Parameter Estimation and Inverse Problems
By Richard C. Aster
 Brian Borchers
 Clifford H. Thurber
Parameter Estimation and Inverse Problems, 2e 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 authorsâ 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, 2e 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. A companion website features computational examples (including all examples contained in the textbook) and useful subroutines using MATLAB.
Audience
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.
Hardbound, 376 Pages
Published: January 2012
Imprint: Academic Press
ISBN: 9780123850485
Reviews

"A few years ago, it was my pleasure to review for the TLE this bookâs first edition, published in 2005â¦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 pseudocode, then backed up by a collection of MATLAB codes downloadable from an Elsevier Web siteâ¦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
Contents
Preface
Chapter 1: 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 Hard
1.6 Exercises
1.7 Notes and Further ReadingChapter 2: 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 ReadingChapter 3: 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 IllPosed Problems
3.6 Exercises
3.7 Notes and Further ReadingChapter 4: Tikhonov Regularization
4.1 Selecting a Good Solution
4.2 SVD Implementation of Tikhonov Regularization
4.3 Resolution, Bias, and Uncertainty in the Tikhonov Solution
4.4 HigherOrder Tikhonov Regularization
4.5 Resolution in HigherOrder Tikhonov Regularization
4.6 The TGSVD Method
4.7 Generalized CrossValidation
4.8 Error Bounds
4.9 Exercises
4.10 Notes and Further ReadingChapter 5: Discretizing Inverse 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 ReadingChapter 6: 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 ReadingChapter 7: 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 ReadingChapter 8: 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 ReadingChapter 9: Nonlinear Regression
9.1 Newton's Method for Solving Nonlinear Equations
9.2 The GaussNewton and LevenbergMarquardt Methods for Solving Nonlinear Least Squares Problems
9.3 Statistical Aspects of Nonlinear Least Squares
9.4 Implementation Issues
9.5 Exercises
9.6 Notes and Further ReadingChapter 10: 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 ReadingChapter 11: 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 (MCMC) Method
11.5 Analyzing MCMC Output
11.6 Exercises
11.7 Notes and Further ReadingChapter 12: Epilogue
A Review of Linear Algebra
A.1 Systems of Linear Equations
A.2 Matrix and Vector Algebra
A.3 Linear Independence
A.4 Subspaces of Rn
A.5 Orthogonality and the Dot Product
A.6 Eigenvalues and Eigenvectors
A.7 Vector and Matrix Norms
A.8 The Condition Number of a Linear System
A.9 The QR Factorization
A.10 Complex Matrices and Vectors
A.11 Linear Algebra in Spaces of Functions
A.12 Exercises
A.13 Notes and Further ReadingB Review of Probability and Statistics
B.1 Probability and Random Variables
B.2 Expected Value and Variance
B.3 Joint Distributions
B.4 Conditional Probability
B.5 The Multivariate Normal Distribution
B.6 The Central Limit Theorem
B.7 Testing for Normality
B.8 Estimating Means and Confidence Intervals
B.9 Hypothesis Tests
B.10 Exercises
B.11 Notes and Further ReadingC Review of Vector Calculus
C.1 The Gradient, Hessian, and Jacobian
C.2 Taylor's Theorem
C.3 Lagrange Multipliers
C.4 Exercises
C.5 Notes and Further ReadingD Glossary of Notation
Bibliography
Index