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Geophysical Data Analysis

by: Menke, William

Geophysical Data Analysis

by: Menke, William

ISBN 13:
9780123971609
ISBN 10:
0123971608
Edition: 3rd
Format: Hardcover
Copyright: 06/21/2012
Publisher: Academic Pr

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Summary

Please use extracts from reviews of first edition Key Features * Updated and thoroughly revised edition * additional material on geophysical/acoustic tomography * Detailed discussion of application of inverse theory to tectonic, gravitational and geomagnetic studies

Introduction	p. xv
Describing Inverse Problems
Formulating Inverse Problems	p. 1
Implicit Linear Form	p. 2
Explicit Form	p. 2
Explicit Linear Form	p. 3
The Linear Inverse Problem	p. 3
Examples of Formulating Inverse Problems	p. 4
Example 1: Fitting a Straight Line	p. 4
Example 2: Fitting a Parabola	p. 5
Example 3: Acoustic Tomography	p. 6
Example 4: X-ray Imaging	p. 7
Example 5: Spectral Curve Fitting	p. 9
Example 6: Factor Analysis	p. 10
Solutions to Inverse Problems	p. 11
Estimates of Model Parameters	p. 11
Bounding Values	p. 12
Probability Density Functions	p. 12
Sets of Realizations of Model Parameters	p. 13
Weighted Averages of Model Parameters	p. 13
Problems	p. 13
Some Comments on Probability Theory
Noise and Random Variables	p. 15
Correlated Data	p. 19
Functions of Random Variables	p. 21
Gaussian Probability Density Functions	p. 26
Testing the Assumption of Gaussian Statistics	p. 29
Conditional Probability Density Functions	p. 30
Confidence Intervals	p. 33
Computing Realizations of Random Variables	p. 34
Problems	p. 37
Solution of the Linear, Gaussian Inverse Problem, Viewpoint 1: The Length Method
The Lengths of Estimates	p. 39
Measures of Length	p. 39
Least Squares for a Straight Line	p. 43
The Least Squares Solution of the Linear Inverse Problem	p. 44
Some Examples	p. 46
The Straight Line Problem	p. 46
Fitting a Parabola	p. 47
Fitting a Plane Surface	p. 48
The Existence of the Least Squares Solution	p. 49
Underdetermined Problems	p. 51
Even-Determined Problems	p. 52
Overdetermined Problems	p. 52
The Purely Underdetermined Problem	p. 52
Mixed-Determined Problems	p. 54
Weighted Measures of Length as a Type of A Priori Information	p. 56
Weighted Least Squares	p. 58
Weighted Minimum Length	p. 58
Weighted Damped Least Squares	p. 58
Other Types of A Priori Information	p. 60
Example: Constrained Fitting of a Straight Line	p. 62
The Variance of the Model Parameter Estimates	p. 63
Variance and Prediction Error of the Least Squares Solution	p. 64
Problems	p. 67
Solution of the Linear, Gaussian Inverse Problem, Viewpoint 2: Generalized Inverses
Solutions Versus Operators	p. 69
The Data Resolution Matrix	p. 69
The Model Resolution Matrix	p. 72
The Unit Covariance Matrix	p. 72
Resolution and Covariance of Some Generalized Inverses	p. 74
Least Squares	p. 74
Minimum Length	p. 75
Measures of Goodness of Resolution and Covariance	p. 75
Generalized Inverses with Good Resolution and Covariance	p. 76
Overdetermined Case	p. 76
Underdetermined Case	p. 77
The General Case with Dirichlet Spread Functions	p. 77
Sidelobes and the Backus-Gilbert Spread Function	p. 78
The Backus-Gilbert Generalized Inverse for the Underdetermined Problem	p. 79
Including the Covariance Size	p. 83
The Trade-off of Resolution and Variance	p. 84
Techniques for Computing Resolution	p. 86
Problems	p. 88
Solution of the Linear, Gaussian Inverse Problem, Viewpoint 3: Maximum Likelihood Methods
The Mean of a Group of Measurements	p. 89
Maximum Likelihood Applied to Inverse Problem	p. 92
The Simplest Case	p. 92
A Priori Distributions	p. 92
Maximum Likelihood for an Exact Theory	p. 97
Inexact Theories	p. 100
The Simple Gaussian Case with a Linear Theory	p. 102
The General Linear, Gaussian Case	p. 104
Exact Data and Theory	p. 107
Infinitely Inexact Data and Theory	p. 108
