Parameter Estimation and Inverse Problems

Parameter Estimation and Inverse Problems, Second Edition provides geoscience students and professionals with answers to common questions like how one can derive a physical model from a finite set of observations containing errors, and how one may determine the quality of such a model. This book takes on these fundamental and challenging problems, introducing students and professionals to the broad range of approaches that lie in the realm of inverse theory. The authors present both the underlying theory and practical algorithms for solving inverse problems. The authors’ treatment is appropriate for geoscience graduate students and advanced undergraduates with a basic working knowledge of calculus, linear algebra, and statistics.

Parameter Estimation and Inverse Problems, Second Edition introduces readers to both Classical and Bayesian approaches to linear and nonlinear problems with particular attention paid to computational, mathematical, and statistical issues related to their application to geophysical problems. The textbook includes Appendices covering essential linear algebra, statistics, and notation in the context of the subject.

• Includes appendices for review of needed concepts in linear, statistics, and vector calculus.
• Accessible to students and professionals without a highly specialized mathematical background.

The book is primarily used as a textbook for graduate and advanced undergraduate students taking courses in geophysical inverse problems. It is also used as a reference for geoscientists and researchers in academe and industry.

Preface

Chapter One. Introduction

1.1. Classification of Parameter Estimation and Inverse Problems

1.2. Examples of Parameter Estimation Problems

1.3. Examples of Inverse Problems

1.4. Discretizing Integral Equations

1.5. Why Inverse Problems Are Difficult

1.6. Exercises

1.7. Notes and Further Reading

Chapter Two. Linear Regression

2.1. Introduction to Linear Regression

2.2. Statistical Aspects of Least Squares

2.3. An Alternative View of the 95% Confidence Ellipsoid

2.4. Unknown Measurement Standard Deviations

2.5. L1 Regression

2.6. Monte Carlo Error Propagation

2.7. Exercises

2.8. Notes and Further Reading

Chapter Three. Rank Deficiency and Ill-Conditioning

3.1. The SVD and the Generalized Inverse

3.2. Covariance and Resolution of the Generalized Inverse Solution

3.3. Instability of the Generalized Inverse Solution

3.4. A Rank Deficient Tomography Problem

3.5. Discrete Ill-Posed Problems

3.6. Exercises

3.7. Notes and Further Reading

Chapter Four. Tikhonov Regularization

4.1. Selecting Good Solutions to Ill-Posed Problems

4.2. SVD Implementation of Tikhonov Regularization

4.3. Resolution, Bias, and Uncertainty in the Tikhonov Solution

4.4. Higher-Order Tikhonov Regularization

4.5. Resolution in Higher-order Tikhonov Regularization

4.6. The TGSVD Method

4.7. Generalized Cross-Validation

4.8. Error Bounds

4.9. Exercises

4.10. Notes and Further Reading

Chapter Five. Discretizing Problems Using Basis Functions

5.1. Discretization by Expansion of the Model

5.2. Using Representers as Basis Functions

5.3. The Method of Backus and Gilbert

5.4. Exercises

5.5. Notes and Further Reading

Chapter Six. Iterative Methods

6.1. Introduction

6.2. Iterative Methods for Tomography Problems

6.3. The Conjugate Gradient Method

6.4. The CGLS Method

6.5. Resolution Analysis for Iterative Methods

6.6. Exercises

6.7. Notes and Further Reading

Chapter Seven. Additional Regularization Techniques

7.1. Using Bounds as Constraints

7.2. Sparsity Regularization

7.3. Using IRLS to Solve L1 Regularized Problems

7.4. Total Variation

7.5. Exercises

7.6. Notes and Further Reading

Chapter Eight. Fourier Techniques

8.1. Linear Systems in the Time and Frequency Domains

8.2. Linear Systems in Discrete Time

8.3. Water Level Regularization

8.4. Tikhonov Regularization in the Frequency Domain

8.5. Exercises

8.6. Notes and Further Reading

Chapter Nine. Nonlinear Regression

9.1. Introduction to Nonlinear Regression

9.2. Newton's Method for Solving Nonlinear Equations

9.3. The Gauss-Newton and Levenberg-Marquardt Methods for Solving Nonlinear Least Squares Problems

9.4. Statistical Aspects of Nonlinear Least Squares

9.5. Implementation Issues

9.6. Exercises

9.7. Notes and Further Reading

Chapter Ten. Nonlinear Inverse Problems

10.1. Regularizing Nonlinear Least Squares Problems

10.2. Occam's Inversion

10.3. Model Resolution in Nonlinear Inverse Problems

10.4. Exercises

10.5. Notes and Further Reading

Chapter Eleven. Bayesian Methods

11.1. Review of the Classical Approach

11.2. The Bayesian Approach

11.3. The Multivariate Normal Case

11.4. The Markov Chain Monte Carlo Method

11.5. Analyzing MCMC Output

11.6. Exercises

11.7. Notes and Further Reading

Chapter Twelve. Epilogue

Appendix A. Review of Linear Algebra

Appendix B. Review of Probability and Statistics

Appendix C. Review of Vector Calculus

Appendix D. Glossary of Notation

Bibliography

Index