Menke W. Geophysical Data Analysis: Discrete Inverse Theory
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GEOPHYSICAL DATA ANALYSIS:
DISCRETE INVERSE THEORY
This
is
Volume
45
in
INTERNATIONAL GEOPHYSICS SERIES
A series
of
monographs and textbooks
Edited by RENATA DMOWSKA and JAMES R. HOLTON
A
complete
list
of
the
books
in this
series
appears at the end
of
this
volume.
GEOPHYSICAL DATA ANALYSIS:
DISCRETE
INVERSE
THEORY
Revised Edition
William
Menke
Lamont-Doherty Geological Observatory and
Department
of
Geological Sciences
Columbia University
Palisades, New York
Formerly
of
College
of
Oceanography
Oregon State University
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INC.
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Jovanovich,
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I
70X
CONTENTS
PREFACE
xi
INTRODUCTION
I
1
DESCRIBING INVERSE PROBLEMS
1.1
Formulating Inverse Problems
7
1.2 The Linear Inverse Problem 9
1.3 Examples of Formulating Inverse Problems
10
1.4 Solutions to Inverse Problems
17
2
SOME COMMENTS ON PROBABILITY THEORY
2.
I
Noise and Random Variables 21
2.2 Correlated Data 24
2.3 Functions of Random Variables 27
2.4 Gaussian Distributions 29
2.5 Testing the Assumption of Gaussian Statistics 31
2.6 Confidence Intervals 33
3
SOLUTION
OF
THE
LINEAR, GAUSSIAN INVERSE PROBLEM,
VIEWPOINT
1
:
THE
LENGTH METHOD
3.1 The Lengths
of
Estimates 35
3.2 Measures of Length 36
3.3 Least Squares for a Straight Line 39
V
vi Contents
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.1
1
3.12
The Least Squares Solution of the Linear Inverse
Problem 40
Some Examples 42
The Existence of the Least Squares Solution
The Purely Underdetermined Problem 48
Mixed- Determined Problems 50
Weighted Measures of Length as a Type
of
A
Priori
Information 52
The Variance
of
the Model Parameter Estimates
Variance and Prediction Error of the Least Squares
So
1
u
t i
o
n
45
Other Types of
A
Priori Information
55
58
58
4
SOLUTION
OF
THE
LINEAR, GAUSSIAN INVERSE PROBLEM,
VIEWPOINT
2:
GENERALIZED INVERSES
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
Solutions versus Operators 61
The Data Resolution Matrix 62
The Model Resolution Matrix 64
The Unit Covariance Matrix 65
Resolution and Covariance
of
Some Generalized
Inverses 66
Measures of Goodness of Resolution and Covariance
Generalized Inverses with Good Resolution and
Covariance 68
Sidelobes and the Backus-Gilbert Spread Function
The Backus-Gilbert Generalized Inverse for the
Underdetermined Problem 73
Including the Covariance Size 75
The Trade-off
of
Resolution and Variance
67
71
76
5
SOLUTION
OF
THE LINEAR, GAUSSIAN INVERSE
PROBLEM,
VIEWPOINT
3:
MAXIMUM LIKELIHOOD METHODS
5.1 The Mean
of
a Group
of
Measurements 79
5.2
5.3
A
Priori Distributions 83
5.4
Maximum Likelihood for an Exact Theory 87
5.5 Inexact Theories 89
Maximum Likelihood Solution
of
the Linear Inverse
Problem 82
Contents vii
5.6 The Simple Gaussian Case with a Linear Theory 91
5.7 The General Linear, Gaussian Case 92
5.8 Equivalence of the Three Viewpoints 95
5.9 The
F
Test of Error Improvement Significance 96
5.10 Derivation of the Formulas
of
Section 5.7 97
6
NONUNIQUENESS AND LOCALIZED AVERAGES
6.1 Null Vectors and Nonuniqueness
101
6.2 Null Vectors of a Simple Inverse Problem 102
6.3 Localized Averages of Model Parameters 103
6.4 Relationship to the Resolution Matrix 104
6.5 Averages versus Estimates 105
6.6 Nonunique Averaging Vectors and A Priori
Information 106
7
APPLICATIONS
OF
VECTOR SPACES
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
Model and Data Spaces 109
Householder Transformations
111
Transformations That Do Not Preserve Length
The Solution of the Mixed-Determined Problem I18
Singular-Value Decomposition and the Natural Generalized
Inverse 119
Derivation of the Singular-Value Decomposition
Simplifying Linear Equality and Inequality
Constraints 125
Inequality Constraints 126
Designing Householder Transformations 115
117
124
8
LINEAR INVERSE PROBLEMS AND NON-GAUSSIAN
DISTRIBUTIONS
8.1
L
,
Norms and Exponential Distributions 133
8.2
8.3 The General Linear Problem I37
8.4 Solving
L,
Norm Problems 138
8.5 TheL-Norm 141
Maximum Likelihood Estimate of the Mean of an Exponential
Distribution 135
...
