Parameter Estimation and Inverse Problems

2nd Edition

Authors: Richard C. Aster Brian Borchers Clifford H. Thurber
Hardcover ISBN: 9780123850485
eBook ISBN: 9780123850492
Imprint: Academic Press
Published Date: 21st January 2012
Page Count: 376
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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.

Key Features

  • 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.

Table of Contents


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. Itera


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About the Author

Richard C. Aster

Affiliations and Expertise

New Mexico Institute of Mining and Technology, Socorro, USA

Brian Borchers

Affiliations and Expertise

New Mexico Institute of Mining and Technology, Socorro, USA

Clifford H. Thurber

Affiliations and Expertise

University of Wisconsin-Madison, USA


"A few years ago, it was my pleasure to review for the TLE this book’s first edition, published in 2005…The present revised version is some 60 pages longer and contains several significant modifications.  As is true of the original, the book continues to be one of the clearest as well as the most comprehensive elementary expositions of discrete geophysical inverse theory.  It is ideally suited for beginners as well as a fine resource for those searching for a particular inverse problem.  Each algorithm is presented in the form of pseudo-code, then backed up by a collection of MATLAB codes downloadable from an Elsevier Web site…All examples in the book are beautifully illustrated with simple, easy to follow "cartoon" problems, and all painstakingly designed to illuminate the details of a particular numerical method."--The Leading Edge, July 2012, page 860