Introductory geophysical inverse theory

208 Pages·2012·1.682 MB·English

by  Scales J.A., Smith M.L., Treitel S.

Introductory Geophysical Inverse Theory
John A. Scales, Martin L. Smith and Sven Treitel
Samizdat
Press
2
Introductory Geophysical Inverse Theory
John A. Scales, Martin L. Smith and Sven Treitel
Samizdat Press Golden White River Junction
(cid:1)
3
Published by the Samizdat Press
Center for Wave Phenomena
Department of Geophysics
Colorado School of Mines
Golden, Colorado 80401
and
New England Research
76 Olcott Drive
White River Junction, Vermont 05001
c Samizdat Press, 2001
(cid:13)
Samizdat Press publications are available via FTP
from samizdat.mines.edu
Or via the WWW from http://samizdat.mines.edu
Permission is given to freely copy these documents.
Contents
1 What Is Inverse Theory 1
1.1 Too many models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 No unique answer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Implausible models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Observations are noisy . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 The beach is not a model . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.7 Beach Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 A Simple Inverse Problem that Isn’t 11
2.1 A First Stab at (cid:26) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.1 Measuring Volume . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.2 Measuring Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.3 Computing (cid:26) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 The Pernicious E(cid:11)ects of Errors . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Errors in Mass Measurement . . . . . . . . . . . . . . . . . . . . 13
2.3 What is an Answer? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.1 Conditional Probabilities . . . . . . . . . . . . . . . . . . . . . . 15
2.3.2 What We’re Really (Really) After . . . . . . . . . . . . . . . . . 16
2.3.3 A (Short) Tale of Two Experiments . . . . . . . . . . . . . . . . 16
ii CONTENTS
2.3.4 The Experiments Are Identical . . . . . . . . . . . . . . . . . . 17
2.4 What does it mean to condition on the truth? . . . . . . . . . . . . . . 20
2.4.1 Another example . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Example: A Vertical Seismic Pro(cid:12)le 25
3.0.2 Travel time (cid:12)tting . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 A Little Linear Algebra 33
4.1 Linear Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.1.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.1.2 Matrices With Special Structure . . . . . . . . . . . . . . . . . . 38
4.2 Matrix and Vector Norms . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3 Projecting Vectors Onto Other Vectors . . . . . . . . . . . . . . . . . . 42
4.4 Linear Dependence and Independence . . . . . . . . . . . . . . . . . . . 45
4.5 The Four Fundamental Spaces . . . . . . . . . . . . . . . . . . . . . . . 46
4.5.1 Spaces associated with a linear system Ax = y . . . . . . . . . . 47
4.6 Matrix Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.7 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . 49
4.8 Orthogonal decomposition of rectangular matrices . . . . . . . . . . . . 52
4.9 Orthogonal projections . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.10 A few examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5 SVD and Resolution in Least Squares 59
5.0.1 A Worked Example . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.0.2 The Generalized Inverse . . . . . . . . . . . . . . . . . . . . . . 61
5.0.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.0.4 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
CONTENTS iii
6 A Summary of Probability and Statistics 71
6.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.1.1 More on Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.2 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.2.1 A De(cid:12)nition of Random . . . . . . . . . . . . . . . . . . . . . . 75
6.2.2 Generating random numbers on a computer . . . . . . . . . . . 75
6.3 Bayes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.4 Probability Functions and Densities . . . . . . . . . . . . . . . . . . . . 79
6.4.1 Expectation of a Function With Respect to a Probability Law . 82
6.4.2 Multi-variate probabilities . . . . . . . . . . . . . . . . . . . . . 83
6.5 Random Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.5.1 The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . 87
6.6 Expectations and Variances . . . . . . . . . . . . . . . . . . . . . . . . 89
6.7 Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.8 Correlation of Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.9 Random Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.10 Probabilistic Information About Earth Models . . . . . . . . . . . . . . 98
6.11 Other Common Analytic Distributions . . . . . . . . . . . . . . . . . . 101
6.12 Computer Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7 Linear Inverse Problems With Uncertain Data 107
7.0.1 Model Covariances . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.1 The World’s Second Smallest Inverse Problem . . . . . . . . . . . . . . 109
