Table Of ContentLecture Notes in Earth Sciences 39
:srotidE
.S Brooklyn Bhattacharji,
.G .M Troy and Brooklyn Friedman,
.J .H Bonn Neugebauer,
.A Tuebmgen Sellacher,
Sven-Erik Hjelt
Pragmatic "srevnI
i o n
of Geophysical Data
galreV-regnirpS
Berlin Heidelberg NewYork
London Pans Tokyo
Hong Kong Barcelona
Budapest
Au~or
Sven-Erik Hjelt
Department of Geophysics, University of Oulu
P. O. Box 400, SF-90571 Oulu, Finland
"For all Lecture Notes in Earth Sciences published till now please see final page of the
book"
ISBN 3-540-55622-2 Sprlnger-Verlag Berhn Heidelberg New York
ISBN 0-387-55622-2 Springer-Verlag New York Berlin Heidelberg
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PREFACE
The systematic development of geophysical quantitative interpretation opened up for a wide
audience with the appearance of the famous book by Grant and West in 1965. The revolutionary
papers of Backus and Gilbert soon followed. Computerized inversion techniques were about to start
to develop. In 1973 I had spent more than a year working on computer implementation of magnetic
inversion using non-linear optimization. This topic had become popular in the geophysical literature.
The optimism over making the interpretation procedure automatic and "objective" soon turned into
more realistic concepts of interactivity, the increasingly important role of the human interpreter - the
geophysicists. The computer was really only a tool.
Many, even now, fundamental papers on geophysical and more general inversion appeared at
that time, many papers repeating the same basic principles over and over again. It was a time for
unifying ideas and developing an introductory course on the general principles of geophysical
inversion. My first lecture series "Geophysical Interpretation Theory" was given at the Helsinki
University of Technology in the fall of 1973. It concentrated on non-linear inversion with optimi-
zation and the technical aspects of writing geophysical computer codes.
Generalized inversion and linearized problems started to become fashionable at the same time.
After having moved to Oulu in 1975, introducing the same course there, I had a chance to visit Aar-
hus in 1976. A group of young enthusiastic geophysicists, Laust B. Pedersen, Kurt S6rensen,
Hans-Kurt Johannesen and others had been exposed to generalized inverse theory by Professors
Ulrich Schmucker and Peter Weidelt. Via the discussions in Aarhus, via lecture notes which Kurt
and later also Peter kindly made available, generalized inverse theory become a standard part of my
lectures, too, both in Oulu and Helsinki. They form the solid background for the text in Chapter 3.
Many excellent books have been written (also primarily for a geophysical audience) since
1965. Many books appear to get so involved with deep mathematical aspects of inversion theory that
the main aim of geophysical modelling and interpretation, to explain the wide and complex
phenomena and structures of Nature itself, seems to fall aside. Therefore, I believe there is still
room for an introductory text, which puts the emphasis not so much on mathematical elegance and
completeness, but on the very basic concepts of inversion, not forgetting the human aspects, and,
what is more important, on describing how and why geophysicists have selected and used various
techniques, as well as the limitations and pitfalls. In a single phrase, a text on pragmatic in-
version.
I am grateful to a great number of colleagues who, over the years, shared my interest in
geophysical inversion, many of them coming from the electromagnetic induction community. The
feedback from students has kept the course on inversion alive and under constant development. One
young fellow, Jarkko Jokinen deserves special thanks for his enthusiasm in helping to scan and
manipulate the figures of my lecture slides, thus presenting a form suitable for electronic redrawing.
