Geophysical Data Analysis: Discrete Inverse Theory

Geophysical Data Analysis: Discrete Inverse Theory

MATLAB Edition

3rd Edition - June 21, 2012

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  • Author: William Menke
  • Paperback ISBN: 9780128100486
  • eBook ISBN: 9780123977847

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Since 1984, Geophysical Data Analysis has filled the need for a short, concise reference on inverse theory for individuals who have an intermediate background in science and mathematics. The new edition maintains the accessible and succinct manner for which it is known, with the addition of: MATLAB examples and problem sets Advanced color graphics Coverage of new topics, including Adjoint Methods; Inversion by Steepest Descent, Monte Carlo and Simulated Annealing methods; and Bootstrap algorithm for determining empirical confidence intervals

Key Features

  • Additional material on probability, including Bayesian influence, probability density function, and metropolis algorithm
  • Detailed discussion of application of inverse theory to tectonic, gravitational and geomagnetic studies
  • Numerous examples and end-of-chapter homework problems help you explore and further understand the ideas presented
  • Use as classroom text facilitated by a complete set of exemplary lectures in Microsoft PowerPoint format and homework problem solutions for instructors


Graduate students and researchers in solid earth geophysics, seismology, atmospheric sciences and other areas of applied physics (e.g. image processing) and mathematics.

Table of Contents

  • Chapter 1. Describing Inverse Problems

    1.1 Formulating Inverse Problems

    1.2 The Linear Inverse Problem

    1.3 Examples of Formulating Inverse Problems

    1.4 Solutions to Inverse Problems

    1.5 Problems


    Chapter 2. Some Comments on Probability Theory

    2.1 Noise and Random Variables

    2.2 Correlated Data

    2.3 Functions of Random Variables

    2.4 Gaussian Probability Density Functions

    2.5 Testing the Assumption of Gaussian Statistics

    2.6 Conditional Probability Density Functions

    2.7 Confidence Intervals

    2.8 Computing Realizations of Random Variables

    2.9 Problems


    Chapter 3. Solution of the Linear, Gaussian Inverse Problem, Viewpoint 1: The Length Method

    3.1 The Lengths of Estimates

    3.2 Measures of Length

    3.3 Least Squares for a Straight Line

    3.4 The Least Squares Solution of the Linear Inverse Problem

    3.5 Some Examples

    3.6 The Existence of the Least Squares Solution

    3.7 The Purely Underdetermined Problem

    3.8 Mixed-Determined Problems

    3.9 Weighted Measures of Length as a Type of A Priori Information

    3.10 Other Types of A Priori Information

    3.11 The Variance of the Model Parameter Estimates

    3.12 Variance and Prediction Error of the Least Squares Solution

    3.13 Problems


    Chapter 4. Solution of the Linear, Gaussian Inverse Problem, Viewpoint 2: Generalized Inverses

    4.1 Solutions Versus Operators

    4.2 The Data Resolution Matrix

    4.3 The Model Resolution Matrix

    4.4 The Unit Covariance Matrix

    4.5 Resolution and Covariance of Some Generalized Inverses

    4.6 Measures of Goodness of Resolution and Covariance

    4.7 Generalized Inverses with Good Resolution and Covariance

    4.8 Sidelobes and the Backus-Gilbert Spread Function

    4.9 The Backus-Gilbert Generalized Inverse for the Underdetermined Problem

    4.10 Including the Covariance Size

    4.11 The Trade-off of Resolution and Variance

    4.12 Techniques for Computing Resolution

    4.13 Problems


    Chapter 5. Solution of the Linear, Gaussian Inverse Problem, Viewpoint 3: Maximum Likelihood Methods

