Stochastic Integral Equations and Rainfall-Runoff Models

400 Pages·1989·18.752 MB·English

by  Prof. Theodore V. Hromadka II, Prof. Robert J. Whitley (auth.)

Theodore V. Hromadka II
Robert J. Whitley
Stochastic Integral Equations
and Rainfall-Runoff Models
With 78 Figures
Springer-Verlag Berlin Heidelberg NewY ork
London Paris Tokyo Hong Kong
Prof. Theodore V. Hromadka II
Williamson & Schmid
17782 Sky Park Blvd.
Irvine, CA 92714
USA
Prof. Robert J. Whitley
Dept. of Mathematics
University of California
Irvine, CA 92717
USA
ISBN 978-3-642-49311-9 ISBN 978-3-642-49309-6 (eBook)
DOl 10.1007/978-3-642-49309-6
Library of Congress Cataloging-in-Publication Data
Hromadka II, Theodore V.
Stochastic integral equations and rainfall runoff models /
Theodore V. Hromadka II, Robert 1. Whitley.
Biography: p.
Includes indexes.
ISBN 978-3-642-49311-9
I. Runoff--Mathematical models.
2. Rain and rainfall--Mathematic models.
3. Stochastic integral equations.
I. Whitley, Robert 1.11. Title.
GB980.H76 1989
551.48'8'015118--dc20 89-32323
This work is subject to copyrighLAIl rights are reserved, whether the whole or part of the material is concerned,
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permitted under the provisions oft he German Copyright LawofSeptember9, 1965,in its version ofJune24,1985,
and a copyright fee must always be paid. Violations fall underthe prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1989
Softcover reprint of the hardcover 1st edition 1989
The use of registered names, trademarks, etc. in this publication does not imply, even in the absence ofa specific
statement, that such names are exempt from the relevant protective laws and regulations and therefore free for
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216113020543210 Printed on acid free paper.
DEDICATION
To Lila
To Mikel
Acknowledgments
The authors pay acknowledgments to Dr. C.C. Yen, of Williamson
and Schmid, Irvine, California, who carefully reviewed the manuscript
several times. Acknowledgments are also paid to Ms. Phyllis Williams,
who typed and modified the various versions of the manuscript. Thanks
are given to Mr. Bill Burchard and Ms. Linda Laurenzi, who carefully
prepared the several figures. And finally, thanks are given to my wife,
Laura, who supported me throughout this project.
Table of Contents
CHAPTER 1: RAINFALL-RUNOFF APPROXIMATION 1
1.1. INTRODUCTION 1
1.1.1.An Analogy to Rainfall-Runoff Modeling 2
1.2. STORMFLOW DETERMINATION METHODS 5
1.3. METHOD FOR DEVELOPMENT OF SYNTHETIC
FLOOD FREQUENCY ESTIMATES 6
1.4. WATERSHED MODELING UNCERTAINTY 17
1.4.1. Some Concerns in Deterministic Rainfall-
Runoff Model Performance 18
1.4.2. Runoff Hydrograph Generation Techniques
(Linear vs. Nonlinear) 37
1.4.3. On Predicting T-Year Return' Frequency
Values of a Criterion Variable 44
1.4.4. The Design Storm/Unit Hydrograph Approach 45
1.5. HYPOTHETICAL FLOODS, BALANCED FLOODS,
AND DESIGN STORM METHODS 46
1.6. A PREVIEW OF THE RAINFALL-RUNOFF MODEL
PREDICTION PROBLEM 52
1. 7. AN OVERVIEW OF RAINFALL-RUNOFF MODEL
STRUCTURES 55
1.7.1. Estimating Effective Rainfall 55
1.7.2. The Physical Processes Involved 56
1. 7 .3. The Phi-Index Method for Estimating
Effective Rainfall 60
1.7 .4. Constant Proportion Loss Rate 61
1.7.5. Coupled Phi-Index and Constant Proportion
Loss Rate Function 64
1.7.6. Horton Loss Rate Function 64
1.7.7. Exponential Loss Rate Function 64
1.7.8. Initial Abstraction Considerations 66
1.7.9. SCS Loss Separation 67
1.7.10. SCS Hydrologic Soil Groups 69
1.7.11. Soil Cover Considerations 71
1.7.12. Generating Runoff Using the Unit
Hydrograph Method 71
1.7.13. Forming Synthetic Unit Hydrographs 81
1.7.14. Synthetic Runoff Hydrograph Development
( Convolution) 86
1.7.15. Detention Basin Routing Procedure
(Modified PuIs Method) 93
1.7.16. Flow-by Channel Model (Runoff Hydrograph
Separation) 98
1.7.17. The Modified Convex Channel Routing
Method 98
IX
1.7.18. Muskingum Channel Routing 103
1.7.19. A Pipeflow Routing Model 105
1. 7 .20. Hydrograph Translation 105
