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An assessment of rainfall-runoff modeling methodology Loague, Keith M. 1986

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AN ASSESSMENT OF RAINFALL-RUNOFF MODELING METHODOLOGY by Keith M. Loague B.S., The University of Michigan, 1978 M.Sc., The University of B r i t i s h Columbia, 1982 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of I n t e r d i s c i p l i n a r y Hydrology) We accept t h i s thesis as conforming to the reauired standard THE UNIVERSITY OF BRITISH COLUMBIA June 1986 (c) Keith Michael Loague, 1986 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department O f I n t e r d i s c i p l i n a r y S t u d i e s The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 16 June 1986 This thesis i s dedicated to the memory of my father CLIFTON J . LOAGUE August 1922 - 30 January 1985) ABSTRACT This study reports model performance c a l c u l a t i o n s f o r three event-based r a i n f a l l - r u n o f f models on both r e a l and synthetic data sets. The models include a regression model, a unit hydrograph model and a qua s i - p h y s i c a l l y based model. The r e a l data sets are f o r small upland catchments from the Washita River Experimental Watershed, Oklahoma; the Mahantango Creek Experimental Watershed, Pennsylvania; and the Hubbard Brook Experimental Forest, New Hampshire. The synthetic data sets are generated with a stochastic-conceptual r a i n f a l l - r u n o f f simulator. Model performance i s assessed f o r a v e r i f i c a t i o n period that i s c a r e f u l l y distinguished from the c a l i b r a t i o n period. Performance assessment was c a r r i e d out both i n for e c a s t i n g mode and i n prediction mode. The r e s u l t s on the r e a l data sets show s u r p r i s i n g l y poor forecasting e f f i c i e n c i e s for a l l models on a l l data sets. The unit hydrograph model and the quasi-physically based model have l i t t l e forecasting power; the regression model i s marginally better. The performance of the models i n prediction mode i s better. The regression model and the unit hydrograph model showed acceptable p r e d i c t i v e power, but the quasi-physically based model produced acceptable predictions on only one of the three catchments. The performance of the regression and unit hydrograph models, i n both forecasting and prediction modes, for synthetic data i s much better than f o r the r e a l catchments. The performance of the quasi-physically based model on a synthetic data set i s s u r p r i s i n g l y poor. i i i Supplemental data gathered from the Oklahoma catchment was used for a s p a t i a l v a r i a b i l i t y a n alysis of steady-state i n f i l t r a t i o n r ates. These data were then used to re-excite the quasi-physically based model; the new information resulted i n improved model performance. The concept of space-time tradeoffs across the hydrologic data sets of competing models i s introduced and tested. Results show the existence of space-time tradeoffs within model data sets but not across model data sets. I t i s the b e l i e f of the author that the primary b a r r i e r to successful a p p l i c a t i o n of p h y s i c a l l y based models i n the f i e l d l i e s i n the scale problems that are associated with the unmeasurable s p a t i a l v a r i a b i l i t y of r a i n f a l l and s o i l hydraulic properties. The f a c t that simpler, l e s s data intensive models provided as good or better predictions than a p h y s i c a l l y based model i s food for thought. The model evaluation and space-time tradeoff experiments reported i n t h i s study are conceptually linked to data-worth an a l y s i s , network-design, and model-choice c r i t e r i a f o r future studies. iv TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v i i LIST OF ILLUSTRATIONS x i i ACKNOWLEDGEMENTS x v i i 1. INTRODUCTION 1 2. STREAMFLOW GENERATION, MODELING TECHNIQUES, AND COMPARATIVE STUDIES 5 Strearaflow Generation 5 Horton Overland Flow 7 Dunne Overland Flow 10 Subsurface Storm Flow . 10 Modeling Techniques 12 Comparative Studies. . 16 3. CALIBRATION, VERIFICATION AND MODEL EFFICIENCY 19 4. RAINFALL-RUNOFF MODELS 28 Regression Model 28 Unit Hydrograph Model 29 Quasi-Physically Based Model 32 5. EXPERIMENTAL CATCHMENTS 38 R-5, Chickasha, Oklahoma 38 WE-38, Klingerstown, Pennsylvania 45 HB-6, West Thornton, New Hampshire 51 Comparison of Catchments 55 6. R-5 REVISITED: SPATIAL VARIABILITY 58 Spatial Variability 61 Methods of Analyzing Spatial Structure 81 Autocorrelation 82 Semi-variogram 84 Kriging 85 Data Collection 87 Infil t r a t i o n 87 Sampling Scheme 89 Laboratory Measurements 94 Summary 98 TABLE OF CONTENTS Page 7. SYNTHETIC DATA SETS 100 Stochastic-Conceptual Rainfall-Runoff Simulator 100 H i l l s l o p e s 105 8 . MODEL VALIDATION: EXPERIMENTAL CATCHMENTS 117 Results 117 Discussion 124 Sample V e r i f i c a t i o n Run and Scattergrams 124 Forecasting E f f i c i e n c i e s 125 P r e d i c t i o n E f f i c i e n c i e s . . . 126 V e r i f i c a t i o n S t a t i s t i c s 126 Regression Model 127 Unit Hydrograph Model 127 Quasi-Physically Based Model 129 Summary 137 9 . R-5 REVISITED: RESULTS OF SPATIAL VARIABILITY ANALYSIS 140 F i e l d Analysis 140 Laboratory Analysis 158 R-5 : A Sand Box? 170 10. MODEL VALIDATION: HYPOTHETICAL HILLSLOPES 177 Results 177 Discussion 181 Forecasting E f f i c i e n c i e s 181 P r e d i c t i o n E f f i c i e n c i e s . . 182 V e r i f i c a t i o n S t a t i s t i c s . 182 Regression Model 183 Unit-Hydrograph Model 187 Quasi-Physically Based Model 194 Summary 199 11. HYPOTHETICAL HILLSLOPES: SENSITIVITY ANALYSES 202 E f f e c t of Hydraulic Conductivity on Rainfall-Runoff Process 202 Impact of P r e c i p i t a t i o n and S o i l Hydraulic Property Data 209 v i TABLE OF CONTENTS Page 12. EXTENSIONS AND FUTURE WORK 237 Scale 237 Model Evaluation 239 Conditional Stochastic Simulation 241 R-5 Water Balance 242 Data Worth 242 Network Design 244 Space-Time Tradeoffs 245 13. CONCLUSIONS 261 Introduction 261 Overview of Methodology 262 Results and Discussion 263 REFERENCES 268 v i i LIST OF TABLES Table Page 5- 1. C h a r a c t e r i s t i c parameters f o r the R-5, WE-38, and HB-6 catchments 41 5.2. Summary variables f o r the R-5, WE-38, and HB-6 r a i n f a l l - r u n o f f events 44 5.3. Comparison of catchments 56 6- 1. Review of research studies u t i l i z i n g data from the R-5 catchment 59 6-2. Review of the s p a t i a l v a r i a b i l i t y l i t e r a t u r e 64 6-3. Review of stochastic-conceptual s p a t i a l v a r i a b i l i t y studies 79 6- 4. Summary of R-5 supplemental data 99 7- 1. SCRRS input parameters f o r case A (adapted from Freeze, 1980a) 106 7-2. Comparison of SCRRS input f o r cases A-J I l l 7-3. Summary var i a b l e s t a t i s t i c s f o r 100 event hypothetical r e a l i t i e s 114 7- 4. R a i n f a l l summary variable s t a t i s t i c s at selected points f o r case A 115 8- 1. Forecasting e f f i c i e n c i e s E f for the three summary output variables f o r the three models on the three data sets 118 8-2. P r e d i c t i o n e f f i c i e n c i e s E for the three summary output variables for^the three models on the three data sets 118 8-3. Summary output variables f o r an observed sample v e r i f i c a t i o n from the R-5 catchment 120 8-4. V e r i f i c a t i o n period s t a t i s t i c s : Means and standard deviations f o r observed and predicted frequency d i s t r i b u t i o n s for the three summary output variables on the three data sets 123 v i i i LIST OF TABLES Table Page 8-5a. Model verification efficiencies for three summary output variables for catchment R-5 with calibrated and uncalibrated quasi-physically based model 133 .8-5b. Verification period stat i s t i c s for three summary output variables for catchment R-5 with calibrated and uncalibrated quasi-physically based model 133 8- 6. Quasi-physically based model calibration summary for the R-5 catchment 136 9- 1. Steady-state i n f i l t r a t i o n v st a t i s t i c s for the R-5 catchment 141 2 9-2. X st a t i s t i c s for the test against a normal distribution, (a) TR1. (b) TR2. (c) grid, (d) Shajma et a l . (1980). N, i s the sample size; X > the observed s t a t i s t i c ; df, the degrees of freedom; and X Q 05 ' t^ i e theoretical s t a t i s t i c at the 95% significance level 144 9-3. Schedule of i n f i l t r a t i o n measurements 147 9-4. Various grid network requirements 151 9-5. Summary statis t i c s for grid i n f i l t r a t i o n measurements based on s o i l type 154 9-6. Split sample statistics for R-5 i n f i l t r a t i o n data. . . 157 9-7. Comparison of i n f i l t r a t i o n studies 159 9-8. Summary statistics for R-5 laboratory determined parameters. Porosity, 0 Q , s o i l moisture, 6 ^ , and saturated hydraulic conductivity, K q 160 2 9-9. x st a t i s t i c s for laboratory determined R-5 parameters 165 9-10. Saturated hydraulic conductivity values used to excite and re-excite the quasi-physically based model. Original data used in Chapter 6, supplemental data used in this chapter 171 i x LIST OF TABLES Table Page 9-11. Forecasting e f f i c i e n c i e s E f for the three summary output v a r i a b l e s f o r the qua s i - p h y s i c a l l y based model on the o r i g i n a l and supplemental R-5 data sets 172 10-1. Forecasting e f f i c i e n c i e s E f for the three summary output variables f o r the three models f o r one r a i n gage on synthetic data set A 179 10-2. P r e d i c t i o n e f f i c i e n c i e s E f or the three summary output variables for^the three models f o r one r a i n gage on synthetic data set A 179 10-3. V e r i f i c a t i o n period s t a t i s t i c s : Means and standard deviations f o r observed and predicted frequency d i s t r i b u t i o n s f o r the three summary output variables f o r one r a i n gage on synthetic data set A 180 10-4. Forecasting e f f i c i e n c i e s E f for the three summary output variables f o r the regression model on the four synthetic data sets, (a) s i n g l e r a i n gage, (b) f i v e r a i n gage average 184 10-5. P r e d i c t i o n e f f i c i e n c i e s E for the three summary output variables for^the regression model on the four synthetic data sets, (a) s i n g l e r a i n gage. (b) f i v e r a i n gage average 184 10-6. Regression model v e r i f i c a t i o n period s t a t i s t i c s : Means and standard deviations f o r observed and predicted frequency d i s t r i b u t i o n s f o r the three summary output variables on the four synthetic data sets 185 2 10-7. C o e f f i c i e n t of determination values r for a l t e r n a t i v e unit hydrograph model $-index r e l a t i o n s h i p s on case A , 189 10-8. Forecasting e f f i c i e n c i e s E f for the three summary output v a r i a b l e s f o r the unit hydrograph model on the synthetic data of case A 192 10-9. P r e d i c t i o n e f f i c i e n c i e s E for the three summary output variables for^the unit hydrograph model on the synthetic data of case A 192 X LIST OF TABLES Table Page 10- 10. Unit hydrograph model v e r i f i c a t i o n period s t a t i s t i c s : Means and standard deviations f o r observed and predicted frequency d i s t r i b u t i o n s f o r the three summary output variables on the synthetic data of case A 193 11- 1. Comparison of SCRRS input parameters f o r cases A-J . . 203 11-2. Correspondence between hypothetical r e a l i t i e s i n t h i s work and the o r i g i n a l work 204 11-3. Comparison of cases showing e f f e c t of hydraulic conductivity on streamflow 206 11-4. Comparison of data-impact experiments on case A. . . . 213 11-5. Comparison of data-impact experiments on case B. . . . 214 11-6. Comparison of data-impact experiments on case C. . . . 215 11-7. Comparison of data-impact experiments on case D. . . . 216 11-8. Comparison of data-impact experiments on case E. . . . 217 11-9. Comparison of data-impact experiments on case F. . . . 218 11-10. Comparison of data-impact experiments on case G. . . . 219 11-11. Comparison of data-impact experiments on case H. . . . 220 11-12. Comparison of data-impact experiments on case I. . . . 221 11-13. Comparison of data-impact experiments on case J . . . . 222 11-14. Comparison of data-impact experiments on case A. . . . 225 11-15. Comparison of data-impact experiments on case B. . . . 225 11-16. Comparison of data-impact experiments on case C. . . . 226 11-17. Comparison of data-impact experiments on case D. . . . 226 11-18. Comparison of data-impact experiments on case E. . . . 227 11-19. Comparison of data-impact experiments on case F . . . 227 x i LIST OF TABLES Table Page 11--20. Comparison of data-impact experiments on case G. . . . 228 11--21. Comparison of data-impact experiments on case H . . . 228 11--22. Comparison of data-impact experiments on case I. . . . 229 11 -23. Comparison of data-impact experiments on case J . . . . 229 11--24. Comparison of unconditional data-impact experiments. 233 11 -25. Comparison of c o n d i t i o n a l data-impact experiments. . . 234 11 -26, Comparison of unconditional data-impact experiments. . 236 11 -27. Comparison of c o n d i t i o n a l data-impact experiments. . . 236 12-1. Forecasting e f f i c i e n c i e s E, for the three summary output variables f o r the regression model on the h i l l s l o p e A data set 254 12-2. Forecasting e f f i c i e n c i e s E f for the three summary output variables f o r the quasi-physically based model on the h i l l s l o p e A data set. The numbers i n brackets represent the numbers of events f o r which the model converged and upon which the E^ r e s u l t s are based 254 12-3. Forecasting e f f i c i e n c i e s E f for the three summary output variables f o r the regression model on the R-5 data set 257 12-4. Forecasting e f f i c i e n c i e s E, for the three summary output variables f o r the quasi-physically based model on the R-5 data set 257 12-5. Split-sample s t a t i s t i c s : Means and standard deviations f o r observed and quasi-physically , based model predicted frequency d i s t r i b u t i o n s for the three summary output variables on the R-5 data set 259 x i i LIST OF ILLUSTRATIONS Figure Page 1-1. Idealized interaction between operational and research hydrologists 2 2-1. Streamflow generation mechanisms for the delivery of r a i n f a l l to a stream channel on a hillslope: (A) Horton overland flow, (B) Dunne overland flow composed of i n f i l t r a t e d water that emerges from the ground (return flow) and direct precipitation onto the resulting saturated zone, (C) subsurface (storm) flow, (D) groundwater flow (adapted from Freeze, 1980b; Dunne, 1980) 2-2. Schematic i l l u s t r a t i o n of the occurrence of various runoff processes in relation to their major environmental controls (after Freeze, 1980a) 8 2-3. Moisture content versus depth profiles for (a) the Horton mechanism and (b) the Dunne mechanism. Overland flow generation for (c) the Horton mechanism and (d) the Dunne mechanism (adapted from Freeze, 1980a) 9 2- 4. Clarke's (1973) hydrologic model classification scheme 1 3 3- 1. Hypothetical rainfall-runoff event showing event summary variables 21 4- 1. Preprocessing of unit hydrograph input data. (a) $-index method of calculating excess r a i n f a l l employed in (4-2b). (b) Hewlett and Hibbert (1967) baseflow separation technique. The slope i s an input parameter in (4-2e) 31 4-2. Schematic i l l u s t r a t i o n of quasi-physically based model. (a) Infiltrating catchment with areally uniform r a i n f a l l (Qi(t) represents lateral inflows), (b) Discreet overland flow planes on which kinematic wave equations are used to describe transient flow, also illustrates discreet channel sections where x's become y's, S 's become S ' s, and n 's become n ' s (adapted from Freeze, 1982a)? ? . . . . 37 x i i i LIST OF ILLUSTRATIONS Figure Page 5-l a . R-5 catchment (G.A. Gander, personal communication, 1981) 39 5-1b. Segments used to transform the catchment in t o overland flow planes 40 5-2. Average s o i l moisture (a) R-5 (G.A. Gander, personal communication, 1981), (b) HB-6 ( C A . Federer, personal communication, 1983) 46 5-3a. WE-38 catchment (W.J. Gburek, personal communication, 1982) 47 5-3b. Segments used to transform.the catchment into overland flow planes (abstracted from Engman, 1974) 48 5-4a. HB-6 catchment ( C A . Federer, personal communication, 1982) 52 5- 4b. Segments used to transform catchment i n t o overland flow planes 53 6- 1. (a) Hypothetical autocorrelograms. (b) Hypothetical semi-variogram showing c h a r a c t e r i s t i c features 83 6-2. Hypothetical i n f i l t r a t i o n curve 88 6-3. Schematic of s i n g l e - r i n g i n f i l t r o m e t e r and components, (a) hard-alloy alluminum sheet r o l l e d into a r i n g and welded at the j o i n t . (b) clear p l a s t i c cover used to reduce: (1) evaporation, and (2) wave action caused by the wind, (c) stand f or uniformly driving rings into the ground, (d) point gage. (e) s i t e water control stand, ( f ) water d i s t r i b u t i o n manif old 91 6-4. Experimental layout for f i v e simultaneous steady-state i n f i l t r a t i o n measurements. Upon completion of experiment #1, experiment #6 i s started (etc.) provided enough water i s av a i l a b l e 93 x i v LIST OF ILLUSTRATIONS Figure Page 6-5a. R-5 catchment: lo c a t i o n s of i n f i l t r a t i o n experiments (and s o i l samples) i n t h i s study 95 6- 5b. R-5 catchment: locatio n s of previously gathered data 96 7- 1. (a) Two-dimensional s p a t i a l h i l l s l o p e g r i d . (b) V e r t i c a l s ection through the h i l l s l o p e (adapted from Freeze, 1980a) 103 7-2. V e r t i c a l cross sections showing topography and water table configurations for ten cases l i s t e d i n Table 7-2 (adapted from Freeze, 1980a) 109 7-3. Case A: (a) Hydraulic conductivity d i s t r i b u t i o n . (b) Source areas f o r overland flow f o r event 1. (c) Source area summary for 100 event hypothetical r e a l i t y 112 7- 4. Case A: Hydrologic conditions on the h i l l s l o p e during event 1. R a i n f a l l i s f a l l i n g on shaded areas; ponding has occurred on hatched areas 113 8- 1. A sample v e r i f i c a t i o n run from catchment R-5 (event 68) 119 8-2. Scattergrams of observed and predicted storm flow depths f o r the 36 R-5 v e r i f i c a t i o n events 122 8-3. Averaged 15-minute unit-hydrographs developed f o r the three catchments 128 8- 4. S e n s i t i v i t y of the quasi-physically based model to changes i n the saturated hydraulic conductivity parameter f o r R-5 c a l i b r a t i o n event 4. The c o n d u c t i v i t i e s associated with cases A, B, C, D, and E are given i n Table 8-6 135 9- 1. Frequency histograms f o r steady-state i n f i l t r a t i o n data, (a) TR1. (b) TR2. (c) g r i d , (d) Sharma et a l . (1980) 142 XV LIST OF ILLUSTRATIONS Figure Page 9-2. S p a t i a l v a r i a t i o n i n steady-state i n f i l t r a t i o n along (a) TR1 and (b) TR2 145 9-3. TR1 experimental functions f or steady state i n f i l t r a t i o n data, (a) autocorrelogram. (b) semi-variogram 148 9-4. TR2 experimental functions f o r steady state i n f i l t r a t i o n data, (a) autocorrelogram. (b) semi-variogram 149 9-5. D i r e c t i o n a l semi-variograms for steady-state i n f i l t r a t i o n based upon the 157 R-5 g r i d observations 152 9-6. R-5 climate, (a) Daily p r e c i p i t a t i o n . (b) Daily temperature 156 9-7. Frequency histograms for TR1 data 162 9-8. Frequency histograms for TR2 data 163 9-9. Frequency histograms f o r g r i d data 164 9-10. Average s o i l water retention curve. Error bars represent one standard deviation 167 9-11. TR1 experimental functions f o r laboratory determined parameters. Autocorrelograms (top), and semi-variograms (bottom) 168 9-12. TR2 experimental functions f o r laboratory determined parameters. Autocorrelograms (top), and semi-variograms (bottom) 169 9-13. • R-5 catchment. . 174 10-1. Case A r a i n f a l l event 64. This i s one of the 14 v e r i f i c a t i o n period events for which the quasi-physically based model was unstable. The hyetograph was determined from a single r a i n gage whose l o c a t i o n i s shown i n Figure 7-la 197 x v i LIST OF ILLUSTRATIONS Figure Page 11-1. Three grid sampling schemes, (a) 50 compiled elements, (b) 8 compiled elements, (c) 2 compiled elements. For each grid the shaded elements represent measurement elements 212 11- 2. Comparison of the size of catchments and h i l l s l o p e s used i n t h i s study 230 12- 1. Hypothetical space-time tradeoff curves resulting within the data set of a single modeling technique 247 12-2. Hypothetical space-time tradeoff resulting across the data sets of two different modeling techniques 249 12-3. H i l l s l o p e representations for assessment of space-time tradeoffs i n comparison of three r a i n f a l l - r u n o f f models. (a) Regression and unit-hydrograph models. (b) Quasi-physically based model (after Freeze, 1982a) 251 ACKNOWLEDGEMENTS F i r s t and foremost I want to acknowledge my partner and best f r i e n d Emily C l a s p e l l . Together we have been able to r e a l i z e many of our dreams. Sharing the completion of t h i s d i s s e r t a t i o n with my wife makes i t a l l the more g r a t i f y i n g . I was extremely fortunate to have A l l a n Freeze f o r a guiding l i g h t during my tenure at UBC. His sage advice and enthusiasm stimulated t h i s research from i t s conception through i t s completion. I t was a rare opportunity to have been a part of Al's hydrogeology group. I'm most g r a t e f u l to Steve Burges for s e l f l e s s l y sharing h i s wisdom and nurturing our f r i e n d s h i p . Schoolmates Grant Garven and Jennifer Rulon have contributed greatly to my success. Grant was my r o l e model as a s c i e n t i s t and Jennifer my best advocate. Andy Black and Dennis Lettenmaier influenced me tremendously through t h e i r zeal and motivation. Portions of t h i s study could not have taken place without the cooperation of the s c i e n t i s t s who provided the data sets described i n Chapter 5. I am indebted to Gene Gander of the USDA-ARS at the Southern P l a i n s Watershed and Water Quality Lab, Chickasha, Oklahoma; B i l l Gburek of the USDA-ARS at the Northeastern Watershed Research Center, University Park, Pennsylvania; and Tony Federer of the USDA-FS at the Northeastern Forest Experiment Station, Durham, New Hampshire. I appreciate t h e i r sharing of t h e i r precious data. I am also g r a t e f u l to J . B. Burford of the USDA-ARS Hydrology Lab i n B e l t s v i l l e , Maryland, who provided considerable information. I also wish to thank Ted Engman of the USDA-ARS x v i i i Hydrology Lab i n B e l t s v i l l e , Maryland, who provided documented copies of the q u a s i - p h y s i c a l l y based model and showed great i n t e r e s t i n t h i s work. The f i e l d component of t h i s research, described i n Chapter 6 , was extremely rewarding. I t s success was due to the collaboration of many i n d i v i d u a l s . F i r s t I wish to express my gratitude to my father-in-law, B i l l C l a s p e l l , f o r l e t t i n g me use h i s jeep and t r a i l e r f o r three months. In Vancouver, Jan deVries and Craig Forster were very h e l p f u l with experimental design. Doug Poison and Ray Rodway b u i l t the equipment used i n t h i s study. Ed Montgomery provided considerable assistance. In Oklahoma, the ARS s t a f f i n both Chickasha and Durant were very supportive of my e f f o r t s . Their contributions are greatly appreciated. My thanks go out to V i r g i l Southwell who provided e x c e l l e n t f i e l d support. I am also very g r a t e f u l to T. R. McCalla for granting access to the R - 5 catchment. F i n a l l y , I would l i k e to acknowledge (again) Gene Gander for h i s p r i n c i p a l r o l e i n the r e a l i z a t i o n of t h i s portion of my t h e s i s . I also wish to thank Gene and h i s family f o r many good times i n Chickasha. I would l i k e to acknowledge my committee members as a group. I am very g r a t e f u l to A l l a n Freeze, Stephen Burges, Andy Black, Jan deVries, B i l l Caselton and L e s l i e Smith for t h e i r i n t e r e s t and h e l p f u l comments throughout my study. Many thanks are extended to Gordon Hodge for d r a f t i n g the f i g u r e s i n t h i s t h e s i s . The external examiners for t h i s i n t e r d i s c i p l i n a r y study were Marshall Moss and David Woolhiser. Support f o r t h i s work came from a grant to A l l a n Freeze from the Natural Science and Engineering Research Council of Canada. 1 CHAPTER 1 Introduction Mathematical models describing the r a i n f a l l - r u n o f f process are commonly employed by hydrologists and engineers f o r a wide var i e t y of problems. However, there i s no universal model appropriate for the s o l u t i o n of a l l problems. The s e l e c t i o n of a s u i t a b l e model for a given s i t u a t i o n by an operational hydrologist i s often a most d i f f i c u l t one. For 15 years, more and more r a i n f a l l - r u n o f f models have been r o l l i n g o f f the assembly l i n e . There have, however, been few unbiased rigorous comparative evaluations of the underlying techniques from which t h i s suite of models were b u i l t . I f a worthwhile comparison i s to be c a r r i e d out, i t i s important to f i r s t recognize the basic structure and operational basis of each model, and then to r e l a t e these to the types of data that are required to excite the models. A comparative evaluation of modeling techniques f a l l s i n t o the domain of the research hydrologist. However, the implications of these comparisons are important to the operational h y d r o l o g i s t . This author's perception of an i d e a l i z e d i n t e r a c t i o n between p r a c t i c a l i t y and theory i n r a i n f a l l - r u n o f f modeling i s symbolized i n Figure 1-1. As shown i n the f i g u r e , the dominion of hydrologic research should be encompassed within the needs of operational hydrology. This t h e s i s i s concerned with the e f f i c i e n t use of mathematical models to predict runoff from r a i n f a l l . The c e n t r a l objective of t h i s study i s to evaluate the r e l a t i v e performances of various underlying r a i n f a l l - r u n o f f modeling techniques using both r e a l and synthetic data. 2 O P E R A T I O N A L ! IIYDR (Model selection and I 1 implementation); R E S E A R C H HYDROLOGIST (Model design and evaluat ion) FIGURE 1-1. Idealized i n t e r a c t i o n between operational and research hydrologists. 3 Intimately r e l a t e d to the performance analyses of the o v e r a l l models are several p a r a l l e l i n v e s t i g a t i o n s on i n d i v i d u a l components of the models. Hydrologic p r a c t i t i o n e r s are c a l l i n g f o r guidelines for s e l e c t i n g r a i n f a l l - r u n o f f models f o r various engineering problems that have s p e c i f i c time frame, budget and performance const r a i n t s . The r e s u l t s of t h i s work provide impetus f o r continued re-examination of r a i n f a l l - r u n o f f methodology that w i l l provide d i r e c t i o n f o r operational hydrologists i n the future. The presentation of material i n t h i s thesis i s spread over 13 chapters. I t i s hoped that these d i v i s i o n s f a c i l i t a t e an easy voyage f o r the reader. In Chapter 2 streamflow generation, r a i n f a l l - r u n o f f modeling techniques and model comparison studies are reviewed. The concepts of c a l i b r a t i o n , v e r i f i c a t i o n and model e f f i c i e n c y are discussed i n Chapter 3. The second and t h i r d chapters together introduce the reader to much of the jargon used i n subsequent chapters. The su i t e of three r a i n f a l l - r u n o f f models used throughout t h i s study i s described i n Chapter 4. The three models are (1) a regression model, (2) a unit hydrograph model, and (3) a quasi-physically based model. In Chapter 5 data sets obtained from three experimental catchments are reviewed. The three data sets come from (1) the Washita River Experimental Watershed, Oklahoma, (2) the Mahantango Creek Experimental Watershed, Pennsylvania, and (3) the Hubbard Brook Experimental Forest, New Hampshire. The catchments selected for t h i s study represent three of the very few North American Catchments that have a s u f f i c i e n t l y broad instrumentation program and data c o l l e c t i o n network to produce a data base compatible with the input requirements of the three modeling procedures. 4 Supplemental data obtained s p e c i f i c a l l y for t h i s study to characterize the s p a t i a l v a r i a b i l i t y of near surface s o i l hydraulic properties on the Oklahoma catchment are described i n Chapter 6. Synthetic r a i n f a l l - r u n o f f data generated at a h i l l s l o p e scale f o r t h i s study are presented i n Chapter 7. In Chapter 8 model v a l i d a t i o n s f or three r a i n f a l l - r u n o f f models on the three small upland catchment data sets are reported and compared. The p o t e n t i a l f o r improving the performance of the quasi-physically based model over that reported i n Chapter 8 by using the supplemental f i e l d data i s explored i n Chapter 9. Model performance for the three r a i n f a l l - r u n o f f models i s further evaluated using the synthetic data sets i n Chapter 10. The organization of Chapter 5 through 10 i s such that a one-to-one correspondence e x i s t s between each of the data Chapters (5, 6, 7) and each of the r e s u l t s Chapters (8, 9, 10). S e n s i t i v i t y analyses f o r the generator of the synthetic data sets are presented i n Chapter 11. The s e n s i t i v i t y experiments address (1) r a i n f a l l - r u n o f f processes and (2) data-impact considerations. In Chapter 12, several p o t e n t i a l extensions to the research reported i n t h i s thesis are discussed. The concept of space-time tradeoffs across hydrologic data sets i s introduced along with the r e s u l t s from preliminary experiments that set a foundation for future study. Chapter 13 provides a summary and conclusions. 5 CHAPTER 2 Streamflow Generation, Modeling Techniques, and Comparative Studies In t h i s chapter streamflow generation concepts, r a i n f a l l - r u n o f f modeling techniques, and model comparison studies are reviewed. The overview presented herein serves as a general foundation for the following chapters. Streamflow Generation In t h i s section three mechanisms of streamflow generation are reviewed. The discussion i s l a r g e l y based upon the review paper of Freeze (1974) and the more recent papers by Freeze (1980a) and Dunne (1982, 1983). The interested reader i s also directed to the c o l l e c t i o n of papers edited by Kirkby (1978) and i n p a r t i c u l a r the paper by Dunne (1978) included i n that c o l l e c t i o n . Figure 2-1 i s a conceptual i l l u s t r a t i o n of streamflow generation on a h i l l s l o p e . The discharge that i s measured at the downstream end of the channel reach (shown i n Figure 2-1) i s supplied by channel inflow at the upstream end of the reach and by the l a t e r a l inflows that enter the channel from the h i l l s l o p e s along the reach. The l a t e r a l inflows may a r r i v e at the stream i n one of three forms: groundwater flow, subsurface stormflow, or overland flow. Groundwater flow provides the baseflow component of streams that sustains t h e i r flow between storm periods. The flashy response of streamflow to i n d i v i d u a l p r e c i p i t a t i o n events may be ascribed to e i t h e r subsurface stormflow or overland flow. However, at most locations the primary source of the l a t e r a l inflows that create peak Time FIGURE 2-1. Streamflow generation mechanisms for the delivery of r a i n f a l l to a stream channel on a h i l l s l o p e : (A) Horton overland flow, (B) Dunne overland flow composed of i n f i l t r a t e d water that emerges from the ground (return flow) and d i r e c t p r e c i p i t a t i o n onto the r e s u l t i n g saturated zone, (C) subsurface (storm) flow, (D) groundwater flow (adapted from Freeze, 1980b; Dunne, 1980). 7 flows during storm runoff events i s overland flow. Overland flow i s generated at a point on a h i l l s l o p e only a f t e r surface ponding takes place. Ponding cannot occur u n t i l the surface s o i l layers become saturated. I t i s now recognized that surface saturation can occur because of two quite d i s t i n c t mechanisms, Horton overland flow and Dunne overland flow. Figure 2-2 summarizes the environmental controls on Horton overland flow, Dunne overland flow, and subsurface stormflow. The concepts of " p a r t i a l areas" and "variable source areas" that are mentioned i n Figure 2-2 and which w i l l f i g u r e i n the following discussions were developed by Betson (1964) and Hewlett and Hibbert (1967) r e s p e c t i v e l y . Both Freeze (1974) and Hewlett (1982) provide d e f i n i t i o n s f or the jargon associated with streamflow generation. Horton Overland Flow The c l a s s i c mechanism, f i r s t espoused by Horton (1933) and placed i n a more s c i e n t i f i c framework by Rubin and Steinhardt (1963), i s for a p r e c i p i t a t i o n rate p that exceeds the saturated hydraulic conductivity K of the surface s o i l . As i s i l l u s t r a t e d i n Figure 2-3a a moisture o ° content versus depth p r o f i l e during such a r a i n f a l l event w i l l show moisture contents that increase at the surface as a function of time. At some point i n time ( t ^ i n Figure 2-3a) the surface becomes saturated, and an inverted zone of saturation begins to propagate downward into the s o i l . I t i s at t h i s time (Figure 2-3c) that the i n f i l t r a t i o n rate drops below the r a i n f a l l rate and overland flow i s generated. The time t ^ i s c a l l e d the ponding time. Dunne overland flow from variable source areas dominates hydrograph; subsurface stormflow less important / 4 Horton overland flow 1 1 I from partial areas Variable dominates hydrograph; Source contributions from sub- Areas surface stormflow are i less important 1 1 1 1 l \ t Subsurface stormflow may dominate hydrograph volumetrically; peaks produced by Dunne overland flow Thin soils; gentle concave footslopes; wide valley bottoms; soils of high to low permeability Topography and Soils Steep, straight or convex hillslopes; deep, very permeable soils; narrow valley bottoms Arid to subhumid climate; thin vegetation; or disturbed by man Humid climate; dense vegetation Climate, Vegetation and Land-use FIGURE 2-2. Schematic i l l u s t r a t i o n of the occurrence of various runoff processes i n r e l a t i o n to t h e i r major environmental controls (after Freeze, 1980a). oo 9 v M o i s t u r e 3.) Gon ten t e O n T i m e b) 9 d ) K 0-0 | | i i I I f 0 f l f 2 f 3 f 4 f 5 T i m e FIGURE 2-3. Moisture content versus depth p r o f i l e s f o r (a) the Horton mechanism and (b) the Dunne mechanism. Overland flow generation f o r (c) the Horton mechanism and (d) the Dunne mechanism (adapted from Freeze, 1980a). The necessary conditions for the generation of overland flow by the Horton mechanism are (1) a r a i n f a l l rate greater than the saturated hydraulic conductivity of the s o i l and (2) a r a i n f a l l duration longer than the required ponding time for a given i n i t i a l moisture profile. The Horton mechanism i s more common on upslope areas. Horton overland flow i s generated from partial areas of the hillslope where surface hydraulic conductivities are lowest. Flow path A in Figure 2-1 represent Horton overland flow. Dunne Overland Flow As a parallel to naming Horton overland flow after i t s discoverer, Freeze [1980a] referred to the second mechanisms as Dunne overland flow. The mechanism (flow path B in Figure 2-1), as described by Dunne (1978), i s illustrated in Figures 2-3b and 2-3d. In this case, p < K q , and the i n i t i a l water table i s shallow. Surface saturation occurs because of a ris i n g water table; ponding and overland flow occur at time t^ when no further s o i l moisture storage i s available. The Dunne mechanism i s more common on near-channel wetlands. Dunne overland flow i s generated from partial areas of the hillslope where water tables are shallowest. Both Horton and Dunne mechanisms lead to variable source areas that expand and contract through wet and dry periods. Subsurface Stormflow The origins of subsurface stormflow were traced to Hursh (1936) and Lowdermilk (1934) by Hewlett (1974). The flowpath labeled C in Figure 2-1 schematically illustrates the occurrence of subsurface stormflow. The impeding subsoil horizon laterally diverts i n f i l t r a t i n g water downslope. Under intense r a i n f a l l events, where the surface s o i l layer becomes saturated to some depth, water i s able to migrate through "preferred pathways" r a p i d l y enough to d e l i v e r contributions to the stream during the peak runoff period. The conditions f o r subsurface stormflow are quite r e s t r i c t i v e . The mechanism i s most l i k e l y to be operative on steep, humid, forested h i l l s l o p e s with very permeable surface s o i l s . Based upon simulations with a mathematical model, Freeze (1972b) concluded that subsurface stormflow by Darcy porous-media flow i s a f e a s i b l e mechanism only on convex h i l l s l o p e s that feed deeply i n c i s e d channels, and then only when the saturated hydraulic conductivity of the s o i l s are very large. The i n t e r e s t e d reader i s directed to the excellent paper by Beven and Germann (1982), i n which the authors review the importance of large continuous openings on water flow i n s o i l s . They use the term "macropores" to describe these openings; t h i s writer fancies the more d e s c r i p t i v e "preferred pathways" used above. The various streamflow generation mechanisms have been c l a r i f i e d over the l a s t two decades. This understanding i s the r e s u l t of both c a r e f u l observations from f i e l d experiments (e.g. Dunne, 1970; Dunne & Black, 1970a, 1970b) and the h e u r i s t i c simulations of hypothetical r e a l i t i e s with rigorous mathematical models (e.g., Freeze, 1972a, 1972b). The interested reader w i l l f i n d numerous studies, concerned with the investigation of the various mechanisms by which r a i n f a l l i s delivered from h i l l s l o p e s to stream channels, referenced i n the reviews by Freeze (1974), Dunne (1978, 1982, 1983), Pearce and McKerchar (1979) and Beven and Germann (1982). Recent f i e l d studies not reported i n the above review papers include Bonell et a l . (1981), Wheater et a l . (1982), Taylor (1982), McCaig (1983), 12 Ando et a l . (1983), and Bonell et a l . (1984). The t h e o r e t i c a l studies of Zaslavsky and S i n a i (1981) are also of i n t e r e s t . A recent Chapman conference that dealt with subsurface contributions to streamflow was reported by H a l l (1981). Modeling Techniques The c l a s s i c review a r t i c l e by Amorocho and Hart (1964) and noteworthy papers by Woolhiser (1973), Clarke (1973) and Klemes (1978) each provide l u c i d a n a l y s i s of mathematical modeling concepts as they are applied i n hydrology. I t i s t h i s author's opinion that the d e f i n i t i v e c l a s s i f i c a t i o n of modeling techniques was proposed by Clarke (1973). His