INVERSE PROBLEMS IN GROUNDWATER MODELING

Transcription

1 INVERSE PROBLEMS IN GROUNDWATER MODELING

2 Theory and Applications of Transport in Porous Media Series Editor: Jacob Bear, Technion - Israel Institute of Technology, Haifa, Israel Volume 6 The titles published in this series are listed at the end of this volume.

3 Inverse Problems in Groundwater Modeling by Ne-Zheng Sun Environmental Science Center, Shandong University, P.R. China Presently, Civil Engineering Department, University of California, Los Angeles, U.S.A. SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

4 Library of Congress Cataloging-in-Publication Data Sun. Ne-Zheng. Inverse problems in groundwater modeling I by Ne-Zheng Sun. p. cm. -- (Theory and applications of transport in porous media; v. 6) Includes bibliographical references and index. ISBN ISBN (ebook) DOI / Groundwater--Mathematical models. 2. Inverse problems (Differential equations) I. Title. II. Series. GB M35S dc ISBN First printed in Reprinted with corrections in Printed on acid-free paper All Rights Reserved 1994,1999 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1999 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

5 To Fang, Yi-Xing and Yi-shan

6 Table of Contents Preface 1 Forward Problems in Groundwater Modeling Mathematical Models Examples Categories of Mathematical Models General Form The Solution of Forward Problems Analytical Solutions Numerical Solutions Well-posedness of the Forward Problems 12 2 An Introduction to Inverse Problems Basic Concepts Model Structure and Model Parameters Model Calibration and Inverse Problem Ill-Posedness of the Inverse Problem Existence Uniqueness Stability The Ill-posedness of Numerical Inverse Solutions The Trial and Error Method Transfer Inverse Problems into Optimization Problems The Indirect Method The Direct Method Prior Information and Constraints Further Considerations Classical Definition of Inverse Problems The Solution of Operator Equations Classical Well-Posedness of Operator Equations Extended Well-Posedness Approximate Solution xi vii

7 viii Table of Contents 3.2 Identifiability Classical Definition of Identifiability Identifiability for Quasisolutions Discretization and Parameterization Parameterization Identifiability of Discretized Models Indirect Methods for the Solution of Inverse Problems S3 4.1 The Output Least Squares Criterion Output Least Squares Formation The Output Least Squares Identifiability Optimization Algorithms Unconstrained Nonlinear Optimization Search Methods Gradient Methods Quasi-Newton Methods Stopping Criteria Program and Examples The Gauss-Newton Method The Fundamental Gauss-Newton Method Modified Gauss-Newton Methods Sensitivity Coefficients The Regularization Method The Regularized OLS Problem The Regularization Coefficient The Lp-Norm Criteria Coupled Inverse Problems A General Definition of Coupled Inverse Problems The Solution of Vector Optimization Problems Inverse Solution Methods S Direct Methods for the Solution of Inverse Problems Equation Error Criteria and the Matrix Method Equation Error Criteria The Matrix Method IIl-posedness in the Matrix Method Data Processing. Constraints and Parameterization Mathematical Programming Methods L1-norm Criterion and Linear Programming L2-Norm Criterion and Quadratic Programming The Quasi-linearization Method

8 Table of Contents ix 6 The Adjoint State Method The Adjoint State Method for Groundwater Flow Problems Green's Theorems The Adjoint Problem for Groundwater Flow in Confined Aquifers The Adjoint Problem for Groundwater Flow in Unconfined Aquifers The Adjoint Problem for Leaky Aquifer Systems The Adjoint State Method for Mass Transport Problems The Adjoint Problem for Advection-Dispersion Equations The Adjoint Problem for Coupled Grol!ndwater Flow and Mass Transport Problems The Solution of Adjoint Problems The Adjoint State Method for General Coupled Problems The Derivation of Adjoint State Equations Adjoint Operation Rules Adjoint State Equations for Oil-Water Two Phase Flow Problems The Discrete Approach for Deriving Adjoint State Equations Applications ofthe Adjoint State Method ~ Parameter Identification The Computation of Sensitivity Matrix Sensitivity Analysis The Stochastic Method for Solving Inverse Problems Defining Inverse Problems in the Statistical Framework A Statistic Statement of Inverse Problems The Measure oflnformation and Uncertainty Prior Information and Posterior Distribution The Transfer of Information Probability Distribution of Errors The Inverse Solution Statistical Properties of Estimators Point Estimation Maximum Likelihood Estimation Generalized Least Squares Estimation Gaussian Prior Distribution Unknown Covariance Parameters... Parameter Estimation Based only on Parameter Observations Spatial Variability of Distributed Parameters

9 x Table of Contents Maximum Likelihood Estimation and Gaussian Conditional Mean The Kriging Estimator for Weakly Stationary Fields The Kriging Estimatorfor Unstationary Fields Applications Parameter Identification Based on Both Parameter and State Variable Observations Stochastic Partial Differential Equations Estimate of Statistical Parameters The Co-kriging Estimate The Adjoint State Method for SPDEs A Numerical Example Experimental Design, Extended Identifiabilities and Model Structure Identification Experimental Design Experimental Design in Groundwater Modeling Experimental Design for Parameter Identification Experimental Design for Prediction Experimental Design for Decision Making Extended Identifiabilities Define Prediction and Decision Spaces Interval Identifiability and Experimental Design Prediction Equivalence Identifiability and Experimental Design Management Equivalence Identifiability and Experimental Design Model Structure Identification Identification of Parameter Structure Structure Equivalent Models and Model Structure Reduction Experimental Design for Model Discrimination A Geological Parameterization Approach Conclusion 249 Appendix A AppendixB AppendixC References Index Mapping, Space and Norm Probabilities and Random Fields A Fortran Program