Oulu, Finland, 15 January, 1992. Sven-Erik Hjelt
CONTENTS
Chapter 1 INTRODUCTION
1.1 Definition of Inversion 3
1.2 Geophysical Models 10
1.3 Model Parameters 16
1.4 Fitting Model Fields to Data 17
1.5 Factors Affecting Inversion 18
1.6 Computational Aspects 25
1.7 Psychological Aspects of Inversion 29
1.8 What then is PragmaItnivce rsion? 31
References 32
Chapter 2 INTERPRETATION USING NOMOGRAMS
2.1 Characteristic Points 35
2.2 Nomograms and Their Use 38
2.2.1 Gravity Nomograms 38
2.2.2 Nomogram for Interpretation of VLF Resistivity Data 41
2.2.3 Nomograms for MagneticD ipole (Slingram) Profiling 42
2.3 Computerized "Nomograms" 44
2.4 On the Accuracy of Characteristic Point Inversion 46
References 46
Chapter 3 LINEAR PARAMETERS
3.1 Defining the Linear Problem 15
3.1.1 Genuine Linear Problems 51
3.1.2 Discretization of the Linear Model 52
3.1.3 IAnearization 53
3.1.4 Iterative Approaches 55
3.2 Fitting Linear Models to Data 57
3.2.1 Systems of Linear Equations 57
3.2.2 Exact Fit O6
3.2.3 Least Squares (LSQ) 60
3.3 Iterative Solutions: The Method of Gauss-Seidel 61
3.4 Generalized Inversion 63
3.4.1 Singular Value Decomposition (SVD) 64
3.4.1.1 Zero Eigenvalues 66
3.4.1.2 Zero Eigenvalues, a Least-Squares Approach 67
3.4.1.3 Why are Small Eigenvalues Dangerous? 69
3.4.2 SVD of a RectanguMlaatrr ix 72
3.4.3 Geophysical Examples of the Use of SVD 73
3.4.3.1 Gravity 73
3.4.3.2 Earth Density Model 78
3.4.3.3 Magnetometry 81
3.4.3.4 Magnetometry, Theoretical Plate Models 82
3.4.3.5 Electromagnetism O9
3.4.3.6 Seismology 94
IIIV
3.4.3.7 Oceanography 96
3.4.3.8 Joint Inversion 97
3.4.4 Ridge Regression 100
3.5 The Backus-Gilbert Approach 101
3.5.1 Definitions I01
3.5.2 Parameter Trade-Off 301
3.6 Geophysical Tomography 104
3.6.1 Seismic Tomography 105
3.6.2 Tomography and the Radon Transform 107
3.6.3 Example on Seismic Regional Tomography 110
References 112
Chapter 4 NON-LINEAR PARAMETERS
4.1 Def'mitions 117
4.1.1 Objective Functions and Norms 118
4.1.1.1 Exact Fit 118
4.1.1.2 Least Squares 118
4.1.1.3 Robust Methods. The Minimax Solution 119
4.1.1.4 Weighted Objective Functions 119
4.1.2 Properties of Objective Functions 121
4.1.3 Constrained Optimization 127
4.1.3.1 Penalty Functions 127
4.1.3.2 Lagrangian Multipliers 128
4.1.4 Stopping Criteria in Iterative Optimization 129
4.2 One-Dimensional Optimization 131
4.2.1 Golden Cut 132
4.2.2 Fibonacci Search 133
4.2.3 Speedup by Parabolic Fit 134
4.2.4 The Secant Method 136
4.2.5 The GradiMeentth od 136
4.2.5.1 tOhne Instability of the Newton Iteration 137
4.3 Multidimensional Search 138
4.3.1 Direct Search 142
4.3.1.1 Sequential Sear(cShe arch by Parameters) 142
4.3.1.2 Hyperparabolic Fit 142
4.3.1.3 Pattern Search 144
4.3,2 Multivariate Search 147
4.3.2.1 The Simplex Method 147
4.3.2.2 Steepest Descent Methods: the Gradient Method 149
4.3.2.3 Steepest Descent Methods: the Method of Conjugate
Directions 150
4.3.2.4 The Levenberg-Morrison-Marquardt Algorithm 152
4.3.3 Random SeaMrecthh ods 351
4.4 Examples 155
4.4.1 Magnetic 2D Profiling 156
4.4.2 Gravity Inversion 165
4.4.3 Seismic Refraction 166
4.4.4 Magnetotellurics 167
References 171
XI
Chapter 5 MAX/MUM LIKELIHOOD DNA MAXIMUM ENTROPY
1.5 Introduction dna Definitions 771
5.1.1 Density Probability 771
5.1.2 Measure of Information 971
5.2 Formulations Probabilistic of Problem Inversion the 181
5.2.1 The Principle of Likelihood Maximum 181
5.2.2 The Method Entropy Maximum 481
5.2.3 Bayesian noitamitsE 681
5.2.4 Expressions Useful Some for Approach Probabilistic the
of Inversion 781
5.2.4.1 Distribution Normal 781
5.2,4.2 Smoothness 881
5.2.4.3 Squares Least Weighted 981
5.2.4.4 Parameter seiradnuoB 981
5.2.4.5 More on Information Priori a 091
References 291
Chapter 6 ANALYTIC INVERSION
1.6 Principles General 791
6.1.1 Combination of Components 791
6.2 Examples of Inversion Analytic 200