scheme i s summarized i n Figure 2-4 and discussed i n the following paragraph. Clarke (1973) describes the general r a i n f a l l - r u n o f f model as: q t = f(P t_ 1»P t_2»•••;1 t_l» (l t_2'••• ; al' a2••••) + (2-!) where, p^ are input v a r i a b l e s , q f c are output variables, are system parameters, £ ' i s the r e s i d u a l error, and f i s the f u n c t i o n a l form of the model. Encoded i n t h i s r e l a t i o n s h i p i s a fundamental d i s t i n c t i o n between model elements: variables change with time; parameters remain constant. The f u n c t i o n a l form of the r e l a t i o n s h i p f can be e i t h e r conceptual or empirical. The input and output variables as well as the system parameters and r e s i d u a l error can be either stochastic or d e t e r m i n i s t i c . Clarke categorizes mathematical models into four major groups: Stochastic-conceptual, s t o c h a s t i c - e m p i r i c a l , deterministic-conceptual, and d e t e r m i n i s t i c - e m p i r i c a l . In h i s view a model i s regarded as stochastic i f any of the variables i n i t s mathematical expression are described by a p r o b a b i l i t y d i s t r i b u t i o n . A model i s termed deterministic i f a l l the HYDROLOGIC MODELS 1 1 PHYSICAL ANALOG MATHEMATICAL MODELS STOCHASTIC-CONCEPTUAL STOCHASTIC-EMPIRICAL DETERMINISTIC-CONCEPTUAL DETERMINISTIC-EMPIRICAL LINEAR IN THE SYSTEMS THEORY SENSE NON-LINEAR IN THE SYSTEMS THEORY SENSE LINEAR IN THE STATISTICAL REGRESSION SENSE NON-LINEAR IN THE STATISTICAL REGRESSION SENSE Same Sub-Classifications as Linear in the Systems Theory Sense PROBABILITY DISTRIBUTED GEOMETRICALLY DISTRIBUTED Same Sub-Classiticattons as Linear in the Statistical Regression Sense Same Classification as Stochastic -Conceptual FIGURE 2-4. Clarke's (1973) hydrologic model c l a s s i f i c a t i o n scheme. v a r i a b l e s are viewed as f r e e from random v a r i a t i o n s . Models are c a l l e d conceptual i f t h e i r f u n c t i o n a l form, f, i s derived from consideration of the physical processes, and empirical i f i t i s not. In Clarke's assessment, there are several sub-categorizations as w e l l . A model i s l i n e a r i n the systems-theory sense i f the p r i n c i p a l of superposition holds and l i n e a r i n the s t a t i s t i c a l - r e g r e s s i o n sense i f l i n e a r i n the parameters to be estimated. Clarke f u r t h e r i d e n t i f i e s three sub-categories in v o l v i n g s p a t i a l v a r i a b i l i t y of input v a r i a b l e s . These are ( 1 ) lumped models, that do not account f o r s p a t i a l d i s t r i b u t i o n , (2) p r o b a b i l i t y - d i s t r i b u t e d models, that describe s p a t i a l v a r i a b i l i t y without reference to geometrical configuration i n the measurement network, and (3) geometrically d i s t r i b u t e d models that express s p a t i a l v a r i a b i l i t y i n terms of o r i e n t a t i o n within the measurement network. Research hydrologists have been p r o l i f i c i n t h e i r production of a continuum of r a i n f a l l - r u n o f f models that embrace every niche and cranny of Figure 2-4 many times over. For t h i s study a s u i t e of three well established r a i n f a l l - r u n o f f models, of progressively increasing complexity, was selected f o r comparative analyses. The models include a regression model, a unit hydrograph model and a quasi-physically based model. The models are f u l l y described i n Chapter 4. These models are representative of underlying modeling techniques discussed by Amorocho and Hart (1964): C o r r e l a t i o n a n a l y s i s (regression model), p a r t i a l system synthesis with l i n e a r a n a l y s i s (unit hydrograph model), and system synthesis (quasi-physically based model). In terms of Clarke's model c l a s s i f i c a t i o n t h i s s u i t e of models i s c l a s s i f i e d as: Stochastic-empirical (regression model), deterministic-empirical (unit hydrograph model), and deterministic-conceptual (quasi-physically based model). The generator used to bui l d synthetic data sets for t h i s study, described i n Chapter 7, i s c l a s s i f i e d as a stochastic-conceptual model. There i s a vast l i t e r a t u r e associated with mathematical modeling of r a i n f a l l - r u n o f f r e l a t i o n s h i p s . Introductory hydrology texts that cover modeling include Dunne and Leopold (1978), Viessman et a l . (1977) and L i n s l e y et^ a!L. (1982). A more advanced analysis i s given by Eagleson (1970). An excellent summary of current knowledge can be gleaned from the following books (Overton & Meadows, 1976; Chow, 1964; Haan et a l . , 1982; Biswas, 1976; Chapman & Dunin, 1975; Geophysical Study Committee, 1982; and Anderson & Burt, 1985). The papers by Freeze (1974), Woolhiser (1982), and Dunne (1982, 1983) review modeling techniques that have been employed to simulate the various streamflow generation mechanisms described i n the previous se c t i o n . A new generation of models recently developed by Smith and Hebbert (1983), Bernier (1982, 1985), Burke and Gray (1983), and Humphrey (1981) may be of i n t e r e s t to the reader. Beven (1982a, 1982b) i s a leader i n applying kinematic wave theory to subsurface stormflow. I t i s the author's wish that the three models selected f or t h i s study be viewed as generic rather than brand-name. I t was not the purpose of t h i s t h e s i s to develop new models but to evaluate existi n g techniques. I t has, of course, been necessary to use a p a r t i c u l a r regression package, a p a r t i c u l a r unit hydrograph program and a p a r t i c u l a r form of the qua s i - p h y s i c a l l y based representation of the hydrologic cycle. However, i t was not the int e n t i o n of t h i s study to compare or assess these p a r t i c u l a r models. This writer believes that the sense of t h i s study's r e s u l t s would hold f o r any set of s p e c i f i c models of s i m i l a r type. Perhaps the most c o n t r o v e r s i a l model s e l e c t i o n i s the q u a s i - p h y s i c a l l y based model of Engman (1974). I t i s the f i r m b e l i e f of t h i s author that Engman's model was the most su i t a b l e f o r the p r a c t i c a l l e v e l of s o p h i s t i c a t i o n demanded i n t h i s study. A number of writers have described the unique properties of the model (Freeze, 1980a; Dunne, 1982; Betson & Ardis, 1978; Quimpo, 1984). Comparative studies During the past two decades there have been hundreds of r a i n f a l l - r u n o f f models described i n the l i t e r a t u r e . Renard et a l . (1982) have compiled a summary of many of the currently a v a i l a b l e models. However, there have been very few comparative studies of model e f f i c i e n c y . Of course, every model i s evaluated to some degree by i t s creator and users. These informal evaluations, however, are not always objective because they often do not preserve the independence of the data records used f o r c a l i b r a t i o n and v e r i f i c a t i o n . In f a c t , close scrutiny of many of the r e s u l t s purporting to demonstrate model accuracy shows that these r e s u l t s are often based s o l e l y on c a l i b r a t i o n s runs. In t h i s study the c a l i b r a t i o n of models i s c a r r i e d out using events taken from a p a r t i c u l a r period of record, and model e f f i c i e n c i e s are calculated on the basis of events taken from a v e r i f i c a t i o n period d i s t i n c t from the c a l i b r a t i o n period. To the best of t h i s w r i t e r ' s knowledge, t h i s study i s one of the f i r s t to attempt model comparisons across a broad s u i t e of model types. Most other comparative studies have l i m i t e d t h e i r comparisons to a set of model species taken from the same genus." For example, Burges and Lettenmaier (1977) provided a comparison of several time s e r i e s models f o r annual streamflow predictions; Singh (1976) compared unit hydrograph models derived by l i n e a r programming and l e a s t squares; Curwick and Jennings (1982) compared f i v e conceptual r a i n f a l l - r a i n o f f models of which four were conceptual and one was a recursive time seri e s algorithm; and the well-known World Meteorological Organization (WMO) study (WMO, 1976; S i t t n e r , 1976) compared 10 operational models. One study that does compare a broad su i t e of models and does so on a catchment scale s i m i l a r to t h i s i n v e s t i g a t i o n i s that of Osborn et a l . (1982). Their r e s u l t s appear to be more favorable than those of t h i s study with respect to p r e d i c t i v e e f f i c i e n c i e s ; however, i t should be noted that t h e i r study does not c a r e f u l l y d i s t i n g u i s h between c a l i b r a t i o n and v e r i f i c a t i o n periods and that there was ongoing parameter adjustment with several of the models throughout the course of the study. Chery et a l . (1979) reported the r e s u l t s of a comparison between a quasi-physically based model and a simpler systems model based on i t . Their r e s u l t s also show more favorable p r e d i c t i v e e f f i c i e n c i e s than the ones we present, but i t i s not c l e a r from the paper whether c a l i b r a t i o n was c a r r i e d out with events subsequently used for v e r i f i c a t i o n or whether a c l e a r separation was maintained. Those readers pursuing a d d i t i o n a l references dealing with comparative model (hydrologic) studies may f i n d the following papers of i n t e r e s t : (Papadakis & Preul, 1973; Heeps & Mein, 1974; Lara, 1974; Shrader et a l . , 1980; Hawley et a l . , 1980; Colon, 1985; Baker & Rogers, 1983; Pitman, 1978; Brandstetter, 1974, 1975; Beasley et a l . , 1979; Sloan & Moore, 1984; Hetrick et a l . , 1985; Nicks' et a l . , 1979a, 1979b; Loague et a l . , 1983; Lane et a l . , 1978; Dickey, et a l . , 1979; Fleming & Franz, 1971; Jawed, 1973; WMO, 1982). In a recent paper of p a r t i c u l a r i n t e r e s t (Task Committee, 1985), a committee of 31 prominent hydrologists judged the c a p a b i l i t i e s of 28 surface hydrology models. They found most models capable of providing "good" accuracy. The reasons f o r t h e i r confidence were based upon personal experience and were admittedly tempered by b e l i e f i n the model o r i g i n a t o r s . The Task Committee recognized the need for continued model intercomparison. Methods of model evaluation are presented by Nash and S u t c l i f f e (1970), Aitken (1973), McCuen and Snyder (1975) and Garrick et a l . (1978). James and Burges (1982) and P i l g r i m (1975) each present l u c i d discussion on model evaluation i n hydrology. The papers by Fox (1981), Willmott (1982) and Willmott et a l . (1985), from the atmospheric sciences l i t e r a t u r e , also proffer excellent discourse on the evaluation of model performance that i s equally pertinent to hydrologic modeling. In t h i s chapter streamflow generation concepts, modeling techniques and model comparison studies were reviewed. The information presented here serves to set the stage f o r future chapters. CHAPTER 3 C a l i b r a t i o n , V e r i f i c a t i o n , and Model E f f i c i e n c y In t h i s chapter the concepts of c a l i b r a t i o n , v e r i f i c a t i o n and model e f f i c i e n c y are reviewed. The d e f i n i t i o n s and notation presented here p e r s i s t throughout the remainder of the t h e s i s . I t i s common i n engineering hydrology to d i f f e r e n t i a t e between p r e d i c t i o n and fo r e c a s t i n g . Clarke (1973) u t i l i z e d the term " p r e d i c t i o n " to r e f e r to s u i t e s of simulated hydrographs that are to be used f o r the purposes of engineering design. The term " f o r e c a s t i n g , " on the other hand, r e f e r s to simulated hydrographs of s p e c i f i c future events to be used i n making operational decisions. These terms are used i n the same sense i n t h i s study, but the reason f o r d i s t i n c t i o n here l i e s not i n the nature of the engineering a p p l i c a t i o n but rather i n the nature of the two modes of performance assessment. In the pr e d i c t i o n mode, i t i s acceptable to judge model performance i n a s t a t i s t i c a l sense: In the forecasting mode i t i s necessary to judge model performance on an event-by-event basis. The i n i t i a l a n a l y s i s need not d i f f e r e n t i a t e between the two approaches. Consider a small catchment with one r a i n gage and one stream gage. Over a period of years, hyetographs f o r a set of r a i n f a l l events are observed P ^ t ) , P 2 ( t ) , . . . , P n ( t ) , P n + 1 ( t ) , . . . , P m ( t ) and hydrographs f o r the corresponding streamflow events are observed: Q ^ t ) , Q 2 ( t ) Q n ( t ) , Q n + 1 ( t ) , . . . , Q m ( t ) One such event i s i l l u s t r a t e d i n Figure 3-1. As i s noted there, the f u l l hyetograph P(t) may i n some cases be represented by the summary va r i a b l e s ^D' ^MX' fcMX^ a n ( * t* i e hydrograph Q(t) may i n some cases be represented by the summary variables {Q^, Qp^, tp^} where t o t a l r a i n f a l l depth [ L ] ; P j ^ maximum short duration r a i n f a l l i n t e n s i t y (120 s f o r experimental catchments reported i n Chapter 4 and 100 s f o r synthetic h i l l s l o p e s reported i n Chapter 7) [L/T]; tMX t i m e t o e n d o r PMJ[ f r o m t n e beginning of the event [ T j ; t o t a l stormflow depth calculated as the volume of flow divided by the t o t a l catchment area [ L ] ; Qpg peak stormflow rate [L /T]; t p ^ time to Qp^ from the beginning of the event [ T ] . Let us assume that the f i r s t n events [P,(t)...P ( t ) ; 1 n Q-^(t).. ,Q n(t) ] have been selected as the c a l i b r a t i o n events and that the remaining events [ P n + j ( t ) . . . P m ( t ) ; Q n + ^ ( t ) . . , Q m ( t ) ] are the v e r i f i c a t i o n events. A mathematical model for the p r e d i c t i o n of a hydrograph Q ^ j C t ) given a hyetograph P ^ ^ C t ) (i«e., f o r a design event or a future operational event that was not included i n the c a l i b r a t i o n or v e r i f i c a t i o n process) would then take one of the forms Q m + 1 ( t ) = F [ P m + 1 ( t ) ; alf a 2 , .... a n ; b ^ b 2 > b j (3-1) or [Q D, Q P K, tpKlpj+x = fnpD, PMJ, ai» a 2 V bi» b2» bn ] (3-2) 21 -J >" -J I— £ 1 2 LU < H DC 2 Q r Li- I DC co O r-CO MX J PK T P MX TIME FIGURE 3 - 1 . H y p o t h e t i c a l r a i n f a l l - r u n o f f event showing event summary v a r i a b l e s . where f represents the fu n c t i o n a l form of the model, [a-,, a2» a ] represents a vector of c a l i b r a t i o n parameters, and [b^, b2» ..., b ] represents a vector of p h y s i c a l l y based parameters. In t h i s study the regression model takes the form of (3-2), and the unit hydrograph and qua s i - p h y s i c a l l y based models take the form of (3-1). The c a l i b r a t i o n parameters a^ are determined from the n c a l i b r a t i o n events f o r which the P(t) and Q(t) are known. They are lumped, single-valued parameters, and they have no physical s i g n i f i c a n c e . The p h y s i c a l l y based parameters b^ are d i s t r i b u t e d parameters such as s o i l h y d r aulic conductivity or channel slope that can be measured or estimated at various points on the catchment. Some models such as the regression model or the unit hydrograph model have only a^ i n th e i r structure with no b^. Some models such as the quasi-physically based model have only b^ with no a^. In the l a t t e r case i t i s recognized that the p h y s i c a l l y based parameters vary as a function of pos i t i o n on the catchment. However, because only a f i n i t e number of measurement locations are possible, there i s always considerable uncertainty as to the representativeness of the measured values. In cases where the use of measured values has not produced good model e f f i c i e n c y or i n cases where measured values are not a v a i l a b l e , i t i s quite common to c a l i b r a t e the model over the n c a l i b r a t i o n events against one or more of the b^. In such cases the conceptual diffe r e n c e between the a^ and the b^ becomes a b i t fuzzy, but the d i s t i n c t i o n s t i l l has merit. Some models such as the regression model require a number of c a l i b r a t i o n events to provide an estimate of the a^. Some models such as the un i t hydrograph model or the quasi-physically based model require 23 only one c a l i b r a t i o n event to provide an estimate of the a^ and b^. For these l a t t e r models, with n c a l i b r a t i o n events, the various parameter values » a^2' a i 3 ' **"' a i n a n C * ^ i l ' ^ i 2 ' ^ i 3 ' "**' ^ i n c a n ^ e a v e r a g e c * to produce more representative values a^ and b\. Once a set of parameter values, a^ or b^, has been determined, one can then use the model i n the form of (3-1) or (3-2) to predict Q(t) for a given P ( t ) . I t i s proper to f i r s t apply the model to a v e r i f i c a t i o n period where the P(t) and Q(t) are known and to carry out some form of comparative a n a l y s i s between observed and predicted values. I t i s usual to carry out t h i s a nalysis i n terms of summary variables such as Qn, Q w , and tp^. In cases where the model i s i n the form of (3-2) these variables are the d i r e c t output; i n cases where the model i s i n the form of (3-1) these summary variables can be obtained from the f u l l Q(t). I t i s now necessary to reintroduce the d i s t i n c t i o n between prediction mode and forecasting mode. Of the two, the fore c a s t i n g mode requires the more stringent measure of model performance. Model e f f i c i e n c i e s must be based on a comparison of the observed and predicted values of the summary variables f o r i n d i v i d u a l v e r i f i c a t i o n events on an event-by-event basis. Model e f f i c i e n c i e s f o r the prediction mode, on the other hand, can be based on a comparison of the observed and predicted values of the summary var i a b l e s over the f u l l s u i t e of v e r i f i c a t i o n events or on a s t a t i s t i c a l comparison of the observed and predicted frequency d i s t r i b u t i o n s . The forecasting case w i l l be considered f i r s t . Let any one of the output summary variables be denoted by Q and l e t predicted value f o r event i ; Q. observed value f o r event i ; I Cj mean value of over the v e r i f i c a t i o n events i = n + l t o i + m. From the s u i t e of possible c r i t e r i a reviewed by James and Burges (1982) the c r i t e r i o n of Nash and S u t c l i f f e (1970) was selected, and w i l l be c a l l e d the forecasting e f f i c i e n c y : (3-3) m - 2 m - 2 m - ? -1 E f = (Z (Q.-Q) -Z (Q,-Q.) ) ' {Z (Q.-Q) 2} 1 i=n+l i=n+l i=n+l 1 I f a l l Q i = Q i f then = 1, For any r e a l i s t i c case, E^ < 1. I t i s possible f o r E^ to become negative; a negative e f f i c i e n c y i n f e r s that the model's predicted value i s worse than simply using the observed mean. Large events exert more leverage i n (3-3); thus, an e f f i c i e n c y can be biased when a large range of events are evaluated. In contrast to t h i s study's a p p l i c a t i o n of E^ to summary variables from many selected events, the Nash and S u t c l i f f e c r i t e r i o n i s most often used to evaluate the complete form of a single event or continuous simulation. Used i n t h i s form, James and Burges (1982) suggested that E^ should exceed 0.97 for model acceptance. The model e f f i c i e n c y E^ i s used i n t h i s study as the quantitative measure by which the three models are compared i n forecasting mode. In p r a c t i c e , i t would seem reasonable to set a minimum E^ that must be achieved by a c a l i b r a t e d model during v e r i f i c a t i o n before i t could be placed i n use f o r operational p r e d i c t i o n s . Alternately, the E^ can be viewed as a measure of confidence i n the ensuing predictions, which should be taken i n t o account i n the estimation of r i s k and the s e t t i n g of safety f a c t o r s . I t i s possible to turn the model back on i t s e l f to predict f o r the c a l i b r a t i o n period i = 1 to i = n, using the measured as input. The c a l i b r a t i o n e f f i c i e n c y provides a quantitative i n d i c a t i o n of the proportion of the v a r i a b i l i t y i n the that i s being explained by the ca l i b r a t e d model. The c a l c u l a t i o n of e f f i c i e n c i e s with (3-3) during c a l i b r a t i o n , though u s e f u l , does not constitute a v e r i f i c a t i o n procedure. When models are used i n prediction mode rather than i n forecasting mode, i t i s possible to consider a l e s s stringent measure of model performance. A measure that i s d i r e c t l y commensurate with the forecasting e f f i c i e n c y i s the pr e d i c t i o n e f f i c i e n c y E^ (M. E. Moss, personal communication, 1984). To c a l c u l a t e t h i s measure, one takes a l l of the observed discharge values f o r a given catchment and ranks them i n order of decreasing discharge. Then one takes a l l the predicted discharge values and ranks them independently i n the same order. Then (3-3) i s used with i defined by the ranked events rather than by the sequential events to compute the pr e d i c t i o n e f f i c i e n c y E^. For any given set of observed and predicted values, E^ >_E^; that i s , p r e d i c t i o n e f f i c i e n c i e s w i l l always be better than or equal to forecasting e f f i c i e n c i e s . I t i s also possible to assess model performance i n prediction mode on the basis of a comparison of the f u l l frequency d i s t r i b u t i o n of the predicted and observed events. In general, i t i s d i f f i c u l t to accept hypotheses at a reasonable l e v e l of confidence about the nature of the d i s t r i b u t i o n s themselves under the small-sample conditions that usually p r e v a i l . However, i t i s reasonably straightforward to te s t hypotheses about the mean and standard deviations of the observed and predicted d i s t r i b u t i o n s by using the usual s t a t i s t i c a l t e s t s f o r normal d i s t r i b u t i o n s (Haan, 1977). In an e a r l i e r report of t h i s research (Loague & Freeze, 1985) the t e s t s t a t i s t i c was used to t e s t for differences i n means of observed and predicted d i s t r i b u t i o n s , and the F t e s t s t a t i s t i c was used to t e s t for differences i n the variances. However, the s u i t a b i l i t y of these t e s t s as a framework f o r an a c c e p t a b i l i t y c r i t e r i o n i s open to question. These hypothesis t e s t s have the disturbing property that as the sample s i z e increases ( i . e . , as the number of predicted events increases), the l i k e l i h o o d of r e j e c t i n g the stated hypothesis with the same model increases (M. E. Moss, personal communication, 1984). In t h i s study, where the s i z e of the samples i s small, the recorded lengths are not uniform, the confidence l e v e l i s a r b i t r a r y , and no analysis was c a r r i e d out to develop a quantitative c r i t e r i o n f o r a c c e p t a b i l i t y , no hypothesis t e s t i n g was c a r r i e d out for quantitative comparison of model performance. Instead, summary s t a t i s t i c s (means and standard deviations) f o r two frequency d i s t r i b u t i o n s of observed and predicted summary output va r i a b l e s are reported for v i s u a l inspection. This q u a l i t a t i v e measure of model performance i n p r e d i c t i o n mode i s intended to lend support to the more rigorous E^ r e s u l t s . There are three sources of error inherent i n r a i n f a l l - r u n o f f models (Lettenmaier, 1984): model error, input error, and parameter error. Model error r e s u l t s i n the i n a b i l i t y of a r a i n f a l l - r u n o f f model to predict runoff accurately, even given the correct estimates and input. Input err o r i s the r e s u l t of errors i n the source term ( r a i n f a l l i n t h i s study). I t can a r i s e due to measurement error, juxtaposition error (between r a i n gages and storm patterns), or synchronization errors (between r a i n f a l l and streamflow gages). Parameter error has two possible connotations. For models with c a l i b r a t i o n parameters i t i s usually the r e s u l t of model parameters that are highly interdependent. In some cases, there may not be a set of unique parameter estimates that can reproduce recorded runoffs. For models with p h y s i c a l l y based parameters, parameter e r r o r r e s u l t s from our i n a b i l i t y to represent a r e a l d i s t r i b u t i o n s on the basis of point measurements. The aggregate of model, input, and parameter e r r o r s i n the c a l i b r a t i o n and v e r i f i c a t i o n periods of model assessment are c a l l e d c a l i b r a t i o n error and v e r i f i c a t i o n e r ror, r e s p e c t i v e l y . The r e l a t i v e importance of the v e r i f i c a t i o n error i s q u a n t i t a t i v e l y indicated by and E^. Unless parameters have been o v e r f i t or v e r i f i c a t i o n i s attempted outside the c a l i b r a t e d range, a model's c a l i b r a t i o n error and v e r i f i c a t i o n error should be s t a t i s t i c a l l y s i m i l a r . In the r e s u l t s that are presented i n Chapters 8 and 10 only v e r i f i c a t i o n performance i s reported. In t h i s chapter c r i t e r i a were selected to evaluate the r e l a t i v e e f f i c i e n c y of r a i n f a l l - r u n o f f models. The concepts discussed here are the foundation for the model evaluations reported i n the following chapters. CHAPTER 4 Rainfall-Runoff Models In t h i s chapter the three r a i n f a l l - r u n o f f models used i n t h i s study-are presented. Regression model The regression model comprises a set of three simple l i n e a r regression equations: Q D = a l P D + a 2 ( 4 " l a > QPK = a3 PMX + a4 ( 4" l b> fcPK = a5 tMX + a 6 ( 4 ~ l c ) The regression model requires only r a i n f a l l and streamflow records. The c a l i b r a t i o n parameters a^ through a^ are estimated from three separate regressions on the s i x summary variables f o r the n c a l i b r a t i o n events f o r each catchment. While the output of the model does not provide a f u l l hydrograph Q(t), i t can be used to produce a simple t r i a n g u l a r representation of i t . The s e l e c t i o n of these three regression equations i s based on an analysis of the f u l l matrix of simple l i n e a r regression c o e f f i c i e n t s f o r the s i x summary variables as well as an analysis of several selected multiple l i n e a r regressions on each of the data sets used i n t h i s study. Regression equations constrained through the o r i g i n , which might be considered to be conceptually more sound (with the exception of 4-1C), were also considered f o r the regression model. The e f f i c i e n c y values f o r the three selected equations were among the best of the simple l i n e a r regression p o s s i b i l i t i e s , and t h e i r form has some physical l o g i c . The regression analysis was c a r r i e d out with program TRP, a regression package av a i l a b l e through the U n i v e r s i t y of B r i t i s h Columbia computer center. Unit Hydrograph Model The l i n e a r response model i s the instantaneous unit hydrograph: t ( 4 - 2 a ) Q ( t ) = fo° U(t-A) p'(X) dX where Q'(t) stormflow hydrograph; P'(t) excess r a i n f a l l hyetograph of duration t ; U(t) instantaneous unit hydrograph. The excess r a i n f a l l hyetograph P'(t) i s r e l a t e d to the measured hyetograph by P'(t) = P(t) - $ ( 4 - 2 b ) where $ i s known as the $-index. I t i s defined as the amount of r a i n f a l l that i s retained by the catchment divided by the duration of the rainstorm. I t provides a means of repl a c i n g the time-varying i n f i l t r a t i o n f unction by an average value. Unfortunately, the index cannot be treated as a c a l i b r a t i o n parameter d i r e c t l y because i t i s a function of the properties of P(t) for each event; the value of the $-index was ca l c u l a t e d from the r a i n f a l l and runoff data f o r each event during the c a l i b r a t i o n period. A multiple l i n e a r regression was carried out to estimate the parameters ay and ag i n the equation * - a 7 P D + a8 PMX ( 4 " 2 c ) This equation was used to specify the value of $ i n the c a l i b r a t i o n and v e r i f i c a t i o n runs. The stormflow hydrograph Q ( t ) i s r e l a t e d to the measured hydrograph Q ( t ) by Q ' ( t ) = Q ( t ) - Q b ( t ) (4-2d) The baseflow Q ^ C t ) i s given by Q b ( t ) = Q q + a 9 t Q b ( t ) < Q ( t ) (4-2e) where t i s the time from the beginning of the event and Q q i s the i n i t i a l flow. This separation technique was f i r s t proposed by Hewlet and Hibbert (1967). The parameter a^ could be treated as a c a l i b r a t i o n parameter, but for the upland catchments under study where baseflow i s very small or nonexistent the value was simply s p e c i f i e d as input. The 3 -1 -2 -1 baseflow separation slope i n t h i s work was taken as 0.0006 m s km h ( a f t e r Hewlett & Hibbert, 1967). Baseflow separation i s merely a convenient f i c t i o n and there i s no phys i c a l reason why a catchment should respond as suggested by (4-2e). However, such a r b i t r a r y but consistent baseflow separation techniques are commonly employed as a preprocessor i n the f i n a l a p plication of the unit hydrograph technique. The $-index and baseflow separation mechanics are i l l u s t r a t e d i n Figure 4-1. The u n i t hydrograph analysis was c a r r i e d out with program UNIT (Morel-Seytoux et a l . , 1980), which works with a discreet form of (4-2a). I t uses a unity-constrained l e a s t squares matrix approach to c a l c u l a t e the c o e f f i c i e n t s f o r a set of l i n e a r equations used to i d e n t i f y the unit hydrograph ordinates i n the d i s c r e t e formulation. These c o e f f i c i e n t s c o n s t i t u t e the c a l i b r a t i o n parameters f o r t h i s model. The l e a s t squares de r i v a t i o n i s without a non-negativity co n s t r a i n t . The p o s s i b i l i t y of a) H Y E T O G R A P H T I M E Inf i l trat ion >-CO z LU Z < L L Z < E x c e s s Ra infa l l ( s t o r m f low) FIGURE 4-1. Preprocessing of unit hydrograph input data, (a) $-index method of c a l c u l a t i n g excess r a i n f a l l employed i n (4-2b). (b) Hewlett and Hibbert (1967) baseflow separation technique. The slope a q i s an input parameter i n (4-2e). negative u n i t hydrograph ordinates derived with t h i s method are a t t r i b u t e d to model er r o r , input error, and parameter error. C a l i b r a t i o n over n events provides n unit hydrographs. The un i t hydrograph used as the p r e d i c t i v e model i s determined by arithmetic ordinate-by-ordinate averaging of the n unit hydrographs. Quasi-Physically Based Model The quasi-physically based model used i s that of Engman (1974) and Engman and Rogowski (1974a). The model i s referred to as " p h y s i c a l l y based" because i t i s based upon the coupled p a r t i a l d i f f e r e n t i a l equations that describe the components of hydrologic response on a catchment; the "quasi-" i s appended because the model uses a n a l y t i c a l solutions of these equations as operating algorithms rather than d i r e c t numerical simulations f o r some of the components. T o t a l l y p h y s i c a l l y based models such as the simulators developed by Freeze (1972) and Beven (1977) require f a r too much data and computer time to be employed routinely by operational hydr o l o g i s t s . In any case, catchments with a s u f f i c i e n t data base could not be located to include such models i n t h i s comparative study. The quasi-physically based model used here has three components: (1) an i n f i l t r a t i o n algorithm that allows c a l c u l a t i o n of the r a i n f a l l excess by d i f f e r e n c e , (2) a kinematic-routing algorithm that t r a n s l a t e s r a i n f a l l excess generated on the overland flow planes into a l a t e r a l inflow hydrograph at the stream channel, and (3) a kinematic-routing algorithm f o r routing the streamflow hydrograph through the channel system. The model allows p a r t i a l source areas to expand and contract during a storm and i s therefore a r e l a t i v e l y r e a l i s t i c Horton runoff simulator. The i n f i l t r a t i o n algorithm i s similar to Philip's (1969) two-parameter i n f i l t r a t i o n equation, which i s based on a solution to Richard's equation for one-dimensional flow in a saturated-unsaturated system. The i n f i l t r a t i o n rate v i s given by v = 1/2 S t " 1 / 2 + K /2 (4-3a) o where S i s the sorptivity and K q i s the saturated hydraulic conductivity. The f i n a l term in (4-3a) i s based on Bouwer's (1966) representation of the hydraulic conductivity at air entry K G , which i s thought to reflect the conductivity in the advancing wetting front. The sorptivity i s computed from Parlange's (1972) approximation: fl„ , (4-3b) s = {2/"° ( 9 - e . ) D ( 6 ) der* e i where 8 i s the volumetric s o i l water content, 8 . i s the antecedent s o i l water content, and 8 q i s the porosity. The diffusivity of the s o i l , D( 8 ) , i s computed on the basis of Rogowski's (1972a, b) parametric method for estimating unsaturated-characteristic curves. The required input for his equations includes the hydraulic conductivity at air entry K G , the porosity 6 ^ , the air entry water content 8 e , the water content at a pressure head of -1.5 MPa 8., ^, and the pressure head at air entry . In this study, 6 e i s estimated as 0.9 8 q (Rogowski, 1971) and K G as 0.5 K q (Bouwer, 1966). In principle, (4-3a) and (4-3b) hold only for uniform nonswelling s o i l profiles and for i n f i l t r a t i o n events that produce incipient surface ponding, but they are widely used as i n f i l t r a t i o n algorithms under broader circumstances. With the distributed model the catchment i s divided into L areal segments. The model allows for a two-layer representation of the s o i l p r o f i l e , so that f or each segment the s o i l type must be s p e c i f i e d and the t o p s o i l and s u b s o i l depths d,p and dg must be given. For each s o i l type i t i s necessary to specify K Q , 6 Q, 6^ ,-, and i j j e . Under i d e a l circumstances these parameters would be measured for each s o i l , but i n cases where measurements are not a v a i l a b l e , they can often be estimated from e x i s t i n g l i t e r a t u r e . The overland flow algorithm i s based on a numerical solution to the kinematic form of the shallow water equations f o r one-dimensional flow across a plane. I t makes use of the Manning stage discharge r e l a t i o n s h i p f o r normal, turbulent flow. For each of the L segments of the i d e a l i z e d representation of a catchment i t i s necessary to specify the slope S Q and the Manning roughness c o e f f i c i e n t n Q . Overland flow i s not generated from a segment u n t i l the volume of r a i n f a l l excess f i r s t exceeds the depression storage capacity H. The depression capacity i s estimated on the basis of land use and slope from the curves of Hiemstra ( 1 9 6 8 ) . The channel-routing algorithm i s based on a numerical solution to the kinematic form of the shallow water equations for one-dimensional flow i n a channel. For each of the reaches i n the i d e a l i z e d representation of the channel system i t i s necessary to spec i f y the slope S C > the roughness c o e f f i c i e n t n^, and the channel c r o s s - s e c t i o n a l geometry. Late r a l inflow to the stream arises both due to overland flow and due to r a i n f a l l i n g d i r e c t l y on the stream channel. The average channel width i s s p e c i f i e d as w"c. For the quasi-physically based model, the vector (b^, b2,*"*, b^} of p h y s i c a l l y based parameters i n ( 3 - 1 ) i s given by the vector {K Q, 0 , 6, „ ^ , d T, d Q, S , n , H) f o r each areal segment and {S , n , W } f o r each linear reach. In the simulation philosophy presented in the original descriptions o f this model (Engman, 1974; Engman & Rogowski, 1974a) i t was held that calibration of this type of model would not be required. It was hoped that the selection of parameter values could be made entirely on the basis of simple f i e l d measurements or with estimates based on values available in the literature. However, experience has shown that calibration of physically based models against one or more of the parameters usually increases modelling efficiencies. In this study the hypothesis that uncalibrated modeling efficiencies could be improved by calibrating the model against i t s most sensitive parameter, the saturated hydraulic conductivity K q , i s investigated. These results are presented in Chapter 8. The use of the sorptivity concept in (4-3b) requires specification of antecedent s o i l water content 8. for each of the soils within each of l the L segments of a catchment. Where s o i l moisture data are available in the f i e l d , these data can be used to estimate 6. for calibration and verification runs. For prediction runs or for calibration and verification runs on catchments where spatial s o i l moisture data are scanty, i t may be necessary to estimate antecedent moisture contents as average values for given dates as taken from annual time series of moisture contents from several years of record at whatever measurement sites are available in the catchment. An alternative approach that has proved useful when data are insufficient to produce annual time series (Engman & Rogowski, 1974b) involves the use of seasonal moisture frequency d i s t r i b u t i o n s to estimate antecedent moisture contents. The numerical solution f o r the quas i - p h y s i c a l l y based model tends to be unstable f o r abrupt changes i n excess p r e c i p i t a t i o n . Engman (1974) a t t r i b u t e d t h i s problem to the e x p l i c i t numerical scheme used i n the model. The interested reader should see Engman (1974) f or a discussion of the numerical s o l u t i o n technique and the scheme employed to prevent the propagation of large errors. In general, mathematical s t a b i l i t y of the qua s i - p h y s i c a l l y based model i s maintained by a proper combination of the time step and space increment. For t h i s study volume balance c a l c u l a t i o n s were performed f o r experiments designed to insure that the model was neither producing or los i n g water beyond reasonable l i m i t s . For those readers who may be interes t e d i n gaining a better appreciation of the model dynamics, see Rogowski (1972a), who addresses the s e n s i t i v i t y of the i n f i l t r a t i o n algorithm to variations i n K , 8 , 3 6 o o 6^ ,-, and i>e, and Engman (1974), who t r e a t s the s e n s i t i v i t y of the model output to v a r i a t i o n s i n 6^, n Q , n^, and H. The quasi-physically based model i s schematically i l l u s t r a t e d i n Figure 4-2. Each model described i n t h i s chapter was excited with data reviewed i n Chapters 5, 6 and 7. The simulation r e s u l t s are reported i n Chapters 8, 9, 10 and 12. FIGURE 4-2. Schematic i l l u s t r a t i o n of quasi-physically based model, (a) I n f i l t r a t i n g catchment with a r e a l l y uniform r a i n f a l l (Q^(t) represents l a t e r a l i n f l o w s ) , (b) discrete overland flow planes on which kinematic wave equations are used to describe transient flow, also i l l u s t r a t e s discrete channel sections where x's become y's, S 's become S 's, and n 's become n c's (adapted from Freeze, 1982a). CHAPTER 5 Experimental Catchments In t h i s chapter the data sets from three small upland catchments, made a v a i l a b l e to the author by cooperating s c i e n t i s t s , are discussed. R-5, Chickasha, Oklahoma The f i r s t data set, obtained from the A g r i c u l t u r a l Research Service of the U.S. Department of A g r i c u l t u r e (USDA-ARS), i s f o r a small catchment 2 known as R-5. This 0.1 km catchment i s located near Chickasha, Oklahoma, i n r o l l i n g p r a i r i e grassland t e r r a i n within the Washita River Experimental Watershed and has been subjected to continuous well-managed grazing of beef c a t t l e . Hydrologic data from R-5 have been widely used i n research studies (see table 6-1). P r e c i p i t a t i o n and streamflow records i n breakpoint form, covering the 9-year period 1966-1974, were used. Locations of the r a i n gage and weir are shown i n Figure 5-la. S o i l types and surface contours for R-5 are also shown i n Figure 5-la. There i s a gentle land slope of about 3%. The average t o p s o i l and subsoil thickness f o r each s o i l (Burford, 1972) i s shown i n Table 5-1. Parameters used to construct s o i l c h a r a c t e r i s t i c curves, shown i n Table 5-1, were abstracted from Sharma and Luxmoore (1979). Note that the saturated hydraulic c o n d u c t i v i t i e s for the three s o i l s on the catchment are i d e n t i c a l . Sharma and Luxmore (1979) reported that based on standard s t a t i s t i c a l t e s t s , no difference between the s o i l s can be shown. In t h e i r i n v e s t i g a t i o n of the s p a t i a l v a r i a b i l i t y of i n f i l t r a t i o n at R-5, Sharma et_ a l . (1980) found the difference between s o i l s to be marginal for FIGURE 5-la. (a) R-5 catchment (G.A. Gander, personal communication, 1981). 