No A Priori Knowledge of the Model Parameters	p. 108
Relative Entropy as a Guiding Principle	p. 108
Equivalence of the Three Viewpoints	p. 110
The F-Test of Error Improvement Significance	p. 111
Problems	p. 113
Nonuniqueness and Localized Averages
Null Vectors and Nonuniqueness	p. 115
Null Vectors of a Simple Inverse Problem	p. 116
Localized Averages of Model Parameters	p. 117
Relationship to the Resolution Matrix	p. 117
Averages Versus Estimates	p. 118
Nonunique Averaging Vectors and A Priori Information	p. 119
Problems	p. 121
Applications of Vector Spaces
Model and Data Spaces	p. 123
Householder Transformations	p. 124
Designing Householder Transformations	p. 127
Transformations That Do Not Preserve Length	p. 129
The Solution of the Mixed-Determined Problem	p. 130
Singular-Value Decomposition and the Natural Generalized Inverse	p. 132
Derivation of the Singular-Value Decomposition	p. 138
Simplifying Linear Equality and Inequality Constraints	p. 138
Linear Equality Constraints	p. 139
Linear Inequality Constraints	p. 139
Inequality Constraints	p. 140
Problems	p. 147
Linear Inverse Problems and Non-Gaussian Statistics
L₁ Norms and Exponential Probability Density Functions	p. 149
Maximum Likelihood Estimate of the Mean of an Exponential Probability Density Function	p. 151
The General Linear Problem	p. 153
Solving L₁ Norm Problems	p. 153
The L_∞ Norm
Problems	p. 160
Nonlinear Inverse Problems
Parameterizations	p. 163
Linearizing Transformations	p. 165
Error and Likelihood in Nonlinear Inverse Problems	p. 166
The Grid Search	p. 167
The Monte Carlo Search	p. 170
Newton's Method	p. 171
The Implicit Nonlinear Inverse Problem with Gaussian Data	p. 175
Gradient Method	p. 180
Simulated Annealing	p. 181
Choosing the Null Distribution for Inexact Non-Gaussian Nonlinear Theories	p. 184
Bootstrap Confidence Intervals	p. 185
Problems	p. 186
Factor Analysis
The Factor Analysis Problem	p. 189
Normalization and Physicality Constraints	p. 194
Q-Mode and R-Mode Factor Analysis	p. 199
Empirical Orthogonal Function Analysis	p. 199
Problems	p. 204
Continuous Inverse Theory and Tomography
The Backus-Gilbert Inverse Problem	p. 207
Resolution and Variance Trade-Off	p. 209
Approximating Continuous Inverse Problems as Discrete Problems	p. 209
Tomography and Continuous Inverse Theory	p. 211
Tomography and the Radon Transform	p. 212
The Fourier Slice Theorem	p. 213
Correspondence Between Matrices and Linear Operators	p. 214
The Frechet Derivative	p. 218
The Frechet Derivative of Error	p. 218
Backprojection	p. 219
Frechet Derivatives Involving a Differential Equation	p. 222
Problems	p. 227
Sample Inverse Problems
An Image Enhancement Problem	p. 231
Digital Filter Design	p. 234
Adjustment of Crossover Errors	p. 236
An Acoustic Tomography Problem	p. 240
One-Dimensional Temperature Distribution	p. 241
L₁ L₂, and L_∞ Fitting of a Straight Line	p. 245
Finding the Mean of a Set of Unit Vectors	p. 246
Gaussian and Lorentzian Curve Fitting	p. 250
Earthquake Location	p. 252
Vibrational Problems	p. 256
Problems	p. 259
Applications of Inverse Theory to Solid Earth Geophysics
Earthquake Location and Determination of the Velocity Structure of the Earth from Travel Time Data	p. 261
Moment Tensors of Earthquakes	p. 264
Waveform "Tomography"	p. 265
Velocity Structure from Free Oscillations and Seismic Surface Waves	p. 267
Seismic Attenuation	p. 269
Signal Correlation	p. 270
Tectonic Plate Motions	p. 271
Gravity and Geomagnetism	p. 271
Electromagnetic Induction and the Magnetotelluric Method	p. 273
Appendices
Implementing Constraints with Lagrange multipliers	p. 277
L₂ Inverse Theory with Complex Quantities	p. 278
Index	p. 281
Table of Contents provided by Ingram. All Rights Reserved.

Amazon no longer offers textbook rentals. We do!

Geophysical Data Analysis

Geophysical Data Analysis

9780123971609

0123971608

Summary

Table of Contents

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