# Geophysical Data Analysis: Discrete Inverse Theory

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Geophysical Data Analysis: Discrete Inverse Theory is an introductory text focusing on discrete inverse theory that is concerned with parameters that either are truly discrete or can be adequately approximated as discrete. Organized into 12 chapters, the book’s opening chapters provide a general background of inverse problems and their corresponding solution, as well as some of the basic concepts from probability theory that are applied throughout the text. Chapters 3-7 discuss the solution of the canonical inverse problem, that is, the linear problem with Gaussian statistics, and discussions on problems that are non-Gaussian and nonlinear are covered in Chapters 8 and 9. Chapters 10-12 present examples of the use of inverse theory and a discussion on the numerical algorithms that must be employed to solve inverse problems on a computer. This book is of value to graduate students and many college seniors in the applied sciences.

## Table of Contents

Preface

Introduction

1 Describing Inverse Problems

1.1 Formulating Inverse Problems

1.2 The Linear Inverse Problem

1.3 Examples of Formulating Inverse Problems

1.4 Solutions to Inverse Problems

2 Some Comments on Probability Theory

2.1 Noise and Random Variables

2.2 Correlated Data

2.3 Functions of Random Variables

2.4 Gaussian Distributions

2.5 Testing the Assumption of Gaussian Statistics

2.6 Confidence Intervals

3 Solution of the Linear, Gaussian Inverse Problem, Viewpoint 1: The Length Method

3.1 The Lengths of Estimates

3.2 Measures of Length

3.3 Least Squares for a Straight Line

3.4 The Least Squares Solution of the Linear Inverse Problem

3.5 Some Examples

3.6 The Existence of the Least Squares Solution

3.7 The Purely Underdetermined Problem

3.8 Mixed-Determined Problems

3.9 Weighted Measures of Length as a Type of A Priori Information

3.10 Other Types of A Priori Information

3.11 The Variance of the Model Parameter Estimates

3.12 Variance and Prediction Error of the Least Squares Solution

4 Solution of the Linear, Gaussian Inverse Problem, Viewpoint 2: Generalized Inverses

4.1 Solutions versus Operators

4.2 The Data Resolution Matrix

4.3 The Model Resolution Matrix

4.4 The Unit Covariance Matrix

4.5 Resolution and Covariance of Some Generalized Inverses

4.6 Measures of Goodness of Resolution and Covariance

4.7 Generalized Inverses with Good Resolution and Covariance

4.8 Sidelobes and the Backus-Gilbert Spread Function

4.9 The Backus-Gilbert Generalized Inverse for the Underdetermined Problem

4.10 Including the Covariance Size

4.11 The Trade-off of Resolution and Variance

5 Solution of the Linear, Gaussian Inverse Problem, Viewpoint 3: Maximum Likelihood Methods

5.1 The Mean of a Group of Measurements

5.2 Maximum Likelihood Solution of the Linear Inverse Problem

5.3 A Priori Distributions

5.4 Maximum Likelihood for an Exact Theory

5.5 Inexact Theories

5.6 The Simple Gaussian Case with a Linear Theory

5.7 The General Linear, Gaussian Case

5.8 Equivalence of the Three Viewpoints

5.9 The F Test of Error Improvement Significance

5.10 Derivation of the Formulas of Section 5.7

6 Nonuniqueness and Localized Averages

6.1 Null Vectors and Nonuniqueness

6.2 Null Vectors of a Simple Inverse Problem

6.3 Localized Averages of Model Parameters

6.4 Relationship to the Resolution Matrix

6.5 Averages versus Estimates

6.6 Nonunique Averaging Vectors and A Priori Information

7 Applications of Vector Spaces

7.1 Model and Data Spaces

7.2 Householder Transformations

7.3 Designing Householder Transformations

7.4 Transformations That Do Not Preserve Length

7.5 The Solution of the Mixed-Determined Problem

7.6 Singular-Value Decomposition and the Natural Generalized Inverse

7.7 Derivation of the Singular-Value Decomposition

7.8 Simplifying Linear Equality and Inequality Constraints

7.9 Inequality Constraints

8 Linear Inverse Problems and Non-Gaussian Distributions

8.1 L1 Norms and Exponential Distributions

8.2 Maximum Likelihood Estimate of the Mean of an Exponential Distribution

8.3 The General Linear Problem

8.4 Solving L1 Norm Problems

8.5 The L∞ Norm

9 Nonlinear Inverse Problems

9.1 Parameterizations

9.2 Linearizing Parameterizations

9.3 The Nonlinear Inverse Problem with Gaussian Data

9.4 Special Cases

9.5 Convergence and Nonuniqueness of Nonlinear L2 Problems

9.6 Non-Gaussian Distributions

9.7 Maximum Entropy Methods

10 Factor Analysis

10.1 The Factor Analysis Problem

10.2 Normalization and Physicality Constraints

10.3 Q-Mode and R-Mode Factor Analysis

10.4 Empirical Orthogonal Function Analysis

11 Sample Inverse Problems

11.1 An Image Enhancement Problem

11.2 Digital Filter Design

11.3 Adjustment of Crossover Errors

11.4 An Acoustic Tomography Problem

11.5 Temperature Distribution in an Igneous Intrusion

11.6 L1, L2, and L∞ Fitting of a Straight Line

11.7 Finding the Mean of a Set of Unit Vectors

11.8 Gaussian Curve Fitting

11.9 Earthquake Location

11.10 Vibrational Problems

12 Numerical Algorithms

12.1 Solving Even-Determined Problems

12.2 Inverting a Square Matrix

12.3 Solving Underdetermined and Overdetermined Problems

12.4 L2 Problems with Inequality Constraints

12.5 Finding the Eigenvalues and Eigenvectors of a Real Symmetric Matrix

12.6 The Singular-Value Decomposition of a Matrix

12.7 The Simplex Method and the Linear Programming Problem

Appendix A: Implementing Constraints with Lagrange Multipliers

Appendix B: Inverse Theory with Complex Quantities

References

Index

## Product details

- No. of pages: 272
- Language: English
- Copyright: © Academic Press 1984
- Published: January 28, 1984
- Imprint: Academic Press
- eBook ISBN: 9780323141284

## About the Author

### William Menke

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