40 Q Flow plane L Left flow plane segment R Right flow plane segment Sections M Main channel FIGURE 5-lb. (b) Segments used to transform catchment into overland flow planes. TABLE 5-1. Characteristic Parameters for the R-5, WE-38 and HB-6 Catchments Average Thickness Soil Soil d^, dg, Number Name Layer* m Conductivity m/s xlO -6 Porosity vo Soil Water Content at -1.5 MPa e l 5* v o r % Pressure Head at Air Entry V mm of water R-5 1 Kingfisher T 0.15 7.55 46 11 -60 S i l t loam S 0.86 5.22 44 15 -60 2 Grant T 0.28 7.55 46 11 -60 S i l t loam S 1.27 5.22 44 15 -60 3 Renfrow T 0.28 7.55 46 11 -60 S i l t loam S 1.40 5.22 44 15 -60 WE-38 38 Shelmadine T 0.25 2.82 44 15 -60 S i l t loam S 0.76 0.71 44 15 -60 54 Hartleton T 0.25 14.8 42 13 -60 S i l t loam S 0.76 9.88 42 13 -60 57 Alvira T 0.25 9.88 38 15 -60 S i l t loam S 0.76 2.82 38 15 -60 66 Leek K i l l T 0.20 14.8 37 12 -60 S i l t Loam S 0.71 24.7 37 12 -60 69 Meckensville T 0.23 9.88 39 18 -60 S i l t loam S 0.61 2.82 39 18 -60 TABLE 5-1. Characteristic Parameters for the R-5, WE-38 and HB-6 Catchments (continued) Soil Water Pressure Content Head at Soil Number Soil Name Layer* Average Thickness d T, d s, m Conductivity K , m/s , xlO" 6 Porosity e , vo? % at -1.5 MPa 61 5' vol % Air Entry V mm or water 71 Albrights T 0.23 9.88 43 13 -60 S i l t loam S 0.51 2.82 43 13 -60 73 Conyngham T 0.23 9.88 40 17 -60 S i l t loam S 0.76 9.88 40 -60 145 Berk T 0.23 24.7 47 11 -60 S i l t loam S 0.64 24.7 47 1 1 -60 149 Klimersville T 0.15 24.7 47 11 -60 S i l t loam S 0.46 24.7 47 1 1 -60 166 Calvin T 0.23 14.8 47 11 -60 S i l t loam S 0.46 24.7 47 1 1 -60 HB-6 701 Becket T 0.15 155.3 56 9 -12 Sandy loam S 0.48 155.3 59 9 -12 720 Berkshire T 0.15 155.3 56 9 -12 Sandy loam S 0.48 155.3 59 9 -12 *T, topsoil; S, subsoil. hydrologic consideration and assumed one s o i l type f o r the whole catchment. The a i r entry pressure head f o r each layer of a l l three R-5 s o i l s was estimated at 60 mm of water based on the Engman and Rogowski (1974a) estimate f o r s i m i l a r s i l t loam s o i l s . Several c r i t e r i a were used for s e l e c t i n g i n d i v i d u a l r a i n f a l l - r u n o f f events from the R-5 data base. Events were chosen only i f they (1) showed an obvious cause-and-effeet r a i n f a l l - r u n o f f r e l a t i o n s h i p , (2) did not have a snowmelt component, (3) had a r a i n f a l l duration l e s s than 24 hours, and (4) had a stormflow depth of at least 0.0025 mm. On the basis of these c r i t e r i a , 72 events were selected for a n a l y s i s . Summary variable s t a t i s t i c s f o r the selected r a i n f a l l - r u n o f f events are shown i n Table 5-2. The f i r s t 36 events were used as c a l i b r a t i o n events and the second 36 as v e r i f i c a t i o n events. The four c r i t e r i a used f o r event s e l e c t i o n undoubtedly r e s t r i c t the evaluation of r a i n f a l l - r u n o f f models over the e n t i r e s u i t e of r a i n f a l l occurrences. The observed hydrographs from R-5 e x h i b i t a flashy response (Luxmoore & Sharma, 1980). At the beginning and end of each selected event the observed stream discharge i s zero. I t appears that baseflow contributions are of l i t t l e importance for R-5, and f o r t h i s reason no baseflow separation was necessary. S c i e n t i s t s at R-5 believe that overland flow i s the dominant source of streamflow generation f o r the catchment (G. A. Gander, personal communication, 1984). Because there i s only one r a i n gage on the catchment, i t was necessary to assume that r a i n f a l l i s uniformly d i s t r i b u t e d over the entire catchment. The locati o n s of s i t e s where s o i l moisture data are available f or t o p s o i l and s u b s o i l layers from neutron-scattering measurements are shown 44 TABLE 5-2. Summary Variable S t a t i s t i c s f o r R-5, WE-38 and HB-6 Rainfall-Runof Events R-5 WE-38 HB-6 Summary Standard Standard Standard Variable Mean Deviation Mean Deviation Mean Deviation PD,mm 32.5 20.8 18.5 12.4 26.9 17.3 P j ^ , mm/hour 70.4 45.0 41.4 25.1 21.8 21.6 t j ^ , hours 1.95 3.34 1.83 2.96 5.42 4.82 Q D, mm 4.6 8.4 1.3 2.5 6.9 10.2 Qp K, l i t . / s e c . 100.2 243.8 423.1 648.2 29.2 54.7 tp£, hours 3.60 2.80 3.36 2.74 20.76 4.35 45 i n Figure 5-la. Data are av a i l a b l e at 34 s i t e s on 169 measurement dates over the same 9-year period as the r a i n f a l l - r u n o f f records. Averaged time s e r i e s of s u b s o i l and t o p s o i l water content were constructed for the catchment. Figure 5-2a shows these time s e r i e s . The antecedent moisture contents used with the quasi-physically based model f o r the R-5 events were estimated from Figure 5-2a. Depression storage was estimated as 5.3 mm, as determined from Hiemstra's (1968) plots of depression storage as a function of land slope and use. This value i s large compared to the runoff threshold for selected events. Values of Manning's n were taken as 0.35 for overland flow on grass (Woolhiser, 1975) and 0.2 f o r channel flow (Luxmoore & Sharma, 1980). The channel geometry was taken as trapezoidal a f t e r Luxmoore and Sharma (1980). The average channel width was estimated as 1.5 m. The channel slope i s constant at 2%. The segments used to transform the R-5 catchment i n t o overland flow planes for use with the quasi-physically based model are shown i n Figure 5-lb. The en t i r e subcatchment was taken as a contributing area. WE-38, Klingerstown, Pennsylvania The second data set was also obtained from USDA-ARS. I t i s for 2 WE-38, a 7.2 km catchment located i n the ridge and valley region near Klingerstown, Pennsylvania. I t i s a part of the Mahantango Creek Experimental Watershed. The major land uses are permanent pasture and c u l t i v a t e d f i e l d s . P r e c i p i t a t i o n and streamflow records i n breakpoint form, covering the 8-year period 1968-1975, were used i n t h i s study. The loc a t i o n s of the r a i n gages and weirs f o r WE-38 are shown i n Figure 5-3a. 47 LEGEND 38 Shelmadine Silt Loam 54 Hartleton Channery Silt Loam 57 Alvira Silt Loam 66 Leek Kill Channery Silt Loam 69 Meckesville Channery Silt Loam 71 Albrights Silt Loam 73 Conyngham Silt Loam 145 Berks Silt Channery Silt Loam 149 Klinesville Shaly Silt Loam 166 Calvin Channery Silt Loam 8 Basher Silt Loam 32 Dekalb Very Stony Sandy Loam 47 Weikert Channery Silt Loam 70 Meckesville Very Stony Loam 75 77 Laidig Gravelly Loam Laidig Very Stony Loam 132 Dekalb Extremely Stony Sandy Loam SYMBOLS • Soil Moisture • Rain Gage • Weir Intensive Study Area 100 0 100 300 600 700 900 Scale in meters Slope % A 0 B C D EF 3 3 - 8 8 - 1 5 15 - 25 25 - 60 Area: 7.2 km2 Contour Interval 100 ft. 1 foot = 0.3048 meters Datum is sea level FIGURE 5-3a. WE-38 catchment (W.J. Gburek, personal communication, 1982). FIGURE 5-3b. Segments used to transform catchment into overland flow planes (abstracted from Engman, 1974). S o i l types, land sloppes, and surface contours are also shown i n Figure 5-3a. The average t o p s o i l and sub s o i l thicknesses assumed for each s o i l are l i s t e d i n Table 5-1 (Rogowski et a l . , 1974). Parameters used to construct s o i l c h a r a c t e r i s t i c curves are also shown i n Table 5-1. A l l the WE-38 values i n Table 5-1 are from Rogowski et a l . (1974) and Engman and Rogowski (1974a), who determined them i n the inte n s i v e study area on the eastern slope of the catchment (Figure 5-3a). The same c r i t e r i a used to se l e c t i n d i v i d u a l r a i n f a l l - r u n o f f events from the R-5 data base were used for WE-38. Only records from April-November of each year were considered to avoid runoff due to snowmelt. The summary variable s t a t i s t i c s f o r the 144 events selected f o r t h i s study are l i s t e d i n Table 5-2. The f i r s t 72 events were used as c a l i b r a t i o n events and the second 72 events as v e r i f i c a t i o n events. The 1972 t r o p i c a l storm Agnes produced an extreme WE-38 event (Engman et a l . , 1974) that was not included i n t h i s analysis due to i t s duration. There was one other large storm not included f or the same reason (Gburek 3 - 1 et a l . , 1977). A baseflow separation slope, a Q i n (4-2e), of 0.004 m s _1 y h was used f o r WE-38. This value i s based on the c r i t e r i a of Hewlett and Hibbert (1967). Although WE-38 has four r a i n gages i n place, only one of them (RE37 on Figure 5-3a) was used, and r a i n f a l l was assumed to be uniformly d i s t r i b u t e d over the enti r e catchment. Streamflow hydrographs from WE-38 show rapid response i n d i c a t i v e of overland flow. However, observations reported by Rawitz et a l . (1970), Engman (1974) and Gburek et a l . (1977) suggest that overland flow generation may be li m i t e d to p a r t i a l areas that represent only a small percentage of the t o t a l watershed. These small p a r t i a l areas, co n t r o l l e d by s o i l hydraulic properties and dynamic saturated zones near stream channels, are believed to dominate storm hydrographs from the WE-38 catchment. The locations of the l i m i t e d number of s o i l moisture s i t e s a v a i l a b l e f o r t h i s study are shown i n Figure 5-3a. Antecedent s o i l moisture contents used i n the quasi-physically based model for WE-38 events were estimated from the 50% point on the s o i l water frequency d i s t r i b u t i o n (W. J . Gburek, personal communication, 1982) prepared for each s o i l . The basic assumption underlying the use of these d i s t r i b u t i o n s i s that the same (or very s i m i l a r ) d i s t r i b u t i o n s w i l l apply i n years other than those i n which the data were gathered. I t was not possible to use the time s e r i e s approach followed i n catchment R-5 because records were not s u f f i c i e n t l y complete. Henninger e_t a l . (1976) discuss surface s o i l moisture within the WE-38 catchment. Depression storage f o r WE-38 was estimated as a catchment constant of 4.1 mm, a f t e r Engman and Rogowski (1974a). Values of Manning's n of 0.35 and 0.03 f o r overland and channel flow, respectively, were assumed (Engman & Rogowski, 1974a). The channel geometry was taken as prismatic t r i a n g u l a r (W. J . Gburek, personal communication, 1982). The average channel width was estimated as 3 m. Channel slopes were abstracted from topographic contour maps. The segments used to transform the WE-38 catchment into overland flow planes compatible with the quasi-physically based model are shown i n Figure 5-3b. The flow plane geometry was taken from Engman (1974). The segment delineation i s much more subjective for WE-38 than f o r R-5. If a very large number of segments were chosen, the s u b j e c t i v i t y would be 51 reduced, but the amount of input data required to describe the catchment would be ponderous. The reaches and t r i b u t a r i e s describing the channel-routing network f o r the i d e a l i z e d WE-38 are i l l u s t r a t e d i n Figure 5- 3b. Stormflow depths simulated with the quasi-physically based model for WE-38 events were calculated as the volume of stormflow divided by the t o t a l area of the overland flow plane. Note that the en t i r e catchment was not taken as a contributing area. HB-6, West Thornton, New Hampshire The t h i r d data set used was obtained from the Forest Service (USDA-FS). I t i s for a catchment to be referred to here as HB-6. This 2 forested 0.13 km catchment i s located i n the White Mountain National Forest near West Thornton, New Hampshire (Pierce et a l . , 1970). Hydrologic studies on HB-6 are part of the ongoing Hubbard Brook Ecosystem Study (Likens et a l . , 1977). P r e c i p i t a t i o n and streamflow records i n breakpoint form, covering the 6- year period 1975-1980, were used i n t h i s study. Locations of the r a i n gage and the weir are shown i n Figure 5-4a. S o i l types and surface contours for HB-6 are also shown i n Figure 5-4a. There i s a steep slope of about 30% over most of the catchment. The average t o p s o i l and subsoil thicknesses assumed f o r each s o i l are shown i n Table 5-1 (Federer, 1979). Parameters used to construct s o i l c h a r a c t e r i s t i c curves for the HB-6 s o i l s are also shown i n Table 5-1. These values were estimated from Pierce (1967), Federer and Lash (1978a), Federer (1979), and C. A. Federer (personal communication, 1983). A l i t t e r layer averaging 0.1 m i n thickness e x i s t s across the catchment. I t was assumed to play no ro l e i n 720D ^ 0 - 2 3 0 0 -720E ,0° -2250 --2200 -- 2 1 5 0 -L E G E N O 701 Becke t Slightly Stony Sandy Loam 720 Berkshire Slightly Stony Sandy Loam S Y M B O L S • Rain Gage • Weir Slope % B 0 - 1 5 D 1 5 - 3 5 E 3 5 - 6 0 Area: 0.14 k m 2 Contour Interval: 50 ft. 1 foot = 0.3048 meters Datum is s e a level 100 50 100 200 S c a l e in meters J* 3 Note: Nest of f iberg lass res is tance units approx imate ly 700 meters southeast of weir. , 1 8 0 0 -FIGURE 5-4a. HB-6 catchment ( C A . Federer, personal communication, 1982). FIGURE 5-4b. Segments used to transform catchment into overland flow planes. the i n f i l t r a t i o n process i n the quasi-physically based model simulations reported i n Chapter 8. The same c r i t e r i a were used to se l e c t i n d i v i d u a l r a i n f a l l - r u n o f f events from the HB-6 data base as f o r R-5 and WE-38. As with WE-38, only records from April-November of each year were considered to avoid runoff due to snowmelt. The summary va r i a b l e s t a t i s t i c s f o r the 53 events selected f o r t h i s analysis are shown i n Table 5-2. The f i r s t 27 events are c a l i b r a t i o n events; the second 26 events are v e r i f i c a t i o n events. 3 -1 -1 A baseflow separation slope of 0.00007 m s h was used f o r HB-6. R a i n f a l l was assumed to be uniformly d i s t r i b u t e d over the e n t i r e catchment. The lo c a t i o n of s o i l moisture data used i n t h i s study, as obtained from f i b e r g l a s s resistance units (C. A. Federer, personal communication, 1981), i s indicated i n Figure 5-4a. Figure 5-2b shows a time s e r i e s of the HB-6 s o i l moisture data. The antecedent s o i l water content for each HB-6 event was estimated from Figure 5-2b. Depression storage was assumed to be zero for t h i s catchment due to the steep land slope and high i n f i l t r a b i l i t y . The value of Manning's n used f o r overland flow was estimated at 0.35. This value has very l i t t l e p h y s i c a l j u s t i f i c a t i o n f o r HB-6 and was simply selected to be equal to the R-5 and WE-38 value for uniformity. The Manning c o e f f i c i e n t f o r channel flow i n the cobble-and-boulder streambed was taken as 0.04. The channel geometry was assumed to be rectangular with an average channel width of 1 m (C. A. Federer, personal communication, 1983). The average channel slope i s 30%. The channel, although taken as uniform i n t h i s a n a l y s i s , i s characterized by organic debris dams (Bilby, 1982). The segments used to transform the HB-6 catchment i n t o overland flow planes compatible with the quasi-physically based model are shown i n Figure 5-4b. The en t i r e subcatchment was taken as a contributing area. Comparison of Catchments The three catchments included i n t h i s study are compared i n Table 5-3. The c l i m a t i c terms i n Table 5-3 are based upon the Thornthwaite c l a s s i f i c a t i o n . Attention should be drawn to the f a c t that there i s some uncertainty associated with the information compiled f o r each of the catchments. The land phase of the runoff process i s dominant for R-5 and HB-6. The land and channel phases together are important f o r WE-38. The dominant runoff processes are believed to be Horton overland flow for R-5, Horton and Dunne overland flow for WE-38 and subsurface stormflow for HB-6. Channel flow on WE-38 and HB-6 i s continuous; channel flow on R-5 i s i n t e r m i t t e n t . The R-5 s p a t i a l data a c q u i s i t i o n network i s the densest of the three catchments. The measurement of i n f i l t r a t i o n and s o i l moisture across the e n t i r e subcatchment makes the R-5 data base one of the f i n e s t i n North America. R a i n f a l l - r u n o f f events from each of the catchments have been simulated previously with quasi-physically based models: Luxmoore (1983) used AGTEEM (Hetric et a l . , 1982) f o r R-5, Federer and Lash (1978a) used BROOK (Federer & Lash, 1978b) for HB-6, and Engman (1974) used the same quasi - p h y s i c a l l y based model employed i n t h i s study f o r WE-38. The R-5 events are the best suited to simulation with the quasi-physically based model i n t h i s a n a l y s i s ; the HB-6 events are the lea s t well suited. 56 TABLE 5-3. Comparison of Catchments 1 Data Set Catchment name Agency Location Climate Area, km2 Land use S o i l s Saturated hydraulic conductivity, m/s x 10 Slopes Streamflow Streamflow generation mechanism R-5 USDA-ARS Washita River Experimental Watershed, Oklahoma subhumid 0.1 range s i l t loams 4.94-7.44 gentle intermittent overland flow: Horton mechanism WE-38 USDA-ARS Mahantango Experimental Watershed, Pennsylvania humid 7.2 pasture and cu l t i v a t e d s i l t loams 0.71-24.7 gentle to moderate continuous overland flow: Horton and Dunne mechanisms HB-6 USDA-FS Hubbard Brook Experimental Forest, New Hampshire humid 0.13 fo r e s t sandy loams 155.0 steep continuous subsurface stormflow The HB-6 catchment i s believed to be characterized by macropores and subsurface stormflow (C. A. Federer, personal communication, 1983). Overland flow, which i s the primary mechanism represented by the physically based model, i s probably a rare event on HB-6. The ramifications of this mismatch between model and reality are discussed in Chapter 8. The three data sets reviewed in this chapter were used to excite the suite of three models described in Chapter 4. The simulation results are reported in Chapter 8. CHAPTER 6 R-5 R e v i s i t e d : S p a t i a l V a r i a b i l i t y " . . . o l d beer cans do not make good i n f i l t r o m e t e r s ! " H. Bouwer, 1986 In t h i s chapter supplemental data from the R-5 catchment, gathered by the author f o r t h i s study, are discussed. Also presented i s a b r i e f review of s p a t i a l v a r i a b i l i t y and methods of analysis related to near-surface hydrologic processes. In 1966 the A g r i c u l t u r a l Research Service (ARS) established four experimental catchments i n the East B i t t e r Creek watershed. This work was spearheaded by Edd D. Rhoades. These grassland catchments are located approximately 11 miles northeast of Chickasha, Oklahoma and were operated continuously by ARS f o r 12 years. The small catchments are i d e n t i f i e d as R-5, R-6, R-7 and R-8 and have been u t i l i z e d i n paired catchment studies. R-5 and R-6 are native grassland pastures that have never been plowed. R-7 and R-8 were c u l t i v a t e d during the early 1900s u n t i l erosion became severe about the time of the dust bowl. The copious data c o l l e c t e d f o r R-5 have earned the catchment pre-eminence among many i n the hydrologic community. Data from R-5 have been comprehensively reported (U.S. A g r i c u l t u r a l Research Service, 1966, 1967, 1968, 1969, 1970, 1971, 1972, 1973, 1974, 1975, 1976, 1977, 1983; Jackson et a l . , 1980; Jackson et a l . , 1982; Bogard et a l . , 1978). Hydrologic data from t h i s catchment have been widely used i n research studies (Table 6-1). TABLE 6-1. Review of research studies u t i l i z i n g data from the R-5 catchment. Author and Date Content Ahuja et a l . (1984a) Ahuja et a l . (1984b) Clapp et a l . (1984) Coleman et a l . (1977) Hawley et a l . (1983) Hetrick et a l . (1985) Jackson et a l . (1981) Loague et a l . (1983) Loague (this study) Luxmoore & Sharma (1980) Luxmoore (1983) Luxmoore & Sharma (1984) Naney et a l . (1983) Nicks et a l . (1979a) Nicks et a l . (1979b) Spatial distribution of saturated hydraulic conductivity Scaling s o i l water properties and i n f i l t r a t i o n modeling Estimating spatial variability of s o i l moisture Spatial and temporal variability of s o i l moisture Surface s o i l moisture variation Comparison of two hydrologic models Soil moisture updating using remote sensing Evaluation of three rainfall-runoff modeling techniques Comparative evaluation of underlying rainfall-runoff modeling techniques, spatial variability of near surface s o i l hydraulic properties Simulation of annual water budgets and daily streamflows Simulation of i n f i l t r a t i o n and runoff Effects of s o i l variability on evapotranspiration Variability of several s o i l properties Evaluation of hydrologic models Evaluation of three chemical transport models Olness et a l . (1975) Nutrient and sediment discharge TABLE 6-1. Review of research studies u t i l i z i n g data from the R-5 catchment, (continued) Author and Date Content Olness et a l . (1980) Rhoades et a l . (1975) Ritchie et a l . (1976) Rogowski (1980) Sharma & Luxmoore (1979) Sharma & Seely (1979) Sharma et a l . (1980) Sharpley et a l . (1981) Fe r t i l i z e r nutrient losses in surface runoff Sediment yield Evapotranspiration Structural analysis of i n f i l t r a t i o n discharges Effects of s o i l variability on simulated water budgets Spatial variability of i n f i l t r a t i o n Spatial variability of i n f i l t r a t i o n Sorption of soluble phosphorus in runoff In Chapter 8 the performance of three rainfall-runoff modeling techniques on the R-5 catchment w i l l be discussed. From a distance of greater than 2000 miles this author anticipated that the R-5 data set would be among the best available data sets to excite the quasi-physically based model. The poor performance of the model that i s reported in Chapter 8 was not totally unexpected but i t was disheartening. One possible explanation for the failure of the quasi-physically based model was linked to a need for additional information describing the spatial v a r i a b i l i t y of near surface s o i l hydraulic properties on R-5. During the f a l l of 1984 physical parameters were measured across the R-5 catchment by the author, Gene A. Gander, and Emily L. Claspell. This supplemental information was amassed for the purpose of re-exciting the . quasi-physically based model with an amended R-5 data set. It was anticipated that any improved estimate of a parameter surface would also improve the model's performance. Spatial Variability The concept of spatial va r i a b i l i t y in hydrology must be viewed in terms of the scale under investigation. For a global water balance a single element may be the size of an entire continent or ocean. Near the opposite end of the hydrologic spectrum are detailed s o i l column experiments where s o i l grain size distributions are important. In this chapter the scale of concern i s a small rangeland catchment and the spatial variability of various s o i l properties is investigated at that scale. Analyses of rainfall-runoff processes with physics-based mathematical models requires considerable information. The heterogeneity of near surface hydrogeologic regimes and the irregularities of precipitation in both time and space, even at the catchment scale, weave a complex web of information that can never be f u l l y obtained or transmitted to a deterministic-conceptual rainfall-runoff model. The best that can be hoped for i s that a r e a l i s t i c sample of information w i l l be sufficient to produce acceptable model results. Knowledge of the spatial distribution of input parameters i s fundamental to the employment of any physically-based rainfall-runoff model. However, the number of samples and the configuration of the sampling networks required to determine the spatial variability of an input parameter i s never known apriori. It i s therefore necessary to characterize the spatial distributions of s o i l hydraulic properties and precipitation to define a sampling scheme that w i l l give the best estimate of distributed hydrologic parameters. In this study i t was possible to address the spatial variability of the near surface s o i l hydraulic properties but not the precipitation because the R-5 catchment had only one rain gage. There i s now a substantial literature concerned with the estimation of spatial distributions. Spatial variability and methods of analysis have been discussed by many authors (Matern, 1960; Beckett & Webster, 1971; Courtney & Nortcliff, 1977; Warrick & Nielsen, 1980; Nortcliff, 1984; Bouma, 1985; Wagenet, 1985). At least four conferences in the last decade have been devoted to the study of spatial variability (Freeze e_t a l . , 1978; Woolhiser & Morel-Seytoux, 1982; Bouma & Bell, 1983; Nielsen & Bouma, 1985). Table 6-2 l i s t s over 100 papers that address the spatial variability of near-surface parameters and variables of hydrologic interest at various scales. These investigations each employed real data to discern spatial distributions and are therefore of special interest to this study. The interested reader i s directed to the original papers for information (when reported) concerning probability density functions and scales of spatial dependence. Although this i s an imposing l i s t of papers, i t certainly i s not all-inclusive. The papers included in the l i s t reflect this author's literature search and unquestionably there are some omissions. Studies of spatial variation are also prominent in the groundwater and mining literature but these are not listed on Table 6-2. The hydrologic consequences of the spatial variability of s o i l properties have been investigated with stochastic techniques by many authors. Table 6-3 l i s t s some of these studies. The international popularity of the R-5 catchment i s illustrated by the number of papers list e d in Table 6-1 which are also found in either Table 6-2 or Table 6-3. Many of the studies listed in Table 6-3 u t i l i z e scaling factors to describe s o i l variability. Tillotson and Nielsen (1984) recently reviewed scale factors in s o i l science. In the remaining studies s o i l heterogeneity i s described by assuming that a random variable (such as the saturated hydraulic conductivity) has a particular probability density function. The effects of spatial variations in precipitation on runoff have been addressed by many authors including Beven and Hornberger (1982), TABLE 6-2. Review of the s p a t i a l v a r i a b i l i t y l i t e r a t u r e 64 References Parameters/ Variables* Measured** Method of Anal y s i s * * * Achouri & G i f f o r d , 1984 Ahuja et a l . , 1984a Ahuja et a l . , 1984b Ahuja et a l . , 1985) Naney et a l . , 1983 Green et a l . , 1982 Sharma et a l . , 1980 T Pb 8» C T G C S C S A l j i b u r y & Evans, 1961 Amegee & Cuenca, 1983 e ( ^ ) pb ET Anderson et j a l . , 1984 DO Andrew & Stearns, 1963 T e c> ) Babalola, 1978 PI B a l l & Williams, 1968 Baker, 1978 I B e l l et a l . , 1980 TABLE 6-2. Review of the s p a t i a l v a r i a b i l i t y l i t e r a t u r e (continued) References Parameters/ Va r i a b l e s * Measured** Method of Analysis*** Biggar & Nielsen, 1976 SD B i l o n i c k , 1984 T G Bressler & Green, 1982 Green et a l . , 1982 Bresler jat a l . , 1984 Russo & Bresler, 1980a C EC C G Burgess jet a l . , 1981 Burgess & Webster, 1980b Burgess & Webster, 1980a C z Burgess & Webster, 1980b Burgess & Webster, 1980a G Burrough et jajl., 1985 Byers & Stephens, 1983 PR C T G Campbell, 1977 T pH W Campbell, 1978 T pH C G Carter & Pearsen, 1985 pH C C G Carvallo jet a l . , 1976 I TABLE 6-2. Review of the s p a t i a l v a r i a b i l i t y l i t e r a t u r e (continued) Parameters/ V a r i a b l e s * Method of References Measured** Analysis*** Cassel, 1983 Pfa C e PR % K s Cassel & Bauer, Pfe C 1 9 7 5 eo jo Chameu, 1982 6 C Pb G Chu & Bras, 1982 P G Cipra et a l . , 1972 C C OM pH Clapp et a l . , 1983 6 C S Coelho, 1974 T P, s Coleman et al., 9 1977 Delhomme & De l f i n e r , P 1973 Devaurs, 1983 I T pb e Downes & Bechwith, pH 1951 Duffy et a l . , 1981 I b e (ij/ ) K TABLE 6-2. Review of the s p a t i a l v a r i a b i l i t y l i t e r a t u r e (continued) References Parameters/ Variables* Measured** Method of Analysis*** Edmonds et a l . , 1982 Egbert & Lettenmaier, 1985 T G Eynon & Switzer, 1983 Folorunso & Rolston, 1984a e c c T Folorunso & Rolston, 1984b C T Gajem et a l . , 1981 6 (ijj ) pH EC C T G 6 SA Gelhar et j a l . , 1983 IRQ I T EC C C T G Grah, 1983 C G Green et a l . , 1982 I e eojo TABLE 6-2. Review of the spatial variability literature (continued) References Parameters/ Variables* Measured** Method o Analysis Greminger et a l . , 1985 Grieve et a l . , 1984 Guma'a, 1978 6 T e ) c T Hammond e_t ed., 1958 Hajrasuliha et a l . , 1980 Hawley et a l . , 1983 Hendrick & Comer, 1970 H i l l s & Reynolds, 1969 Hjelmfelt & Burwell, 1984 Huff, 1955 Huff, 1970 Huff & Shipp, 1968 Huff & Shipp, 1969 Hutchinson, 1970 Jackson, 1969 C 6 PH EC e p p p p p p p c G C C C C C C TABLE 6-2. Review of the s p a t i a l v a r i a b i l i t y l i t e r a t u r e (continued) Parameters/ Variables* Method of References Measured** Analysis*** Kachanoski et a l . , z C 1985a p b T Kachanoski et a l . , z C 1985b P b T Kachanoski et a l . , p, C 1985c z T C K e i s l i n g , et a l . , 6 ) C 1 9 7 7 Pb T I Khan & N o r t c l i f f , C C 1982 Kinniburgh & Beckett, C C 1983 G Lanjon & H a l l , pH T 1981 C z T Lauren, 1984 K C G Lims, 1985 Q C Luxmoore et a l . , I C 1981 G Merzougui, 1982 I C T G McBratney & Webster, T C 1981 W T C G pH TABLE 6-2. Review of the s p a t i a l v a r i a b i l i t y l i t e r a t u r e (continued) Parameters/ Va r i a b l e s * Method of References Measured** Analysis*** McBratney & Webster, T C 1983a G McBratney & Webster, pH C 1983b G G z Naney et a l . , 1983 K s C 6 ) Nielsen et a l . , I C 1973 eojo T pb P r i c e & Bauer, 1984 e C pb 9o T OM Rao et a l . , 1983 T C P K s $ 6(I/J) Raupach, 1951a pH C C Raupach, 1951b C C pH H CD Reynolds, 1984 p C C TABLE 6-2. Review of the s p a t i a l v a r i a b i l i t y l i t e r a t u r e (continued) References Parameters/ Variables* Measured** Method of Anal y s i s * * * Rogowski, 1972c Rogowski, 1980 Russo, 1983 Pb e (ty ) K I ty C G C G Russo, 1984b Russo, 1984c Russo & Bressler, 1980a Russo & Bressler, 1980b Russo & Bresler, 1981a Russo & Bresler, 1982 Saddiq et a l . , 1985 ty F I e ty I b ty e T e (ty) Russo & Bressler, 1980a Russo & Bressler, 1980a Russo & Bressler, 1980a ty C G C G C T C G C T G Sharma & Seely, 1979 Sharma et a l . , 1980 TABLE 6-2. Review of the spatial variability literature (continued) References Parameters/ Variables* Measured** Method of Analysis*** Sharma et a l . , 1980 C S Sharma et a l . , 1983 I K C G Simmons e_t a l . , 1979 e i c s Singh et a l . , 1985 Sisson & Wierenga, 1981 C I C T Smith, 1981 Soulie, 1984 s a P d PR C T C G Springer & Gifford, 1980 Stockton & Warrick, 1971 Tabor et a l . , 1985a 6 010 Pb c G Tabor et a l . , 1985b pH T C EC C G Taylor et a l . , 1971 IH TABLE 6-2. Review of the spatial variability literature (continued) References Parameters/ Variables* Measured** Method of Analysis*** Ten Berge et al_., 1983 Ti s d a l l , 1951 Towner, 1968 Trickier, 1981 Uehara £t a l . , 1984 Vachaud et a l . , 1984 Van de Pol et ad., 1977 Vauclin et a l . , 1982 Vauclin et a l . , 1983 Vieira et a l . , 1981 Vieira et a l . , 1983 6 (ij> ) Pb T e °c TIE e i e i pH c T e SD 6 Or T e eojo c G C C G C C C T G C G C T G C TABLE 6-2. Review of the spatial variability literature (continued) References Parameters/ Variables* Measured** Method of Analysis*** Vieira & Hatfield, 1984 C T G Villeneuve elt a l . , 1979 Wagenet, 1981 Wagenet & Jurinak, 1978 T I 6 (ty ) Pb EC C G Wagenet & Rao, 1983 Walker & Brown, 1983 I 9 ty IH C S Warrick et a l . , 1977a Coelho, 1974 Keisling et a l . , 1977 Nielsen et a l . , 1973 Wauchope et a l . , 1977 IH Webster, 1973 Webster & Wong, 1969 C T Webster, 1977 Kb EC pH C C T Webster, 1978 Webster & Wong, 1969 Webster & Cuanalo, 1975 C T 75 TABLE 6-2. Review of the spatial variability literature (continued) References Parameters/ Variables* Measured** Method of Analysis*** Webster & Wong, 1969 z T G Webster & Cuanalo, 1975 T pH G z C C T Webster & Burges, 1980 ER Webster & Nortcliff, 1984 Khan & Nortcliff, 1982 Whittle, 1954 Wollum & Cassel, 1984 F V C G Xu & Webster, 1984 0M C T pH C G Yeh et a l . , 1984 ty C T G Yost et al.,1982a C P C G Yost et a l . , 1982b Yost et a l . , 1982a C G Zawadzki, 1973 Zirschky jelt _al., 1985 PPI HW C G TABLE 6-2. Review of the spatial v a r i a b i l i t y literature (continued) Key Measured Parameters/Variables: c s o i l chemistry °c s o i l temperature DO dissolved oxygen Measured Parameters/Variables: ER electrical r e s i s t i v i t y ET evapotranspiration EC ele c t r i c a l conductivity F agricultural crop G stones/gravel HW hazardous waster 1 infiltration/drainage IH insecticide/herbicide IRQ irrigation return flow water quality K s saturated hydraulic conductivity OM s o i l organic matter P precipitation , P d s o i l particle diameter P c precipitation chemistry PR s o i l penetration resistance PI s o i l plastic index pH acidity Q discharge SA particle surface area TABLE 6-2. Review of the spatial variability literature (continued) Key (continued) Measured Parameters/Variables: SD solute displacement T s o i l texture TIE thermal infrared emission V vegetation W s o i l color z s o i l thickness 6 o s o i l porosity 6 s o i l moisture 6 (ij, ) s o i l moisture retention s o i l water tension a s o i l compaction pb s o i l bulk density PPI radar images of precipitation * For most references the listed data were measured directly. In a few cases data from an unrelated study was transformed into the parameters/variables shown. ** In many papers data were recorded at various depths and/or through time. *** Methods of Analysis: C Classical st a t i s t i c s S Scaling theory T Time series analysis Spectral analysis TABLE 6-2. Review of the spatial variability literature (continued) Key (continued) *** Methods of Analysis (continued): G Geostatistics Analysis i s often performed on transformed data. As an example, Sharma et a l . (1980) describe the spatial variability of S for the R-5 catchment. Sorptivity values were determined from i n f i l t r a t i o n measurements based on Philip's two parameter equation. Table 6-3. Review of stochastic-conceptual spatial variability studies Hydrologic Phenomena Author & Date Simulated* ** Ahuja et a l . , 1984b i n f i l t r a t i o n Bresler et a l . , 1979 solute transport Bresler & Dagan, 1979 solute transport Bresler et a l . , 1981 crop yield Bresler et a l . , 1982 crop yield Bresler et a l . , 1983 water quality Bresler & Dagan, 1983a i n f i l t r a t i o n Bresler & Dagan, 1983b solute transport Clapp et a l . , 1983 s o i l moisture Cundy, 1982 i n f i l t r a t i o n Dagan & Bresler, 1979a solute transport Dagan & Bresler, 1983 i n f i l t r a t i o n Feinerman et a l . , 1985 crop yield Freeze, 1980a rainfall-runoff processes Gelhar et a l . , 1983 irrigation return flow water quality Gurovich & Ramos, 1985 crop yield Koch, 1985 runoff Luxmoore, 1983 runoff Luxmoore & Sharma, 1980 water budgets Luxmoore & Sharma, 1984 evapotranspiration Mailer & Sharma, 1981 i n f i l t r a t i o n Moore & Clarke, 1981 runoff 80 Table 6-3. Review of stochastic-conceptual spatial v a r i a b i l i t y studies (continued) Author & Date Hydrologic Phenomena Simulated* ** Peck et a l . , 1977 Persaud, 1985 Rao et a l . , 1977 Russo & Bresler, 1981b Sharma & Seely, 1979 Sharma & Luxmoore, 1979 Smith & Hebbert, 1979 Warrick et a l . , 1977b Warrick & Amoozegar-Fard 1979 Warrick & Gardner, 1983 Yeh et a l . , 1985a Yeh et a l . , 1985b Yeh et a l . , 1985c water budgets solute transport i n f i l t r a t i o n solute transport i n f i l t r a t i o n water balances i n f i l t r a t i o n i n f i l t r a t i o n i n f i l t r a t i o n crop yield i n f i l t r a t i o n i n f i l t r a t i o n i n f i l t r a t i o n * In many of these studies more than one hydrologic phenomenon was simulated, eg., Bresler & Dagan, 1983b combine i n f i l t r a t i o n and solute transport investigation. ** The rigor of i n f i l t r a t i o n analyses in these papers range from Richard's equation for transient flow through an unsaturated porous medium to various analytical and empirical approximations. Dawdy and Bergmann (1960), Troutman (1983), and Wilson et a l . (1979). The spatial variability of precipitation has also received considerable attention related to the design of effective r a i n f a l l gaging networks (eg. Eagleson, 1967; Rodriguez-Iturbe & Mejia, 1974; Shih, 1982). Methods of Analyzing Spatial Structure Matheron (1971) coined the term "regionalized variable" to describe variables that can be characterized from a certain number of measurements which identify spatial structure. The so i l properties measured across the R-5 catchment for this study are considered regionalized variables. A random variable i s any numerical quantity whose value i s determined by the outcome of a random experiment (Hadley, 1969). Every hydrologic variable can be taken as either a discrete or continuous random variable (Haan, 1977). There are many excellent references dealing with the interpretation and modeling of spatially distributed random hydrologic variables that were not mentioned in the previous section (Agterberg, 1974; Bartlett, 1975; Box & Jenkins, 1976; Bras & Rodriguez-Iturbe, 1985; Clark, 1979; C l i f f & Ord, 1973, 1981; C l i f f et a l . , 1975; David, 1977; Davis, 1973; Delhomme, 1978; Gutjahr, 1985; Sharma, 1983; Jenkins & Watts, 1968; Journel & Huijbregts, 1978; Matheron, 1971; Olea, 1975, 1977; Rendu, 1978; Ripley, 1981; Vanmarcke, 1983; Nielsen et a l . , 1983; Webster & Burgess, 1983; Webster, 1984, 1985). Alldredge and Alldredge (1978) have compiled an extensive bibliography of the early geostatistical literature. A brief review of the spatial analysis techniques proposed for this study now follows. Autocorrelation Autocorrelation i s the process of self —comparison that expresses the linear correlation between the members of a spatial series and other members of the same series separated by a fixed interval of space (Davis, 1973). A space series i s defined here as a 1-D vector of values on a grid (Zp Z^, ... Z^ . . . ) . The spatial dependency i s characterized by the autocorrelation function. The one-dimensional autocorrelation function i s defined by: PZ(h) - C 0 V ( Z i ' Zi+h> (6-1) VAR Z. 1 where h i s the lag, Z^ i s the value of Z at the i * " * 1 position, and Z^^ i s the value of Z at the (i+h) position. The graphical representation of (6-1) i s called the autocorrelogram. A hypothetical autocorrelogram i s shown in Figure 6-la. The autocorrelogram has a value of +1.0 at 0 lag, has a range from +1.0 to -1.0 and i s dimensionless. If Z^ i s strongly dependent upon i t s neighbouring values then the autocorrelogram w i l l decay slowly. If spatial dependence i s weak the autocorrelogram w i l l decay rapidly. Webster and Cuanalo (1973) suggest sampling at intervals no smaller than one-half the distance from the origin to the point where the autocorrelogram flattens after a steady decay. If there are two space series measured on the same grid, the one-dimensional cross-correlation function i s defined by: a) FIGURE 6-1. (a) Hypothetical autocorrelograms. (b) Hypothetica semi-variogram showing characteristic features. * Pv „(h) X,Y' COV(X. s Y . + h ) (VAR X. ' VAR Y.) 1 1 (6-2) where the two spatial series are represented by X and Y. Semi-variogram The semi-variogram expresses the distribution of variance as a function of separation between two points. The estimator for the one-dimensional semi-variogram for a stationary f i e l d i s given by: * 1 N ( h ) 2 (6-3) where N(h) i s the number of experimental pairs separated by the vector h. The semi-variogram dimensions are those of the measured variable squared. A hypothetical semi-variogram i s illustrated in Figure 6-lb. By definition y (h) = 0 when h = 0. However, a "nugget" i s often found in an experimental semi-variogram. The discontinuity at the origin i s due either to spatial variability at a scale smaller than the spacing of the data, or to measurement error (Delhomme, 1978). The semi-variogram may reach a limiting value called the " s i l l " that i s equal to the variance of the data. The distance between the origin and the point at which the s i l l i s reached i s called the "range". Observations separated by distances greater than the range are not correlated. If measurements are totally without correlation the experimental semi-variogram w i l l exhibit a pure nugget effect. To use an experimental semi-variogram to describe spatial structure i t i s necessary to f i t a theoretical model to i t . The most commonly used semi-variogram models in the hydrosciences are discussed by Delhomme (1978). Directional and two-dimensional average semi-variograms were calculated for the R-5 data set with the computer program RANDOM.IR provided to the author by Gary H. Giroux. Kriging Kriging i s a method of interpolation and spatial averaging with sparse data from a random f i e l d . The method was named for Daniel Krige by his friend and colleague, Georges Matheron, who details the development of his "theory of regionalized variables" in a series of books and papers (eg. Matheron, 1963; 1971; 1973). Kriging gives the best linear unbiased estimate for the surface of a random variable. In addition to the unbiased estimation of a surface, Kriging provides an estimation variance. The square root of this variance i s the Kriging standard deviation which i s one measure of the precision of the interpolation. The point Kriging estimation for a random f i e l d i s given by: Z*(X ) = Z A. Z(X.) ( 6 ~ 4 ) x=l where Z*(X ) i s the estimated value at the X location; the Z(X.) are the measured values at X^, i = 1,2,..., n where X^ represents the pair of X-Y coordinates for each measurement location; and \ are weights that must be unbiased and optimal such that E { Z * ( X Q ) - Z ( X Q ) } = 0 ( 6 - 5 ) and E { [ Z * ( X Q ) - Z ( X q ) ] 2 } a minimum ( 6 - 6 ) Equation ( 6 - 6 ) i s known as the Kriging or estimation variance. When there i s a total lack of structure in the data the linear-interpolation technique of Kriging cannot be used. This observation i s quite pertinent to this study as w i l l be shown in Chapter 9. The geostatistical software used for this study includes Kriging with generalized covariances. The theory was developed by Matheron ( 1 9 7 3 ) and expanded by Delfiner (1975) and Delhomme ( 1 9 7 8 ) . The method i s also described by Davis and David ( 1 9 7 8 ) , Hughes and Lettenraaier ( 1 9 8 1 ) and Bras and Rodriguez-Iturbe ( 1 9 8 5 ) . Kitanidis ( 1 9 8 3 , 1985) in an important study has evaluated various parameter estimation techniques for generalized covariance functions. Computer routines were made available to the author, while a vis i t i n g graduate student at the University of Washington, by Wendy A. Rice of Battelle Pacific Northwest Laboratories. These routines were modified and adapted to the UBC computer by the author. The routines were originally developed by James P. Hughes ( 1 9 8 0 ) , based on Delfiner ( 1 9 7 5 ) , and subsequently modified by Rice ( 1 9 8 2 ) . The interested reader i s directed to the aforementioned references for a more complete development of geostatistical theory. Data Collection The purpose for measuring near surface s o i l hydraulic properties across the R-5 catchment in this study was to determine their spatial structure. The hydrologic parameter of principal interest in the f i e l d study was the steady state i n f i l t r a t i o n rate. Optimal sampling strategies and network design c r i t e r i a are of considerable interest to hydrologists (eg. Langbein, 1979; Bras, 1978; Bras & Rodriguez-Iturbe, 1976b; Bresler & Green, 1982; Hughes & Lettenmaier, 1981; O'Connell, 1982; Olea, 1984; Rouhani, 1985; Rodriguez-Iturbe & Mejia, 1974; Russo, 1984a; Russo & Bressler, 1982; Virdee & Kottogoda, 1984). A series of papers (Burgess & Webster, 1980a, b; Webster & Burgess, 1980; Burgess et a l . , 1981; McBratney & Webster, 1983a) present a sampling strategy related in s p i r i t to this study. The recent papers by Burgess & Webster (1984a, b) discuss boundary spacing distributions and risk functions for optimal s o i l sampling strategies. Infiltration I n f i l t r a t i o n i s defined as the entry of water from the surface into the s o i l profile (Skaggs & Khaleel, 1982). Excellent reviews of the i n f i l t r a t i o n processes have been presented by H i l l e l (1980), Morel-Seytoux (1973) and Philip (1969). A recent conference (ASAE, 1983) was devoted entirely to theoretical and experimental advances in i n f i l t r a t i o n . Figure 6-2 shows a hypothetical i n f i l t r a t i o n curve. When i n f i l t r a t i o n rates are measured for a sufficiently long time a constant rate i s approached. This "steady-state i n f i l t r a t i o n rate" (v ) i s often 88 FIGURE 6-2. Hypothetical i n f i l t r a t i o n curve. assumed to be equal to the saturated hydraulic conductivity (KQ)» but i s actually somewhat less due to entrapped a i r . Miller and Bresler (1977) coined the.term "satiated hydraulic conductivity" (K ) to identify this s reduced conductivity. S o i l properties that control i n f i l t r a t i o n rates include: (a) particle size distribution, (b) porosity, (c) layering (homogeneity), (d) colloid content, (e) colloid swelling, (f) salt content, (g) organic matter, (h) shrinkage cracks, (i) root and animal activity, (j) structure, (k) temperature, (1) surface cover, (m) surface slope, and (n) surface conditions (eg., raindrop impact, inwash of particles and compaction). The early stages of i n f i l t r a t i o n also depend upon the i n i t i a l s o i l water content ( 0 ^ ) . For i n i t i a l l y dry soils i n f i l t r a t i o n rates are higher but i f i n f i l t r a t i o n i s continued indefinitely the rate w i l l eventually approach K regardless of 0.. I n f i l t r a t i o n rates have also been shown to be seasonally dependent (eg., Chery, 1976; Constantz, 1982), variable with r a i n f a l l intensity (Hawkins, 1982), and influenced by the temperature and chemistry of i n f i l t r a t i n g water (Nielsen & Biggar, 1982). Sampling Scheme The following three-step procedure was used to identify the spatial v a r i a b i l i t y of steady-state i n f i l t r a t i o n across R-5. 1) Measure steady-state i n f i l t r a t i o n rates along a line transect at close intervals. Determine the spatial structure of these data based upon (6-1) and (6-3). 2) Measure steady-state i n f i l t r a t i o n rates across a grid covering the entire catchment. The basic configuration of the grid i s based upon the scale of spatial dependency found in part one. Employ Kriging to estimate the steady-state i n f i l t r a t i o n structure and the Kriging standard deviation surface. 3) Measure steady-state i n f i l t r a t i o n rates at locations earmarked by the Kriging standard deviation surface as poorly interpolated. Repeat step two including the new measurement. Step three of the scenario i s repeated until the precision of the entire estimated surface i s acceptable. Using the Kriging variance as a guideline for optimal sampling design i s a relatively simple method of variance reduction analysis. By re-Kriging the entire system after each acquisition of targeted data the influence upon the estimation variance for a l l sites i s measured. The geostatistical components of the interactive network-design strategy described here were accomplished from the f i e l d on the UBC computer via the Telenet commercial network. An objective of the f i e l d portion of this study was to collect as much i n f i l t r a t i o n data as possible without redundancy. Simplicity of the experiment and equipment was of the utmost importance. To f u l f i l l these c r i t e r i a the cylinder- or flooding-type infiltrometer (Bouwer, 1986; Johnson, 1963; H i l l s , 1970) was selected. Based on the discussion of Bouwer (1986), a large diameter single-ring infiltrometer was assumed to be capable of providing the best measure of true vertical i n f i l t r a t i o n for this study. Figure 6-3 schematically illustrates one of the infiltrometers used in this study along with related components. The FIGURE 6-3. Schematic of single-ring infiltrometer and components, (a) hard-alloy alluminum sheet rolled into a ring and welded at the joint, (b) clear plastic cover used to reduce: (1) evaporation, and (2) wave action caused by the wind, (c) stand for uniformly driving rings into the ground, (d) point gage, (e) site water control stand, (f) water distribution manifold. experimental layout for five simultaneous steady-state i n f i l t r a t i o n measurements i s shown in Figure 6-4. On each selected site, the grass was trimmed down to about 2 cm and the ring uniformly driven approximately 5 cm into the ground. Before driving an infiltrometer, a narrow s l i t was cut into the ground directly beneath the cutting edge of the ring and then irrigated to reduce disturbance of the site during installation. The number of i n f i l t r a t i o n experiments was constrained by the ava i l a b i l i t y of water which had to be transported from Chickasha to R-5, and the time required to complete a f u l l set of measurements. The placement of an infiltrometer was occasionally offset due to local topography or animal activity. The steady-state i n f i l t r a t i o n measurements for the R-5 catchment were assumed to be time-invariant. The technique used in this study to measure i n f i l t r a t i o n does not enjoy the same rigor as the method described by Ahuja et_ al. (1976) but does provide a good estimate of the steady-state i n f i l t r a t i o n rate for an undisturbed site. For this study the steady-state i n f i l t r a t i o n rates were measured with five single ring infiltrometers each 1 m in diameter. During the irrigation phase of every experiment a 5 cm ponding depth was maintained for a minimum of two hours (Sharma et a l . , 1980 reported steady-state i n f i l t r a t i o n rates were attained within 60 minutes for R-5 s o i l s ) . At the beginning of every measurement, irrigation was terminated and the average i n f i l t r a t i o n rate determined for a 10 minute period. The decay of the ponded water surface during the measurement phase was determined with a point gage. After each measurement the ponded water surface was restored to the original depth and another measurement made. Infiltration Experiment S i ng l e - r i n g Inf i l t rometer Ma in F l ow Con t r o l Man i f o l d S i t e F l ow Con t r o l F a u c e t #10 Not to Scale FIGURE 6-4. Experimental layout for five simultaneous steady-state i n f i l t r a t i o n measurements. Upon completion of experiment #1, experiment #6 is started (etc.) provided enough water is available. experiments were repeated until two consecutive measurements agreed to within 5.0 x 10 ^  m/s of each other. The f i n a l i n f i l t r a t i o n rate measured was assumed to be the steady-state i n f i l t r a t i o n rate for that location. A total of 247 steady-state i n f i l t r a t i o n measurements were made between 27 September 1984 and 16 November 1984 across the R-5 catchment. Figure 6-5a shows the locations of these experiments. The measurements were completed in the following order: 1) Transect #1, made up of 50 measurements on a traverse that dissects the catchment near i t s mid-point at 5 m intervals. 2) A square pattern of 157 measurements that essentially covers most of the catchment at a 25 m square r i d spacing. 3) Transect #2, located within a single s o i l mapping unit (see Figure 5-la) and composed of 50 measurements spaced 2 m apart. Penetration resistance was also measured across the R-5 catchment at i n f i l t r a t i o n sites with a 30° 2.54 cm2 cone penetrometer on 30 November 1984. Laboratory Measurements Soil core samples were taken from each of the 247 i n f i l t r a t i o n sites across the R-5 catchment during the period November 20-25, 1984. A l l cores were obtained with a hand operated s o i l core sampler from the 2 to 9 cm interval in the surface layer. Thin walled brass cylinders, 5.4 cm in diameter and either 3 or 6 cm high, retained the s o i l samples. The cores were carefully wrapped and packed, then shipped in mass to UBC for analyses at the hydrogeology laboratory by the author. 95 O LEGEND Grid, 157 sites, AX=AY=25m Transect # 1 , 50 sites AX = 5m, hatches are grid sites Transect *2, 50 sites, AX = 2m Fence Rain Gage V-notch weir FIGURE 6-5a. R-5 catchment: locations of i n f i l t r a t i o n experiments (and s o i l samples) in this study. IS n v • a • m m m v si is 11 • V IS m • is is 34 40 26 8 • Soil moisture (G.A. Gander,. personal communication, 1982) IS Soil survey (a.A. Gander, personal communication, 1984) V Infiltration and physical properties (Sharma, et al., 1880) T Infiltration and physical properties (S.T. Chu, personal communication, 1986) FIGURE 6-5b. R-5 catchment: locations of previously gathered data. Between 23 February 1985 and 19 April 1985 the s o i l samples from the R-5 catchment were analyzed to determine the following characteristics: 1) Bulk density, p^, calculated for each core from the dry s o i l mass and core volume. 2) Porosity, 6 , determined by the relation o 6 = 1 - P b (6-7) P where i s the particle density. For this study P^ was assumed to be 2.65 for a l l R-5 soi l s (G.A. Gander, personal communication, 1984). 3) Texture, determined as the percent sand, s i l t , and clay was determined by the hydrometer method of particle-size analysis (Day, 1965). 4) Saturated hydraulic conductivity, K q, determined by the relation K 0 = _Q L _ (6-8) At H where Q i s the volume of water that passes through the sample in a known time ( t ) , A i s the cross sectional area of the sample, L i s the length of the sample, and H i s the constant head differential. For this study s o i l cores were inserted into modified Tempe pressure cells and connected to a constant-head source. 5) Soil moisture, 8 , measured by gravimetric methods. 6) Water retention, 6 0J0 , measured with standard pressure plate apparatus at pressures of 0.01, 0.03, 0.05, 0.1, 0.3 and 1.5 MPa. The generating algorithm of Mejia and Rodriguez-Iturbe (1974) i s used to generate a l l of the two-dimensional fields of autocorrelated variables and parameters. S t a t i s t i c a l parameters representing the mean, u , the standard deviation, a , and the autocorrelation parameter, a , are necessary to generate each f i e l d . The generated values are assumed to represent element properties rather than point measured properties. The spatial autocorrelation structure can be represented by the relation P(h) = e* a (7-1) where p(h) i s the autocorrelation function described in Chapter 6. SCRRS uti l i z e s a stochastic r a i n f a l l generator (Bras & Rodriguez-Iturbe, 1976a) and allows for the generation of overland flow by both the Horton and Dunne mechanisms. The i n f i l t r a t i o n algorithm i s similar to Smith's (1972) i n f i l t r a t i o n equation for time-varying r a i n f a l l . The routing of overland flow, from the element in which i t is generated, to the stream course, i s not carried out with a kinematic wave. Rather, the time delay between generation and lateral inflow to the stream i s treated as a simple stochastic travel time. For a defense of this simplification of the actual physical system, see Freeze (1980a). To apply the SCRRS i n f i l t r a t i o n algorithm three parameters are used to describe the hydraulic properties of hillslope soils. These include: saturated hydraulic conductivity, K , porosity, n, and a s o i l storage parameter B. For a description of the s o i l storage parameter, see Freeze (1980a). The r a i n f a l l generation scheme generates a storm that moves laterally across the hillslope at a velocity U. 102 To characterize the SCRRS application i t i s necessary to introduce both time-independent hillslope parameters and time-dependent hillslope and r a i n f a l l variables. The time-independent hillslope parameters are defined on a two-dimensional spatial grid (Figure 7-la) with subscripts i = 1, 2, ..., 20 in the x direction and j = 1, 2, 10 in the y direction. For example, the saturated hydraulic conductivity K at a point with coordinates (x^.y^) i s denoted by K^j. The time-independent hillslope parameters remain constant through a l l of the 60 time steps of each of the 100 simulated r a i n f a l l events. The time-dependent hillslope and r a i n f a l l variables which are calculated at each time step of each event are defined over time intervals with superscripts m = 1, 2, 60. For example, the i n f i l t r a t i o n f at a point with coordinates (x^, y .) at time t m i s denoted by f™.. The time interval between time steps i s 100 s. For any hypothetical reality SCRRS can be represented by the following six-step procedure. 1. Generate the time-independent hillslope parameters. These include the topographic elevation the overland flow travel times g T „ , and three s o i l properties: saturated hydraulic conductivity porosity n^j! a n d a s o i l storage parameter 2. Generate the external storm properties for each r a i n f a l l event. These include the time since the previous storm, T ; the storm duration D; the storm velocity U; and the total storm r a i n f a l l depth H. 3. Generate the i n i t i a l hillslope conditions for each event. These include (Figure 7-lb) the water table elevation z^j'» the unsaturated s o i l depth z. .", the i n i t i a l moisture content 6 ? . , and the i n i t i a l moisture d e f i c i t s?.. 103 ( a ) Element number 20 19 Q V 3 2 1 20 200 v r 5 ^ - X 5 (X v  Y y 10 1 / 9 • 8 7 6 5 / 5 v. / 5 ' ^ 4 3 2 1 21 41 61 81 101 121 141 161 181 D. to (1) « c '5 k_ a> O) a > o "D a> to 3 10 c <u E UJ O. <B O) o a! >. c ra c '5 t_ c 0) E a; ^ - 7 1 2 3 9 10 X = X Y = Y i FIGURE 7-1. (a) Two-dimensional spatial hillslope grid, (b) Vertical section through the hillslope (adapted from Freeze, 1980a). 4. Generate the internal r a i n f a l l intensity pattern . for each time step of each event. 5. Calculate the i n f i l t r a t i o n rate f™. and the r a i n f a l l excess i j m r. . for each time step of each event. 6. Calculate the streamflow hydrograph Qm, m = 1, 2, 60 for each event. With this procedure, 100 rainfall-runoff events were simulated for each of ten hypothetical hillslopes. The values of z. ., z. .', z. .", 9° ., s 0 K.., n.., B.., T. ., and s. . are known for each of the 200 elements of each i j i j i j i j i j hillslope. The values of p™., f™., and r™. are known for each element for each of the 60 time steps of each event. If i t i s assumed that only certain of these variables and parameters are actually measured at a few specific locations, i t w i l l be possible to compare the performance of the three rainfall-runoff models, described in Chapter 4, on this representative limited data set against the "perfect" information available for these hypothetical events. For each hillslope the f i r s t 50 events are designated for calibration of the three models the second 50 for verification. For this study SCRRS was modified to produce and store the information required to excite each of the rainfall-runoff models described in Chapter 4. Stormflow hydrographs were generated at the downstream outlet (Figure 7-la). Rainfall hyetographs were recorded for both a single rain gage assumed to have been placed on a particular hillslope element and for a five rain gage average (Figure 7-la). Hillslope parameters and variables Kf^, n „ , and 9? ^ were a l l stored. For the application of the quasi-physically based model, i t was necessary to calculate s o i l sorptivity (4-3b). The sorptivity was estimated as B. . (7-2) based on the work of Smith and Parlange (1978). Topsoil and subsoil water content at -1.5 MPa was not required as sorptivity was calculated directly with (7-2). hydrograph analysis i n i t i a l s o i l moisture contents for a single hillslope element and a five element average were stored. These i n i t i a l s o i l moisture locations correspond to the hyetograph locations (Figure 7-la). Hillslopes Ten synthetic data sets, cases A-J, were created for this study. Cases A-D were used to excite the suite of rainfall-runoff models described in Chapter A. The results of these simulations are reported in Chapter 10. Cases A-J were designed to show the effects of the hydraulic conductivity distribution on s t a t i s t i c a l parameters of streamflow. The sensitivity analysis results are presented in Chapter 11. The input parameters for Case A are listed in Table 7-1. These values (Freeze, 1980a) are also used for cases B-J except for changes in the hydraulic conductivity distribution parameters. The hillslope geometries for a l l systems are indicated in Figure 7-2. Cases A-D In order to incorporate additional information into the unit TABLE 7-1. SCRRS input parameters for case A (adapted from Freeze, 1980a) Input Parameters* Symbol Dimensions Values for Case A: [L], m; [T]. s Sources of Data Hillslope Parameters Topographic slope Topographic residual, standard deviation Topographic residual, auto-correlation parameter Log hydraulic conductivity, mean Log hydraulic conductivity, standard deviation Log hydraulic conductivity, a u t o c o r r e l a t i o n parameter Porosity, mean Porosity, standard deviation Porosity, autocorrelation parameter Soil storage parameter, mean Soil storage parameter, standard Soil storage parameter, autocorrelation parameter Surface runoff travel time, overland flow parameter a z a °B ° B 8 [L/L] [L] 0 [log L/T] [log L/T] 0 [L J/L J] 0 [L] [L] 0 [T/L] 0.08 0.05 0.30 -5.0 0.80 0.30 0.30 0.10 0.30 0.10 0.02 0.30 3.00 Mitchell & Jones (1978), Stone & Dugundji (1965) Freeze (1975), Rogowski (1972c), Willardson & Hurst (1965), Neilsen et a l . (1973), Smith (1972) Bakr (1976), Smith (1978), Delhomme (1976) Freeze (1975) Freeze (1975) Webster & Cuanalo (1975) Smith & Parlange (1978) Pilgrim (1976), Emmett (1978) TABLE 7-1. SCRRS input parameters for case A (adapted from Freeze, 1980a) (continued) Input Parameters* Symbol Values for Case A: Dimensions [L], m; [T], s Sources of Data Surface runoff travel time, channel flow parameter Surface runoff travel time residual, standard deviation Surface runoff travel time residual, autocorrelation I n i t i a l Moisture Conditions I n i t i a l moisture content, decay parameter I n i t i a l moisture content, decay parameter I n i t i a l moisture content residual, standard deviation h 0L„ [T/L] [T] 0 [1/L] [1/T] [L 3/L 3] 10.0 50.0 0.30 0.30 6.0 x 10 0.02 -7 Leopold & Maddock (1953) Nielsen et a l . (1973), Rogowski (1972) Storm and Rainfall Parameters Time between storms, mean Storm duration, mean Total storm r a i n f a l l depth, mean D^ [T] [T] [L] 3.0 x 10" 1.0 x 10~ 2.0 x 10 -2 Eagleson (1970, 1972, 1978b) Eagleson (1970, 1972, (1978b) Eagleson (1970, 1972, 1978b) TABLE 7-1. SCRRS input parameters for case A (adapted from Freeze, 1980a) (continued) Input Parameters* Symbol Dimensions Values for Case A: [L], m; [T]. Sources of Data Storm velocity, mean Storm velocity, standard deviation Depth-duration relationship Rainfall intensity pattern, y a ( t ) , a a ( t ) a [L/T] [L/T] 0 0.10 0.02 see Figure 7 from Freeze (1980a) 0.15 Bras & Rodriquez-Iturbe (1976a) * The saturated hydraulic conductivity, K was taken as log normally distributed. Therefore, a parameter Y was defined such that Y = log K . The values of Y. . were generated from the distribution N(j^ 5 Q j o^. ) • The values of _. were determined from K5? _. = exp (2.3 Yn. _.). 109 a) C a s e s A, E, F, G, H, I, J Horton G e n e r a t i o n b) C a s e B Dunne G e n e r a t i o n c) C a s e C Horton/Dunne Genera t ion r 10 < > UJ _J 111 100 d) C a s e D Horton Genera t ion D I S T A N C E (m) 100 FIGURE 7-2. Vertical cross sections showing topography and water table configurations for ten cases listed in Table 7-2 (adapted from Freeze, 1980a). 110 represent four different hillslope and water table configurations. The dominant streamflow-generation mechanism for the hillslopes are also l i s t e d in Figure 7-2. The properties of the input hydraulic conductivity distributions ( u^, a^, ) shown in Table 7-2, distinguish cases A and E-J from each other. Table 7-2 also l i s t s the maximum unsaturated s o i l depths (z„") made for each of the ten cases. Figure 7-3a shows the distribution of hydraulic conductivity values on the hillslope generated from input data for case A. The reader should be forewarned that this conductivity surface (as well as other parameter and variable surfaces) differs from the one(s) reported by Freeze (1980a) due to separate random number generations. However, the findings of Freeze as related to this study are uncorrupted. The hatched portions of Figure 7-3b show parts of the case A hillslope on which ponding occurred during event 1. Figure 7-3c i s a summary of the overland flow source areas for case A. Snapshots of event 1 from case A are displayed in Figure 7-4. The summary-variable s t a t i s t i c s for the ten sets of 100 rainfall-runoff events simulated for this study are presented in Table 7-3. The means and standard deviations of the total r a i n f a l l depth, P^, were designed to be the same for each case. Freeze (1980a) has already investigated the influence of external storm properties and r a i n f a l l distribution parameters on the s t a t i s t i c a l properties of streamflow with SCRRS. Statistics for the two other r a i n f a l l summary variables, P ^ and tMX' a r e n o t r e P o r t e ^ f° r e a ° h of the 200 hillslope elements. However, a l l three r a i n f a l l summary variables are listed in Table 7-4 for a single hillslope element and a five element average (Figure 7-la) for case A. TABLE 7-2. Comparison of SCRRS input parameters for cases A-J Hydraulic Conductivity Case Geometry (z. .") max, J m K S m/s y Y °Y A 2.25 i o - 5 -5.0 0.8 . 0.3 B 0.60 i o " 4 -4.0 0.8 0.3 C 0.60 IO"5 -5.0 0.8 0.3 D 6.20 10~4 -4.0 0.8 0.3 E 2.25 IO" 5 -5.0 0.1 0.3 F 2.25 IO" 5 -5.0 0.1 1.0 G 2.25 IO"5 -5.0 0.0 H 2.25 IO"5 -5.0 0.8 0.9 I 2.25 IO"5 -5.0 0.8 0.6 J 2.25 IO"5 -5.0 1.6 0.3 K S = exp {2.3u y} 112 FIGURE 7-3. Case A: (a) Hydraulic conductivity distribution, (b) Source areas for overland flow for event 1. (c) Source area summary for 100 event hypothetical reality. 113 FIGURE 7-4. Case A: Hydrologic conditions on the hillslope during event 1. Rainfall i s f a l l i n g on shaded areas; ponding has occurred on hatched areas. TABLE 7-3. Summary variable statistics for 100 event hypothetical realities Case mm mm Q p K, l i t . / sec. T P K ' hours M SD M SD M SD M SD A 18.5 27.0 9.5 18.0 91.0 160.0 0.42 0.19 B 18.5 27.0 11.5 22.5 98.0 190.0 0.61 0.23 C 18.5 27.0 14.0 24.5 120.0 200.0 0.47 0.19 D 18.5 27.0 2.3 5.5 29.0 63.0 0.44 0.28 E 18.5 27.0 10.0 20.5 84.0 170.0 0.42 0.28 F 18.5 27.0 10.0 21.0 87.0 170.0 0.36 0.22 G 18.5 27.0 10.0 21.0 87.0 170.0 0.31 0.23 H 18.5 27.0 9.0 17.0 75.0 140.0 0.67 0.25 I 18.5 27.0 9.0 17.0 77.0 140.0 0.64 0.31 J 18.5 27.0 9.5 16.5 96.0 150.0 0.44 0.14 M = mean SD = standard deviation TABLE 7-4. Rainfall summary variable s t a t i s t i c s at selected points for case A Pn, mm P.™, mm/hour trirv, hours „ . , D M X M A Number , of Elements M S D M S D M S D 1 17.4 25.9 128.4 126.6 0.41 0.10 5* 18.1 27.0 64.1 70.1 0.32 0.15 * Average values The grid shown in Figure 7-la also describes the overland flow planes used for the quasi-physically based model (Chapters 10 and 12). The 20 sections perpendicular to the channel (Figure 7-la), composed of ten segments each, define 20 overland flow planes. The simulation of ten synthetic data sets was reviewed in this chapter. The data sets defined by case A, and to lesser extents cases B through D, are used to excite the three rainfall-runoff models described in Chapter 4. The results of these simulations are reported in Chapter 10. A l l ten hypothetical rea l i t i e s are employed in various sensitivity analyses that are recounted in Chapter 11. 117 CHAPTER 8 Model Validation: Experimental Catchments This chapter i s closely linked with Chapters 4 and 5. The results of rainfall-runoff simulations carried out with f i e l d data are discussed. A suite of three rainfall-runoff models i s comparatively evaluated. The suite includes a regression model, a unit hydrograph model, and a quasi-physically based model. Each model was described in Chapter 4. The model performance calculations were carried out with data sets from three small upland catchments. The catchments (R-5, WE-38, HB-6) and their respective sets of data were presented in Chapter 5. A model's performance in both prediction and forecasting modes, as defined in Chapter 3, i s assessed based upon a b i l i t y to simulate observed output summary variables. The three output summary variables used throughout this study (Q n, QnT/, t ™ ) were introducted i n Chapter 2. Forecasting (E^) and prediction (Ep) efficiencies reported in this chapter were determined with (3-3). Results Table 8-1 summarizes the forecasting efficiencies E^ for the three summary output variables Q^ , Qp^, and tp^ for verification runs with the three models on the three data sets. Table 8-2 summarizes the prediction efficiencies E^ for the three summary output variables for verification runs with the three models on the three data sets. The reader could easily drown in the sea of E^ and E^ presented in Tables 8-1 and 8-2. To keep our feet on firm ground the results of one specific verification event are presented in Figure 8-1 and Table 8-3. The f u l l hydrographs are plotted as simulated for the quasi-physically TABLE 8-1. Forecasting efficiencies E^ for the three summary output variables for the three models on the three data sets Catchment Number of Events Model % QPK tPK R-5 36 regression unit hydrograph quasi-physically based 0.24 0.21 0.26 0.24 0.07 0.44 0.30 -0.23 -0.20 WE-38 72 regression unit hydrograph quasi-physically based 0.41 0.01 -0.16 0.04 -0.28 -0.05 0.32 -0.84 0.11 HB-6 26 regression unit hydrograph quasi-physically based 0.73 -0.34 -0.32 -0.03 -0.04 -0.18 0.69 -0.19 -10.11 TABLE 8-2. Prediction efficiencies E for output variables for the Ehree data sets the three models on summary the three E P Catchment Number of Events Model % QPK fcPK R-5 36 regression unit hydrograph quasi-physically based 0.40 0.50 0.48 0.31 0.11 0.47 0.70 0.65 0.66 WE-38 72 regression unit hydrograph quasi-physically based 0.57 0.61 -0.15 0.17 0.75 -0.02 0.89 0.10 0.67 HB-6 26 regression unit hydrograph quasi-physically based 0.84 0.64 -0.32 0.08 0.54 -0.18 0.76 0.21 -9.97 6TI TABLE 8-3. Summary output variables for an observed sample verification from the R-5 catchment Event 68 w A i QD ..?PK , tPK Model mm l i t . / s e c . hours Observed 10.2 220.3 2.97 Regression 5.8 84.7 3.85 Unit hydrograph 5.3 49.6 2.75 Quasi-physically based 5.6 148.9 2.87 121 based model and the unit hydrograph. For the regression model the triangular representation of the hydrograph i s based on the (Qp^, tpg) coordinate and a base width determined from our knowledge of the runoff volume as calculated from the values. This particular event i s taken from R-5. It i s neither the best nor the worst of the simulations from that catchment. It i s representative of the model performances on individual events from a l l three catchments. L i t t l e attention should be paid to the performance of any one model on this specific event. In particular, the performance of the quasi-physically based model when calibrated against one of i t s parameters (K Q) should be disregarded pending discussion later in the thesis. As a further graphical indication of the nature of the results, one can consider the scattergrams plotted in Figure 8-2, which show the predicted versus observed values for for each of the models for the 36 R-5 verification events. These plots are similar to those that arise for WE-38 and HB-6. Table 8-4 summarizes the means and standard deviations for the observed and predicted frequency distributions for the three summary output variables for the verification events on each of the three data sets. Assessments of model performance in a forecasting mode should be based on consideration of the sequential event-based analysis of Table 8-1; assessment of model performance in a prediction mode should be based on consideration of the ranked event-based analysis of Table 8-2. The time step and space increment used for the quasi-physically based model simulations for a l l three catchments included in this study were 122 REGRESSION MODEL UNIT HYDROGRAPH MODEL QUASI-PHYSICALLY-BASED MODEL QUASI-PHYSICALLY-BASED MODEL (calibrated against K 0 ) 0 13 26 39 52 65 Observed Q D (mm) FIGURE 8-2. Scattergrams of observed and predicted storm flow depths for the 36 R-5 verification events. TABLE 8-4. Verification period sta t i s t i c s : means and standard deviations for observed and predicted frequency distributions for the three summary output variables on the three data sets. Mean Standard Deviation Number of Catchment Events Model mm lit./s e c , , tPK hours QD ,.?PE . fcPK mm l i t . / s e c . hours R-5 36 WE-38 72 HB-6 26 observed 6.1 128.3 3.55 10 .2 316.9 3.02 regression 2.5 80.7 3.96 3 .6 71.6 1.79 unit hydrograph 2.8 25.8 2.05 4 .1 37.9 2.41 quasi-physically based 3.3 132.2 3.11 11 .7 523.3 4.21 observed 1.5 453.1 3.38 3 .3 813.5 2.77 regression 1.0 385.4 3.80 1 .3 104.8 2.05 unit hydrograph 1.0 566.1 1.94 1 .5 894.5 1.52 quasi-physically based 0.03 36.5 1.89 0 .1 111.9 2.82 observed 6.4 28.9 20.59 11 .2 68.5 4.56 regression 7.1 26.6 21.73 8 .6 12.5 2.75 unit hydrograph 9.9 43.0 18.78 9 .4 35.7 7.25 quasi-physically based 0.01 0.2 5.81 0 .01 0.2 4.74 124 120 s and 6.1 m, respectively. The unit hydrograph duration for a l l three catchments was 15 min. Discussion Sample Verification Run and Scattergrams Let us begin by looking at the R-5 sample verification run shown on Figure 8-1 and summarized on Table 8-3. Compared to most published diagrams of similar type, the model performance looks very poor. Perhaps this i s because most published diagrams of this type represent calibration rather than verification. The point should be made that each of these models can easily be made to f i t a single event during calibration. However, once a single parameter set was chosen on the basis of calibration over a l l calibration events, the f i t for individual events during verification suffered tremendously. The surprisingly poor predictive capabilities of the models for this particular event are not unrepresentative, although the relative performance varies from event to event. Having accepted the i n i t i a l shock of poor overall model performance on individual events, we are s t i l l able to see some positive features. The predicted hydrographs from both the unit hydrograph model and the quasi-physically based model, while quite poor with respect to peak flows, do exhibit a reasonable shape. In addition, i f we recognize that these are a l l small events on a small catchment, we see that the predictions are correct in a qualitative sense: Runoff for these events from R-5 w i l l be small. These are small consolations, perhaps, but they deserve mention. The failure of the models to perform well on individual events i s also clearly indicated by the wide scatter of the data displayed in Figure 125 8-2. A l l models appear to underestimate the runoff in the middle range of observed runoff. It i s also clear that there i s an overabundance of small events and a shortage of large events in the verification period. For the one large event the quasi-physically based model outperformed the regression and unit hydrograph models. Forecasting Efficiencies A perusal of the E^ values on Table 8-1 leads to the following comments: 1. The poor model performance reflected in the sample verification run and the scattergrams i s confirmed by the low E^ values throughout Table 8-1. Recall that for perfect efficiency, E^ = 1. Most of the positive values on Table 8-1 l i e in the range of 0.-0.5. The negative values on Table 8-1 infer that one would be better off using the.observed means than the model predictions. 2. In general, E^ values are higher for prediction of Q D and Qp^ lower for tp^. The time to peak stormflow for the unit hydrograph and quasi-physically based models was set to zero for events where no runoff was simulated. 3. In general, E^ values are highest for R-5, lower for WE-38 and lowest for HB-6. 4. In general, E^ values are highest for the regression model, lower for the unit hydrograph and quasi-physically based models. In summary, i t i s clear that for these small catchments the unit hydrograph and quasi-physically based models have l i t t l e predictive power with respect to individual events. The regression model i s only marginally better. 