Geophysical Inverse Theory

408 Pages·1994·15.709 MB·English

by  Robert L. Parker

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GEOPHYSICAL INVERSE THEORY

GEOPHYSICAL
INVERSE
THEORY
ROBERT L. PARKER
PRINCETON UNIVERSITY PRESS
PRINCETON, NEW JERSEY
Thomas J. Bata Library
TRENT
UNIVERSITY
PETERBOROUGH, ONTARIO
Qi,C 608^
Copyright © 1994 by Princeton University Press
Published by Princeton University Press, 41 William Street,
Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press,
Chichester, West Sussex
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Parker, Robert L. (Robert Ladislav), 1942-
Geophysical Inverse Theory / by Robert L. Parker.
p. cm. — (Princeton series in geophysics)
Includes bibliographical references and index.
ISBN 0-691-03634-9
1. Inversion (Geophysics). 2. Geophysics—measurement.
3. Functional analysis. 4. Mathematical optimization.
I. Title. II. Series.
QC808.5.P37 1994
550’.1’515—dc20 93-44915
CIP
Printed on recycled paper
Princeton University books are printed on
acid-free paper and meet the guidelines for
permanence and durability of the Committee on
Production Guidelines for Book Longevity
of the Council on Library Resources
Printed in the United States of America
2468 10 97531
CONTENTS
Preface
Chapter 1: Mathematical Precursor
1.01 Linear Vector Spaces 3
1.02 Subspaces, Linear Combinations,
and Linear Independence 7
1.03 Bases and Dimension 8
1.04 Functionals 10
1.05 Norms 12
1.06 Some Norms on Finite-Dimensional Spaces 13
1.07 Some Norms on Infinite-Dimensional Spaces 18
1.08 Convergence, Cauchy Sequences,
and Completeness 20
1.09 Completion 25
1.10 Inner Products 27
1.11 Hilbert Space 31
1.12 Two Simple Minimization Problems 34
1.13 Numerical Aspects of Hilbert Space 38
1.14 Lagrange Multipliers 46
1.15 Convexity 51
Chapter 2: Linear Problems with Exact Data
2.01 An Example 55
2.02 Linear Inverse Problems 58
2.03 Existence of a Solution 62
2.04 Uniqueness and Stability 65
2.05 Some Special Solutions 72
2.06 The Magnetic Anomaly Profile 79
2.07 Interpolation 88
2.08 Seismic Dissipation in the Mantle 104
Chapter 3: Linear Problems with Uncertain Data
3.01 Tolerance and Misfit 119
3.02 Fitting within the Tolerance 131
3.03 The Spectral Approach 144
3.04 The Magnetic Anomaly Problem Revisited 153
3.05 The Theory in Numerical Dress 163
3.06 Return to the Seismic Dissipation Problem 174
3.07 Large Numbers of Observations 186
Chapter 4: Resolution and Inference
4.01 The Central Question 199
4.02 Resolution 201
CONTENTS
VI
4.03 Bounding Linear Functionals in Hilbert Space 214
4.04 Bounding Functionals in Practice 226
4.05 Ideal Bodies 241
4.06 Linear and Quadratic Programming 253
4.07 Examples using LP and QP 268
4.08 A Statistical Theory 278
Chapter 5: Nonlinear Problems
5.01 Some Familiar Questions 293
5.02 Two Examples 295
5.03 Functional Differentiation 304
5.04 Constructing Models 308
5.05 The Gravity Profile 319
5.06 The Magnetotelluric Problem I 333
5.07 Resolution in Nonlinear Problems 345
5.08 The Magnetotelluric Problem II 351
5.09 Coda 368
Appendix A: The Dilogarithm Function 371
Appendix B: Table for 1-norm Misfits 373
References 375
Index 381