Vlll
Contents
9
NONLINEAR INVERSE PROBLEMS
9.1 Parameterizations 143
9.2 Linearizing Parameterizations 147
9.3 The Nonlinear Inverse Problem with Gaussian Data 147
9.4 Special Cases 153
9.5
9.6 Non-Gaussian Distributions 156
9.7 Maximum Entropy Methods 160
Convergence and Nonuniqueness
of
Nonlinear
L,
Problems 153
10
FACTOR ANALYSIS
10.1 The Factor Analysis Problem 161
10.2 Normalization and Physicality Constraints 165
10.3 &-Mode and R-Mode Factor Analysis 167
10.4
Empirical Orthogonal Function Analysis 167
11
CONTINUOUS INVERSE THEORY AND TOMOGRAPHY
11.1
The Backus-Gilbert Inverse Problem 171
1
1.2 Resolution and Variance Trade-off 173
1 1.3
1 1.4 Tomography and Continuous Inverse Theory 176
1 1.5 Tomography and the Radon Transform 177
1
1.6 The Fourier Slice Theorem 178
1
1.7 Backprojection 179
Approximating Continuous Inverse Problems as Discrete
Problems 174
12
SAMPLE INVERSE PROBLEMS
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
12.9
12.10
An image Enhancement Problem 183
Digital Filter Design 187
Adjustment
of
Crossover Errors 190
An Acoustic Tomography Problem 194
Temperature Distribution in an Igneous Intrusion
Finding the Mean of a Set of Unit Vectors
Earthquake Location
213
Vibrational Problems 217
198
L,
,
L,,
and
L,
Fitting of a Straight Line
Gaussian Curve Fitting 210
202
207
Contents
ix
13
NUMERICAL ALGORITHMS
13.1
Solving Even-Determined Problems
222
13.2
Inverting a Square Matrix
229
1
3.3
Solving Underdetermined and Overdetermined
Problems
23
1
13.4
L,
Problems with Inequality Constraints
240
13.5
Finding the Eigenvalues and Eigenvectors of
a
Real
Symmetric Matrix
25
1
13.6
The Singular-Value Decomposition of
a
Matrix
254
13.7
The Simplex Method and the Linear Programming
Problem
256
14
APPLICATIONS
OF
INVERSE THEORY TO GEOPHYSICS
14.1
14.2
14.3
14.4
14.5
14.6
14.7
14.8
Earthquake Location and the Determination
of
the Velocity
Structure
of
the Earth from Travel Time Data
Velocity Structure from Free Oscillations and Seismic Surface
Waves
265
Seismic Attenuation
267
Signal Correlation
267
Tectonic Plate Motions
268
Gravity and Geomagnetism
269
Electromagnetic Induction and the Magnetotelluric
Method
270
Ocean Circulation
27
1
26
1
APPENDIX
A:
Implementing Constraints with Lagrange
Multipliers
273
APPENDIX
B:
L2
Inverse Theory with Complex
Quantities
275
REFERENCES
277
INDEX
28
1
INTERNATIONAL GEOPHYSICS
SERIES
288