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... A diskette with the updated programme of Appendix C and examples is available through the author at a small fee.
email: nezheng@ucla.edu
fax: 1--310--825--5435 ...
This book systematically discusses basic concepts, theory, solution methods and applications of inverse problems in groundwater modeling. It is the first book devoted to this subject.
The inverse problem is defined and solved in both deterministic and statistic 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. For a stochastic field, the maximum likelihood estimation and co-kriging techniques are used to estimate unknown parameters. The ill-posed problem of inverse solution is highlighted through the whole book. The importance of data collection strategy is specially emphasized. Besides the classical design criteria, the relationships between decision making, prediction, parameter identification and experimental design are considered from the point of view of extended identifiabilities. The problem of model structure identification is also considered.
This book can be used as a textbook for graduate students majoring in hydrogeology or related subjects. It is also a reference book for hydrogeologists, petroleum engineers, environmental engineers, mining engineers and applied mathematicians.

### Chapter 1. Forward Problems in Groundwater Modeling

Abstract
What is a mathematical model? Why do we need mathematical models in planning and management of groundwater resources? How do we construct a model for groundwater systems and solve it with a computer? These problems have been expounded in detail by many authors over the past two decades (Domenico, 1972; Bear, 1972, 1979; Sun, 1981; Huyakorn and Pinder, 1983; Van der Heijde et al., 1985; De Marsily, 1986; Kinzelbach, 1986; Bear and Veruijt, 1987; Willis and Yeh, 1987; Sun, 1989a, 1994a).
Ne-Zheng Sun

### Chapter 2. An Introduction to Inverse Problems

Abstract
Once a simulation model, as shown in Figure 1.2.1, is built for a groundwater system, the forecast problem can be solved, i.e., we can forecast the response (Status u) of the system for different excitations (Control variables q). As a result, different management decisions can be compared and an optimal decision can be selected based on certain criteria. Thus, the management problem can be solved by incorporating a simulation model with an optimization program (Bear, 1979).
Ne-Zheng Sun

### Chapter 3. Classical Definition of Inverse Problems

Abstract
It is known that various mathematical models in groundwater modeling can be represented commonly by operator equations relating state variables and parameters. In forward problems we solve for state variables when parameters are given, while in inverse problems we solve parameters when state variables are measured. Thus, the general theory of inverse problems should be based on the solution of operator equations. In this section, we will introduce accurate and approximate solutions of operator equations and discuss their well-posedness. Introducing the classical definition of inverse problems not only makes these concepts more general and clear, but also helps to understand their origin and developments in depth. If the reader is not familiar with the terminologies used in this section, such as mapping, vector space, norm and etc., it is suggested to read Appendix A of the book first. In this section, we assume the model to be free of structure error. Therefore, it is unnecessary to differentiate between a “real system” and a “model”.
Ne-Zheng Sun

### Chapter 4. Indirect Methods for the Solution of Inverse Problems

Abstract
In a special case of defining the quasisolution, the observation space is furnished with the L 2-norm. The corresponding performance is called the output least squares (OLS) criterion which is the most popular tool for parameter identification.
Ne-Zheng Sun

### Chapter 5. Direct Methods for the Solution of Inverse Problems

Abstract
In this section, a general form of equation error criteria is introduced. The inverse solution is then obtained by solving a system of superdeterministic equations. The ill-posedness of inverse problems, however, may strongly manifest itself in this case through the ill-condition of coefficient matrix. Data processing, constrains imposition and parameterization are always necessary in this case for improving the stability of inverse solutions.
Ne-Zheng Sun

### Chapter 6. The Adjoint State Method

Abstract
The adjoint state method based on the variational theory has been used in groundwater modeling for more than two decades (Carter et al., 1974; Chavent et al., 1975; Seinfeld and Chen 1978; Neuman et al., 1980; Sun and Yeh, 1985; Townley and Wilson, 1985; Ahlferd et al., 1988; Sun and Yeh, 1990a; and etc.). Its applications include not only parameter identification, but also sensitivity analysis, reliability estimates, observation design, and so forth. In this section, we will show how to derive the adjoint equations for groundwater flow and mass transport problems and how to solve them.
Ne-Zheng Sun

### Chapter 7. The Stochastic Method for Solving Inverse Problems

Abstract
To solve inverse problems, we must have some prior information, field observations, and a model relating the observations to unknown parameters. Since observation error and model structure error always exist and depend on some uncontrollable factors, it is necessary to consider the nature of inverse problems under the framework of stochastics.
Ne-Zheng Sun

### Chapter 8. Experimental Design, Extended Identifiabilities and Model Structure Identification

Abstract
The accuracy of identified model parameters and thus the reliability of model predictions depend on the quantity and quality of observation data obtained in the field according to predetermined experimental designs. A good experimental design should give enough information for model calibration while save experimental expenses. Basic ideas and methods of experimental design have been well established in statistics and extensively applied to various scientific and engineering fields (Silvey, 1980; Pâzman, 1986). For groundwater modeling, however, there are some difficulties associated with the design of experiments. First, the observed state variables, such as the head and concentration, are always nonlinear functions with respect to the unknown hydrogeological parameters. Second, the model structure can never be known exactly. Third, the cost of experiments is usually high and the number of observation wells is very limited. Fourth, the experimental scale is generally small to compare with the aquifer scale. Therefore, in the field of groundwater modeling, special considerations should be given to the design of experiments.
Ne-Zheng Sun

### Conclusion

Abstract
We have introduced basic concepts, theories and methods of inverse problems in groundwater modeling. Both deterministic and stochastic approaches were considered. The basic problem that runs through the whole book is how to overcome or avoid the ill-posedness of inverse solutions. The fundamental way out is to gain sufficient observations, both in quantity and quality, based on predetermined criteria and model applications. It is also important to incorporate all existing geological and hydrogeological information into the inverse solution procedure.
Ne-Zheng Sun