterns; Water Resources Research, v. 25, n. 2, p. 319-332. Schumm, S.A., Mosley, M.P., and Weaver, W.E., 1987, Experimental Fluvial Geomorphology; John Wiley, New York, 413 pp. Singh, V.P., 1996, Kinematic Wave Modeling in Water Resources: Surface Water Hydrology; John Wiley and Sons, Inc., New York, 1399 pp. 215 Singh, V.P., 2002, Is hydrology kinematic? Hydrological Processes, v. 16, P. 667-716. Smith, R.E., Goodrich, D.C., Woolhiser, D.A., and Unkrich, C.L., KINEROS — a kinematic runoff and erosion model; Chapter 20 in Computer Models of Watershed Hydrology, Singh, V.P., Ed., Water Resources Publications, Highlands Ranch, Colorado, p. 697-732. Song, C., Zheng, Y., and Yang, C., 1995, Modeling of river morphologic changes; International Journal of Sediment Research, v. 10, n. 2, p. 1-20. Song, C.C.S., and Yang, C.T., 1980, Minimum stream power: theory; Journal of the Hydraulics Division, ASCE, v. 106, n. HY9, p. 1477-1487. Syed, K.H., 1999, The impacts of digital elevation model data type and resolution on hydrologic modeling. Unpublished Doctoral Dissertation, University of Arizona, Department of Hydrology. 268 pp. Szel, S., and Gaspar, C., 2000, On the negative weighting factors in the MuskingumCunge scheme; Journal of Hydraulic Research, v. 38, n. 4., p. 299-306. USACE, 1994, Flood-Runoff Analysis; U.S. Army Corps of Engineers Report EM 11102-1417, http://www.usace.army.mil/inet/usace-docs/eng-manuals/em1110-2-1417/ USCB, 2003, Large Suburban Cities in West Are Fastest-Growing; U.S. Census Bureau Report CB03-106; http://www.census.gov/Press-Release/www/2003/cb03-106.html Vieira, D.A. and Wu, W., 2002, CCHElD — Flow and Channel Morphology Model for Channel Networks; Proceedings of the Second Federal Interagency Hydrologic Modeling Conference, July 28-August 1, 2002, Las Vegas, Nevada. Vieira, D.A., and Wu, W., 2000, One-dimensional channel network model CCHElD 2.0 — User's Manual; Report NCCHE-TR-2000-2, University of Mississippi Press, Oxford, Mississippi. Weinmann, P.E., and Laurenson, E.M., 1979, Approximate flood routing methods: a review. Journal of the Hydraulics Division, ASCE, v. 105, n. HY12, P. 1521-1536. Werritty, A., 1997, Short-term changes in channel stability; in Applied Fluvial Geomorphology for River Engineering and Management, Thorne, C.R., Hey, R.D., and Newson, M.D., eds., John Wiley and Sons Ltd., 376 pp. White, W.R., Bettess, R., and Paris, E., 1982, Analytical approach to river regime; Journal of the Hydraulics Division, ASCE, v. 108, n. HY10, p. 1179-1192. 216 Willgoose, G., and Kuczera, G., 1995, Estimation of subgrid scale kinematic wave parameters for hillslopes; in Scale Issues in Hydrological Modeling, Kalma, J.D., and Sivapalan, M., eds., John Wiley and Sons, New York, 489 pp. Willgoose, G., Bras, R.L., and Rodriguez-Iturbe, I., 1991a, A coupled channel network growth and hillslope evolution model, 1. Theory; Water Resources Research, v. 27, n. 7, p. 1671-1684. Willgoose, G., Bras, R.L., and Rodriguez-Iturbe, I., 1991b, A coupled channel network growth and hillslope evolution model, 2. Nondimensionalization and applications; Water Resources Research, v. 27, n. 7, p. 1685-1696. Woolhiser, D.A., 1996, Search for a physically based runoff model - a hydrologic El Dorado?; Journal of Hydraulic Engineering, v. 122, n. 3, p. 122-129. Woolhiser, D.A., and Ligget, J.A., 1967, Unsteady, one-dimensional flow over a plane the rising hydrograph; Water Resources Research, v. 3, n. 3, p. 753-771. Woolhiser, D.A., Smith, R.E., and Goodrich, D.C., 1990, KINEROS, A kinematic runoff and erosion model: documentation and user manual; USDA-Agricultural Research Service, ARS-77, 130 pp. Wu, W., and Vieira, D.A., 2000, One-dimensional channel network model CCHElD 2.0 — Technical Manual; Report NCCHE-TR-2000-1, University of Mississippi Press, Oxford, Mississippi. Yang, C.T. and Song, C.C.S., 1979, Theory of minimum rate of energy dissipation; Journal of the Hydraulics Division, ASCE, v. 105, n. HY7, p. 769-784. Yang, C.T., and Simôes, F.J.M., 1998, Simulation and prediction of river morphologic changes using GSTARS 2.0; Proceedings of the 3 rd International Conference on Hydro-Science and —Engineering, Cottbus/Berlin, Germany, Aug. 31-Sep. 3. Yang, C.T., and Song, C.C.S., 1986, Theory of minimum energy and energy dissipation rate; in Encyclopedia of Fluid Mechanics, Cheremisinoff, I., Ed., Gulf Publishing Company, Houston, Texas, p. 353-399. Yang, C.T., Song, C.C.S., and Woldenberg, M.H., 1981, Hydraulic geometry and minimum rate of energy dissipation; Water Resources Research, v. 17, n. 4, p. 10141018. Yen, B.C., 1979, Unsteady flow mathematical modeling techniques; in Modeling of Rivers, Shen, H.W., ed., John Wiley and Sons, New York. 217 Zoppou, C., O'Neill, I.C., 1982, Criteria for the choice of flood routing methods in natural channels; Proceedings of the Hydrology and Water Resources Symposium, May 11-13, 1982, Melbome, Austrailia.
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