    5.1 The Mean of a Group of Measurements

    5.2 Maximum Likelihood Applied to Inverse Problem

    5.3 Relative Entropy as a Guiding Principle

    5.4 Equivalence of the Three Viewpoints

    5.5 The F-Test of Error Improvement Significance

    5.6 Problems


    Chapter 6. Nonuniqueness and Localized Averages

    6.1 Null Vectors and Nonuniqueness

    6.2 Null Vectors of a Simple Inverse Problem

    6.3 Localized Averages of Model Parameters

    6.4 Relationship to the Resolution Matrix

    6.5 Averages Versus Estimates

    6.6 Nonunique Averaging Vectors and A Priori Information

    6.7 Problems


    Chapter 7. Applications of Vector Spaces

    7.1 Model and Data Spaces

    7.2 Householder Transformations

    7.3 Designing Householder Transformations

    7.4 Transformations That Do Not Preserve Length

    7.5 The Solution of the Mixed-Determined Problem

    7.6 Singular-Value Decomposition and the Natural Generalized Inverse

    7.7 Derivation of the Singular-Value Decomposition

    7.8 Simplifying Linear Equality and Inequality Constraints

    7.9 Inequality Constraints

    7.10 Problems


    Chapter 8. Linear Inverse Problems and Non-Gaussian Statistics

    8.1 L1 Norms and Exponential Probability Density Functions

    8.2 Maximum Likelihood Estimate of the Mean of an Exponential Probability Density Function

    8.3 The General Linear Problem

    8.4 Solving L1 Norm Problems

    8.5 The L∞ Norm

    8.6 Problems


    Chapter 9. Nonlinear Inverse Problems

    9.1 Parameterizations

    9.2 Linearizing Transformations

    9.3 Error and Likelihood in Nonlinear Inverse Problems

    9.4 The Grid Search

    9.5 The Monte Carlo Search

    9.6 Newton’s Method

    9.7 The Implicit Nonlinear Inverse Problem with Gaussian Data

    9.8 Gradient Method

    9.9 Simulated Annealing

    9.10 Choosing the Null Distribution for Inexact Non-Gaussian Nonlinear Theories

    9.11 Bootstrap Confidence Intervals

    9.12 Problems


    Chapter 10. Factor Analysis

    10.1 The Factor Analysis Problem

    10.2 Normalization and Physicality Constraints

    10.3 Q-Mode and R-Mode Factor Analysis

    10.4 Empirical Orthogonal Function Analysis

    10.5 Problems


    Chapter 11. Continuous Inverse Theory and Tomography

    11.1 The Backus-Gilbert Inverse Problem

    11.2 Resolution and Variance Trade-Off

    11.3 Approximating Continuous Inverse Problems as Discrete Problems

    11.4 Tomography and Continuous Inverse Theory

    11.5 Tomography and the Radon Transform

    11.6 The Fourier Slice Theorem

    11.7 Correspondence Between Matrices and Linear Operators

    11.8 The Fréchet Derivative

    11.9 The Fréchet Derivative of Error

    11.10 Backprojection

    11.11 Fréchet Derivatives Involving a Differential Equation

    11.12 Problems


    Chapter 12. Sample Inverse Problems

    12.1 An Image Enhancement Problem

    12.2 Digital Filter Design

    12.3 Adjustment of Crossover Errors

    12.4 An Acoustic Tomography Problem

    12.5 One-Dimensional Temperature Distribution

    12.6 L1, L2, and L∞ Fitting of a Straight Line

    12.7 Finding the Mean of a Set of Unit Vectors

    12.8 Gaussian and Lorentzian Curve Fitting

    12.9 Earthquake Location

    12.10 Vibrational Problems

    12.11 Problems


    Chapter 13. Applications of Inverse Theory to Solid Earth Geophysics

    13.1 Earthquake Location and Determination of the Velocity Structure of the Earth from Travel Time Data

    13.2 Moment Tensors of Earthquakes

    13.3 Waveform “Tomography”

    13.4 Velocity Structure from Free Oscillations and Seismic Surface Waves

    13.5 Seismic Attenuation

    13.6 Signal Correlation

    13.7 Tectonic Plate Motions

    13.8 Gravity and Geomagnetism

    13.9 Electromagnetic Induction and the Magnetotelluric Method


    Chapter 14. Appendices

    14.1 Implementing Constraints with Lagrange multipliers

    14.2 L2 Inverse Theory with Complex Quantities

Product details

  • No. of pages: 330
  • Language: English
  • Copyright: © Academic Press 2012
  • Published: June 21, 2012
  • Imprint: Academic Press
  • Paperback ISBN: 9780128100486
  • eBook ISBN: 9780123977847

About the Author

William Menke

William Menke is a Professor of Earth and Environmental Sciences at Columbia University. His research focuses on the development of data analysis algorithms for time series analysis and imaging in the earth and environmental sciences and the application of these methods to volcanoes, earthquakes, and other natural hazards. He has thirty years of experience teaching data analysis methods to both undergraduates and graduate students. Relevant courses that he has taught include, at the undergraduate level, Environmental Data Analysis and The Earth System, and at the graduate level, Geophysical Inverse Theory, Quantitative Methods of Data Analysis, Geophysical Theory and Practical Seismology.

Affiliations and Expertise

Professor of Earth and Environmental Sciences ,Columbia University

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  • MiaoZhang Tue Feb 04 2020

    Excellent book

    It is an excellent book! I like it very much.