6.2.1 Magnetometry 200
6.2.2 EM Dipole Magnetic dna na Conducting Well Infinitely
Half-Plane 702
References 112
Chapter 7 ADVANCED INVERSION METHODS
1.7 Methods Analytical Functional 215
7.1.1 Methods of Points Singular 216
7.1.2 Method of Contours Tightening 216
7.1.3 Method fo Functions Finite 219
7.2 Cont~uation of Fields 222
7.3 Migration 226
References 226
8 Chapter ERROR ANALYSIS
1.8 Introduction 132
8.2 Errors Linearized On 232
8.3 Inversion Least-Squares in Minimal Error 332
8.4 Between Correlation sretemaraP 234
8.5 Concepts Advanced of Analysis Error 240
References 342
Chapter 9 PARALLEL MODELLING IN COMPUTATION INVERSION AND
1.9 Parallelism in Problems Geophysical 248
9,2 Problems Forward 249
3.9 Inversion by noitazimitpO 250
9.4 Systems Large of Equations 352
References 254
SUBJECT INDEX 257
1
Chapter !
I ~,TR()I)[, ('TIO
J
"When you have eliminated
the impossible,
whatever remains,
however improbable,
must be the truth."
(A.C. Doyle: A study in Scarlet)
CHAPTER 1
INTRODUCTION
1.1 DEFINITION OF INVERSION
Geophysical measurements are not done for the sake of art only. The ultimate goal is to solve
some well-defined geological, tectonic or structural problem. For this purpose the data, the
measurements have to be interpreted, that is translated into a physical model of the subsurface.
Geophysical measurements depend on and are actually designed for variations of physical properties
of the bedrock, its structures, minerals etc. Geophysical interpretation - or as it is more often called
today - inversion of geophysical measurements can be therefore defined as the construction of a
physical model of the subsurface. The physical model, depending on whether it is based on densities,
magnetic properties, electrical conductivity or differences in seismic velocities often differs from the
distribution of geological minerals. Models based on different physical properties may thus differ
quite considerably, since they describe the subsurface variation of different physical properties.
However, the models, more often than not, complete each other and a joint interpretation or complex
interpretation, as it has been called especially in eastern European literature, will lead to the best
description of the Earth.
This book tries to describe some of the most important common features of inverting different
geophysical data sets. At all stages the emphasis will be on the practical, pragmatic, aspects of the
interpretation process. For many practical purposes the virtue of a good inversion system or
algorithm is its interactivity and speed. In prospecting interpretation it is expected to be as operational
as possible. Being at its best an operative, an interactive data inversion system enables geophysicists
to direct the next steps in a prospecting campaign. If a geophysicist equipped with a proper inversion
system can bring waiting costs for drilling rigs down, the resources put into the development of such
a system will pay its costs back quickly -- and with high interest.
Fundamentally there is no difference between approaching various sets of geophysical
measurements. Consider a rectangular coordinate system, where traditionally z is positive down into
the Earth, and x along the profile. One can speak of following subdivision:
Type of Inversion Dimension
measurement produces of model
a) sounding )z(marap D1
b) prof'ding )x(marap partially 2D
c) arrays x(marap and y) 2D horizontal
d) combination of a and b param(x and z) 2D cross-section
e) combination of a and c param(x, y and z) 3D
Some definitions of 2.5-dimensional models have been introduced in the literature, but their use is not
recommended in order to avoid confusion with the newly risen concept of fractals. Another source of
confusion are the different definitions of 2.5-dimensionality for different geophysical methods. In