126 Prediction Efficiencies An item-by-item comparison of the prediction efficiencies on Table 8-2 and the forecasting efficiencies on Table 8-1 shows that E^ i s always greater than E^, as i t must be by definition. Some of the E^ values approach levels that might be considered acceptable. However, there are s t i l l many values below 0.5 and several negative values. The patterns noted in the previous section with respect to E^ also hold for E . P Verification Statistics Consideration of the verification st a t i s t i c s on Table 8-4 for the frequency distributions of observed and predicted events provides a somewhat rosier picture than has been afforded by the event-by-event analysis. Throughout Table 8-4 there i s a much closer f i t between the observed and predicted means for a l l models on a l l watersheds than there was on Table 8-3 for a representative individual event. We thus gain considerably more confidence in the application of these types of models on these types of data sets in prediction mode than we do in forecasting mode. A perusal of the s t a t i s t i c s displayed on Table 8-4 leads to the following more detailed comments: 1. The relationship between predicted and observed values i s much closer for the means than i t i s for the standard deviations. 2. There i s no obvious difference in the performance of the models with respect to the predicton of Qn, Cw and t p„. 127 3. In general, i t appears that the predicted values are closer to the observed for the regression and the unit hydrograph models than for the quasi-physically based model. 4. On HB-6 and WE-38 the regression and unit hydrograph models performed quite well, whereas the quasi-physically based model performed very poorly. On R-5 the quasi-physically based model performed quite well. In summary, the models performed better under s t a t i s t i c a l verification than under event-by-event verification. The quasi-physically based model, however, produced acceptable predictions on only one of the three watersheds. In general, the regression and unit hydrograph models provided more reliable results. Regression Model The regression model makes no use of antecedent s o i l moisture content data. In principle, i t would be possible to develop a suite of regressions for varying moisture contents, but in the current study the number of events available and the sparse s o i l moisture content data would not allow this extension. The regression model used in this study requires minimal information but i s applicable only to the catchment for which i t was developed. Unit Hydrograph Model The unit hydrograph technique i s s t i l l the most widely applied technique for online forecasting (Soorooshian, 1983). The unit hydrograph models developed for the three basins are shown in Figure 8-3. No attempt was made to modify or massage the unit hydrographs in any way. Each unit hydrograph i s an ordinate-by-ordinate averaged representation of the 128 "O C o o <D CO CO 300 - R - 5 5 O _i u. cc o t-» 4 8 12 TIME (hours) 16 **t. 20 WE-38 C 15000-o o 10000-5 o 5000 -or O r-w 10 15 TIME (hours) 20 c O o a> CO CO CD 5 o _ l u. or o r-w 200 HB-6 20 TIME (hours) FIGURE 8-3. Averaged 15-rainute unit-hydrographs developed for the three catchments. output from program UNIT (Morel-Seytoux et a l . , 1980) for the calibration events on the watershed. The absence of a non-negativity constraint i n the least squares approach i s conceptually disturbing. The oscillating trace and the negative ordinates on the unit hydrograph for HB-6 are a result of the algorithms contained in program UNIT. Eagleson et a l . (1966) discuss physically realizable unit hydrographs. There are a few comments that should be made with respect to the poor performance of the unit-hydrograph model on individual events in this study: 1. The method i s probably handicapped by the small size of the catchments studied. The nonlinearities in these small systems may not be averaged out as they are in larger catchments. It i s worth noting that the largest catchment studied, WE-38, has the most reasonable' hydrograph shape. 2. The superimposed average of unit hydrograph ordinates may result in the average unit hydrograph model peak being substantially suppressed. 3. The greatest weakness in the application of unit hydrograph models l i e s in the necessary preprocessing: the calculation of effective r a i n f a l l and the baseflow separation procedure. The $-index method used in this study to calculate effective r a i n f a l l i s a rather weak method, but data needs often rule out more sophisticated methods based on a knowledge of antecedent moisture conditions. Quasi-Physically Based Model The quasi-physically based model performed poorly in both performance assessment modes on a l l three catchments. There i s , however, a definite pattern in relative performance with respect to the three catchments. 130 Performance was best on R-5, less good on WE-38 and worst on HB-6. This ranking corresponds to the order of model suitability to the three catchments. The quasi-physically based model i s best suited to range and agricultural conditions; i t i s i l l - s u i t e d to forested environments. The wretched performance of the model on HB-6 could therefore have been anticipated (and perhaps the application of this inappropriate, albeit commonly used, type of model on HB-6 was ill-advised). It i s the poor performance of the model on WE-38, and especially on R-5, that deserves our attention. There are several reasons that can be advanced as possible explanations for the relatively poor performance of the quasi-physically based model used: 1. There i s no subsurface flow component in the quasi-physically based model, yet i t i s lik e l y that subsurface stormflow i s a runoff-generating mechanism on some parts of the catchments in some events. At HB-6 i t i s almost certainly the primary streamflow-generating mechanism. Engman (1974) has also suggested that interflow may be present at WE-38. Sharma and Luxmoore (1979) have even suggested that possibility for R-5. 2. The model calculates the runoff that arises from r a i n f a l l directly onto the stream using the estimated average water surface width to do so. There i s no consideration of channel expansion or the growth of near-stream source areas caused by surface saturation by rising water tables (Dunne, 1978; Freeze, 1980a). The Dunne mechanism may be important. 3. Inspection of individual simulations revealed that in many selected events i n which runoff was observed in the f i e l d the quasi-physically based model failed to generate any lateral inflows to the channel. There may be a threshold problem whereby the model cannot provide good predictions for events below a certain size. If so, the failure i s probably due to the method of regionalizing precipitation intensities and hydraulic conductivity values across the distributed representation of the catchment. 4. There may be a systematic bias in the r a i n f a l l measurements. 5. Estimates of antecedent s o i l water contents from averaged annual time series and/or frequency distributions may lead to modeled antecedent conditions quite different from actual conditions. However, sensitivity analyses with the model to variation in antecedent moisture contents were carried out within the observed ranges for the catchments studied, and the results do not show great sensitivity; certainly the effects are not sufficiently strong to explain the low model efficiencies. 6. Kinematic sheet flow, as simulated by the quasi-physically based model, may not be representative of the type of surface runoff generated on nonsmooth agricultural and forest s o i l s , where microrelief often leads to quick channelization. 7. The cascade overland flow geometry of the quasi-physically based model may not be representative of the catchments used in this study which have some converging slopes. 8. Interception and evapotranspiration may be important components of the hydrologic process during the events simulated in this work as well as between them. If so, the model i s not conceptually complete. 9. The i n f i l t r a t i o n algorithm i s a relatively simple one that does not f u l l y represent the complexities of nonponded i n f i l t r a t i o n into nonuniform s o i l s . 10. The time invariance assumption that underlies a l l three models does not take into account the influence of wet and dry seasons. In terms of Lettenmaier's (1984) sources of rainfall-runoff modeling error noted earlier, items 1, 2, 6, 7, 8, 9 and 10 constitute possible model error; item 4 i s input error; item 5 i s parameter error; and item 3 i s a combination of input error and parameter error. As described thus far, the application of the quasi-physically based model has not involved calibration of any kind. It could be argued that an uncalibrated model i s at some disadvantage when compared against the regression and unit hydrograph models, both of which allow calibration. In addition, when the consistently low model efficiencies for the quasi-physically based model f i r s t became apparent during this study, i t was expected that the situation would be much improved by calibration against one or more of the more sensitive input parameters. A sensitivity analy-sis was therefore carried out on the R-5 data. It indicated that the most sensitive parameter in the model i s the saturated hydraulic conductivity of the s o i l , K q . The model was then calibrated against K q . However, i t was found that the calibrated conductivities (that i s , the values that produced the highest E^) were very similar to the uncalibrated (measured) values and that calibration against K q only slightly improved the model efficiencies during verification. On Table 8-5a the Er and E values are recorded both for the f P straightforward application of the quasi-physically based model and for 133 TABLE 8-5a. Model verification efficiencies for three summary output variables for catchment R-5 with calibrated and uncalibrated quasi-physically based model Quasi-Physically Based Model Q D Q p R t p R Q D Q p K t p K Uncalibrated 0.26 0.44 -0.20 0.40 0.47 0.66 Calibrated against K 0.25 0.71 -0.24 0.36 0.72 0.65 ° o TABLE 8-5b. Verification period s t a t i s t i c s for three summary output variables for catchment R-5 with calibrated and uncalibrated quasi-physically based model Mean Standard Deviation Quasi-Physically Q D Q ^PK QD , .QPK > K Based Model mm l i t . / s e c . hours mm l i t . / s e c . hours Uncalibrated 3.3 132.2 3.11 11.7 523.3 4.21 Calibrated against K Q 2.3 94.9 3.05 10.2 433.0 4.22 Observed 6.1 128.3 3.55 10.2 316.9 3.02 the case where the model has been calibrated against K q . There i s no significant improvement in the model's performance with each calibration. This can also be seen on Table 8-5b, where the calibrated s t a t i s t i c s for the summary output variables do not show any particular improvement over the uncalibrated st a t i s t i c s when compared against the observed values. This can be graphically observed on the scattergrams on Figure 8-2. The results of the trial-and-error calibration runs are quite instructive in themselves. Recall that on R-5 the K values are assumed o to be uniform across the three s o i l types on the catchment but that different values are assumed for the topsoil and subsoil as given on Table 5-1. Figure 8-4 illustrates the a b i l i t y to f i t the quasi-physically based model to the single event (R-5, event 4) by adjusting the saturated hydraulic conductivity parameter. The conductivities associated with cases A, B, C, D, and E are given on Table 8-6. Table 8-6 also summarizes the impact on of the five calibration runs for five of 36 R-5 calibration events. It i s clear that there i s usually a K q pair that provides a good match between the predicted and observed values of Q^ , but i t i s a different K q pair for different events. The "calibrated" K q pair for a l l 36 R-5 calibration events i s the one that somehow "best" represents the whole suite of events. In this study the definition of "best" i s maximum E^. This turns out to be case D on Table 8-6 (only slightly better than case E for a l l 36 events), which provides, as an example, a reasonably good f i t for events 29 and 31 but a very poor f i t for events 3, 22 and 35 when applied to individual events. The calibrated model sometimes provides a better match to the observed values than the uncalibrated model, sometimes worse. For the particular 12 April, 1967 T I M E ( h o u r s ) FIGURE 8-4. Sensitivity of the quasi-physically based model to changes in the saturated hydraulic conductivity parameter for R-5 calibration event 4. The^conductivities associated with cases A, B, C, D, and E are given in Table 8-6. U) U l TABLE 8-6. Quasi-physically based model calibration summary for the R-5 catchment Saturated Predicted Storm Flow Depth, mm Case Conductivity* K / .^-0 o m/s x 10 Layer* Event 35 Event 22 Event 3 Event 31 Event 29 A 1.83 T 49.4 15.4 10.8 7.9 4.9 1.27 S B 7.55 T 21.58 6.3 4.1 2.9 1.6 5.22 S C 9.52 T 6.9 0.8 0.6 0.4 0.3 6.63 S D 11.57 T 3.0 0.3 0.2 0.1 0.04 8.04 S E 15.03 T 0.7 0.03 0.02 0.01 0.005 10.37 S Observ ed storm flow depth, mm 25.3 14.2 4.5 0.3 0.1 *T, topsoil; S, subsoil 137 example shown in Figure 8-1 the calibrated model performed worse than the uncalibrated model. The conclusion one i s forced to draw from this analysis i s that the quasi-physically based model does not represent reality very well; in other words, there i s considerable model error present. Summary 1. The assessment of model performance in a forecasting mode requires the calculation of model efficiencies on a sequential event-by-event basis. The results of model efficiency calculations for three rainfall-runoff models on 269 events from three small upland catchments show surprisingly poor efficiences for a l l models on a l l catchments. The unit hydrograph and quasi-physically based models have l i t t l e predictive power. The regression model i s marginally better. 2. The assessment of model performance in a prediction mode allows the use of less stringent c r i t e r i a involving the analysis of ranked events. The performance of the models under predictive mode i s better than under forecasting mode. In predictive mode the regression and unit hydrograph models showed acceptable predictive power, but the quasi-physically based model produced acceptable predictions on only one of the three catchments. 3. More favorable model performance reported by many authors for specific event-based models on specific catchments can often be traced to a failure to distinguish carefully between calibration and verification. A. It i s believed that the results of this study are generic rather than brand name and that similar results would arise with any set of similar event-based models on similar data sets. It i s f e l t by this 138 author that more comparative evaluations are needed of the predictive power of underlying technique rather than comparisons of different brand name models of the same genus. 5. The results do not point to any one modeling approach as being superior to a l l others for a l l catchments. However, the fact that simpler, less data intensive models in the form of regressions and unit hydrographs provided as good or better predictions than a more physically based model i s food for thought. 6. The poor performance of the quasi-physically based model is probably due to a combination of model error, input error and parameter error. Among the factors contributing to model error are the failure of the model to consider subsurface storm flow and near-stream source area expansion. No one physically based model i s currently available that simulates a l l the processes that are involved in streamflow generation. Even i f such a model were available, i t would be of questionable value to the operational hydrologist due to data requirements and computer limitations. Among the factors contributing to input error and parameter error are problems in regionalizing precipitation intensities and s o i l properties and specification of antecedent moisture conditions. The primary barrier to the application of physically based models l i e s in the spatial variability in r a i n f a l l and in near-surface s o i l hydraulic properties. There are problems of scale that are d i f f i c u l t to solve. 7. The performance of. the quasi-physically based model was only slightly improved by calibration against i t s most sensitive parameter, the saturated hydraulic conductivity K q . There i s usually a spatial K Q pattern that provides a good match between predicted and observed runoff . 139 variables for a single storm, but i t seldom provides the best match over a l l storms. 8. The performance of the unit hydrograph model was probably handicapped by the small size of the catchments studied; nonlinearities in the response of these small systems may not be averaged out as they are in larger catchments. However, the greatest weakness in the application of such models at any scale l i e s in the arbitrariness of the necessary preprocessing: the calculation of effective r a i n f a l l and the baseflow separation procedure. 9. The results reported here are limited to rainfall-runoff events on the scale of small upland catchments. The quasi-physically based model i s probably limited in application to this scale. Conclusions drawn in this study with respect to any of the three models should not be extended to larger scales without further comparative studies at those scales. This chapter reports a set of model performance calculations for three event-based rainfall-runoff models (Chapter 4) on three data sets involving 269 events from small upland catchments (Chapter 5). In Chapter 9 similar calculations are carried out for the enhanced R-5 data (Chapter 6) and in Chapter 10 for the hypothetical hillslopes (Chapter 7). 140 CHAPTER 9 R-5 Revisited: Results of Spatial Variability Analysis "Can i t be that the vast labor of characterizing these systems, combined with the vast labor of analyzing them, once they are adequately characterized, i s wholly disproportionate to the benefits that could conceivably follow?" J. R. Philip, 1980 In this chapter results are summarized for the R-5 data collection described f u l l y in Chapter 6. The reader i s reminded that the data collection program was undertaken because this author anticipated that the performance of the quasi-physically based model could be improved over that reported in Chapter 8 by acquisition of supplemental data describing the spatial variability of near surface s o i l hydraulic properties. The chapter i s divided into three sections. F i r s t , the results of the f i e l d i n f i l t r a t i o n measurements are reviewed in some detail with respect to the interactive network-design strategy proposed in Chapter 6. Next, the findings from the laboratory analysis are briefly examined. Finally, the implications of this detailed f i e l d study are discussed with regard to the redeployment of the quasi-physically based model. Field Analyses Table 9-1 summarizes the means and standard deviations for steady-state i n f i l t r a t i o n rates measured across R-5. Frequency histograms for the data are plotted in Figure 9-1. The sample distributions were 2 tested for normality or log normality using an X test at the 95% 141 TABLE 9-1. Steady state i n f i l t r a t i o n , v , stat i s t i c s for the R-5 catchment Transect #1 (TRl) Transect #2 (TR2) Regular Grid (Grid) Irregular Grid Sharma^ et a l . , 1980 Network Design This Study Number of measurements 50 50 157 26 Spacing between measurements 5 m 2 m 25 m 60 m (average) Mean x 10"° (m/s) 2.12 0.65 1.58 1.22 Standard deviation x 10 * (m/s) 1.03 0.29 1.16 0.73 Based upon i n f i l t r a t i o n rates measured at 60 min. (v f i n ) . 142 a) b ) o c d> r> IT CD >» O c CO CD 2 0 1 5 1 0 -5 -2 0 1 5 1 0 t . I I . -I 1 . 0 7 2 . 6 2 4 . 1 8 5 . 7 3 c) d) 6 0 5 0 4 0 > o c Q D 3 0 cr 0 2 0 0 . 2 6 0 . 6 5 1 . 0 4 1 . 4 3 S t e a d y Sta te Inf i l t rat ion Rate x 1CT 5 ( m / s ) 1 0 1 5 -O 1 0 c CD 3 cr CO 0 . 4 9 2 . 0 6 3 . 6 3 5 . 2 0 0 . 4 2 1 . 5 9 2 . 7 5 S t e a d y S t a t e In f i l t ra t ion Rate x 1 0 " 5 ( m / s ) FIGURE 9-1. Frequency histograms for steady-state i n f i l t r a t i o n data, (a) TR1. (b) TR2. (c) grid, (d) Sharma et. a l . (1980). 143 2 significance level. The results are reported in Table 9-2. The x s t a t i s t i c s provides only an approximate test of the true nature of the distributions. An alternative approach to identify the sample distribution i s a combination of a f r a c t i l e diagram and cumulative probability plot (eg., Warrick e_t a l . , 1977b). Also included in Table 9-1, Table 9-2 and Figure 9-1 are comparable findings from an earlier R-5 i n f i l t r a t i o n study (Sharma et a l . , 1980). In that study the authors found no obvious pattern to i n f i l t r a t i o n with respect to s o i l type or position in the catchment. Figure 6-5 (a and b) shows the locations of a l l i n f i l t r a t i o n measurements. The spatial distribution of the steady-state i n f i l t r a t i o n rates for the two line transects i s shown in Figure 9-2. The summary statis t i c s reported in Table 9-1 are well within an order of magnitude of each other. The a b i l i t y of this study to reproduce the results from the earlier study i s very pleasing. Closer inspection of Table 9-1 and Figure 9-2 shows that the magnitude and spatial variation i s greater for transect #1 (TRl) than transect #2 (TR2). The saturated hydraulic conductivity and steady-state i n f i l t r a t i o n rate are usually thought to be log normally distributed (Freeze, 1975; Cundy, 1982; 2 Hoeksema & Kitanidis, 1985). A perusal of the x values presented in Table 9-2 and a visual inspection of the histograms shown in Figure 9-2 generally lend support for log normality. The one exception i s from the 2 grid network, which w i l l be addressed later. In Table 9-2, i f x > 2 2 2 X Q Q5 the hypothesis i s rejected. If x <. X Q the hypothesis, although not accepted, cannot be rejected. 144 TABLE 9-2. x stati s t i c s for the test against a normal distribution, (a) TRl. (b) TR2. (c) Grid, (d) Sharma e£ a l . (1980). N i s the sample size; X » the observed s t a t i s t i c ; df, the degrees of freedom; and XQ QC» the theoretical s t a t i s t i c at the*93% significance level. Parameter N 2 X df 2 ^0.05 Hypothesis (a) V s 50 6.1 2 6.0 R log v s 50 5.9 2 (b) 6.0 C V s 50 8.1 3 7.8 R log v s 50 3.5 3 (c) 7.8 C V s 157 29.9 3 7.8 R log v s 157 14.1 5 (d) 11.1 R V60 26 6.3 1 3.8 R log v 6 0 26 5.2 2 6.0 C C, cannot reject; R, reject TR1 i s the foundation for the interactive network-design scenario proposed in Chapter 6. The transect was located (Figure 6-5a) in a relatively undisturbed section of the catchment and i s perpendicular to the channel. The lowest elevation along the transect i s at i t s center. On each side of the channel the line traverses overland flow planes f e l t to be characteristic of the entire catchment. The 5 m spacing used for TR1 was adopted out of practicality. The schedule of i n f i l t r a t i o n measure-ments for each of the three networks i s list e d in Table 9-3. It w i l l be shown later in this section that the experimental time frame had significant influence upon the results. Figures 9-3 and 9-4 show the autocorrelograms and semi-variograms calculated for TR1 and TR2 respectively. These experimental functions were determined using (6-1) and (6-3). Each of the points in Figures 9-3 and 9-4 represent a minimum of 30 data pairs. Bresler and Green (1982) and Russo (1984) each recommend that sample sizes should be greater than or equal to 30 observations. The TR1 autocorrelation function (Figure 9-3a) does not have a simple form. It decays to zero within three lags and then increases to a positive correlation for lag distances between 60 and 95 m. Based upon the criterion suggested by Webster and Cuanalo (1976) the minimum sampling distance i s approximately 5 m. The TR1 semi-variogram (Figure 9-3b) has the same cyclic pattern described for the autocorrelogram. In general there appears to be a range of approximately 25 m. Assuming that the semi-variogram i s true, the minimal sampling interval should be between 10 and 15 m. 147 Table 9-3. Schedule of i n f i l t r a t i o n measurements Network Dates Transect #1 24 September - 7 October Transect #2 6-13 November Grid 11 October - 5 November 14-16 November 148 FIGURE 9-3. TRl experimental functions for steady state i n f i l t r a t i o n data, (a) autocorrelogram. (b) semi-variogram. FIGURE 9-4. TR2 experimental functions for steady state i n f i l t r a t i o n data, (a) autocorrelogram. (b) semi-variogram. 1 5 0 The spatial structure of steady-state i n f i l t r a t i o n for TR1 indicates that the i n i t i a l grid spacing should be as tight as 5 or 10 m. This i s assuming that i n f i l t r a t i o n across the catchment i s an isotropic process and that TR1 i s an accurate marker. Table 9-4 l i s t s the number of measurements approximately required to cover R-5 with various square grids. Time constraints prevented the rigorous application of the three-step sampling scheme proposed in Chapter 6. To insure that the entire catchment was covered a 25 m grid spacing was adopted. The grid was developed by projecting parallel lines off TR1. Each of the i n f i l t r a t i o n sites from the three networks were surveyed (coordinates and elevations) by Tom Boswell after a l l measurements were completed. Summary information for the 157 grid i n f i l t r a t i o n measurements i s found in Tables 9-1 and 9-2 and Figure 9-1. Directional and two-dimensional average semi-variograms for steady-state i n f i l t r a t i o n are shown in Figure 9-5. A l l points in this figure are based upon at least 30 measurement pairs from the R-5 grid. Close inspection of these semi-variograms suggests that the isotropic assumption was not a bad one for R-5. However, there appears to be l i t t l e correlation between measurements spaced 25 m apart as was prognosticated from the TR1 analysis. Due to a lack of persistence or spatial structure between the grid i n f i l t r a t i o n measurements, Kriging i s not an effective network design tool. The best f i t generalized covariance function for the measured data i s described by a pure nugget effect at scales greater than 25 m. This random nature i s indicated in the two-dimensional semi-variograms TABLE 9-4. Various grid network requirements Spacing, m ( A X = A Y ) Number of Measurements Days j Required 1 96,000 19,200 2 24,000 4,800 5 3,840 768 10 960 192 15 427 85 20 240 48 25 154 31 30 107 21 35 78 16 40 60 12 45 47 9 50 38 8 assuming 5 measurements per day •* North-South Semi-Variogram •* East-West Semi-Variogram • Northeast-Southwest Semi-Variogram •• Northwest-Southeast Semi-Variogram • Average Semi-Variogram i i i i i 1 1 1 1 1 1 1— 0 4 0 8 0 1 2 0 1 6 0 2 0 0 2 4 0 2 8 0 3 2 0 3 6 0 4 0 0 4 4 0 h - l ag (m) Directional semi-variograms for steady-state i n f i l t r a t i o n based upon the 157 R-5 grid observations. discussed above. Delhomme (1978) provides the basis of the c r i t e r i a used here to evaluate possible generalized covariance models. Forty i n f i l t r a t i o n measurements with a spacing of 50 m were abstracted from the total grid to i l l u s t r a t e the method of variance reduction for network design. Just as reported earlier for the entire grid, the best f i t generalized covariance function describing the smaller data set i s a pure nugget. The value of the nugget i s reduced from 1.52 x -5 —5 2 10 to 1.18 x 10 (m/s) going to the more closely spaced data. This reduction i s due to the smaller variance of the larger data set. To again halve the spacing distance would require approximately 614 total measurements or an increase of almost 75% over the grid data collected for this study. This would require more than 120 days of f i e l d work using the methods described in Chapter 6. This additional data collection was deemed infeasible. Even at the new spacing the results of TRl suggest that there may s t i l l not be enough information to describe the spatial v a r i a b i l i t y of steady-state i n f i l t r a t i o n across R-5. In Table 9-5 mean and standard deviation values for the grid steady-state i n f i l t r a t i o n measurements are summarized based upon the three R-5 s o i l types shown in Figure 5-la. Using this subdivision of the data, and with f u l l regard for the small and varied sample sizes, i t appears that the Grant s i l t loam i s more than twice as permeable as the Kingfisher s i l t loam. Recall that Sharma et a l . (1980) found no difference in i n f i l t r a t i o n with s o i l type. The ranking of the steady-state i n f i l t r a t i o n rates relative to s o i l types agrees with the observations of the author. The TR2 steady-state i n f i l t r a t i o n measurements were made in order to study spatial variability over a more homogeneous section of R-5. Results TABLE 9-5. Summary st a t i s t i c s for grid i n f i l t r a t i o n measurements based on s o i l type. Steady-State Infil t r a t i o n Rate Standard Soil Soil Number of Mean _^ Deviatio: Number Name Measurements m/s x 10 m/s x 10 1 Kingfisher 6 9.5 9.3 2 Grant 61 19.9 12.8 3 Renfrow 90 13.4 9.9 from TR2 have already been presented in parallel to the TRl and grid measurements. Figure 9-4 (a and b) exhibits much the same form observed * for the early TRl logs. The decay of P z(h) over h=0 to 15 m i s cleaner for TR2 than TRl but the conclusion i s unchanged. The range for the R-5 soils i s small (<20 m) and very closely spaced measurements would be needed to take advantage of Kriging type variance reduction. The penetrometer measurements made across the R-5 grid were not useful for determining topsoil and subsoil depths and are not reported. An attempt was also made to relate the resistence data to the i n f i l t r a t i o n rates. This, too was not revealing and i s not recounted here. Climate was found to have a major effect upon i n f i l t r a t i o n experiments in this study. Figure 9-6 shows daily values for temperature and precipitation from stations located near R-5 for a period preceding the i n f i l t r a t i o n measurements to their completion. The catchment was very dry when the i n f i l t r a t i o n experiments were initiated, but became quite wet during the course of the f i e l d study. Table 9-6 reports grid i n f i l t r a t i o n summary st a t i s t i c s for the wet and dry periods. The periods are divided at October 16 (Figure 9-6a). Also summarized in Table 9-6 i s another s p l i t of the measurements based upon a pre- and post-storm division with respect to the major storm on October 28. The time invariance assumption linked to the steady-state i n f i l t r a t i o n rate in Chapter 6 appears to be quite suspect. This time variability i s obviously linked to the changing climate and poses a serious problem for collection of a homogeneous data set. To study the spatial variability of i n f i l t r a t i o n at the catchment scale many measurements are needed requiring considerable time. However, a) 75 E ^ 50 i c o •*-» CO o CD k_ 0_ > 25 'co Q |— Infiltration Measurements - * j Dry Period Wet Period L ik 1 -4- I EL b ) 50 i O 40 H 3 30 "5 k. CD °- 20 CD r -> 10 CO Q 0 H - 1 0 H Infiltration Measurements Maximum August 31, 1984 October 15 November 15 Date December FIGURE 9-6. R-5 climate. (a) Daily precipitation, (b) Daily temperature. TABLE 9-6. Split sample sta t i s t i c s for R-5 i n f i l t r a t i o n data Number of measurements Mean ^ m/s x 10 Standard Deviation^ m/s x 10 Dry Period"^ 36 1.74 0.86 Wet Period 121 1.53 1.23 2 Pre-Storm, Storm 91 1.80 1.04 Post-Storm 66 1.27 1.26 A l l grid measurements 157 1.58 1.16 see Figure 9-6a for division of wet and dry periods. measurements before October 28 are pre-storm and storm, those after are post-storm. as described here the longer the measurement time frame the more li k e l y i t i s that the catchment conditions controlling i n f i l t r a t i o n (see Chapter 6) w i l l change. To complicate matters even further the response of i n f i l t r a t i o n to changes in the catchment are probably hysteretic. As a f i n a l observation, the non-stationarity of the grid data explains why the larger data set was not described by a theoretical distribution. The geostatistical analysis of the 247 measurements made for this study provides only slightly better understanding of the spatial v a r i a b i l i t y of steady-state i n f i l t r a t i o n across R-5 than does the study of Sharma et a l . (1980) with an order of magnitude less information. However, i t i s possible with the larger data set to identify different mean steady-state i n f i l t r a t i o n rates for the three R-5 s o i l types. Table 9-7 summarizes a few i n f i l t r a t i o n studies and places this work in perspective. Reported for each study i s a range that refers to the maximum distance of separation between correlated measurements. There are two possible reasons why the range i s not reported for some studies. Either the range i s smaller than the sample spacing or (less likely) the range was not reached. The large range reported by Vieira et a l . (1981) i s probably due to the homogeneous isotropic nature of a prepared agricultural f i e l d . The smaller range observed for R-5 i s more typical of a naturally heterogeneous anisotropic rangeland catchment. Laboratory Analysis Table 9-8 summarizes the means and the standard deviations for parameters determined in the laboratory from TRl, TR2 and grid s o i l samples. The parameters include porosity, s o i l moisture, texture and saturated hydraulic conductivity. Frequency histograms for these TABLE 9-7. Comparison of infiltration studies Study Number Investigator(s) Number of Measurements Spacing^ (m) 2 Distribution Range^ (m) Surface Soil Texture Activity 1 . Achouri and Gifford (1984) 70 2 LN NR loam grazing 2. Grah (1983) 120 3 LN 17 clay loam grazing 3. Luxmoore et a l . (1981) 48 2 LN NR clay solid waste disposal area 4. Merzousui (1982) 104 2 LN NR loamy sand grazing 5. Sharma et a l . (1983) 55 226 100 1 LN N NR NR coarse coarse grazing forest 6. Sharma et a l . (1980) and Rogowski (1980) 26 50 LN NR s i l t loam grazing 7. Vieira et a l . (1981) 1,280 1 N 50 s i l t loam experimental farm 8. Wagenet (1981) 133 1 LN NR sandy loam experimental farm 9. This study 247 2 LN 10 loam grazing refers to the closest pair of measurements. LN = log normal. N = normal. NR = not reported. 160 TABLE 9-8. Summary s t a t i s t i c s for R-5 laboratory determined parameters. Porosity, 6Q, s o i l moisture, 6., and saturated hydraulic conductivity, K . Parameter (50 TR1 samples) TR2 (50 samples) Grid (157 samples) Mean Standard Deviation Mean Standard Deviation Mean Standard Deviation e 1 o 48.5 3.3 47.1 2.2 47.9 3.1 e 1 i 30.0 2.6 33.9 1.6 29.7 2.9 % Sand 43.7 5.0 42.3 2.5 46.0 5.0 % S i l t 43.0 4.4 44.3 3.0 41.6 5.2 % Clay 13.3 8.2 13.4 3.0 12.4 3.6 2 -? Ko X 1 0 (m/s) — — 4.64 9.46 — — percent by volume. 49 samples. 161 parameters are plotted in Figures 9-7 (TR1), 9-8 (TR2) and 9-9 (grid). Table 9-9 reports the results of distribution tests for the same parameters. The summary information presented for the laboratory-determined parameters i s very similar for TR1, TR2 and the grid. For porosity, neither of the normal or log normal distributions could be rejected at the 95% significance level for any of the three sampling networks. The time variability of s o i l moisture is illustrated by the distribution tests. For both TR1 and TR2 neither the normal or log normal distribution can be rejected. From each transect the s o i l samples were taken on a single day. In the case of the grid where samples were collected over four days both distributions are rejected suggesting that the total sample i s not stationary. The textural classification for almost a l l 247 s o i l samples, based upon sedimentation and the U.S. Department of Agriculture classification, i s loam. No difference in texture was found for the three s o i l types shown in Figure 5-la. The small diameter of the s o i l cores biased the saturated hydraulic conductivity results obtained for TR2 with a constant head permeameter. The mean K q value seen in Table 9-8 i s three orders of magnitude greater than the v g value reported in Table 9-1. Preferred pathways through the length of the s o i l cores resulted in the high values. This i s a problem of scale. The diameter of the cores i s 19 times smaller than the i n f i l t r a t i o n rings. Also, the cores represent only a thin section of s o i l versus the entire profile. n Frequency n CD a w w vO I CD •Q C CD a o CO rt O 00 i-J CO 3 CO l-n O i-t H (X 03 rt 03 34.0-40.0-§ 46.0-[ a 52.0 3 6 . 0 - ~ 2* 42.0 -T = 48.0 54.0 -T 7.0-1 1.0-o 15.0-< 19.0-o I Frequency ro O C D Frequency Frequency "0 o -* o w < 43.0-t 5? 48.0 -r % 53.0 § 58.0 c 3 CD I o Frequency c? 21.0-P ^ 25.0 -[ I 290 ^ 33.0 CD o L_ in i ro o Z91 Frequency i—i o M \0 I oo -t CD C (D 3 n p* H-CO r t O OO i-t 0) H i o SO N 5 C L ca rt P) 3 5 . 0 39 .0 H 0) § 43 .0 -f CL 4 7 . 0 -o i to o X Frequency 3 7 . 0 -* 41 .0 g 4 5 . 0 4 9 . 0 Frequency CD o 1 -O 7.o a 1 1 . 0 -» 15.0 1 9 . 0 - T Frequency Frequency ° 4 4 . 0 O 2 .47.0 1 5 0 . 0 -Frequency | 3 0 . 0 2 3 3 . 0 O w 3 6 . 0 ^ 3 9 . 0 i _ ro o 3 co I •< Q. E. W o £ o S O (fi II o rv> o 0 3 o o 0 . 0 2 8 0 . 1 4 0 0 .251 -0 . 3 6 2 -" €91 n F requency ~^ I—I o a sa w I M .£3 c CD D n <^ cr H-co rr O cro M 03 3 co M i o M 00 l - l a . 03 rr 03 ro i n o 32 0 40 0 03 48 0 a . 56 0 o 03 7 F r equency ro o 26.0 34.0 CO 42.0 50.0 T 1 fD F r equency O 9.0 15.0 21.0 27.0 4 -7 i F requency ro o cn 38.0 -TJ O 43.0 g 48.0 *< 53.0 F r equency cn O O o 2. 1 4 0 Z. 22.0 30.0 w c 38.0 i CD r o o TABLE 9-9. x statistics for laboratory determined R-5 parameters TRl TR2 Grid Parameter N 2 X df 2 x0.05 N 2 X df 2 x0.05 N 2 X df 2 x0.05 e o 50 2.1 2 6.0 50 0.8 3 7.8 157 6.1 3 7.8 log 9 0 50 1.4 .2 6.0 50 0.6 3 7.8 157 7.8 3 7.8 e. i 50 1.1 2 6.0 50 1.6 3 7.8 157 18.0 1 3.8 log ei 50 2.4 2 6.0 50 1.3 3 7.8 157 33.2 2 6.0 % sand 50 10.1 4 9.5 50 13.5 3 7.8 157 4.6 3 7.8 log % sand 50 10.1 3 7.8 50 16.2 3 7.8 157 9.8 4 9.5 % s i l t 50 12.8 3 7.8 50 4.2 3 7.8 157 11.6 4 9.5 log % s i l t 50 11.6 3 7.8 50 3.9 3 7.8 157 19.8 4 9.5 % clay 50 20.2 4 9.5 50 26.9 4 9.5 157 52.3 2 6.0 log % clay 50 23.0 4 9.5 50 25.6 4 9.5 157 48.3 2 6.0 K o 49 12.7 1 3.8 — — — — The average s o i l water retention curve for TR2 i s shown in Figure 9-10. The "undisturbed" s o i l cores proved to be d i f f i c u l t for abstracting reliable characteristic information using the pressure plate apparatus. This was especially true at higher tensions where long equilibrium periods were required. Figures 9-11 and 9-12 show the autocorrelograms and semi- variograms calculated for the laboratory data from TRl and TR2 respectively. Each of the points in these figures represents a minimum of 30 data pairs. The form of the TRl experimental functions i s strikingly similar to that described for i n f i l t r a t i o n (see Figure 9-3). The TR2 functions appear to be without spatial structure. Experimental cross-correlation functions were calculated with f i e l d measured i n f i l t r a t i o n rates, v and laboratory determined s o i l s J properties (6 ; 9., % sand; % s i l t ; % clay) for TRl and TR2 using o i (6-2). Cross-correlations are useful i f the spatial structure of one variable helps to explain the variability of another. When this i s the case, i t i s possible to develop cross semi-variograms and use co-Kriging techniques. The author developed software for such analyses anticipating, for example, that porosity would provide information about i n f i l t r a t i o n . However, no new information could be gleaned from the cross-correlations concerning the spatial v a r i a b i l i t y of steady state i n f i l t r a t i o n rates for either R-5 transect and further analyses were aborted. The s o i l property results reported in this section are consistent with those reported in previous R-5 studies (see Table 6-1 for references). FIGURE 9-10- Average s o i l water retention curve. Error bars represent one standard deviation. 