Geophysical Data Analysis: Discrete Inverse Theory

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

REFERENCES

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

REFERENCES

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

REFERENCES

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

REFERENCES

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

REFERENCES

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

REFERENCES

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

REFERENCES

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

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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

REFERENCES

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

REFERENCES

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

REFERENCES

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

REFERENCES

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

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Chapter 14. Appendices

14.1 Implementing Constraints with Lagrange multipliers

14.2 L2 Inverse Theory with Complex Quantities