Geophysical inverse theory and regularization problems Cover Contents Preface Part I: Introduction to Inversion Theory Chapter 1. Forward and inverse problems in geophysics 1.1 Formulation of forward and inverse problems for different geophysical fields 1.2 Existence and uniqueness of the inverse problem solutions 1.3 Instability of the inverse problem solution Chapter 2. ILL Posed problems and the methods of their solution 2.1 Sensitivity and resolution of geophysical methods 2.2 Formulation of well posed and ill posed problems 2.3 Foundations of regularization methods of inverse problem solution 2.4 Family of stabilizing functionals 2.5 Definition of the regularization parameter Part II: Methods of the Solution of Inverse Problems Chapter 3. Linear discrete inverse problems 3.1 Linear least squares inversion 3.2 Solution of the purely underdetermined problem 3.3 Weighted least squares method 3.4 Applying the principles of probability theory to a linear inverse problem 3.5 Regularization methods 3.6 The Backus Gilbert Method Chapter 4. Iterative solutions of the linear inverse problem 4.1 Linear operator equations and their solution by iterative methods 4.2 A generalized minimal residual method 4.3 The regularization method in a linear inverse problem solution Chapter 5. Nonlinear inversion technique 5.1 Gradient type methods 5.2 Regularized gradient type methods in the solution of nonlinear inverse problems 5.3 Regularized solution of a nonlinear discrete inverse problem 5.4 Conjugate gradient re weighted optimization Part III: Geopotential Field Inversion Chapter 6. Integral representations in forward modeling of gravity and magnetic fields 6.1 Basic equations for gravity and magnetic fields 6.2 Integral representations of potential fields based on the theory of functions of a complex variable Chapter 7. Integral representations in inversion of gravity and magnetic data 7.1 Gradient methods of gravity inversion 7.2 Gravity field migration 7.3 Gradient methods of magnetic anomaly inversion 7.4 Numerical methods in forward and inverse modeling Part IV: Electromagnetic Inversion Chapter 8. Foundations of electromagnetic theory 8.1 Electromagnetic field equations 8.2 Electromagnetic energy flow 8.3 Uniqueness of the solution of electromagnetic field equations 8.4 Electromagnetic Greens tensors Chapter 9. Integral representations in electromagnetic forward modeling 9.1 Integral equation method 9.2 Family of linear and nonlinear integral approximations of the electromagnetic field 9.3 Linear and non linear approximations of higher orders 9.4 Integral representations in numerical dressing Chapter 10. Integral representations in electromagnetic inversion 10.1 Linear inversion methods 10.2 Nonlinear inversion 10.3 Quasi linear inversion 10.4 Quasi analytical inversion 10.5 Magnetotelluric (MT) data inversion Chapter 11. Electromagnetic migration imaging 11.1 Electromagnetic migration in the frequency domain 11.2 Electromagnetic migration in the time domain CH&# This book presents state of the art geophysical inverse theory developed in modern mathematical terminology. The book brings together fundamental results developed by the Russian mathematical school in regularization theory and combines them with the related research in geophysical inversion carried out in the West. It presents a detailed exposition of the methods of regularized solution of inverse problems based on the ideas of Tikhonov regularization and shows the different forms of their applications in both linear and nonlinear methods of geophysical inversion. This text is the first to treat many kinds of inversion and imaging techniques in a unified mathematical manner. The book is divided in five parts covering the foundations of the inversion theory and its applications to the solution of different geophysical inverse problems including potential field electromagnetic and seismic methods. The first part is an introduction to inversion theory. The second part contains a description of the basic methods of solution of the linear and nonlinear inverse problems using regularization. The following parts treat the application of regularization methods in gravity and magnetic electromagnetic and seismic 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 methods. The book also includes chapters related to the modern technology of geophysical imaging based on seismic and electromagnetic migration. This volume is unique in its focus on providing a link between the methods used in gravity electromagnetic and seismic imaging and inversion and represents an exhaustive treatise on inversion theory