168 FIGURE 9-11. TR1 experimental functions for laboratory determined parameters. Autocorrelograms (top), and semi-variograms (bottom). 169 a) Porosity 10 20 30 40 b) Soil Moisture 10 20 30 40 20 30 h - lag (m) E "e 0.3 10 20 30 40 h - lag (m) c) d) 10 20 30 40 10 20 30 40 % Clay o o +-10 20 30 40 h - lag (m) 10 20 30 40 h - lag (m) h - lag (m) FIGURE 9-12. TR2 experimental functions for laboratory determined parameters. Autocorrelograms (top), and semi-variograms (bottom)• 1 7 0 R-5: A sand box? Following the data collection program, the question was addressed as to whether or not to redeploy the quasi-physically based model for R-5 in l i e u of the supplemental information obtained from the f i e l d program. Based upon the geostatistical analyses reported here the answer i s no. The extensive new data base provides no clearer picture of the spatial v a r i a b i l i t y of i n f i l t r a t i o n across the R-5 catchment. Therefore once again this author was faced with the prospect of using average values to excite the distributed quasi-physically based model. Because the average values of input parameters determined in this study were so similar to the ones discussed in Chapter 6 there was no justification for re-exciting the model. This was a l l very disappointing as i t was anticipated that the performance of the quasi-physically based model reported in Chapter 8 could be improved upon given additional f i e l d data. It was, however, possible to re-excite the quasi-physically based model using the mean steady-state i n f i l t r a t i o n values for the three s o i l types determined from the 157 grid measurements. Table 9-10 shows the saturated hydraulic conductivity values used for both the original R-5 validation (Chapter 8) and for the results to be presented here. Subsoil values of hydraulic conductivity were estimated to be approximately 60% of the topsoil (or measured values). A l l other information used to re-excite the quasi-physically based model was l e f t unchanged. Table 9-11 summarizes the forecasting efficiencies for the three summary output variables for the quasi-physically based model for verification runs with the two R-5 data sets. The efficiencies reported for the original data set have already been presented in Table 8-1. The TABLE 9-10. Saturated hydraulic conductivity values used to excite and re-excite the quasi-physically based model. Original data used i n Chapter 6, supplemental data used in this chapter. Saturated So i l Soil Hydraulic Conductivity, K Number Name Layer* / m - 6 J m/s x 10 Original Supplemental Data Data 1 Kingfisher T 7.6 9.5 S 5.2 5.5 2 Grant T 7.6 19.9 S 5.2 11.5 3 Renfrow T 7.6 13.4 S 5.2 7.8 *T, topsoil; S, subsoil TABLE 9-11. Forecasting efficiencies E^ for the three summary output variables for the quasi-physically based model on the original and supplemented R-5 data sets. E f Number Data Set of Events Q D Q p R t p R Original 36 0.26 0.44 -0.20 Supplemented 36 0.28 0.80 -0.25 173 procedure used to calculate forecasting efficiencies for the quasi-physically based model has already been described (Chapters 3 and 8). An item-by-item comparison of the values on Table 9-11 shows that the supplemental data has the greatest impact on improving prediction of Qpg. The 0.80 E^ value i s higher than any of the values reported on Table 8-1 for three rainfall-runoff models on three catchments. The E^ value for the prediction of i s only slightly improved with additional data. The larger data set and the characterization of distributed parameters based upon s o i l type do not improve the model's a b i l i t y to predict tp^. A single large verification event (see Figure 8-2) exerted considerable leverage in the improved E^ values. In summary, the performance of the quasi-physically based model was improved with supplemental s o i l hydraulic property data. However, the higher E^ values may not warrant the cost of the additional data. The remainder of this section focuses upon the hydrologic nuances of R-5. A number of authors have used data from the R-5 catchment (Table 6-1) with various hydrologic models. In every case i t has been necessary to make tremendous simplifying assumptions concerning the controlling streamflow generation mechanism and the spatial distribution of near surface s o i l hydraulic properties. This author benefited considerably from witnessing the hydrology of R-5 f i r s t hand. As has been maintained throughout this study, overland flow i s the primary streamflow generation mechanism for R-5. It i s now painfully obvious to this author, however, that the small catchment i s not a sand box to be easily modeled. Figure 9-13 illustrates the locations of a FIGURE 9-13- R-5 catchment. number of major hydrologic controls not typically reported for R-5. These include farm roads, buried pipelines and remnant buffalo wallows. Also of significant hydrologic importance across R-5 are many ancient wagon t r a i l ruts. The extent of a semi-defined channel i s portrayed in Figure 9-13. The channel area i s the most active geomorphological part of the catchment. Slumping and erosion failures are prevalent along the channels length. The three s o i l types identified for the R-5 catchment (see Figure 5-la) are principally defined by vegetation. Buffalo and blue grama grasses are found with the Renfrow soils and bluestem grass with the Grant and Kingfisher s o i l s . A l l of the catchment characteristics described in this paragraph help to control the rainfall-runoff process on R-5. During the author's tenure at R-5 the catchment was used by the land owner as a cattle nursery. As described in Chapter 5, R-5 i s a well managed pasture. However, grazing and such side effects as cow t r a i l s unquestionably influence local hydrologic processes. During the f i e l d portion of this study R-5 was also alive with animal activity (armadillos, ants, coyotes, gophers, rabbits, earthworms, snakes, spiders). Each of these animals substantially influenced the micro-hydrology of the catchment with their burrowing. The transient nature of animal holes across R-5 helps to explain the non-stationary grid i n f i l t r a t i o n data discussed in the previous section. Most of the hydrologic controls discussed in this section do not lend themselves easily to physically-based modeling. It i s widely held in hydrology that the use of average s o i l hydraulic properties determined from s o i l texture can be used to effectively excite conceptual rainfall-runoff models. It i s the opinion of this writer that the 176 performance of physics-based models excited with average values based upon s o i l texture w i l l be quite tenuous. The f i e l d observations at R-5 support this conclusion. After spending considerable time observing the dynamic hydrologic nature of R-5 this author feels that both Horton and Dunne overland flow are operative mechanisms on the catchment. The Horton and Dunne mechanisms were observed in conjunction with partial-source and variable source areas respectively. Further f i e l d study i s needed to determine the quantitative importance of each mechanism. The quasi- physically based model i s suited to the R-5 catchment; however, the f i t i s far from perfect as the model does not describe the Dunne mechanism. Infiltration rates on R-5 are strongly influenced by vegetation and climate. Freeze and Banner (1970) have noted that estimates of i n f i l t r a t i o n based only upon s o i l texture can be misleading. They stress that i t i s important to consider the entire system. Small differences in the hydrologic properties of similar f i e l d soils can account for large differences in their reaction to the same hydrologic event as witnessed by this author for R-5. In this chapter the new R-5 data set, reviewed in Chapter 6, has been summarized. The results of geostatistical analysis of these data have been presented. Observations are made concerning the impact of the supplemental data on the performance of the quasi-physically based model. Finally, the results from re-exciting the model with supplemental information are reported. CHAPTER 10 Model Validation: Hypothetical Hillslopes This chapter i s the sequel of Chapter 8. In Chapter 8 three rainfall-runoff models were evaluated using data sets from three experimental catchments. In this chapter the same model suite (Chapter 4) i s re-evaluated, but the data sets used to excite the models this time have been abstracted from the hypothetical hillslopes discussed in Chapter 7. The format of this chapter and the notation are carried over from Chapter 8. Results In Chapter 7 the generator used to create synthetic data sets was described in detail. The reader i s asked to rec a l l that runoff i s simulated with a generator using s t a t i s t i c a l l y specified r a i n f a l l hyetographs that are known for every hillslope node. Also known for each hillslope element are a l l near surface s o i l hydraulic properties (eg., saturated hydraulic conductivity, s o i l moisture). A set of 100 computer generated rainfall-runoff events on a 200 element hypothetical hillslope comprise a single synthetic data set. Four synthetic data sets are uti l i z e d in this chapter. To evaluate the suite of three rainfall-runoff models i t i s necessary to pretend that (as in real l i f e ) runoff i s available only at a single weir and that r a i n f a l l i s available at only a limited number of rain gages (hillslope elements). In this study a single rain gage and a five gage average are used. For the regression and unit hydrograph models only r a i n f a l l and runoff data are needed. However, for the quasi-physically based model s o i l hydraulic property information i s needed as well. For the evaluations presented in this chapter i t i s assumed that a l l hillslope information i s available to excite the quasi-physically based model. The use of the entire synthetic hillslope data set i s equivalent to a f i e l d measurement program that totally describes the near surface s o i l hydraulic properties of a catchment. In Chapter 12 the quasi-physically based model i s re-excited using the same data set but with less hillslope information to investigate data worth. In the following paragraphs the performance of the three rainfall-runoff models i s summarized collectively then separately. Table 10-1 summarizes the forecasting efficiencies for the three summary output variables Q^ , Qp^, and tp^ for verification runs with the three rainfall-runoff models on the hypothetical reality of case A. Table 10-2 summarizes the prediction efficiencies E^ for the three summary output variables for verification runs with the three models on case A. Table 10-3 summarizes the means and standard deviations for the observed and predicted frequency distributions for the three summary output variables for the verification events on case A. Case A was selected for comparative model analysis because i t s principal mechanism of streamflow generation was the Horton mechanism. Each of the three models compared here are thought to be best suited to hydrologic regimes in which runoff i s generated by Horton overland flow. The results reported in Tables 10-1, 10-2 and 10-3 are a l l based upon 50 verification events. TABLE 10-1. Forecasting efficiencies E^ for the three summary output variables for the three models for one rain gage on synthetic data set A. E f Number Model of events fcPK Regression 50 0.97 0.70 0.18 Unit hydrograph 50 0.95 0.97 -2.37 Quasi-physically 36 0.33 -1.30 -1.39 based TABLE 10-2. Prediction efficiencies E for the three summary output variables for the three models for one rain gage on synthetic data set A. E P Number Model of events QPK tPK Regression 50 0.97 0.73 0.41 Unit hydrograph 50 0.97 0.99 -1.89 Quasi-physically 36 0.34 -1.29 -0.87 based TABLE 10-3. Verification period s t a t i s t i c s : Means and standard deviations for observed and predicted frequency distributions for the three summary output variables for one rain gage on synthetic data set A. Mean Standard Deviation Model Number of Events mm l i t . / s e c . , tPK hours mm l i t . / s e c . , tPK hours observed 50 12.4 115.4 0.42 22.5 195.9 0.19 regression 50 11.8 87.4 0.42 19.6 118.7 0.05 unit hydrograph 50 14.2 117.1 0.72 24.3 182.7 0.10 quasi-physically based 36 0.9 17.6 0.65 2.6 63.5 0.32 181 The r a i n f a l l source terms, used to excite the three models and produce the results li s t e d in Tables 10-1 through 10-3 were derived from the hyetographs P ^ t ) (where i = n + l , ..., m; n = 50; m = 100) of a single gage (element number 90, Figure 7-la). The regression model and the unit hydrograph model were calibrated with 50 events ( i = 1, ... n) as described in Chapters 3 and 8. The quasi-physically based model was run in an uncalibrated mode for both calibration and verification events in this chapter. The time step and space increment used for a l l quasi-physically based model simulations reported in this chapter were 100 s and 10 m respectively. A l l unit hydrograph durations were 100 s. Discussion Forecasting Efficiencies A perusal of the E^ values in Table 10-1 leads to the following comments: 1. In general, the performance of the regression model and the unit hydrograph model i s far superior on case A than reported in Table 8-1 for the three experimental catchments. The quasi-physically based model continues to perform poorly. Recall that for perfect efficiency, E^ = 1. 2. In general the E^ values are higher for the prediction of Qp and Qp^, lower for t p j , . This same trend was observed in Table 8-1. 3. The negative efficiencies for the quasi-physically based model infer that Qp^ and t p ^ would be better predicted with the observed means rather than the model predictions. 182 4. The quasi-physically based model was unable to simulate 14 verification period events. This problem i s discussed in detail later in the chapter. In summary, both the regression and unit hydrograph models have demonstrated some predictive power with respect to individual events for the summary variables and Qp^, but l i t t l e prowess for tp^. The quasi-physically based model exhibited predictive powers only for Q^. Prediction Efficiencies An item-by-item comparison of the prediction efficiencies on Table 10-2 and the forecasting efficiencies on Table 10-1 shows that E^ i s always greater than E^ (as expected). In general, the overall performance of the regression model i s best. Verification Statistics Consideration of the verification s t a t i s t i c s on Table 10-3 for the frequency distributions of observed and predicted events paints essential-ly the same picture afforded by the event-by-event and ranked analyses. A perusal of the sta t i s t i c s displayed on Table 10-3 leads to the following more detailed comments: 1. The relationship between predicted and observed values i s much closer for the means than i t i s for the standard deviations. The same observation was made in Chapter 8. 2. The predicted values are much closer to the observed values for the regression and unit hydrograph models than for the quasi-physically based model. 3. For case A, and tp^ were best predicted with the regression model and Q p„ with the unit hydrograph model. 4. In general, the quasi-physically based model under estimated Qp^ by an order of magnitude. The estimate of t p ^ was considerably better, suggesting that individual events were of the correct form but subdued. In summary, and in general, the regression and unit hydrograph models performed quite well under s t a t i s t i c a l verification, just as they had under event-by-event verification. The quasi-physically based model did not produce reliable results for the case A hypothetical hillslope. Regression Model The regression model applied in this chapter i s the same one discussed throughout the study. It would have been possible to redevelop the model and incorporate the antecedent s o i l water content using the synthetic data sets described in Chapter 7. However, this secondary analysis was suppressed to maintain the probity of the entire study. Total knowledge of the rainfall-runoff relation afforded by the hypothetical r e a l i t i e s made i t possible to investigate the influence of additional rain gages upon the output summary variables predicted with the regression model. In this section both a single rain gage and a five rain gage average (see Figure 7-lb) are used to determine the source terms for the regression model. The summary input variables P^, and t j ^ were described in Chapter 3. Table 10-4 summarizes the forecasting efficiencies for the three summary output variables for verification runs with the regression model on cases A, B, C and D (refer to Figure 7-2) based f i r s t upon a single rain rain gage (Table 10-4a) and then upon a five rain gage average (Table 10-4b). Table 10-5 summarizes the prediction efficiencies comparable to Table 10-4. Table 10-6 summarizes the means and standard deviations for 184 TABLE 10-4. Forecasting efficiencies E^ for the three summary output variables for the regression model on the four synthetic data sets of 50 events each, (a) single rain gage, (b) five rain gage average (a) (b) E f E f Case tPK QpK tPK A 0.97 0.70 0.18 0.97 0.87 0.19 B 0.87 0.55 -0.03 0.87 0.69 0.01 C 0.97 0.64 0.16 0.97 0.80 0.22 D 0.89 0.62 0.01 0.89 0.79 0.02 TABLE 10-5. Prediction efficiencies E for the three summary output variables for the regression model on the four synthetic data rain sets of 50 gage, (b) events each, five rain gage (a) single average (a) E P (b) E P Case QPK TPK % QPK TPK A 0.97 0.73 0.41 0.97 0.88 0.45 B 0.88 0.59 0.78 0.88 0.71 0.55 C 0.97 0.69 0.70 0.97 0.82 0.63 D 0.89 0.63 0.02 0.90 0.80 0.50 185 TABLE 10-6. Regression model verification period s t a t i s t i c s : Means and standard deviations for observed and predicted frequency distributions for the three summary output variables on the four synthetic data sets of 50 events each. Mean Standard Deviation Case % mm l i t . / s e c . , tPK hours % mm l i t . / s e c . ^PK hours A Observed 12.4 115.4 0.42 22.5 195.9 0.19 Predicted''' 11.8 87.4 0.42 19.6 118.7 0.05 2 Predicted 11.5 90.1 0.42 19.5 140.7 0.06 B Observed 16.2 135.9 0.62 28.7 237.4 0.22 Predicted''' 12.4 79.7 0.64 20.6 110.2 0.16 2 Predicted 12.1 82.2 0.60 20.6 130.5 0.10 C Observed 18.0 155.3 0.47 30.8 248.4 0.19 Predicted"^ 17.0 106.4 0.51 26.9 136.5 0.11 2 Predicted 16.6 109.7 0.48 26.7 162.3 0.09 D Observed 3.2 38.9 0.44 7.2 80.0 0.29 Predicted^ 3.0 27.2 0.43 5.5 43.0 0.003 2 Predicted 2.9 28.2 0.43 5.5 50.6 0.01 Single rain gage Five rain gage average the observed and predicted frequency distributions for the three summary output variables for the verifications events on cases A through D based upon a single rain gage and a five rain gage source term. A perusal of the E^ values in Table 10-4a leads to the following comments: 1. For a l l four cases, E^ values are very high for and relatively high for Q p j r . The regression models exhibited l i t t l e predictive a b i l i t y for tp^. 2. The overall best performance of a regression model was for Case A. Streamflow generation .on Hillslope A i s dominated by the Horton mechanism. The next most successful model performance was for case C (Horton/Dunne generation), then for case D (Horton generation) and f i n a l l y for case B (Dunne generation). In summary, the performance of four rainfall-runoff regression models i s quite good for and to a lesser extent Qp^« It appears that the modeling technique i s best suited to runoff events generated by the Horton mechanism. An itera-by-item inspection of Table 10-4a and 10-4b indicates that Q p j , i s predicted more ef f i c i e n t l y on an event-by-event basis with a five rain gage average of p ^ than from a single rain gage. There i s very l i t t l e difference in the E^ values for and tp^ in Table 10-4a and Table 10-4b. This suggests that the spatial variability of and t j ^ i s not as important as the spatial structure of short duration maximum r a i n f a l l intensity Pj^« Five gage averages for P^, P ^ and t ^ are used to arrive at the predictions of Q ^ , Q p ^ and t p ^ reported in Table 10-4b. The same pattern that was noted for in Table 10-4 i s in general also observed for E^ in Table 10-5. The only noticeable exception to this observation i s that the E^ value for tp^ on cases B and C are greater for a single rain gage than for a five rain gage average. Case D on the other hand i s much improved by a five gage average. An inventory of the st a t i s t i c s presented on Table 10-6 yields the following remarks: 1. The relationship between predicted and observed values i s much closer for the means than i t i s for the standard deviations. 2. In general, i t appears that the predicted values are closer to the observed values for and tp^. 3. The best overall performance of the regression model was for cases A and D. Both of these hillslopes are dominated by the Horton mechanism. In summary, the performance of the regression model was quite good for both Horton and Dunne hillslopes; each of the eight regression models i s applicable only to the hillslope for which i t was developed. Unit Hydrograph Model The unit hydrograph model used in this study was reviewed in Chapter 4. The $-index preprocessor was selected to calculate effective r a i n f a l l because sparse s o i l moisture data for the experimental catchments described in Chapter 5 did not afford the luxury of a more rigorous technique. The $-index method i s recognized by this author as the Achilles' heel of the unit hydrograph model under evaluation. A number of hydrologic researchers have stressed the point to this author, that (4-2c) must include the antecedent s o i l water content for the $-index to be a reliable predictor. Because the hypothetical reality of case A contains i n i t i a l s o i l moisture values across the entire grid i t was possible to test the aforementioned contention to some degree. A simple linear regression analysis i s used to establish the best 2 $-index relationship. The coefficient of determination r i s used to 2 measure the quality of each relationship. The range of r i s between 0 and 1, with a perfect f i t denoted by 1. The interested reader attempting 2 to build a parallel between r and for regression analysis i s 2 advised that they are only equal for calibration (Aitken, 1973). The r i s used to draw attention to the fact that the *-index relationships are evaluated only as a component of the unit-hydrograph model. The complete unit hydrograph evaluation was not carried out for each $-index relationship. Table 10-7 summarizes the results for seven different $-index relationships for the 50 calibration events for case A. Interestingly 2 enough the relationship that has the highest r , for both the single and five gage networks, includes and P ^ as independent variables but does not include the antecedent s o i l water content. The preferred relationship i s the same as (4-2c). For the $-index analysis, the locations of the rain gage(s) and s o i l moisture sampling site(s) (the same grid element(s)) for both the single measurement and five measurement averages are shown on Figure 7-la. Based upon the information presented here the most important parameter i n determining the ^index i s P J ^ I the least important i s 6_^. This limited test with synthetic data was not designed to show that the i n i t i a l TABLE 10-7. C o e f f i c i e n t of determination values r for alternative unit hydrograph model $-index relationships on case A. 2 r $-index relationships Single Gage Five Gage Average 1. 0.21 0.29 2. $ = a ? P D + a 8 P m x (4-2C) 0.69 0.61 3. $ = a n P D + a 1 2 6. 0.12 0.23 4. * = a12 PD + a14 PMX + a15 6 i 0.64 0.57 5. $ = a16 PMX 0.58 0.57 6. $ " a17 PMX + a18 9 i 0.51 0.53 7. $ - a19 6 i • 0.003 0.04 s o i l moisture i s not needed to employ the 4>-index method. Rather, these results lend credence to the use of (4-2c). The mechanics and application of the unit hydrograph model were discussed in Chapters 4 and 8, respectively. The unit hydrograph analysis in this chapter util i z e s only the synthetic data of case A. This hypothetical reality helps to confirm the importance of input error to rainfall-runoff modeling and the model UNIT in particular. At this point the reader i s reminded of the time- and space-dependent r a i n f a l l information generated across the 200 elements of case A (see Chapter 7). The f u l l data set from the hillslope i s treated as a known when i t i s generated but as an unknown when i t is modeled. For this study only a single hillslope element and a five element average we're selected to record hyetographs as i f they were one rain gage or five" rain gages. The problem encountered with both rain gage network designs in this study was one of juxtaposition between rain gage(s) and storm patterns. The problem was of course caused by storms tracking across the hillslope before being noticed by the rain gage(s). This i s a common problem in the preparation and analysis of f i e l d data. The single rain gage and five rain gage average represent only 0.5 and 2.5 percent (respectively) of the available information. However, compared with the network density of a typical experimental catchment, where instrumentation i s constrained by practicality and economics, the coverage described herein for a small hillslope i s quite good. For the work presented here the juxtaposition error was a r t i f i c i a l l y circumvented by shifting the effective r a i n f a l l hyetographs to commence with the f i r s t time steps of their respective events. The penalty for shifting the r a i n f a l l vectors was to introduce synchronization errors between r a i n f a l l and streamflow gages. As a concerned parent might lecture their less than truthful child, one l i e leads to another. Tables 10-8 and 10-9 summarize the forecasting and prediction efficiencies for the three summary output variables for verification runs with the unit hydrograph model on case A based upon a single rain gage and a five rain gage average. Table 10-10 summarizes the means and standard deviations for the observed and predicted frequency distributions for the three summary output variables for the 50 verification events on case A based upon a single rain gage and five rain gage source term. A perusal of Tables 10-8 through 10-10 leads to the following comments: 1. The performance of the unit hydrograph model was excellent for prediction of and Qp^« but sorrowful for tp^. 2. In general, E^, E^ and verification s t a t i s t i c s are equal for both rain gage networks. 3. The sin of shifting the input vector i s clearly shown in the prediction of tp^. In summary, the performance of the unit hydrograph model using the synthetic data of case A was quite gratifying considering i t s relatively poor performance on the three experimental catchments reported in Chapter 8. The small size of the hillslope did not seem to hinder the performance of the unit hydrograph technique. Case A i s dominated by the Horton mechanism and i s therefore appropriate for unit hydrograph analysis. The $-index method of determining excess r a i n f a l l did not appear to be as detrimental here as surmised in Chapter 8. The importance of time- and TABLE 10-8. Forecasting efficiencies E^ for the three summary output variables for the unit hydrograph model on the synthetic data of case A containing 50 events. No. of Rain Gages Q D Q p K t p R 1 0.95 0.97 -2.37 5 0.96 0.95 -1.97 TABLE 10-9. Prediction efficiencies E for the three summary oStput variables for the unit hydrograph model on the synthetic data of case A containing 50 events. E P^K 'PK No. of Rain Gages 1 0.97 0.99 -1.89 5 0.97 0.96 -0.91 TABLE 10-10. Unit hydrograph model verification period s t a t i s t i c s : Means and standard deviations for observed and predicted frequency distributions for the three summary output variables on the synthetic data of case A containing 50 events. Mean Standard Deviation % mm l i t . / s e c . u t pK hours % mm l i t . / s e c . hours Observed 12.4 115.4 0.42 22.5 195.9 0.19 Predicted"^ 14.2 117.1 0.72 24.3 182.7 0.10 2 Predicted 13.3 110.6 0.66 19.9 161.4 0.20 Single rain gage Five rain gage average space-rainfall variability was illustrated by the poor prediction of tp^ caused by the misalignment of input vectors. The model was quite forgiving in this regard with respect to and Q p j r . Quasi-Physically Based Model The performance of the quasi-physically based model on the seemingly ideal synthetic data of case A was quite p i t i f u l . At the outset of this analysis i t was a distinct concern that the similarity in conceptual foundation between the stochastic-conceptual rainfall-runoff simulator SCRRS (Chapter 7) and the quasi-physically based model (Chapter A) would bias the results. It i s rather easy to see that this did not happen. As the quasi-physically based model i s a Horton simulator only case A was used for i t ' s excitement. However, a number of problems arose in attempting to excite the model with the synthetic data described in Chapter 7. The f i r s t problem was the mismatch in the procedures for routing overland and channel flow. The quasi-physically based model requires three parameters not to be gleaned from the SCRRS synthetic data set. These are the Manning's roughness coefficients for both overland and channel flow (n Q and respectively) and the channel slope S^. These parameters as well as the dimensions of the channel i t s e l f could have been used as calibration parameters. Instead, the values of n c > n , S and the channel dimensions were estimated to be equal to those o c reported in Chapter 5 for the R-5 catchment. This subjective decision was influenced by the calibration experiment described in Chapter 8 for the quasi-physically based model on the R-5 data and the Horton nature of both the R-5 catchment and the synthetic case A. The saturated hydraulic conductivity K q was identified previously as the most sensitive parameter in the quasi-physically based model (Chapter 8 ) . Because the synthetic K q grid from case A offered total knowledge of the quasi-physically based model's most sensitive parameter, i t appeared appropriate apriori to assume values for the less sensitive parameters n Q, n^ and S^. The synthetic data base includes a l l the remaining physically based parameters (K q, S, 6^, S q , d,pf dg) required by the model. Therefore, the f u l l nodal sets of these values were used as input. In Chapter 12 experiments w i l l be presented that use representative values for these parameters as opposed to the f u l l nodal sets. The notation used here was defined in Chapter 4. The depression storage capacity H was set to zero for each hillslope element. A vector of a l l physically based parameters was input to the model for each element describing the hills l o p e . A grid of 200 elements (Figure 7-la) was used to represent (mimic) case A. The quasi-physically based model was designed for only one source terra. In this study a single rain gage was assumed to have been placed on a central hillslope element (Figure 7-la, element number 9 0 ) . The hyetographs recorded at this gage were used to excite the model. A l l r a i n f a l l information from the remaining 199 elements was (in general) ignored for this section of the study. The instability of the quasi-physically based model to abrupt changes in excess precipitation was a major obstacle for i t s implementation and evaluation in this segment of the study. The convergence problem was mentioned in Chapter 4. A thick sheet flow moving down a flow plane leads to the model's instability when adjoining segments have significantly different hydraulic properties. The reader i s reminded that SCRRS routes overland flow by a parametric routine based upon the concept of time-area curves that does not allow for r e i n f i l t r a t i o n . The generator therefore did not suffer from the numerical d i f f i c u l t i e s described for the quasi-physically based model. The quasi-physically based model did not encounter similar problems with the f i e l d data from the three experimental catchments (Chapter 8) because there was insufficient data to map any high-contrast, closely-spaced, permeability variations. Of the 100 events synthesized for case A the quasi-physically based model failed to converge for 24. These events had the greatest r a i n f a l l intensities and total depths. Using the five rain gage average source term utilized by the regression and unit hydrograph models the model • • converged on seven more events. This increase in model performance did not warrant switching rain gage networks as the five gage average only masked the problem for marginal cases by lengthening and smoothing those hyetographs. For the single rain gage the mean value of i s twice as great as for the five gage network (see Table 7-4). This five gage value i s similar to the mean value of Pj^ reported for the R-5 catchment in Table 5-2; the actual P^ mean-value for the 100 case-A events (Table 7-3) i s more closely estimated by the five gage network (Table 7-4). In the verification period the model was stable for only 72% of the 50 events. Many of the 14 events that failed to converge had short duration r a i n f a l l intensities greater than 400 mm/hour. One such event i s illustrated in Figure 10-1. The inability of the quasi-physically based model to simulate large Horton controlled events i s contrary to i t s objective. 197 3 O - C '—. E E < rr 5 0 0 - i 4 0 0 -£ 3 0 0 CO I 2 0 0 1 0 0 P D = 8 4 . 6 m m 12 14 1 6 1 8 2 0 2 2 2 4 T I M E S T E P (At = 1 0 0 s ) 2 6 2 8 FIGURE 10-1. Case A r a i n f a l l event 64. This i s one of the 14 verification period events for which the quasi-physically based model was unstable. The hyetograph was determined from a single rain gage whose location i s shown in Figure 7-la. Tables 10-1 and 10-2 summarize the forecasting and prediction efficiencies for the three summary output variables for verification runs with the uncalibrated quasi-physically based model on case A. Results are reported only for the 36 events in which the model was able to converge. Table 10-3 summarizes the means and standard deviations for the observed and predicted frequency distributions for the three summary output variables for the verification events on case A. Statistics are presented for a l l 50 events in the verification period and the 36 events for which the model was stable. The model displayed some predictive a b i l i t y for on the subset of smaller events that were found stable but none for Q p K or t p K . In summary, the performance of the quasi-physically based model was unexpectedly poor considering the similarity in conceptual foundation between i t and SCRRS. Even for this small synthetic Horton hillslope, with total knowledge of hydraulic properties, the model performance did not equal that reported in Chapter 8 on the R-5 catchment where there was substantial lumping of distributed parameters. The quasi-physically based model was not calibrated in these analyses against the more sensitive s o i l hydraulic properties. By overfitting these parameters to obtain more appealing results the existing data set and the model integrity would have been compromised. However, i t i s not clear that more appealing results would be obtained regardless. It appears that the reason the predictions are so poor i s the lack of representativeness of the r a i n f a l l . In any case, calibrating against K Q to produce values different from the ones that are known to be correct i s s i l l y . This i s of course as opposed to the real world where a l l the K values are never known and i t can be productive to calibrate against the unknown values. Summary Many of the comments made throughout both the discussion and conclusion sections of Chapter 8, concern individual model nuances. For the synthetic data sets analysed in this chapter the following points can be made: 1. The individual performance for the regression and unit hydrograph models on case A synthetic data were in general very good for both forecasting and prediction. The quasi-physically based model had no predictive power in either forecasting or prediction mode. 2. The explicit numerical scheme used in the quasi-physically based model seriously constrains the range of conditions under which the model can be used. It must be remembered that the substance of the code was developed more than a decade ago before some of the more efficient techniques were available. During the selection of a quasi-physically based representation of the hydrologic cycle the code of Engman (1974) was believed by this author to be among the best available. It now i s apparent to the author that a model based upon the latest numerical techniques, but of the same conceptual s p i r i t as Engman1s model, is needed to stretch the envelope identified in this chapter. 3. The quasi-physically based model simulations lend further credence to the importance of understanding r a i n f a l l variability in physically based modeling on small catchments. The spatial and temporal distributions of r a i n f a l l were in essence the only information not transmitted to the model in these analyses and s t i l l the performance was 200 poor. It i s important that we have a model that w i l l not balk at the kind of r e a l i s t i c event shown in Figure 10-1. It must be recognized that the events analyzed in this chapter were generated with a rainfall-runoff simulator that i t s e l f can be questioned as to i t s correspondence with real events. It may be that the consistently poor model performance with a l l models with respect to tp^ i s in fact a reflection on the generator rather than the models. Among the aspects of SCRRS that deserve mention are the following: 1. The simulator does not allow for subsurface storm flow (refer to Chapter 2). For the Horton hillslope concentrated upon in this analysis this i s not considered a factor. 2. The water table configuration described by Freeze (1980a) precludes the formation of a seepage face. Seepage faces can be important controls for variable source areas. 3. The simple travel-time approach to overland flow routing referred to earlier in this chapter and in Chapter 7 does not allow for runon (r e i n f i l t r a t i o n ) . This may be an important conceptual flaw in the simulator as there w i l l not always be a hydraulic connection between a partial area and the channel. 4. The simulator used independent stochastic processes to describe the spatial patterns of s o i l hydraulic parameters. It would be more r e a l i s t i c to generate these values with a multivariate stochastic process. The cross correlation between near surface s o i l hydraulic parameters as well as the autocorrelation within each parameter set was addressed for the R-5 catchment in Chapter 9. Two of Beven's (1981) comments pertaining to Freeze's original employment of SCRRS, and applicable to this work, deserve mention: 1. The hypothetical r e a l i t i e s utilized in this analysis are not generated from a parameter data set of a single (real) catchment. The author hopes to address this problem in future work (see Chapter 12) but agrees with Freeze (1980a) that SCRRS generates rainfall-runoff events that show similarity to those measured in the f i e l d . 