Parameter Estimation and Inverse Problems

Parameter Estimation and Inverse Problems, 2nd Edition

Parameter Estimation and Inverse Problems, 2nd Edition,Richard C. Aster,Brian Borchers,Clifford H. Thurber,ISBN9780123850485

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Key Features

  • Includes appendices for review of needed concepts in linear, statistics, and vector calculus.
  • Companion website contains comprehensive MATLAB code for all examples, which readers can reproduce, experiment with, and modify.
  • Online instructor’s guide helps professors teach, customize exercises, and select homework problems
  • Accessible to students and professionals without a highly specialized mathematical background.


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 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, 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.


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.

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

Parameter Estimation and Inverse Problems, 2nd Edition


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 Reading

Chapter 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 Reading

Chapter 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 Ill-Posed Problems
3.6 Exercises
3.7 Notes and Further Reading

Chapter 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 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 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 Reading

Chapter 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 Reading

Chapter 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 Reading

Chapter 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 Reading

Chapter 9: Nonlinear Regression
9.1 Newton's Method for Solving Nonlinear Equations
9.2 The Gauss-Newton and Levenberg-Marquardt 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 Reading

Chapter 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 Reading

Chapter 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 Reading

Chapter 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 Reading

B 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 Reading

C 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 Reading

D Glossary of Notation



Quotes and reviews

"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

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