10 Preface During the past twenty years, the technique of mathematical modeling has been extensively used in the study of groundwater resources management, seawater intrusion, aquifer remediation and other problems related to groundwater. To build a model for a real groundwater system, two problems, the forward problem (simulation) and its inverse (calibration), must be solved. The former predicts unknown system states by solving appropriate governing equations, while the latter determines unknown physical parameters and other conditions of the system by fitting observed system states. We must first solve the inverse problem to find appropriate model structure and model parameters, and then solve the forward problem to obtain required prediction results. Unfortunately, studies on these two problems are not in balance. The study of forward problems has developed rapidly. At this time, we can accurately simulate three-dimensional multicomponent transport in multiphase flow without any essential difficulty. On the other hand, the study of inverse problems is still limited to consideration of very simple models. In some case studies, we may see a situation where a very fine groundwater model was calibrated only by the primary trial and-error method and insufficient observation data. If large errors are included in model structure, model parameters, sink/source terms and boundary conditions, the model can not be expected to produce accurate prediction results. Clearly, the model quality can not be improved by increasing only the accuracy of forward solutions if the inverse problem is not well solved. Like a person, the technique of groundwater modeling has two legs: the forward solution and the inverse solution. If one leg is long and another is short, how can the technique walk well? The progress of inverse solution techniques is blocked by several inherent difficulties. First, the inverse problem is often ill-posed, i.e., its solution may be non-unique and unstable with respect to the observation error. Second, the quantity and quality of observation data are usually insufficient. Third, the model structure error, which is difficult to estimate, often dominates other errors. If these difficulties are not considered, satisfactory inverse solutions can never be obtained by changing only performance criteria and optimization algorithms. In recent years, some significant developments in this field have been reported. xi

11 xii Preface Methods coming from stochastics and control theory were successfully applied in the study of inverse problems. The concept of extended identifiabilities and the method of optimal data collection strategy may help us to find reliable models for given model applications. To make the "short leg" longer, however, more effort is needed and more difficulties must be overcome. This is the first book on this subject. It systematically introduces basic concepts, theories, and methods, as well as recent developments in this field. The inverse problem is defined and solved in both deterministic and stochastic frameworks. Various direct and indirect methods are discussed and compared. As a useful tool, the adjoint state method and its applications are given in detail. Difficulties with the problem of iil-posedness in the inverse solution are highlighted throughout the book and the importance of data collection strategy is emphasized. The study of inverse problems requires knowledge of forward solution methods, as well as a background in mathematical programming, variational analysis and stochastics. For the reader's convenience, basic mathematical tools used in this book are either introduced in the text or given in Appendices. Exercises following each section are designed to help expand the reader's comprehension of the text. Chapter 1 is an introduction to forward problems in groundwater modeling. Different types of models and their solution methods are reviewed. Chapter 2 is an introduction to inverse problems, in which the ill-posedness of inverse problems is explained using various examples related to groundwater modeling. From this chapter, the reader will obtain an overall understanding of the nature of inverse problems. Chapter 3 presents a general definition of inverse problems from the view-point of solving operator equations. The well-posedness of quasi-solutions and approximate solutions of an operator equation is discussed. If the reader is not interested in mathematical arguments, he or she can skip this Chapter. Various indirect methods of inverse solution are given in Chapter 4. Besides the general least squares criterion, other Lp-norm criteria and regularization methods are considered. Chapter 5 discusses direct methods of inverse solution. The quasi-linearization technique is presented as a special case of using mathematical programming methods. In Chapter 6, adjoint state equations are derived for flow problems, mass transport problems and general coupled problems in groundwater modeling. The adjoint state method is then used for parameter identification and sensitivity analysis. In Chapter 7, the inverse problem is defined and solved in the stochastic framework. The maximum likelihood estimate and other estimators are introduced. Kriging algorithms are derived for both stationary and unstationary stochastic fields, and co-kriging estimates are developed for both steady state and transient flow fields. In Chapter 8, different criteria of experimental design are presented for parameter identification, model prediction and decision making. After defining various extended identifiabilities, experimental designs based on these extended identifiabilities are discussed. In the last section of Chapter 8,

12 Preface Xlll criteria of experimental design for model structure identification are considered. A parameterization method that can directly incorporate geological structure into parameter identification of three- dimensional models is introduced. There is a short conclusion at the end of the text, in which a step by step procedure of building mathematical models for real systems is summarized. This procedure is different from the conventional approach in that relationships between data collection, parameter identification, and model application are systematically considered. Additionally, some open problems in this area are presented that may help readers to develop original research projects. A FORTRAN program developed by the author is given in Appendix C. By changing a few input parameters, it can solve either forward or inverse problems of coupled groundwater flow and mass transport in steady or transient flow fields. This program is able to deal not only with hypothetical but also practical problems. This book is recommended as a text for graduate level courses or seminars. It should also provide a useful reference for hydrogeologists, geochemists, petroleum engineers, environmental engineers and applied mathematicians. I wish to thank professor Jacob Bear, the editor of the book series, who suggested that I write a book on this topic and reviewed the outline. I wish to thank professor William W-G. Yeh for the support that he has provided for my work at the University of California, Los Angeles. Some research results quoted in this book were completed under his support. I wish to thank Dr. C. Wang, the senior engineer of Municipal Water District of Southern California, for his cooperation in the use of lately developed approaches to solve practical inverse problems of groundwater modeling. I wish to thank my friend, professor Julius Glater, who helped in the modification of the manuscript. The whole manuscript was typed by my son, Yi-shan Sun. He also drew figures and corrected all equations for the book. I thank Kluwer editors for their unfailing cooperation. The code and readme files and examples can be downloaded from seas. ucla.edu/ ~nezheng/