1.7.21. A Link-Node Rainfall-Runoff Model 107
Study Problems 108
CHAPTER 2: PROBABILITY AND STATISTICS REVIEW 117
2.1. PROBABILITY SPACES 117
2.2. RANDOM VARIABLES 121
2.3. MOMENTS 125'
2.4. TWO RANDOM VARIABLES 128
2.5. SEVERAL RANDOM VARIABLES 139
2.6. PARAMETER ESTIMATION 144
2.7. CONFIDENCE INTERVALS 153
Study Problems 160
CHAPTER 3: INTRODUCTION TO STOCHASTIC INTEGRAL
EQUATIONS IN RAINFALL-RUNOFF
MODELING 169
3.1. INTRODUCTION 169
3.2. INTRODUCTION TO ANALYSIS OF RAINFALL-
RUNOFF MODEL STRUCTURES 171
3.2.1. Rainfall-Runoff Model #1 171
3.2.2. Rainfall-Runoff Model #2 180
3.3. APPLICATION OF STOCHASTIC INTEGRAL
EQUATIONS TO RAINFALL-RUNOFF DATA 185
3.4. ANOTHER LOOK AT PROBABILISTIC MODELING:
ASSUMING MUTUALLY INDEPENDENT
PARAMETERS 191
Study Problems 21'0
CHAPTER 4: STOCHASTIC INTEGRAL EQUATIONS
APPLIED TO A MULTI-LINEAR RAINF ALL-
RUNOFF MODEL 215
4.1. STOCHASTIC INTEGRAL EQUATION METHOD
(S.I.E. M.) 216
4.1.1. Rainfall-Runoff Model Errors
4.1.2. Developing Distributions for Model
Estimates Using the S.I.E.M. 218
4.1.3. Application 1: Coupling the S.I.E.M.
to a Complex Model 220
4.1.3.1. Rainfall-runoff Model Description
and Data Forms 220
4.1.3.2. Development of the Distribution ['" M(')] 222
4.1.3.3. Functional Operator Distributions 223
x
4.2. SENSITIVITY OF FUNCTIONAL OPERATOR
DISTRIBUTIONS TO SAMPLING ERROR 224
4.2.1. True Distributions 224
4.2.2. Application 2: Development of Total
Error Distributions 226
4.2.2.1. A Translation Unsteady Flow Routing
Rainfall-runoff Model 226
4.2.2.2. Multilinear Unsteady Flow Routing and
Storm Classes 229
4.2.2.3. Multilinear Hydrologic Unsteady Flow
Routing 231
4.2.2.4. Example 234
4.3. A MULTILINEAR RAINFALL-RUNOFF MODEL 234
4.3.1. Generalization of Model 234
4.3.2. Application 3 - Multilinear Rainfall-Runoff
Model 239
4.4. AN APPLICATION OF THE S.I.E.M. 247
Study Problems 255
CHAPTER 5: RAINFALL-RUNOFF MODEL CRITERION
VA RIABLE FREQUENCY DISTRIBUTIONS 262
5.1. PROBABILISTIC DISTRIBUTION CONCEPT 262
5.2. THE DISTRIBUTION OF THE CRITERION VARIABLE 263
5.3. SEQUENCE OF ANNUAL MODEL INPUTS 264
5.4. MODEL INPUT PEAK DURATION ANALYSIS 265
5.5. CRITERION VARIABLE DISTRIBUTION ANALYSIS 267
5.6. ESTIMATION OF T-YEAR VALUES OF THE
CRITERION VARIABLE 271
5.7. T-YEAR ESTIMATE MODEL SIMPLIFICATIONS 272
5.8. DISCUSSION OF RESULTS 274
5.9. COMPUTATIONAL PROBLEM 283
5.10. COMPUTATIONAL PROGRAM 293
Study Problems 323
CHAPTER 6: USING THE STOCHASTIC INTEGRAL
EQUATION METHOD 326
6.1. INTRODUCTION 326
6.2. PROBLEM SETTING 326
6.3. STOCHASTIC INTEGRAL EQUATION METHOD
(S.I.E. M.) 327
6.4. APPROXIMATION OF CRITERION VARIABLE
CONFIDENCE INTERVALS, USING THE S.I.E.M. 329
6.5. RAINFALL-RUNOFF MODELS, AND THE
V ARIANCE IN THE CRITERION VARIABLE
ESTIMATES 330
XI
6.6 RAINFALL-RUNOFF MODEL CALIBRATION 336
6.7. CONFIDENCE INTERVAL ESTIMATES 337
6.8. UNIT HYDROGRAPHS AS A MULTIVARIATE
NORMAL DISTRIBUTION 339
6.8.1. S.I.E.M. Formulation 339
6.8.2. Criterion Variable Value Estimation 342
6.8.3. Computational Problem 344
6.8.4. Discussion of Computational Problem
Results 348
6.8.5. Computer Program 6.1 349
Study Problems 366
REFERENCES 368
AUTHOR INDEX 375
SUBJECT INDEX 378
List of Tables
1.1 STATISTICAL ESTIMATION OF Qp (CATEGORY I)
1.2 STATISTICAL ESTIMATION OF Qp (CATEGORY II)
1.3 INDEX FLOOD ESTIMATE (CATEGORY III)
1.4 TRANSFER METHODS (CATEGORY IV)
1.5 EMPIRICAL EQUATIONS (CATEGORY V)
1.6 SINGLE EVENT SIMULATION (CATEGORY VI)
1.7 MULTIPLE DISCRETE EVENTS (CATEGORY VII)
1.8 CONTINUOUS SIMULATION (CATEGORY VIII)
1.9 DEPRESSION STORAGE ESTIMATES
1.10 TYPICAL SOIL CAPACITIES
1.11 SCS HYDROLOGIC SOIL GROUPS
1.12 MINIMUM INFILTRATION RATES
1.13 SOIL COVER QUALITY DEFINITIONS
1.14 EXAMPLE PROBLEM STORAGE INDICATION CURVE
DEVELOPMENT
1.15 EXAMPLE PROBLEM BASIN ROUTING TABULATION
1.16 CONVEX ROUTING EXAMPLE PROBLEM SOLUTION
1.17 MUSKINGUM ROUTING EXAMPLE PROBLEM
SOLUTION
3.1 DISCRETIZATION IN PARALLEL MODEL RESULTS
3.2 DISCRETIZATION MODELING RESULTS
4.1 WATERSHED CHARACTERISTICS
4.2 PRECIPITATION GAUGES USED IN LOS ANGELES
COUNTY FLOOD RECONSTITUTIONS