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Improving the Reliability of Compartmental Models: Case of Conceptual Hydrologic Rainfall-Runoff Models Item Type text; Technical Report Authors Sorooshian, Soroosh; Gupta, Vijai Kumar Publisher Department of Hydrology and Water Resources, University of Arizona (Tucson, AZ) Rights Copyright © Arizona Board of Regents Download date 23/09/2021 13:48:41 Link to Item http://hdl.handle.net/10150/614011 HWR No. 86-010 Improving the Reliability of Compartmental Models: Case of Conceptual Hydrologic Rainfall- Runoff Models by Soroosh Sorooshian and Vijai Kumar Gupta Department of Hydrology and Water Resources University of Arizona Tucson, Arizona 85721 Research Project Report on IMPROVING THE RELIABILITY OF COMPARTMENTAL MODELS: CASE OF CONCEPTUAL HYDROLOGIC RAINFALL -RUNOFF MODELS Submitted to National Science Foundation Grant No. CEE -8217823 by Soroosh Soroosh Ian Principal Investigator Associate Professor Department of Hydrology and Water Resources and Vi jai Kumar Gupta Post -Doctoral Research Associate University of Arizona Tucson, Arizona 85721 August 1986 ACKNOWLEDGMENTS The Principal Investigator wishes to acknowledge the technical support provided by the Center for Computing Information and Technology at the Uni- versity of Arizona. The typing and editing of this report was done by Ms. Corla Thies. Her patience and creativity are greatly appreciated. i LIST OF PUBLICATIONS AS THE RESULT OF THIS RESEARCH PROJECT Gupta, V.K., and S. Soróoshian, "The Automatic Calibration of Conceptual Catchment Models Using Derivative -Based Optimization Algorithms,' Water Resources Research, 21(4), pp. 473 -486, April 1985. Sorooshian, S., and V.K. Gupta, The Analysis of Structural Identifiability Theory and Application to Conceptual Rainfall -Runoff Models," Water Resources Research, 21(4), pp. 487 -496, April 1985. Gupta, V.K., and S. Sorooshian, "The Relationship Between Data and the Pre- cision of Estimated Parameters," Journal of Hydrology, 81, pp. 57 -77, 1985. Sorooshian, S., "Synthesis of Hydrologic and System Sciences in the Develop- ment of Rainfall -Runoff Models," Invited paper for special issue on sys- tem identification and parameters. In Revue des Sciences de l'eau, 1(1/4), pp. 21 -28, 1985. Sorooshian, S., "Synthesis of Hydrologic and System Sciences in the Develop- ment of Rainfall -Runoff Models," Invited paper for special issue on sys- tem identification and parameters. In Hydrology and Environment for Applied Mathematics and Computation: Modeling the Environment, 17, pp. 279 -298, 1985. Sorooshian, S., "The Impact of Catchment Modeling on Hydrologic Reliabil- ity," NATO Advanced Institute in "Engineering Reliability and Risk in Water Resources," Tucson, Arizona, May 19 -June 2, 1985, Edited by L. Duckstein and E. Plate, to appear. Gupta, V.K., and S. Sorooshian, "The Influence of Data Length, Information Content, and Noise Characteristics on Model Calibration," In: Multivar- iate Analysis of Hydrologic Processes, Edited by H.W. Shen et al., Pub- lished by Colorado State University Press, Fort Collins, Colorado, pp. 434 -449, 1986. ii TABLE OF CONTENTS Page Acknowledgments i List of Publications as the Result of This Research Project ii Table of Contents iii List of Tables vii List of Figures ix 1. CONCEPTUAL MODELING OF THE RAINFALL -RUNOFF PROCESS 1 1.1 Introduction 1 1.2 The General Identification Problem 2 1.2.1 Introduction 2 1.2.2 The Practical Identification Problem 3 1.2.3 Practical Considerations 8 1.3 Review of the Literature 9 1.3.1 The Development of Conceptual Models 9 1.3.2 Specification of the Measure of Closeness 10 1.3.3 Selection of an Informative Data Set 12 1.3.4 Algorithms for Identifying the Best Parameter Values 14 1.3.5 Identifiability of the Model Set 15 1.3.6 Evaluation of the Degree of Success of the Identification Procedure 17 1.4. Organization and Scope of the Research 20 2. SELECTION OF THE MODEL TO BE INVESTIGATED 22 2.1 The Structure of a Typical Conceptual Rainfall - Runoff Model 22 2.2 A Brief Description of the Soil Moisture Accounting Part of the National Weather Service's River Forecast System Model (NWSRFS) 25 2.3 The Model SIXPAR 27 3. INVESTIGATION AND MODIFICATION OF MODEL STRUCTURE 31 3.1 The Issue of Identifiability 31 3.2 Model Structural Identifiability 34 3.2.1 Introduction 34 3.2.2 A Measure of Structural Identifiability 37 iii iv TABLE OF CONTENTS -- Continued Page 3.3 Parameter Sensitivity -Identifiability Analysis 39 3.3.1 Introduction 39 3.3.2 The Region of Indifference 40 3.3.3 Parameter Identifiability Measure 41 3.3.4 Parameter Sensitivity and Precision 45 3.4 Evaluation of the Derivatives 48 3.4.1 Introduction 48 3.4.2 Modality in the Behavior of CRR Models 48 3.4.3 Mathematical Representation of a Modal -Type Model. 