2. The scale of a SCRRS (grid) application should be married to spatial parameter data and the scale of f i e l d measurement. This writer agrees with Beven (1981, p. 432) "that the whole question of the interactions between hydrological processes, measurement techniques and scale, and model structure and parameters i s one that has been barely touched in hydrology". Despite the shortcoming in the SCRRS generator highlighted above, i t s use in this study i s defended on the grounds that only with a synthetic data set can some of the experiments to be reported in Chapters 11 and 12 be carried out. It i s foolhardy to base conclusions on synthetic data sets alone but when their analysis i s coupled with the analysis of f i e l d data, as i s in this thesis, the added insight can be valuable. This chapter reports a set of model performance calculation for three event-based rainfall-runoff models (Chapter 4) on synthetic data sets from small hillslopes (Chapter 7) involving hundreds of events. 202 CHAPTER 11 Hypothetical Hillslopes: Sensitivity Analyses In this chapter sensitivity results are reported for the SCRRS hypothetical r e a l i t i e s presented in Chapter 7. In the f i r s t section, the effect that the saturated hydraulic conductivity distribution has upon the rainfall-runoff process i s shown. In the second section, the impact of both precipitation and s o i l hydraulic property data i s investigated. This chapter i s an extension of Chapters 7 and 10 and parallels Chapter 12. Effect of hydraulic conductivity distribution on rainfall-runoff process In an earlier analysis using SCRRS, Freeze (1980a) studied the effects that various hydraulic conductivity distribution parameters (mean, standard deviation, autocorrelation) have upon the rainfall-runoff process. In Chapter 7 of this thesis the fundamental components of SCRRS were briefly reviewed. The SCRRS parameter values used to generate ten synthetic rainfall-runoff data sets were also discussed in Chapter 7. The same ten hypothetical r e a l i t i e s are utilized in this section. For this study, Freeze's (1980a) sensitivity analysis of hydraulic conductivity distribution parameters i s extended. The analysis in this study i s designed to investigate what effect the autocorrelation parameter has upon the rainfall-runoff process. In the paper by Freeze a similar study was reported incorrectly. Table 11-1 presents the data used with SCRRS for this study. Table 11-2 indicates the correspondence between this work and Freeze's. In the following paragraphs the effect that the hydraulic conductivity distribution has upon the rainfall-runoff process i s discussed. TABLE 11-1. Comparison of SCRRS input parameters for cases A-J Hydraulic Conductivity Case Geometry (z. .") max, J m m/s o Y Oty A 2.25 IO" 5 -5.0 0.8 0.3 B 0.60 IO" 4 -4.0 0.8 0.3 C 0.60 IO" 5 -5.0 0.8 0.3 D 6.20 IO"* -4.0 0.8 0.3 E 2.25 IO" 5 -5.0 0.1 0.3 F 2.25 IO" 5 -5.0 0.1 1.0 G 2.25 IO" 5 -5.0 0.0 H 2.25 IO" 5 -5.0 0.8 0.9 I 2.25 • IO" 5 -5.0 0.8 0.6 J 2.25 IO" 5 -5.0 1.6 0.3 K S = exp {2.3uy} TABLE 11-2. Correspondence between hypothetical r e a l i t i e s in this work and the original work. This Study Freeze (1980a) Case Case A A B B C C D D E E F F G H H — 205 Tables 11-1 and 11-3 summarize the sensitivity analysis. The overland flow summaries presented in Table 11-3 include both Horton and Dunne overland flow (see Chapter 2 for definitions). Figure 7-2 shows the principal streamflow generation mechanisms in each of the ten cases. The value of the ratio Qp/Ppj i s similar to the runoff coefficient described by Dunne (1978). The overbars indicate mean values. The N° notation refers to the number of events in each of the 100-event experiments for which no runoff was generated. The combination of geometry and hydraulic conductivity values outlined in Table 11-1 leads to overland flow generation by the Horton and Dunne mechanisms for cases A and B respectively. The Horton mechanism i s dominant in case A for two reasons: 1. The average r a i n f a l l intensity of 2.0 x 10 ^  m / s exceeds the median hydraulic conductivity of 10 ~* m/s. 2. The depth of the water table (see Figure 7-2) over the greater part of the hillslope i s too large to allow i n i t i a l moisture deficits s? . to be overcome by individual r a i n f a l l events. The Dunne mechanism i s dominant for case B because: 1. The median hydraulic conductivity of 10 4 m/s i s greater than the average r a i n f a l l intensity of 2.0 x 10 ^  m/s. 2. The maximum thickness of the unsaturated zone (0.6 m) i s much smaller than in case A (2.25 m). The influence of the mean hydraulic conductivity u v in controlling the rainfall-runoff process i s clear from the descriptions of Horton runoff (case A) and Dunne runoff (case B). Case C produces runoff events in which the Horton and Dunne mechanism are both operative during the same 206 TABLE 11-3. Comparison of cases showing effect of hydraulic conductivity on streamflow. Overland Flow: Number of Elements in Percentage Range Case 0-25 26-50 51-75 76-100 A 44 58 81 17 0.51 11 B 7 102 84 7 0.62 7 C 0 21 122 57 0.76 5 D 158 31 11 0 0.12 26 E 0 87 113 0 0.54 25 F 0 83 116 1 0.54 19 G 0 97 103 0 0.54 28 H 52 52 74 22 0.49 6 I 55 43 75 27 0.49 5 J 67 32 43 58 0.51 1 event. A comparison of cases A and D in Table 11-3 indicates that when higher hydraulic conductivities are coupled with deeper water tables, the amount of runoff generated by any given climatic regime i s much reduced. The runoff coefficient i s much lower for case D than for case A, and the number of elements that seldom generate overland flow i s greater. The influence of the standard deviation in hydraulic conductivity °Y can be seen by comparing cases E and J with case A. In the f i r s t of these comparisons (case E and case A) the reduced range of hydraulic conductivity results in a more uniform response of overland flow sources. It also results in a reduced median peak flow Qp^ (Table 7-3) and a greater number of events that do not generate runoff N°. In the second comparison (case J and case A) the increased range results in a less uniform response of overland flow sources, an increased median flow rate, and fewer events that do not generate runoff. The influence of the autocorrelation parameter o^ . ±s indicated by comparison of cases H and I with case A. A reduction in the degree of autocorrelation (indicated by an increase in C y ) has a similar effect on the system (cases H and I) to an increase in a v (case J ) . Case G reflects a set of input parameters identical to those in case A, except that a l l the elements on the hillslope have the same hydraulic conductivity value of K s = 10 m/s. In other words, o"y = 0. A comparison of case G (the homogeneous case) with case A (the heterogeneous case) reveals the following. (1) The homogeneous hillslope reacts more uniformly with respect to the pattern of overland flow sources. There are no elements that contribute to overland flow in fewer than 25% of the events or greater than 75% of the events. (2) For the homogeneous case 208 the median peak flow i s slightly reduced and the mean time to peak tp^, i s significantly reduced. (3) The number of events during which no runoff i s generated i s much greater in the homogeneous case than in the heterogeneous case. (A) The runoff coefficient i s nearly the same for both cases. It i s not the total volumes of runoff that differ, i t i s the distribution of their component peaks and duration. In summary, as pointed out by Freeze (1980a), while the homogeneous hillslope reacts more uniformly than the heterogeneous hillslope with respect to the spatial pattern of overland flow sources, i t reacts less uniformly with respect to the distribution of peak flows that are generated. In effect, a homogeneous hillslope propagates the f u l l range of climatic variability to the streamflow peaks, whereas a heterogeneous hillslope acts in a certain sense as an attenuating medium. The cause of this attenuation must l i e in the diversity of interaction between r a i n f a l l intensity and hydraulic conductivity on a heterogeneous hillslope. The combined influence of the standard deviation and autocorrelation parameters i s illustrated by comparing case F with case A. The reduction in Oy i s offset by the increase in ot^ .. The uniform response in the overland flow sources i s similar to that seen for cases E and G. The closer relationship between case E and case F suggests that o"y is more important than ay. The work reported in this section i s an extension of Freeze (1980a). The following paraphrased comments of Freeze deserve repeating. 1. SCRRS produces hillslope runoff events that are individually and s t a t i s t i c a l l y quite representative of actual runoff events reported in the literature from experimental catchments. 209 2. The hydraulic conductivity distribution on a hillslope can be viewed as a spatial stochastic process. Its properties can be represented by the three parameters Uy, Oy and a y , where y y i s the mean of the log-transformed, log normal data, Oy i s the standard deviation and a y i s an autocorrelation parameter. The results indicate that each of these parameters exerts an important influence on the s t a t i s t i c a l properties of runoff events arising from a hillslope under a given climatic regime. The mean value i s the most important parameter; the standard deviation i s quite important; the autocorrelation parameter i s the least important. 3. Hydraulic conductivity influences overland flow generation directly in the Horton mechanism and indirectly, through control of the water table position, in the Dunne mechanism. In a shallow heterogeneous slope with a relatively large standard deviation in hydraulic conductivity, the simulations suggest that i t should be quite common to have both Horton and Dunne generation of overland flow from different points on the slope during the same storm event. 4. The limited set of comparisons outlined in this section do not begin to unravel the f u l l set of relationships between hillslope parameters and rainfall-runoff processes. Impact of precipitation and s o i l hydraulic property data In Chapter 7 a six-step procedure describing the stochastic conceptual rainfall-runoff simulator was presented. In this section that procedure i s modified to study the impact of spatial information using the ten hypothetical realities summarized in Tables 7-2 and 7-3. The philosophy behind these experiments i s to assign a f i n i t e number of 210 measurements to larger areas more representative of the typical level of information available. The f i r s t four steps in the simulation procedure are unchanged. These steps, described i n Chapter 7 and by Freeze (1980a), are: 1. Generation of time-independent hillslope parameters. 2. Generation of external storm properties for each r a i n f a l l event. 3. Generation of i n i t i a l hillslope conditions for each event. 4. Generation of the internal r a i n f a l l intensity pattern for each time step of each event. The modifications to the SCRRS procedure begin with step 5 and are summarized below. 5. Reassign spatial information assumed to have been measured at selected elements to neighboring elements. This includes the internal m' precipitation intensity pattern p „ for each time step of each event, and the three s o i l hydraulic properties: saturated hydraulic conductivity K S' i j , porosity n ^ j ? a n& the s o i l storage parameter 6. Calculate the i n f i l t r a t i o n rate f™. and the r a i n f a l l excess r™. for each time step of each event. m' 7. Calculate the streamflow hydrograph Q , ra= 1,2, 60 for each event. The notation here i s the same as defined in Chapter 7. The primes indicate that spatial information has been reassigned as described in step 5 and that subsequent results are so based. In step 5 three general scenarios were investigated for each of the hypothetical r e a l i t i e s . • 2 1 1 (a) Precipitation information (P™J) reassigned. Soil information (Kf., n. ., B. .) unchanged. (b) Soil information reassigned. Precipitation unchanged. (c) Both precipitation and s o i l information reassigned. Figure 11-1 shows the three hypothetical gaging networks used to reassign precipitation and s o i l information. In Figure 11-la the 200 element grid (Figure 7-la) i s represented by 50 measurement elements (shaded). For this design the appropriate information from each of the 50 observation elements was assigned to three neighboring elements. The grid i s made up of 50 compiled elements of dimension 20 by 20 m. Two additional hypothetical sampling schemes were also studied. Figure 11—lb shows the second grid of eight 50 by 50 m compiled elements, while Figure 11-lc shows the f i n a l grid of two 100 by 100 m compiled elements. Tables 11-4 through 11-13 summarize the data impact experiments performed in this section using SCRRS. The f i r s t case in each of these tables i s a summary of the SCRRS simulations described in Chapter 7. Results from these ten hypothetical r e a l i t i e s are used here as baselines for evaluating the impact of precipitation and s o i l hydraulic property data. In these tables cases 1, 2 and 3 reassign precipitation information, cases 4, 5 and 6 reassign s o i l information, and cases 7, 8 and 9 reassign both precipitation and s o i l information. For the ten tables cases 1, 4 and 7 use the 50 element grid shown in Figure 11-la for reassigning information, cases 2, 5 and 8 use the eight element grid shown in Figure 11—lb, and cases 3, 6 and 9 use the two element grid shown in Figure 11-lc. FIGURE 11-1. Three grid sampling schemes. (a) 50 compiled elements, (b) 8 compiled elements, (c) 2 compiled elements. For each grid the shaded elements represent measurement elements. TABLE 11-4. Comparison of data-impact experiments on case A. Case Number of Elements s Streamflow Summary Statistics Mean Standard Deviation Overland Flow: Number of Elements in Percentage Range P. . ni 1 % ^PK tPK % Qpr t p r B*j (mm) (lit.7Bec.) (hours) (mm) (lit.7sec.) (hours) 0-25 26-50 51-75 76-100 D^ % QPK 'PK A A-l A-2 A-3 A-4 A-5 A-6 A-7 A-8 A-9 200 50 8 . 2 200 200 200 50 8 2 200 200 200 200 50 8 2 50 8 2 9.5 9.5 9.5 9.5 9.0 9.0 9.5 9.5 9.0 9.5 91.0 96.0 96.0 120.0 93.0 110.0 120.0 98.0 120.0 160.0 0.42 0.39 0.44 0.42 0.44 0.44 0.31 0.42 0.47 0.33 18.0 18.5 18.0 18.0 18.0 17.0 18.0 18.0 17.5 18.0 160.0 160.0 170.0 200.0 160.0 170.0 190.0 170.0 180.0 240.0 0.19 0.19 0.21 0.16 0.20 0.18 0.19 0.20 0.19 0.19 44 43 44 44 50 49 79 50 46 82 58 57 55 57 48 70 21 46 73 18 81 78 80 79 93 34 98 93 34 100 17 22 21 20 9 47 2 22 47 0 0.51 0.51 0.51 0.51 0.49 0.49 0.51 0.51 0.49 0.51 1 1 1 9 1 1 6 1 1 20 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.87 1.0 0.98 0.86 0.67 0.72 0.92 0.74 0.92 -0.25 0.99 0.90 0.94 0.77 0.51 -0.25 TABLE 11-5. Comparison of data-impact experiments on case B. Number of Elements Streamflow Summary Statistics Mean Standard Deviation Overland Flow: Number of Elements in Percentage Range Case i j P A J B j^ (mm) ( l i t j s e c . ) (hours) (mm) (lit.7sec.) (hours) 0-25 26-50 51-75 76-100 % QPK fcPK B 200 200 11.5 98.0 0.61 22.5 190.0 0.23 7 102 84 7 0.62 7 — — — B-l 50 200 11.5 100.0 0.58 23.0 190.0 0.23 6 101 85 8 0.62 6 1.0 1.0 0.69 B-2 8 200 11.5 100.0 0.61 22.5 190.0 0.24 9 102 81 8 0.62 6 1.0 0.99 0.49 B-3 2 200 11.5 120.0 0.56 23.0 220.0 0.24 10 102 80 8 0.62 7 1.0 0.93 0.37 B-4 200 50 11.5 98.0 0.64 22.5 190.0 0.24 10 102 82 6 0.62 7 1.0 1.0 0.83 B-5 200 8 12.0 110.0 0.53 23.0 190.0 0.22 4 110 60 26 0.65 11 1.0 0.99 0.26 B-6 200 2 10.5 89.0 0.47 22.0 170.0 0.31 1 118 80 1 0.57 16 0.99 0.99 -0.68 B-7 50 50 11.5 100.0 0.58 23.0 190.0 0.23 8 103 82 7 0.62 7 1.0 1.0 0.85 B-8 8 8 12.0 110.0 0.53 23.0 190.0 0.22 5 102 67 26 0.65 10 1.0 0.99 0.23 B-9 2 2 10.5 120.0 0.44 22.0 210.0 0.28 1 102 96 1 0.57 17 0.99 0.93 -0.77 TABLE 11-6. Comparison of data-impact experiments on case C. Streamflow Summary Statistics Number Overland Flow: of Elements Number of Elements Mean Standard Deviation in Percentage Range 1 J „ „ • ^D E f n-t *i °-n °-PIT tp? % Qpr tpp — „ Case P ± B*j (mm) (lit.7sec.) (hours) (mm) (lit.7sec.) (hours) 0-25 26-50 51-75 76-100 Pp N° Q D Q p K t p R C 200 200 1A.0 120.0 0.47 24.5 200.0 0.19 0 21 122 57 0.76 5 — — — C-1 50 200 14.5 120.0 0.44 25.0 200.0 0.19 0 20 121 59 0.78 5 1.0 1.0 0.76 C-2 8 200 14.0 130.0 0.50 24.5 210.0 0.20 1 22 121 56 0.76 4 1.0 0.99 0.50 C-3 2 200 14.0 150.0 0.47 24.5 240.0 0.20 1 21 122 56 0.76 4 1.0 0.89 0.56 C-4 200 50 14.0 120.0 0.50 24.5 200.0 0.18 0 23 123 54 0.76 3 1.0 1.0 0.56 C-5 200 8 14.0 130.0 0.50 24.5 200.0 0.14 0 27 119 59 0.76 3 1.0 1.0 0.57 C-6 200 2 14.0 130.0 0.39 24.5 200.0 0.22 0 39 93 68 0.76 12 1.0 0.99 -0.23 C-7 50 50 14.5 120.0 0.47 25.0 200.0 0.19 0 20 121 ' 59 0.78 5 1.0 0.99 0.79 C-8 8 8 14.0 140.0 0.50 24.5 210.0 0.15 0 24 120 56 0.76 3 1.0 0.98 0.53 C-9 2 2 14.0 170.0 0.39 24.5 250.0 0.20 0 33 88 79 0.76 12 1.0 0.82 -0.11 TABLE 11-7. Comparison of data-impact experiments on case D. Number of Elements Streamflow Summary Statistics Mean Standard Deviation Overland Flow: Number of Elements in Percentage Range Case P. . n i i % QPK • fcPK % QPK tpR-B*j (mm) (lit.7sec.) (hours) (mm) (lit.7sec.) (hours) 0-25 26-50 51-75 76-100 % QPK tPK D 200 200 .2.3 29.0 0.44 5.5 63.0 0.28 158 31 11 0 0.12 26 — — — D-l 50 200 2.4 31.0 0.42 6.0 66.0 0.28 158 32 10 0 0.13 26 1.0 0.99 0.92 D-2 8 200 2.4 30.0 0.44 5.5 64.0 0.28 157 32 11 0 0.13 24 1.0 0.99 0.73 D-3 2 200 2.4 39.0 0.42 6.0 84.0 0.24 157 31 12 0 0.13 24 1.0 0.85 0.83 D-4 200 50 2.1 29.0 0.47 5.0 63.0 0.28 158 34 7 1 0.11 22 0.99 1.00 0.70 D-5 200 8 3.1 56.0 0.42 6.0 100.0 0.27 150 21 29 0 0.17 27 0.95 0.42 0.70 D-6 200 2 2.6 42.0 0.14 7.5 110.0 0.19 162 38 0 0 0.14 59 0.90 0.34 -1.19 D-7 50 50 2.2 31.0 0.44 5.5 67.0 0.31 159 33 8 . 0 0.12 27 1.0 0.99 0.86 D-8 8 8 3.1 62.0 0.42 6.0 110.0 0.31 150 24 26 0 0.16 31 0.96 0.14 0.67 D-9 2 2 2.5 54.0 0.14 7.0 140.0 0.20 177 23 0 0 0.14 67 0.91 -0.79 -1.33 r o r-> CTv TABLE 11-8. Comparison of data-impact experiments on case E. Number of Elements Streamflow Summary Statistics Mean Standard Deviation Overland Flow: Number of Elements in Percentage Range Case B j^ (mm) (lit./sec.) (hours) (mm) (lit.7sec.) (hours) 0-25 26-50 51-75 76-100 D D^ P^K PK E 200 200 10.0 84.0 0.42 20.5 170.0 0.28 0 87 113 0 0.54 25 — — — E-l 50 200 10.0 88.0 0.39 21.0 180.0 0.27 0 76 124 0 0.54 25 1.0 0.99 0.78 E-2 8 200 9.5 89.0 0.42 20.5 180.0 0.28 0 84 116 0 0.51 25 1.0 0.99 0.67 E-3 2 200 9.5 110.0 0.39 21.0 210.0 0.26 0 95 105 0 0.51 24 1.0 0.88 0.79 E-4 200 50 9.5 83.0 0.42 20.5 170.0 0.28 0 97 103 0 0.51 25 1.0 1.0 0.90 E-5 200 8 10.0 87.0 0.39 21.0 180.0 0.26 0 83 117 0 0.54 26 1.0 1.0 0.81 E-6 200 2 9.5 87.0 0.28 20.5 170.0 0.23 0 132 68 0 0.51 29 1.0 1.0 0.31 E-7 50 50 10.0 88.0 0.39 21.0 180.0 0.28 0 78 122 . 0 0.54 26 1.0 0.99 0.78 E-8 8 8 10.0 93.0 0.39 21.0 180.0 0.27 0 71 129 0 0.54 28 1.0 0.99 0.69 E-9 2 2 9.5 110.0 0.31 20.5 220.0 0.24 0 111 89 0 0.51 29 1.0 0.87 0.39 TABLE 11-9. Comparison of data-impact experiments on case F. Streamflow Summary Statistics Number Overland Flow: of Elements Number of Elements Mean Standard Deviation in Percentage Range E f " i i QD Q PK - t PK QD QPK t PK -3-Case P i J B^J (mm) (lit.7sec.) (hours) (mm) (lit.7sec.) (hours) 0-25 26-50 51-75 76-100 N° Q D QpJ, t p K F 200 200 10.0 87.0 0.36 21.0 170.0 0.22 0 83 116 1 0.54 19 — — — F-l 50 200 10.0 91.0 0.36 21.0 180.0 0.21 0 76 123 1 0.54 17 1.0 0.99 0.83 F-2 8 200 10.0 92.0 0.42 20.5 180.0 0.24 0 75 125 1 0.54 18 1.0 1.0 0.67 F-3 2 200 10.0 110.0 0.42 21.0 220.0 0.23 0 88 111 1 0.54 17 1.0 0.89 0.62 F-A 200 50 9.5 85.0 0.33 20.5 170.0 0.24 0 104 96 0 0.51 25 1.0 1.0 0.30 F-5 200 8 10.0 91.0 0.28 20.5 170.0 0.20 0 117 83 0 0.54 27 1.0 1.0 0.11 F-6 200 2 10.0 100.0 0.31 20.5 180.0 0.19 0 92 108 0 0.54 22 1.0 0.99 0.17 F-7 50 50 10.0 89.0 0.31 21.0 180.0 0.24 0 99 101 . 0 0.54 26 1.0 0.99 0.22 F-8 8 8 10.0 95.0 0.28 20.5 180.0 0.21 0 107 93 0 0.54 27 1.0 0.99 0.10 F-9 2 2 10.0 130.0 0.31 21.0 230.0 0.19 0 94 106 0 0.54 23 1.0 0.79 0.15 TABLE 11-10. Comparison of data-impact experiments on case G. Number of Elements Streamflow Summary Statistics Mean Standard Deviation Overland Flow: Number of Elements in Percentage Range Case n. . Bxq i j t\ • (mm; 1QBK [ lit.1 sec.) (hours) <\ (mm) QBK (lit./sec.) *"P7 (hours) 0-25 26-50 51-75 76-100 *D *D N° E f P^K tPK G 200 200 10.0 87.0 0.31 21.0 170.0 0.23 0 97 103 0 0.54 28 — — — G-l 50 200 10.0 92.0 0.28 21.0 180.0 0.23 0 90 110 0 0.54 28 1.0 0.99 0.93 G-2 8 200 10.0 92.0 0.33 21.0 180.0 0.24 0 104 96 0 0.54 29 1.0 0.99 0.78 G-3 2 200 10.0 110.0 0.31 21.0 220.0 0.23 0 83 117 0 0.54 29 1.0 0.88 0.84 G-4 200 50 10.0 87.0 0.31 21.0 170.0 0.23 0 97 103 0 0.54 28 1.0 1.0 1.0 G-5 200 8 10.0 87.0 0.31 21.0 170.0 0.23 0 97 103 0 0.54 28 1.0 1.0 1.0 G-6 200 2 10.0 87.0 0.31 21.0 170.0 0.23 0 97 103 0 0.54 28 1.0 1.0 1.0 G-7 50 50 10.0 92.0 0.28 21.0 180.0 0.23 0 89 111 • 0 0.54 28 1.0 0.99 0.93 G-8 8 8 10.0 92.0 0.33 21.0 180.0 0.24 0 104 96 0 0.54 29 1.0 0.88 0.84 G-9 2 2 10.0 110.0 0.31 21.0 220.0 0.23 0 83 117 0 0.54 29 1.0 0.88 0.84 TABLE 11-11. Comparison of data-worth experiments on case H. Number of Elements Streamflow Summary Statistics Mean Standard Deviation Overland Flow: Number of Elements in Percentage Range Case P. i j n. . B^. ^PV *"P7 ^f) QpF ''PIT (mm) (lit.7sec.) (hours) (mm) (lit./sec.) (hours) 0-25 26-50 51-75 76-100 <PK PK H 200 200 9.0 75.0 0.67 17.0 140.0 0.25 52 52 74 22 0.49 6 — —' — H-l 50 200 9.0 76.0 0.64 17.5 140.0 0.26 52 52 73 23 0.49 6 1.0 1.0 0.58 H-2 8 200 9.0 78.0 0.69 17.0 140.0 0.28 57 45 77 21 0.49 6 1.0 1.0 0.49 H-3 2 200 9.0 94.0 0.69 17.5 170.0 0.27 55 48 69 28 0.49 6 1.0 0.90 0.38 H-4 200 50 8.5 78.0 0.61 17.5 150.0 0.22 41 67 72 20 0.46 6 1.0 1.0 0.57 H-5 200 8 9.0 76.0 0.58 18.0 140.0 0.39 31 77 79 13 0.49 17 1.0 0.99 -1.11 H-6 200 2 8.0 100.0 0.75 13.0 160.0 0.28 100 0 9 91 0.43 10 0.94 0.91 -0.05 H-7 50 50 9.0 80.0 0.61 17.5 150.0 0.22 40 65 69 • 26 0.49 6 1.0 1.0 0.54 H-8 8 8 9.0 83.0 0.58 18.0 150.0 0.42 30 75 82 13 0.49 17 1.0 0.98 -1.25 H-9 2 2 8.0 130.0 0.72 13.5 200.0 0.31 100 0 0 100 0.43 12 0.93 0.57 -0.31 r-o O TABLE 11-12. Comparison of data-impact experiments on case I. Case Number of Elements Streamflow Summary Statistics Mean Standard Deviation Overland Flow: Number of Elements in Percentage Range n i 1 % QpK fcPK % %K 'PIT B*j (mm) (lit.7sec.) (hours) (mm) (lit.7sec.) (hours) 0-25 26-50 51-75 76-100 P^K UPK I 200 200 9.0 77.0 0.64 17.0 140.0 0.31 55 43 75 27 0.49 5 — 1-1 50 200 9.5 81.0 0.61 17.5 140.0 0.31 56 41 74 29 0.51 5 1.0 1.0 0.65 1-2 8 200 9.0 86.0 0.61 17.0 140.0 0.31 55 41 76 28 0.49 5 1.0 0.99 0.63 1-3 2 200 9.0 100.0 0.72 17.0 170.0 0.28 58 38 73 31 0.49 5 1.0 0.90 0.48 1-4 200 50 9.5 86.0 0.58 17.5 150.0 0.28 55 35 82 28 0.51 5 1.0 0.99 0.59 1-5 200 8 7.5 70.0 0.58 13.5 100.0 0.36 93 32 10 68 0.41 6 0.94 0.92 0.01 1-6 200. 2 7.5 87.0 0.28 16.5 170.0 0.22 68 77 55 0 0.41 29 0.98 0.94 -1.96 1-7 50 50 9.5 90.0 0.56 18.0 150.0 0.31 54 32 81 • 33 0.51 5 1.0 0.97 0.54 1-8 8 8 7.5 78.0 0.56 13.5 110.0 0.36 92 33 5 70 0.41 6 0.94 0.92 -0.09 1-9 2 2 7.5 110.0 0.31 16.5 210.0 0.23 68 43 89 9 0.41 31 0.97 0.56 -1.88 TABLE 11-13. Comparison of data-impact experiments on case J. Number of Elements Streamflow Summary Statistics Case Mean Standard Deviation Overland Flow: Number of Elements in Percentage Range n. . B^ Q r i Q p p ' Q ~ Qpp- t p « (mm) (lit.7sec.) (hours) (mm) (lit.fsec.) (hours) 0-25 26-50 51-75 76-100 PK J J - l J-2 J-3 J-4 J-5 J-6 J-7 J-8 J-9 200 50 8 2 200 200 200 50 8 2 200 200 200 200 50 8 2 50 8 2 9.5 9.5 9.5 9.5 9.5 8.0 9.5 9.5 8.0 9.5 96.0 100.0 100.0 130.0 99.0 120.0 140.0 100.0 130.0 180.0 0.44 0.42 0.44 0.44 0.44 0.47 0.36 0.42 0.47 0.36 16.5 17.0 16.5 16.5 16.0 13.5 15.0 16.5 13.5 15.0 150.0 150.0 160.0 190.0 150.0 160.0 190.0 160.0 180.0 250.0 0.14 0.12 0.14 0.11 0.13 0.12 0.14 0.13 0.12 0.15 67 67 68 68 79 97 96 79 96 96 32 34 33 32 17 26 4 17 27 4 43 42 42 40 50 27 0 50 27 0 58 0.51 57 0.51 57 0.51 60 0.51 54 0.51 50 0.43 100 0.51 54 0.51 50 0.43 100 0.51 1.0 1.0 1.0 1.0 1.0 0.99 0.86 1.0 2 9 1 2 12 0.96 0.96 0.98 0.81 1.0 0.99 0.96 0.91 0.98 0.15 0.56 0.45 0.65 0.63 0.13 -1.50 0.83 0.26 -1.55 N5 to 223 The forecasting efficiencies, E^, reported in each of the ten tables are based upon the c r i t e r i a discussed in Chapter 3 and calculated with (3-3). For every table there are nine comparisons in which the baseline case i s considered to be observed and the case under study to be predicted. In the following paragraphs the impact of spatial precipitation and s o i l property data i s discussed with regard to the E^ analyses. The use of the E^ criterion for the SCRRS sensitivity studies presented in this section maintains a parallelism with the model efficiency results already reported in Chapters 8, 9, and 10 and the space-time tradeoff analyses in Chapter 12. A perusal of the E^ values on Tables 11-4 through 11-13 leads to the following comments: 1. In general, E^ values are excellent for the summary variable in a l l ten cases with a l l three network designs. 2. In general, E^ values are quite high for Qp^ and lower for tp^. 3. In general, the consequence of reassigning hydrologic information i s most pronounced for tp^. 4. In general, E^ values decrease with network density as would be expected. 5. In general, E^ values appear to be more dependent upon s o i l hydraulic property information than precipitation data (depending upon the spatial variability of each hydrologic parameter). The most important feature of cases A through J, in Tables 11-4 through 11-13, are the E^ values for Q^ , Qp^ and t p ^ reported for cases 7, 8 and 9. These results are separated out onto summary Tables 11-14 224 through 11-23. An item-by-item comparison of the E^ values in Tables 11-14 through 11-23 i l l u s t r a t e many of the same patterns commented on for Tables 11-4 through 11-13. Once again, the E^'s for are a l l very good, while the E^'s for Qp^ and t p ^ range from very good to very bad. A comparison of the E^ values from the SCRRS sensitivity analysis in Tables 11-14 through 11-23 with the E^ values for the rainfall-runoff model evaluations reported for real (Table 8-1) and synthetic data (Table 10-1) affords the following comments: 1. In general, the predictions from the SCRRS sensitivity analysis for a l l networks are far superior to those from the three model evaluations on the three real catchments. An explanation for this performance difference may be the small size of the hypothetical hillslopes. Figure 11-2 i l l u s t r a t e s the relative size of each of the three catchments and for the synthetic hillslopes. The predictions from the regression and unit hydrograph models using the SCRRS generated data are similar to the values reported in this chapter for the same size hillslopes. 2. In general, the Qp^ and t p ^ predictions are better for the SCRRS sensitivity analysis for the larger measurement networks than for the rainfall-runoff model evaluations on either the synthetic or real data sets. However, the E^ values are much more comparable between the regression and unit hydrograph models on the synthetic data and the SCRRS sensitivity analysis for the smaller measurement network. In summary, the E^'s for the regression and unit hydrograph model evaluations on the synthetic data set and the E f's for the SCRRS TABLE 11-14. Comparison of data-impact experiments on case A. *f Case A Measurements Qpg tp^ 50 1.0 0.99 0.90 8 1.0 0.94 0.77 2 1.0 0.51 -0.25 TABLE 11-15. Comparison of data-impact experiments on case B. Case B Measurements Q p R tp^ 50 1.0 1.0 0.85 8 1.0 0.99 0.23 2 0.99 0.93 -0.77 226 TABLE 11-16. Comparison of data-impact experiments on case C. Case C Measurements Qp^ tp^ 50 1.0 0.99 0.79 8 1.0 0.98 0.53 2 1.0 0.82 -0.11 TABLE 11-17. Comparison of data-impact experiments on case D. Case D Measurements Qp^ tp^ 50 1.0 0.99 0.86 8 0.96 0.14 0.67 2 0.91 -0.79 • -1.33 227 TABLE 11-18. Comparison of data-impact experiments on case E. Case E Measurements Qp^ tp^ 50 1.0 0.99 0.78 8 1.0 0.99 0.69 2 1.0 0.87 0.39 TABLE 11-19. Comparison of data-impact experiments on case F. Case F Measurements Qp Qp^ t p ^ 50 1.0 0.99 0.22 8 1.0 0.99 0.10 2 1.0 0.79 0.15 TABLE 11-20. Comparison of data-impact experiments on case G. E f Case G : Measurements Qpj, tp^ 50 1.0 0.99 0.93 8 1.0 0.99 0.78 2 1.0 0.88 0.84 TABLE 11-21. Comparison of data-impact experiments on case H. Case H Measurements Qp Qp^ tp^ 50 1.0 1.0 0.54 8 1.0 0.98 -1.25 2 0.93 0.57 -0.31 TABLE 11-22. Comparison of data-impact experiments on case I. Case I Measurements Qp^ tp^ 50 1.0 0.97 0.54 8 0.94 0.92 -0.09 2 0.97 0.56 -1.88 TABLE 11-23. Comparison of data-impact experiments on case J. Case J Measurements Qp^ tp^ 50 1.0 0.99 0.83 8 0.96 0.91 0.26 2 0.98 0.15 -1.55 230 AREA, km 2 Name 0.02 Synthetic hillslopes 0.1 R-5 0.13 HB-6 7.2 WE-38 WE-38 HB-6 R-5 Synthetic hillslopes r ' i 1 1 1 1 1 1—•-—i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r -0.5 1.0 1.5 2.0 2.5 DISTANCE, kilometers FIGURE 11-2. Comparison of the size of catchments and hillslopes used in this study. sensitivity analysis show striking similarity. The poor performance of the quasi-physically based model i s once again illustrated. Finally, there appears to be a relation between model performance and hillslope/catchment scale. The most important summary variable for most engineering applications i s Qp^. Therefore, a perusal of the E^'s for Qp^ in Tables 11-14 through 11-23 lead to the following comments. 1. In general, the E^ values for QpT^ are highest for the two heterogeneous Dunne hillslopes (case B and case C) and the homogeneous Horton hillslope (case G). 2. For a l l ten cases the value of E^ for Qp^ . i s reduced as the number of measurements decreases. For only two measurement points, E^'s vary from 0.93 to -0.79 (case B and case D respectively). The -0.79 value i s for a Horton hillslope with a higher mean hydraulic conductivity and a deeper water table. On this hillslope only the more extreme r a i n f a l l events produce runoff. The sparse sampling network i s not sufficient to characterize the distribution of high intensity r a i n f a l l or the hillslope properties. In summary, i t appears that Qpj, i s best predicted with a sampling design that includes spatially distributed measurements of r a i n f a l l and s o i l hydraulic properties. There appears to be a systematic relationship between the dominant streamflow generation mechanism, controlled by s o i l hydraulic properties and topography, and how effi c i e n t l y Qp^ can be predicted. To study the effects of the deterministic reassigning of hydrologic information described above, two random reassignment strategies were tested. For both of these scenarios the case A information (Tables 7-1 and 7-2) was used to excite SCRRS. The f i r s t strategy can be considered an unconditional reassignment. For each rainfall-runoff event in this experiment the hydrologic data (P™J» %-±y nij» ^ i j ^ ^ o r e a c n member element in a compiled element was randomly reassigned with the value(s) of a single member element. As an example, each of the four member elements in the 50 compiled elements (see Figure 11-la) were assigned the SCRRS generated information from one of the member elements. A random number generator was used to select the contributing member element. Other than the selection of the contributing elements, the simulation procedure follows exactly the same seven steps described earlier in this section. The unconditional approach i s used to study the effect of relaxing the assumption that some near surface s o i l hydraulic properties (eg., saturated hydraulic conductivity) are time-invariant. The second strategy for stochastically reassigning hydrologic information i s a conditional approach. In this experiment the random reassignment of contributing elements i s l e f t unchanged for the entire 100 event ensemble. This approach i s more defensible for real world analogy as there can only be one deterministic realization of each hillslope which can be sampled at specific spots. Tables 11-24 and 11-25 summarize the data impact experiments for the unconditional and conditional reassignment of contributing elements. These findings further i l l u s t r a t e many of the same points discussed above for Tables 11-4 through 11-13. It i s again very clear that the tp^ summary variable i s the most affected by reassigning spatially variable hydrologic information. TABLE 11-24. Comparison of unconditional data-impact experiments. Number of Elements Streamflow Summary Statistics Mean Standard Deviation Overland Flow: Number of Elements in Percentage Range Case l j KS. T^) ^PJf ^PF ^Pl? ^PF (mm) ( l i t j s e c . ) (hours) (mm) (lit.fsec.) (hours) 0-25 26-50 51-75 76-100 % QPK fcPK AU 200 200 9.5 91.0 0.42 18.0 160.0 0.19 44 58 81 17 0.51 11 — — — AU-1 50 200 9.5 93.0 0.42 18.0 160.0 0.20 43 56 82 19 0.51 11 1.0 1.0 0.92 AU-2 8 200 9.5 110.0 0.44 18.5 180.0 0.22 41 59 81 19 0.51 11 1.0 0.96 0.68 AU-3 2 200 9.5 120.0 0.44 18.5 200.0 0.19 43 56. 81 20 0.51 7 0.99 0.84 0.31 AU-4 200 50 9.5 93.0 0.42 18.5 160.0 0.21 31 86 76 7 0.51 13 1.0 1.0 0.69 AU-5 200 8 9.0 89.0 0.50 18.5 160.0 0.28 16 78 106 0 0.49 14 0.98 0.97 -1.29 AU-6 200 2 9.5 100.0 0.39 20.0 190.0 0.28 0 156 44 0 0.51 24 0.89 0.90 -1.03 AU-7 50 50 9.5 95.0 0.42 18.5 160.0 0.21 31 84 76 . 9 0.51 13 1.0 0.99 0.69 AU-8 8 8 9.0 100.0 0.47 18.5 180.0 0.27 16 76 108 0 0.49 17 0.99 0.96 -1.29 AU-9 2 2 9.5 130.0 0.42 20.0 230.0 0.31 0 164 36 0 0.51 22 0.89 0.64 -1.17 ho to TABLE 11-25. Comparison of conditional data-impact experiments. Number of Elements Streamflow Summary Statistics Mean Standard Deviation Overland Flow: Number of Elements in Percentage Range Case i j *D EK PK <D PK (mm) (lit.7sec.) (hours) (mm) (lit.7sec.) (hours) 0-25 26-50 51-75 76-100 *D % <W 'PK AC 200 200 9.5 91.0 0.42 18.0 160.0 0.19 44 58 81 17 0.51 11 — — — AC-1 50 200 9.5 90.0 0.42 18.0 160.0 0.21 43 54 81 22 0.51 11 1.0 1.0 0.88 AC-2 8 200 9.5 130.0 0.39 18.0 200.0 0.18 44 54 76 26 0.51 11 1.0 0.83 0.85 AC-3 2 200 8.5 140.0 0.58 17.0 250.0 0.14 46 65 67 22 0.49 11 0.99 0.58 -0.17 AC-4 200 50 9.0 83.0 0.50 17.5 150.0 0.25 53 46 81 20 0.49 12 1.0 0.99 0.16 AC-5 200 8 9.0 87.0 0.33 18.5 160.0 0.22 27 92 78 3 0.49 15 1.0 0.99 -0.10 AC-6 200 2 3.4 43.0 0.53 8.5 100.0 0.39 100 93 7 0 0.18 32 0.60 0.74 -3.45 AC-7 50 50 9.0 85.0 0.53 17.5 150.0 0.28 53 48 74 . 25 0.49 12 1.0 0.99 -0.26 AC-8 8 8 9.0 120.0 0.39 18.5 210.0 0.24 20 97 83 0 0.49 16 1.0 0.83 0.05 AC-9 2 2 3.3 56.0 0.47 8.0 130.0 0.33 100 91 9 0 0.19 32 0.57 0.88 -2.39 S3 235 In summary, the results presented in this section are generally consistent and revealing. Based upon these analyses i t would appear that the characterization of spatially variable precipitation and near-surface s o i l hydraulic properties i s more important for describing Qp^ and tp^ than Qp. The E^ values for Q^ , and tp^ for cases 7, 8 and 9 for cases AU and AC in Tables 11-24 and 11-25 are separated out onto summary Tables 11-26 and 11-27. Inspection of these two tables shows the same systematic improvement for predictions with the number of measurements that i s seen in Tables 11-14 through 11-23. The same relationship i s not as clear for either Qp^ or tp^. Comparison of Tables 11-14, 11-26 and 11-27 show that for the deterministic, unconditional, and conditional assignment schemes used to assign contributing elements there i s l i t t l e difference in the E^ values for and Qp^« However, there are major disparities for the tpj, values. It i s the belief of the author that the simulations carried out on hillslope A are representative of a l l ten hillslopes. In this chapter the stochastic-conceptual rainfall-runoff simulator (Chapters 7 and 10) was subjected to a sensitivity analysis to investigate (a) the effects that the hydraulic conductivity distribution parameters have upon the rainfall-runoff process and (b) the impact of spatially variable hydrologic data. TABLE 11-26. Comparison of unconditional data-impact experiments. Case AU Measurements Qp^ tp^ 50 1.0 0.99 0.69 8 0.99 0.96 -1.29 2 0.89 0.64 -1.17 TABLE 11-27. Comparison of conditional data-impact experiments. Case AC Measurements Qp^ tp^ 50 1.0 0.99 -0.26 8 1.0 0.83 0.05 2 0.57 0.88 -2.39 237 CHAPTER 12 Extensions and Future Work In this chapter possible directions of future research related to the current work are discussed. Scale One of the most important areas for research in a l l of hydrology concerns the effect scale has upon hydrologic relationships. In rainfall-runoff studies the problems associated with transferring results from small experimental catchments to larger watersheds have been recognized by many authors (eg., Amerman & McGuinness, 1967; Engman et a l . , 1971; Pilgrim et a l . , 1982; Pilgrim, 1983). The identification of c r i t i c a l parameters for the application of a rainfall-runoff model in a large basin i s often based upon small plot (or even laboratory) findings that characterize the f i e l d hydrology poorly . Due to this problem of scale, the estimation of useful rainfall-runoff model parameters can often become an arduous quest. There are many excellent philosophic and review papers addressing the problems and questions of scale in hydrology (eg., Church & Mark, 1980; Dooge, 1983, 1984). There have recently been two conferences devoted entirely to scale problems in hydrology (Rodriguez-Iturbe & Gupta, 1983; Gupta et a l . , 1986). To study scale interrelationships inherent to the rainfall-runoff process two different, but complementary, research strategies are proposed. The f i r s t approach for quantifying the rainfall-runoff scale phenomenon involves the use of f i e l d data. The potential for this type of investigation already exists to some extent. The Washita River Experimental Watershed in Oklahoma (Chapters 5 and 6) has hydrologic data for both small and large catchments. As the smaller catchments are contained within the larger catchments there i s an excellent opportunity here to study scale. Unfortunately many of these stations were mothballed in 1978. However, i t i s promising that there i s renewed interest in these stations and a move for their re-establishment to study mesoscale hydrolog-i c a l and meteorological phenomena i s underway (Brutsaert et a l . , 1985). It would be of particular interest to this author to see a new experimental catchment established in Hawaii (the author's current home). The proposed catchment would have an instrumentation network designed to provide pertinent information for studying hillslope processes and scale relationships as they relate to modeling efforts. Because of the climate a Hawaiian catchment would provide many rainfall-runoff events per year and not be handicapped with cold weather hydrologic phenomena. As with the Oklahoma catchments, however, a long term commitment would be required for such a program. Finally, the proposed f i e l d investigations must seriously address the relationship between preferred pathways and s o i l hydraulic properties. The spatial distribution of preferred pathways w i l l almost certainly result in parameters that are probabilistically quantified for rainfall-runoff modeling. The second approach suggested here to investigate the questions of scale would make use of the stochastic-conceptual rainfall-runoff simulator (SCRRS) described in Chapter 7. It i s the intention of this author to adopt SCRRS for the efficient simulation of much larger hillslopes than reported in this thesis. With larger conditional synthetic hillslopes i t w i l l be possible to group hillslope elements into progressively larger areas and analyze the scale relationship between rainfall-runoff relationships as they are controlled by spatial variations in near surface s o i l hydraulic properties, variations in r a i n f a l l patterns, and hillslope geometry. The f i r s t approach might mistakenly be considered only in the domain of a f i e l d hydrologist while the second approach could wrongly be construed as a concoction of a theoretical hydrologist not willing to face the real world. This i s not the intention, however, as data collected from the f i r s t approach could help to excite SCRRS in the second approach. Also, the conceptual findings from the second approach could provide useful guidelines for further data collection in the f i r s t approach. Thus, by combining the two approaches, basic contributions to the understanding of scale aspects of hydrology can be made. Model Evaluation The evaluation of underlying rainfall-runoff modeling techniques has. only been initiated with this thesis (Chapters 8 and 10, also see Chapter 2 for review of previous comparative studies). There is much work to be done. Only small upland catchments and hypothetical hillslopes were employed in the current work. In the future, rigorous analysis in the same s p i r i t w i l l need to be carried out for much larger watersheds. At the larger hydrologic scales engineering decisions are influenced by the selection of a particular rainfall-runoff model. However, there are no standard c r i t e r i a to base the selection of these models for appropriate problems. Future work needs to be directed at establishing 240 c r i t e r i a to judge model performance by analyzing the economic consequences of simulation error for various engineering applications. To establish the relative strengths and weaknesses of each modeling technique both real and synthetic data sets w i l l be required. The proposed extension of model comparisons i s intimately coupled to the scale investigation experiments described i n the previous section. The rainfall-runoff techniques that would be evaluated in the continuing study would be of the same levels of abstraction used in this study (Chapter 4) but not necessarily the same models. By continued modeling of hypothetical rainfall-runoff r e a l i t i e s (at various scales) based upon synthetic hillslopes and r a i n f a l l patterns that faithfully represent near surface hydrology i t w i l l be possible to evaluate models under known conditions. By testing models outside of their calibrated ranges and synthetic data we can learn more about expected model performance for situations where conditions are unknown (i. e . extreme events). There are already moves in this direction in the hydrologic research world. Steve Burges of the University of Washington has recently initiated a rainfall-runoff evaluation study. The investigation of Burges (1985) parallels the s p i r i t of this thesis and i t s extensions. Burges and his colleagues are evaluating the capabilities of conceptual models using hypothetical catchments developed from the deterministic hillslope segment model of Smith and Hebbert (1983), which i s perhaps the most rigorous quasi-physically based hillslope model currently available. 