Parameter Estimation and Inverse Problems

Parameter Estimation and Inverse Problems

2nd Edition - December 10, 2011

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  • Authors: Richard C. Aster, Brian Borchers, Clifford H. Thurber
  • eBook ISBN: 9780123850492

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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

  • 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



Product details

  • No. of pages: 376
  • Language: English
  • Copyright: © Academic Press 2012
  • Published: December 10, 2011
  • Imprint: Academic Press
  • eBook ISBN: 9780123850492

About the Authors

Richard C. Aster

Professor Aster is an Earth scientist with broad interests in geophysics, seismological imaging and source studies, and Earth processes. His work has included significant field research in western North America, Italy, and Antarctica. Professor Aster also has strong teaching and research interests in geophysical inverse and signal processing methods and is the lead author on the previous two editions. Aster was on the Seismological Society of America Board of Directors, 2008-2014 and won the IRIS Leadership Award, 2014.

Affiliations and Expertise

New Mexico Institute of Mining and Technology, Socorro, USA

Brian Borchers

Brian Borchers
Dr. Borchers’ primary research and teaching interests are in optimization and inverse problems. He teaches a number of undergraduate and graduate courses at NMT in linear programming, nonlinear programming, time series analysis, and geophysical inverse problems. Dr. Borchers’ research has focused on interior point methods for linear and semidefinite programming and applications of these techniques to combinatorial optimization problems. He has also done work on inverse problems in geophysics and hydrology using linear and nonlinear least squares and Tikhonov regularization.

Affiliations and Expertise

New Mexico Institute of Mining and Technology, Socorro, USA

Clifford H. Thurber

Clifford H. Thurber
Professor Thurber is an international leader in research on three-dimensional seismic imaging ("seismic tomography") using earthquakes. His primary research interests are in the application of seismic tomography to fault zones, volcanoes, and subduction zones, with a long-term focus on the San Andreas fault in central California and volcanoes in Hawaii and Alaska. Other areas of expertise include earthquake location (the topic of a book he edited) and geophysical inverse theory.

Affiliations and Expertise

University of Wisconsin-Madison, USA

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  • Andri H. Thu Dec 28 2017

    Parameter estimation and inverse problems

    I enjoy reading this book. Well explained.