52 3.4.4 The Derivative Equations 55 3.5 Analysis of a Linear Reservoir Model with Threshold 56 3.5.1 Introduction 56 3.5.2 Theory 56 3.5.3 A Note on Practical Implementation 65 3.6 The Percolation Equation of the NWSRFS Model 66 3.6.1 Essential Concepts 66 3.6.2 Mathematical Representation of Percolation 69 3.6.3 The ZPERC -REXP Relationship 75 3.6.4 Response Surface Analysis of the ZPERC -REXP Parameter Space Using Real Data 75 3.6.5 The Impact of Percolation "Demand" Computations 80 3.6.6 Implications of the Results to Parameter Observability 83 3.7 Reparameterization of the Percolation Equation 90 3.7.1 Introduction 90 3.7.2 Theoretical Basis for Reparameterization 91 3.7.3 Methods of Applying Reparameterization 92 3.7.4 A Theoretical Look at the Percolation Equation 93 3.7.5 Response Surface Analysis 99 3.7.6 Rescaling of Parameter "a" 102 3.7.7 Experimental Testing of the Reparameterized Percolation Equation 105 3.7.7.1 Structural Identifiability Analysis 105 3.7.7.2 Parameter Estimation 111 4. THE STOCHASTIC PARAMETER ESTIMATION PROCEDURE 113 4.1 Introduction 113 4.2 The Stochastic Nature of the Parameter Identification Problem 119 4.3 Parameter Identification Methodologies Available 122 V TABLE OF CONTENTS -- Continued Page 4.4 The Method of Maximum Likelihood 124 4.5 Likelihood and the Identification of Dynamic Models 128 4.5.1 Introduction 128 4.5.2 Input Measured Without Error 130 4.5.3 Inputs and Outputs Contain Error 133 4.6 Discussion of the Likelihood Approaches Suggested by Sorooshian (1978) and Fulton (1982) 134 4.6.1 Introduction 134 4.6.2 Bias Caused by Fulton's (1982) Proposed Modification 135 4.7 An Alternative Likelihood Procedure for Watershed Models 138 4.7.1 Introduction 138 4.7.2 A Probability Law Model for Streamflow Measurements 142 4.7.3 Implementation of the Likelihood Function for Identification 150 4.8 Experimental Testing 151 4.8.1 Introduction 151 4.8.2 Identifiability of the Transformation Parameter - 152 4.8.3 Sensitivity of the Model Parameters to the Value of y 157 4.8.4 Joint Identification of the Model and Transformation Parameters 163 4.8.5 Conclusions and Comments 169 5. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH 174 APPENDICES 1. Operational Outline of the Model SIXPAR 176 2. Computer Programs 177 3. Conditional Parameter Sensitivity; Concentricity and Interaction 253 4. Parameter Sensitivity Index 257 5. Modification of Horton's Infiltration Equation for Use with a Conceptual Rainfall -Runoff Model 264 6. Measure of Closeness Based on Likelihood 272 vi TABLE OF CONTENTS -- Continued Page 7. Data Sets Used 276 8. Procedure for Creating Error Corrupted Output Data 278 9. Derivation of Derivatives for CRR Models 280 REFERENCES 287 LIST OF TABLES Table Page 3.1 Parameters present in the percolation equation of the SMA -NWSRFS model 71 3.2 State variables present in the percolation equation of the SMA -NWSRFS model 72 3.3 Parameter sets in the region of which the response surface analyses were conducted 78 3.4 Concentricity indices in the two -parameter subspaces for parameterization P1 106 3.5 Interaction indices in the two -parameter subspaces for parameterization P1 106 3.6 Indices of marginal confidence (MCIi) and confidence (CIi) at the 95% level and the sensitivity (ni) for each parameter of P1 107 3.7 Parameter correlation matrix for parameterization P1 107 3.8 Concentricity indices in two -parameter subspaces for parameterization P2 109 3.9 Interaction indices in two -parameter subspaces for parameterization P2 109 3.10 Indices of marginal confidence (MCIi) and confidence (CIi) at the 95% level and the precision ratio (ni) for each parameter of P2 110 3.11 Parameter correlation matrix for parameterization P2 110 3.12 Algorithm convergence results comparing parameteriza- tions P1 and P2 112 4.1 Measures of closeness associated with different kinds of models and data 123 a2 4.2 a estimate for different atrue and values 139 4.3 Minimum apriori information choices for error probability distribution functions 145 4.4 Maximum likelihood estimates for the transformation parameter - 154 vii viii LIST OF TABLES -- Continued Table Page 4.5 Maximum likelihood estimates for the transformation parameter y 155 4.6 Maximum likelihood estimates for the transformation parameter y 158 4.7 Optimal parameter values using the likelihood function 166 4.8 Optimal parameter values using the SLS function 167 4.9 Sum of squares of differences between model output and true flows for the parameters of Tables 4.7 and 4.8 168 4.10 Optimal parameter values using the likelihood function 170 4.11 Optimization runs using the likelihood function started at the -20% perturbation parameter values 171 4.12 Optimization runs using the SLS function started at the -20% perturbation parameter values 171 4.13 Sum of squares of differences