241 Conditional Stochastic Simulation The stochastic-conceptual analysis of rainfall-runoff processes (SCRRS) carried out by Freeze (1980a) was a foundation for much of the work in this thesis (see Chapters 7, 10, and 11). It must therefore be noted that Beven (1981) raised important queries concerning the differences between parameter values measured at various grid scales and their interaction with different grid scales used in simulation. In his comment on Freeze's paper Beven (1981, p. 431) argues that "the multiple realizations of the Monte Carlo analysis make no reference to any original parameter data set. A real catchment i s not a collection of multiple samples from an underlying stochastic process; i t i s rather a single specific realization of such a process. In fact, the real stochastic process may be nonstationary in space and time but must reflect the integrative nature of the geomorphological, pedological, and biological history of the catchment under natural conditions". The argument being made by Beven i s in effect an argument for conditional rather than unconditional simulation (DeMarsily, 1984). The hydrologic information available from the R-5 catchment (Chapters 5, 6 and 9) would allow an extension of the analysis to address Beven's comment. However, i t would be necessary to employ summary s t a t i s t i c a l parameters that represent the areas of grid elements modeled and not the point measurements themselves. It was suggested in Chapter 10 that the SCRRS hillslopes would have held more rigor i f the spatial patterns of s o i l parameters were generated from a multivariate process. However, the R-5 supplemental data (Chapters 6 and 9) revealed no cross-correlation between near surface parameters (eg., saturated hydraulic conductivity and porosity) at the scale of measurement. This lack of cross-correlation structure within the R-5 data set indicates that the inclusion of a multivariate generator for the proposed analysis i s not necessary. Because SCRRS incorporates both Horton and Dunne streamflow generation mechanisms (Chapter 7) the work suggested in this section would be an excellent opportunity to investigate the practical usefulness of stochastic-conceptual analysis in the f i e l d . R-5 Water Balance The research proposed in the proceeding sections is designed to address fundamental concepts. The analysis suggested in this section lingers as somewhat of a loose end. Although peripheral to this thesis, maximum evapotranspiraton (ET) rates should be determined for the R-5 rainfall-runoff events reported in Chapter 5. The data required for this analysis with the Priestley-Taylor (1972) equation i s available. The Priestley-Taylor model should be suitable for the R-5 grassland catchment. The estimation of ET combined with rainfall-runoff data would enable water balance calculations designed to identify R-5 i n f i l t r a t i o n as the residual. The i n f i l t r a t i o n and ET results might help to characterize the performance of the quasi-physically based rainfall-runoff model reported in Chapter 8. The proposed ET analysis would therefore be supplemental but complimentary to the existing model evaluation. Data-Worth The central thread of this thesis has been the evaluation of underlying rainfall-runoff modeling techniques. It should be obvious however, that rainfall-runoff models and the data used to excite them are intimately related. Therefore, as noted by Freeze (1982a), model assessment implies a concomitant assessment of data availability and worth. The very collection of information suggests that the data have some use and therefore worth (Dawdy, 1979). However, the economic worth of hydrologic data (James and Lee, 1971) i s not always easy to assess. A particular data set may have significant value for one type of model and l i t t l e or no value to another. The collection of hydrologic data for rainfall-runoff modeling may reflect the model users marginal u t i l i t y for additional information. It has often been shown that the f i r s t data collected yields the greatest information and therefore had the greatest value. It i s subsequently up to the user to determine when enough information has been collected for a given problem. The u t i l i t y ascribed by a data user to additional information depends upon the cost of the supplemental data (including the cost of project delay), relative to the value of the data. The marginal u t i l i t y concept is illustrated in Chapter 9. We saw that a large increase in the quantity of near surface s o i l hydraulic property data contained l i t t l e new information for exciting the quasi-physically based rainfall-runoff model. Therefore, the marginal u t i l i t y for additional information of this type i s quite low. A cost-benefit analysis i s outside the scope of this thesis. However, the model evaluation experiments designed for this study set a foundation for possible future work that would incorporate economic analysis. Network-Design Langbein (1979) described network-design in terms of planning the investment of capital and s k i l l s . Langbein characterized data planning choices and decisions as: How much to expend for what type of data, of what quality, when, where, and for how long. Network-design is a decision analysis problem. The optimal network configuration (or sampling scheme) i s the one for which established objectives are satisfied more effici e n t l y than for any other configuration. Optimization i s usually accomplished (formally or informally) by maximizing or minimizing an objective function subject to some constraints. Both the objective function and the constraints are usually couched in economic terms. Hydrologic networks need to be carefully planned to provide useful information, often for more than one user, that i s not laced with redundancy. The data from an efficient network should also have long term uses and not become obsolete as methods of hydrologic analysis go in and out of style. The rain gage network located near Chickasha, Oklahoma that was established in 1961 (Nicks, 1966) i s an example of a well planned network that continues to provide useful information for current research (Brutsaert et a l . , 1985). Clearly, data-worth and network-design characteristics are closely associated. Data-worth and network-design philosophy related to the f i e l d component of this research was briefly discussed in Chapters 6 and 9. The coupling of economic analysis with network-design strategy i s also outside the scope of this thesis. The use of cost-benefit analysis to identify optimal network-design i s a logical and important next step for future study. Space-Time Tradeoffs The concepts of data-worth and network-design are well entrenched in the hydrologic literature. Perhaps less well established i s the concept of space-time tradeoffs. In 1954 Walter B. Langbein described "geographic sampling" and "time sampling". The concept of space-time tradeoffs has i t s roots in Langbein's early paper. Concepts of space-time tradeoffs have since been employed by Moss and coworkers (Moss and Karlinger, 1974; Tasker and Moss, 1979; Moss 1979a, b) and Rodriguez-Iturbe and coworkers (Rodriguez-Iturbe and Mejia, 1974; Bras and Rodriguez-Iturbe, 1976b; Lenton and Rodriguez-Iturbe, 1977). Recently, Freeze (1978, 1982a, b) proposed using the space-time tradeoff concept as a linch-pin between data-worth considerations and the rigorous evaluation of various underlying rainfall-runoff modeling techniques. Following Freeze (1982a, b), a space-time tradeoff w i l l be defined as the relative increase in model efficiency that can be achieved through an increase in the density of measurement points as opposed to a lengthening of records. Most documented examples of space-time tradeoffs have been for a single data set using only one modeling technique. It i s the belief of this writer that there i s some promise for extending the idea of space-time tradeoffs across the hydrologic data sets of various modeling techniques. The possibility of improving the efficiency of rainfall-runoff predictions by increasing geometrically-distributed measurements of spatially-variable, time-invariant catchment parameters on a one-time collection basis, and thereby reducing the need for long continuous rainfall-runoff records, i s seductive. The concept of data-worth i s married to cost-benefit analysis. ! Hence, increases in model efficiency due to improvements in the data acquisition network should be subject to economic justification. If space-time tradeoffs do exist across hydrologic data sets, then the demands of increasing the efficiency of a rainfall-runoff model can be evaluated economically based upon the cost of obtaining required data and the penalty paid for delay. Thus, the economic parameters of a particular application may dictate the modeling technique and the corresponding validation data. The worth of these data w i l l be directly related to the size and cost of a project. The possible delays of a project to obtain additional data w i l l be characterized by the decreasing marginal u t i l i t y of the model. A suite of hypothetical space-time tradeoff curves for a single data set are shown in Figure 12-1. These curves represent combinations of spatial and temporal information that result in levels of equal efficiency for a single model. For any combination of spatial coverage and length of record in a data set there i s an associated cost. Superimposed on the curves in Figure 12-1 are parallel lines of equal cost. The intersection of equal cost lines with E^ lines provide a measure of space-time tradeoffs for a single mqdel. Presumably there would be a diagram like Figure 12-1 for each type of model. To prove the concept of space-time tradeoffs across model data sets one would have to compare a 247 t \ \ \ |\ \ \ Q rr o o n LU -t. DC % 0 ^ 1 ° \— * o w i l l SPATIAL COVERAGE (# of g a g e s ) FIGURE 12-1. Hypothetical space-time tradeoff curves resulting within the data set of a single modeling technique. suite of such diagrams. The hypothetical curves shown in Figure 12-2 represent a summary of two Figure 12-1 type diagrams. To further c l a r i f y the economic basis of space-time tradeoffs one might define two possible objectives with respect to Figure 12-1. One objective function would minimize costs given a target E^. In the alternative approach the objective function would maximize E^ given a constraint on cost. The technical analyses in the current study set the stage for possible future studies that would include cost-benefit analysis. Figure 12-2 i s presented to i l l u s t r a t e the concept of a space-time tradeoff across the hydrologic data sets of two rainfall-runoff models. The efficiency of model one i s assumed to improve with increasing spatial data (more measurement locations). The performance of model" two i s assumed to increase with more temporal data (longer records). In this simple il l u s t r a t i o n the two modeling techniques require different types of data but provide equal efficiency where their performance curves cross. This intersection (point T) represents the tradeoff across data sets. Before point T the selection of model two would be the optimal choice for an operational hydrologist wishing to maximize predictive efficiency but minimize costs. Beyond the tradeoff point the selection of model one i s optimal. At point T either model provides the same efficiency for an equivalent allocation of resources. Resources can represent, for example, people-hours, instrumentation outlays, project delay penalties, and operational and maintenance costs. 249 FIGURE 12-2. Hypothetical space-time tradeoff resulting across the data sets of two different modeling techniques. It should be obvious for the three rainfall-runoff models (Chapter A) employed in this study (Chapters 8, 9, 10) that space-time tradeoffs have multivariate dimensions. The spatial distribution of measurements, the length of measurement records, and the level of model performance are a l l important degrees of freedom for the three models. Figure 12-3 i s a schematic i l l u s t r a t i o n of the space-time tradeoff concept for the three rainfall-runoff modeling techniques used in this thesis. For the regression and unit-hydrograph models (Figure 12-3a) records of r a i n f a l l P(t) and runoff Q(t) are used to calibrate the models. An increase in the efficiency of these models can only be attained by extending the P(t) and Q(t) records or by adding additional rain gages (not shown in Figure 12-3) to better estimate the space-time behavior of r a i n f a l l . For both models, either approach requires a waiting period while new data are collected. This waiting period can be thought of as project delay time or as an opportunity loss. For the quasi-physically based model (Figure 12-3b) i t may also be possible to increase the efficiency by extending r a i n f a l l records and (or) by adding more gages. However, i t may also be possible to improve the performance of the quasi-physically based model by making additional measurements of physically-based model parameters. If near-surface s o i l hydraulic properties are considered to be time invariant then intensive data collection programs can be designed to generate physically-based model parameters that might lead to improved model efficiencies without long project delays. It i s the opinion of this author that the best diagnostic information of this type is in the form of spatially-distributed saturated hydraulic conductivity measurements. The \ s ^ \ reader i s asked to recall that the saturated hydraulic conductivity was shown i n Chapter 11 to be the most important near-surface s o i l hydraulic property for controlling the rainfall-runoff process. For this study two sets of preliminary space-time tradeoff experiments were performed: one for a stochastic-conceptual "hypothetical r e a l i t y " and one for a f i e l d problem. Only the regression and quasi-physically based models were considered in these experiments. The SCRRS-generated hillslope-A data set (200 hillslope elements, 100 rainfall-runoff events) i s discussed in detail in Chapter 7. Information i s abstracted from this synthetic data set for space-time tradeoff experiments using the same c r i t e r i a established earlier in parallel validation (Chapter 10) and sensitivity analyses (Chapter 11). Rain gage locations are discussed in Chapter 10. Two network scenarios are used for the regression model, a single gage and a five gage average (see Figure 7-1). This coverage represents only 0.5 and 2.5 percent respectively of the available data. Only the single gage network i s used for the quasi-physically based model. Soi l hydraulic property measurement sites are described in Chapters 10 and 11. The information reassignment c r i t e r i a established in Chapter 11 for the SCRRS sensitivity analysis i s employed here to define compiled hillslope elements that characterize the overland-flow-plane structure of the quasi-physically based model. Four network densities are used. They contain 2, 8, 50, and 200 sampling elements (see Figures 11-1 and 7-1). They represent 1, 4, 25, and 100 percent respectively of the available information. In the model validation experiments reported i n Chapter 10 the quasi-physically based model was unable to simulate approximately 25% of hillslope-A events due to numerical in s t a b i l i t y (see Chapter 10 for discussion). The results reported here (as in Chapter 10) are based solely on successful simulations. It must also be mentioned that the quasi-physically based model i s not calibrated in the space-time tradeoff experiments on hillslope A. A l l model parameters were gleaned directly from the synthetic data set as described above. Table 12-1 summarizes the forecasting efficiencies for the three summary output variables for the regression model for various combinations of spatial and temporal precipitation data on hillslope A. Table 12-2 summarizes the forecasting efficiencies for the three summary output variables for the quasi-physically based model for various combinations of record length (number of events) and spatial density (number of s o i l hydraulic property measurements) on hillslope A. A perusal of the values in Tables 12-1 and 12-2 leads to the following comments: 1. In general, for the regression model, the prediction of Q^ , Qp^, and tp^ i s very slightly improved with more rain gages and/or longer rainfall-runoff records. The number of events used to calibrate the model has a slightly greater influence on model performance than does the number of rain gages. Improvements in a l l cases are very small. The ab i l i t y of the regression model to predict Q^ , and i t s inability to predict t p F , are confirmed for a l l data sets. TABLE 12-1. Forecasting efficiencies E^ for the three summary output variables for the regression model on the hillslope A data set. *f Number of Number Case Rain Gages of Events Q p£ t p ^ 1 1 50 1 100 2 5 50 5 100 0.94 0.82 0.06 0.97 0.79 0.12 0.95 0.82 0.09 0.98 0.91 0.15 TABLE 12-2. Forecasting efficiencies E f for the three summary output variables for the quasi-physically based model on the . hillslope A data set. The numbers in brackets represent the number of events for which the model converged and upon which the E- results are based. Number E^ of Soil Measurement Number Case Locations of Events Qp Qp^ t p ^ 1 2 50(39) 0.08 0.05 -0.15 2 100(73) 0.17 0.15 -0.19 2 8 50(34) 0.30 0.22 0.27 8 100(64) 0.37 0.19 0.32 3 50 50(39) 0.31 0.36 0.02 50 100(74) 0.45 0.38 -0.04 4 200 50(40) 0.23 0.23 -0.48 200 100(76) 0.28 -0.52 -0.88 255 2. In general, for the quasi-physically based model, the values for QJJ and Qp^ are improved with more spatial and/or temporal information. This generality i s lost in the f i n a l case where model performance declined despite complete knowledge of the hillslope properties. In fact, the greatest improvement in forecasting efficiency for the three parameters appears to occur when the number of measurements i s increased from two to eight. In general, the model shows greater improvement in performance with more spatial information than i t does with more temporal information. Several of the E^ values for tp^ (and one for Qpjr) are negative. Recall once again that for perfect efficiency, = 1. The negative values in Table 12-2 infer that one would be better off using the observed means than the model predictions in these cases. 3. It appears there are some space-time tradeoffs within the data sets of the two modeling techniques. However, i t i s impossible to show a space-time tradeoff across Tables 12-1 and 12-2. This i s due to the fact that the values for the regression model are so much higher than those for the quasi-physically based model. Space-time tradeoffs across modeling techniques are probably not of any significance unless the models are more competitive. The superiority of the regression model over the quasi-physically based model was totally unanticipated in the early stages of this study. The fact that the regression model did perform better made i t very d i f f i c u l t to prove the existence of space-time tradeoffs across these two models. A comparison between the unit hydrograph and quasi-physically based models would probably yield results similar to those reported here. The reader i s asked to recall that the E-. values for the regression and unit hydrograph models reported in Table 10-1 are very similar. 4. The effect of additional information on values for both models was rather small. For the small scale hillslope this would seem to infer that small data networks may be optimal and that the law of diminishing returns comes into play earlier than would be expected. Further study incorporating economic analysis i s needed to address this observation. For a complete description of model performance with regard to individual summary variables the reader i s referred to Chapter 10. Validation results for hillslope A events are discussed for both the regression and quasi-physically based models. The discourse on the limitations of the quasi-physically based model i s equally relevant to the results reported here. Turning now to the f i e l d experiments, Table 12-3 summarizes the forecasting efficiencies of the three summary output variables for calibration runs of two different lengths for the regression model for the R-5 data set. Table 12-4 summarizes the forecasting efficiencies for the three summary output variables for verification runs for the quasi-physically based model for various combinations of record length and spatial density of hydraulic conductivity measurements. A perusal of the E^ values in Tables 12-3 and 12-4 lead to the following comments: 1. In general, the overall performance of the regression model i s improved with longer rainfall-runoff records. TABLE 12-3. Forecasting efficiencies E^ for the three summary output variables for the regression model on the R-5 data set. Case Number of Events PK 36 72 0.45 0.40 0.19 0.36 0.17 0.24 TABLE 12-4. Forecasting efficiencies E f for the three summary output variables for the quasi-physically based model on the R-5 date set Case Number of Measurements Number of Events TK "PK 26 26 36 72 -0.31 0.15 -0.79 0.26 -0.19 -0.20 2 157 36 0.06 0.03 -0.26 175 72 0.26 0.68 -0.25 2. In general, the overall performance of the quasi-physically based model i s improved with more spatial and/or temporal information. As has already been discussed the model f a i l s to predict tp^. 3. In general, as shown with the synthetic data set, space-time tradeoffs are observed within the model data sets but not across the model data sets. For more competitive models, and short records, space-time tradeoffs probably exist but this study has not unequivocally answered the question. An item-by-item comparison of the E^ values in Tables 9-11 and 12-4 illustrates that the performance of the quasi-physically based model i s far superior for events 37 through 72 than for events 1 through 36. The split-sample procedure used to investigate space-time tradeoffs i s obviously handicapped by the problem of s t a t i s t i c a l l y dissimilar calibration errors for equal sample sizes. To further i l l u s t r a t e this problem, split-sample stat i s t i c s for the frequency distributions of the observed and predicted events are summarized in Table 12-5. It should be obvious that the events in the split-sample periods have different properties. It i s pleasing to note that the overall performance of the quasi-physically based model i s far superior for the period that includes the larger Horton type events. What can not be shown in Table 12-5 i s the antecedent s o i l moisture conditions for the R-5 events that might help to explain the nonstationarity from period to period. The preliminary experiments completed for this section show that some space-time tradeoffs exist within the data sets of competing rainfall-runoff modeling techniques. However, i t i s clear from this 259 TABLE 12-5. Split sample st a t i s t i c s : Means and standard deviations for observed and quasi-physically based model predicted frequency distributions for the three summary output variables on the R-5 data set. Mean Standard Deviation % FCPK % QPK FCPK Events Case mm l i t . / s e c . hours mm l i t . / s e c . hours 1-36 Observed 3.1 72.0 3.65 5.8 136.2 2.61 Predicted 1.0 32.3 1.84 3.3 97.4 2.05 37-72 Observed 6.1 128.3 3.55 10.2 316.9 3.02 Predicted 3.3 132.2 3.11 11.7 523.3 4.21 1-72 Observed 4.6 100.2 3.60 8.3 243.8 2.80 Predicted 1.3 55.2 2.37 5.8 270.7 3.33 i n i t i a l analysis that much more evidence i s needed before a jury could decide whether the space-time tradeoff concept i s valid across data sets. It i s not yet possible to state that space-time tradeoffs represent a practical foundation for the data-worth, network-design, model-choice scenario proposed earlier in the section. Clearly the establishment of optimal modeling strategies for operational forecasting and engineering design requires the definition of objective functions specific to the type of problem at hand. The economic analysis for the sizing of a small culvert w i l l be very different from that for the operation of a large dam and reservoir. The economic analysis of space-time tradeoffs i s outside the scope of this thesis but i t represents a logical and important next step. It w i l l also be necessary to consider the policy side of selecting a data collection program to match model performance requirements. The effectiveness of a data network and subsequently the performance of a model are the result of policies set by resource managers. In this chapter several areas of potential research directly related to the current thesis have been proposed, and a preliminary analysis of the concept of space-time tradeoffs across data sets has been presented. The work suggested for future investigation i s far from inclusive; i t reflects the interests of the writer. CHAPTER 13 Conclusions Introduction The previous twelve chapters contain many concepts and experimental results related to the assessment of rainfall-runoff modeling methodology. To re-establish the thread that runs through this dissertation the chapters are briefly outlined in this introduction to the f i n a l chapter. The f i r s t three chapters are devoted to introduction and review. In Chapter 4 a suite of three rainfall-runoff models are described. The three models include: a regression model, a unit hydrograph model and a quasi-physically based model. The evaluation of these models i s the central theme of this study. The data sets used to excite the rainfall-runoff models are reviewed in Chapters 5, 6 and 7. The data sets include: Three small upland experimental catchments (R-5, WE-38, HB-6); a supplemental f i e l d study for R-5; and a series of synthetic data sets. The results from model evaluation experiments are presented in Chapters 8, 9 and 10. The data sets from Chapters 5, 6 and 7 are related to the results in Chapters 8, 9 and 10 respectively. The results in Chapter 9 include an analysis of the spatial variability of f i e l d measured i n f i l t r a t i o n rates on the R-5 catchment. In Chapter 11, a sensitivity analysis of the synthetic data generator i s presented. Some of many possible extensions of this work are discussed in Chapter 12. 262 The major contributions of this thesis can be outlined as follows: 1. This study i s one of the f i r s t to attempt an objective comparative evaluation of underlying event-based rainfall-runoff modeling techniques on both real and synthetic data sets. 2. This study provides a large new data set of steady state i n f i l t r a t i o n rates for a rangeland catchment. 3. This study links the concepts of space-time tradeoffs, data-worth, network design and model choice to form a potential decision analysis strategy for future operational hydrologists. Overview of Methodology This study i s concerned with the use of mathematical models to predict runoff from r a i n f a l l . Modeling efficiencies are calculated and compared for three rainfall-runoff models on real and synthetic data. The models selected for comparison represent a suite of underlying techniques. The model comparisons are carried out at catchment (real data sets) and hillslope (synthetic data sets) scales. The small upland catchments used in this study are the only scale at which real data sets are available to conduct model comparisons across the selected suite of techniques. Locating data sets that meet the needs of a broad suite of modeling techniques i s not an easy task. This author contacted personnel at 13 North American experimental watersheds. Most had adequate rainfall-runoff records, but few had measurement programs for spatially variable physical parameters such as hydraulic conductivity. There are those who w i l l chide this study over the applicability of the selected models on the selected catchments. The reader i s reminded that when i t comes to models and data sets, there i s a surprisingly small 263 intersecting set. The catchments selected for this study represent three of the very few North American catchments that have a sufficiently broad instrumentation program and data collection network to produce a data base compatible with the input requirements of the three modeling procedures. The three models that were compared in this study are a l l thought to be best suited to hydrologic regimes in which runoff i s generated by Horton overland flow. It was the author's hope that i t would be possible to select catchments for analysis in which the primary mechanism of streamflow generation i s Horton overland flow. However, i t i s quite clear that a l l the selected catchments do not meet the Horton c r i t e r i a to an equal degree. The problems of matching models and data sets are alleviated somewhat with the use of synthetic data. The model evaluations and comparisons in this study were carried out with an event-based analysis rather than an analysis of continuous hydro-logic records. This approach leads to a model comparison that rewards the a b i l i t y of a model to predict stormflow from r a i n f a l l events and does not penalize any lack of a b i l i t y to predict antecedent conditions following long interstorm periods. In short, i t avoids the messy problem of simulating evapotranspiration and i t s effect on s o i l moisture conditions. Results and Discussion In the following paragraphs a brief summary of the results from this study are presented along with some philosophical comments. The reader i s directed back to the specific chapters for detailed comments on individual experiments. In Chapter 8, the results of the model efficiency calculations for three rainfall-runoff models on 269 events from three small upland 264 catchments show surprisingly poor efficiencies for a l l models on a l l catchments. The unit hydrograph and quasi-physically based models had . l i t t l e predictive power. The regression model was marginally better. For this model evaluation the quasi-physically based model was l e f t uncalibrated. However, this study did test the hypothesis of whether the uncalibrated modeling efficiencies could be improved by calibrating the model against i t s most sensitive parameter, the saturated hydraulic conductivity K q . It was found that they could not. The performance of the quasi-physically based model on the obviously i l l suited HB-6 catchment i s actually better than indicated by the values. The model did not simulate Horton overland flow for the catchment and in that sense functioned correctly. The experiments in Chapter 8 do not point to any one modeling approach as being superior to a l l others for a l l catchments. However, the fact that simpler, less data intensive models in the form of regressions and unit hydrographs provided as good or better predictions than a more physically based model, i s food for thought. In Chapter 9, the analysis of steady-state i n f i l t r a t i o n data from R-5 exposed a range too small to characterize the spatial structure across the entire catchment. It was therefore not possible to apply such ideas to an interactive network design program. It was possible, however, to re-excite the quasi-physically based model using average i n f i l t r a t i o n rates for different s o i l types and improve model performance. The steady-state i n f i l t r a t i o n rate was found to be seasonably variable. Therefore, i t appears to be very d i f f i c u l t to collect a sizable stationary data set at the catchment scale. The analysis of the R-5 i n f i l t r a t i o n measurements also brings to the front the importance of scale in simulating catchment response with physically based rainfall-runoff models. It w i l l never be possible to collect sufficient information to excite a deterministic model without some uncertainty. Even i f a model i s conceptually true, the transfer of point measurements to the areas of grid elements causes information to be lost and model performance i s reduced. There i s a need for relating point measurements to the grid elements of deterministic distributed conceptual models. : In Chapter 10 the results of the model efficiency calculations for the regression and unit hydrograph models on 100 events from a hypothetical Horton hillslope are far superior to those reported in Chapter 8 for the three catchments. The quasi-physically based model performance on a seemingly ideal data set was surprisingly poor. | The model comparison studies in Chapters 8 and 10 are a foundation for future studies in the same s p i r i t that w i l l hopefully incorporate larger catchments and continuous simulation models. In future work a suite of model evaluation methods should be used, including both s t a t i s t i c a l c r i t e r i a and graphical techniques. Such an approach may have greater model assessment- powers than the sole reliance on model efficiencies used in this study. In Chapter 11, by way of sensitivity analysis, i t was possible to il l u s t r a t e that the spatial variability of near-surface s o i l hydraulic properties, the v a r i b i l i t y of r a i n f a l l in time and space, and topography combine to control the mechanisms of streamflow generation. It was also possible to show that measurements of saturated hydraulic conductivity have greater impact than similar precipitation data for a particular h i l l s l o p e , that exhibits runoff due to the Horton mechanism. Future studies need to establish i f this i s true for a l l Horton hillslopes and whether i t applies also to hillslopes which exhibit runoff due to the Dunne mechanism. In Chapter 12, the results of a preliminary investigation show that space-time tradeoffs exist within the data sets of both the regression and quasi-physically based models: It was not possible in this study to establish the existence of space-time tradeoffs across the data sets of competing models. However, the results here are promising and in the opinion of the author establish a firm toehold for future investigations. Much more analysis is needed, with both real and synthetic data sets to quantify the concept. In the future, building upon space-time tradeoffs, i t may be possible to establish the best use for specific modeling techniques. Perhaps the biggest value of physically based models i s in the development of concepts. Hopefully what we learn from these models w i l l lead us to simpler and more efficient procedures for operational hydrologists. Application of the trickle-down philosophy could possibly be targeted in the areas of flood forecasting, reservoir management, nonpoint sources of agriculture pollution, and s o i l erosion. The type of model evaluation described in this thesis may eventually lead to the unification of the most efficient modeling techniques in hydrology. Most decisions in engineering hydrology are carried out on watersheds much larger than the small catchments considered in this study. Engineering hydrologists rarely have to make predictions for small catchments, and when they do, the economic consequences of simulation error would usually be small. Because of this, this thesis i s viewed by the author as a research contribution rather than as a direct contribution to the methodology of operational forecasting or engineering design. However, i t i s hoped that this study w i l l provide impetus for continued re-examination of rainfall-runoff methodology for engineering purposes., Hydrologic modeling i s both art and science, and i t i s likely to remain so. Predictive hydrologic modeling i s normally carried out on a given catchment using a specific model under the supervision of an individual hydrologist. The usefulness of the results depends in large measure on the talents and experience of the hydrologist and his/her understanding of the mathematical nuances of his/her particular model and the hydrologic nuances of his/her particular catchment. 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