7.1.1 The Damped Least Squares Problem . . . . . . . . . . . . . . . 112
8 Tomography 117
8.1 The X-ray Absorber . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
iv CONTENTS
8.1.1 The Forward Problem . . . . . . . . . . . . . . . . . . . . . . . 118
8.1.2 Linear Absorption . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.1.3 Model Representation . . . . . . . . . . . . . . . . . . . . . . . 121
8.1.4 Some Numerical Results . . . . . . . . . . . . . . . . . . . . . . 122
8.2 Travel Time Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . 124
8.3 Computer Example: Cross-well tomography . . . . . . . . . . . . . . . 125
9 From Bayes to Weighted Least Squares 131
10 Bayesian versus Frequentist Methods of Inference 135
10.0.1 Bayesian Inversion in Practice . . . . . . . . . . . . . . . . . . . 136
10.0.2 Bayes vs Frequentist . . . . . . . . . . . . . . . . . . . . . . . . 138
10.1 What Di(cid:11)erence Does the Prior Make? . . . . . . . . . . . . . . . . . . 139
10.1.1 Bayes Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
10.1.2 What is the Most Conservative Prior? . . . . . . . . . . . . . . 141
10.2 Example: A Toy Inverse Problem . . . . . . . . . . . . . . . . . . . . . 141
10.2.1 Bayes Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
10.2.2 The Flat Prior is Informative . . . . . . . . . . . . . . . . . . . 142
10.3 Priors in High Dimensional Spaces: The Curse of Dimensionality . . . . 144
11 Iterative Linear Solvers 151
11.1 Classical Iterative Methods . . . . . . . . . . . . . . . . . . . . . . . . . 151
11.2 Conjugate Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
11.2.1 Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
11.2.2 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 154
11.2.3 Quadratic Minimization . . . . . . . . . . . . . . . . . . . . . . 155
11.2.4 Computer Exercise: Steepest Descent . . . . . . . . . . . . . . . 159
11.2.5 The Method of Conjugate Directions . . . . . . . . . . . . . . . 160
CONTENTS v
11.2.6 The Method of Conjugate Gradients . . . . . . . . . . . . . . . 162
11.2.7 Finite Precision Arithmetic . . . . . . . . . . . . . . . . . . . . 164
11.2.8 CG Methods for Least-Squares . . . . . . . . . . . . . . . . . . 166
11.2.9 Computer Exercise: Conjugate Gradient . . . . . . . . . . . . . 167
11.3 Practical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 168
11.3.1 Sparse Matrix Data Structures . . . . . . . . . . . . . . . . . . . 168
11.3.2 Data and Parameter Weighting . . . . . . . . . . . . . . . . . . 169
11.3.3 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
11.3.4 Jumping Versus Creeping . . . . . . . . . . . . . . . . . . . . . 171
11.3.5 How Smoothing A(cid:11)ects Jumping and Creeping . . . . . . . . . 172
11.4 Sparse SVD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
11.4.1 The Symmetric Eigenvalue Problem . . . . . . . . . . . . . . . . 174
11.4.2 Finite Precision Arithmetic . . . . . . . . . . . . . . . . . . . . 176
11.4.3 Explicit Calculation of the Pseudo-Inverse . . . . . . . . . . . . 179
12 More on the Resolution-Variance Tradeo(cid:11) 183
12.1 A Surfer’s Guide to Backus-Gilbert Theory . . . . . . . . . . . . . . . . 183
12.2 Using the SVD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
vi CONTENTS
List of Figures
1.1 We think that gold is buried under the sand so we make measurements
of gravity at various locations on the surface. . . . . . . . . . . . . . . . 2
1.2 Inverse problems usually start with some procedure for predicting the
response of a physical system with known parameters. Then we ask:
how can we determine the unknown parameters from observed data? . 3
1.3 An idealized view of the beach. The surface is (cid:13)at and the subsurface
consists of little blocks containing either sand or gold. . . . . . . . . . . 3
1.4 Our preconceptions as to the number of bricks buried in the sand. There
is a possibility that someone has already dug up the gold, in which case
the number of gold blocks is zero. But we thing it’s most likely that
there are 6 gold blocks. Possibly 7, but de(cid:12)nitely not 3, for example.
Since this preconception represents information we have independent of
the gravity data, or prior to the measurements, it’s an example of what
is called a priori information.. . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Pirate chests were well made. And gold, being rather heavy, is unlikely
to move around much. So we think it’s mostly likely that the gold bars
are clustered together. It’s not impossible that the bars have become
dispersed, but it seems unlikely. . . . . . . . . . . . . . . . . . . . . . . 6
1.6 The path connecting nature and the corrected observations is long and
di(cid:14)cult. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.7 The true distribution of gold bricks. . . . . . . . . . . . . . . . . . . . . 9
1.8 An unreasonable model that predicts the data. . . . . . . . . . . . . . . 10
2.1 A chunk of kryptonite. Unfortunately, kryptonite’s properties do not
appear to be in the handbooks. . . . . . . . . . . . . . . . . . . . . . . 11
2.2 A pycnometer is a device that measures volumes via a calibrated beaker
partially (cid:12)lled with water. . . . . . . . . . . . . . . . . . . . . . . . . . 12