Preface

Michael S. Zhdanov, Salt Lake City, Utah

Inverse solutions are key problems in many natural sciences. They form the basis of our understanding of the world surrounding us. Whenever we try to learn something about physical laws, the internal structure of the earth or the nature of the Universe, we collect data and try to extract the required information from these data. This is the actual solution of the inverse problem. In fact the observed data are predetermined by physical laws and by the structure of the earth or Universe. The method of predicting observed data for given sources within given media is usually referred to as the forward problem solution. The method of reconstructing the sources of some physical, geophysical, or other phenomenon, as well as the parameters of the corresponding media, from the observed data is referred to as the inverse problem solution.

In geophysics, the observed data are usually physical fields generated by natural or artificial sources and propagated through the earth. Geophysicists try to use these data to reconstruct the internal structure of the earth. This is a typical inverse problem solution.

Inversion of geophysical data is complicated by the fact that geophysical data are invariably contaminated by noise and are acquired at a limited number of observation points. Moreover, mathematical models are usually complicated, and yet at the same time are also simplifications of the true geophysical phenomena. As a result, the solutions are ambiguous and error-prone. The principal questions arising in geophysical inverse problems are about the existence, uniqueness, and stability of the solution. Methods of solution can be based on linearized and nonlinear inversion techniques and include different approaches, such as least-squares, gradient-type methods (including steepest-descent and conjugate-gradient), and others.

A central point of this book is the application of so-called "regularizing" algorithms for the solution of ill-posed inverse geophysical problems. These algorithms can use a priori geological and geophysical information about the earth's subsurface to reduce the ambiguity and increase the stability of the solution.

In mathematics, we have a classical definition of the ill-posed problem: a problem is ill-posed, according to Hadamard (1902), if the solution is not unique or if it is not a continuous function of the data (i.e., if to a small perturbation of data; there corresponds an arbitrarily large perturbation of the solution). Unfortunately, from the point of view of classical theory, all geophysical inverse problems are ill-posed, because their solutions are either nonunique or unstable. However, geophysicists solve this problem and obtain geologically reasonable results in one of two ways. The first is based on intuitive estimation of the possible solutions and selection of a geologically adequate model by the interpreter. The second is based on the application of different types of regularization algorithms, which allow automatic selection of the proper solution by the computer using a priori geological and geophysical information about the earth's structure. The most consistent approach to the construction of regularization algorithms has been developed in the works of Tikhonov and Arsenin (1977) (see also Strakhov, 1968, 1969; Lavrent'ev et al., 1986; Dmitriev, 1990). This approach gives a solid basis for the construction of effective inversion algorithms for different applications.

In the usual way, we describe the geophysical inverse problem by the operator equation:

m=d,m?M,d?D,

where D is the space of geophysical data and M is the space of the parameters of geological models; A is the operator of the forward problem that calculates the proper data d ? D for a given model m ? M. The main idea of the regularization method consists of approximating the ill-posed problem with a family of well-posed problems Aa depending on a scalar regularization parameter a. The regularization must be such that as a vanishes, the procedures in the family Aa should approach the accurate procedure A. It is important to emphasize that regularization does not necessary mean "smoothing" of the solution. Regularization may include "smoothing," but the critical element of this approach is in selecting the appropriate solution from a class of models with the given properties. The main basis for regularization is an implementation of a priori information in the inversion procedure. The more information we have about the geological model, the more stable is the inversion. This information is used for the construction of the "regularized family" of well-posed problems Aa.

The main goal of this book is to present a detailed exposition of the methods of regularized solution of inverse problems based on the ideas of Tikhonov regularization, and to show different forms of their applications in both linear and nonlinear geophysical inversion techniques.

The book is arranged in five parts. Part I is an introduction to inversion theory. In this part, I formulate the typical geophysical forward and inverse problems and introduce the basic ideas of regularization. The foundations of regularization theory described here include: (1) definition of the sensitivity and resolution of geophysical methods, (2) formulation of well-posed and ill-posed problems, (3) development of regularizing operators and stabilizing functionals, (4) introduction of the Tikhonov parametric functional, and (5) elaboration of principles for determining the regularization parameter.

In Part II, I describe basic methods of solution of the linear inverse problem using regularization, paying special attention to iterative inversion methods. In particular, Chapter 4 deals with the classical minimal residual method and its generalizations based on different modifications of the Lanczos method. The important result of this chapter is that all iterative schemes, based on regularized minimal residual methods, always converge for any linear inverse problem. In Part II, I discuss the major techniques for regularized solution of nonlinear inverse problems using gradient-type methods of optimization. Thus, the first two parts outline the general ideas and methods of regularized inversion.

In the following parts, I describe the principles of the application of regularization methods in gravity and magnetic Part III, electromagnetic (Part IV), and seismic (Part V) inverse problems. The key connecting idea of these applied parts of the book is the analogy between the solutions of the forward and inverse problems in different geophysical fields. The material included in these parts emphasizes the mathematical similarity in constructing forward modeling operators, sensitivity matrices, and inversion algorithms for different physical fields. This similarity is based on the analogous structure of integral representations used in the solution of the forward and inverse problems. In the case of potential fields, integral representations provide a precise tool for linear modeling and inversion. In electromagnetic or seismic cases, these representations lead to rigorous integral equations, or to approximate but fast and accurate solutions, which help in constructing effective inversion methods.

The book also includes chapters related to the modern technology of geophysical imaging, based on seismic and electromagnetic migration. Geophysical field migration is treated as the first iteration in the iterative solution of the general inverse problem. It is also demonstrated that any inversion algorithm can be treated as an iterative migration of the residual fields obtained on each iteration. From this point of view, the difference between these two separate approaches to the interpretation of geophysical data-inversion and migration-becomes negligible.

In summary, this text is designed to demonstrate the close linkage between forward modeling and inversion methods for different geophysical fields. The mathematical tool of regularized inversion is the same for any geophysical data, even though the physical interpretation of the inversion procedure may be different. Thus, another primary goal of this book is to provide a unified approach to reconstructing the parameters of the media under examination from observed geophysical data of a different physical nature. It is impossible, of course, to cover in one book all the variety of modern methods of geophysical inversion. The selection of the material included in this book was governed by the primary goals outlined above. Note that each chapter in the book concludes with a list of references. A master bibliography is given at the end of the text, for convenience.

Portions of this book are based on the author's monograph "Integral Transforms in Geophysics" (1988), where the general idea of a unified approach to the mathematical theory of transformation and imaging of different geophysical fields was originally introduced. The corresponding sections of the book have been written using research results originated by the author in the Institute of Terrestrial Magnetism, Ionosphere and Radio-Wave Propagation (IZMIRAN) and later in the Geoelectromagnetic Research Institute of the Russian Academy of Sciences in 1980-1992....