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Parameter Estimation and Inverse Problems
Richard C. Aster, Brian Borchers, and Clifford Thurber
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June 9, 2004
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2002-2004, Aster, Borchers, and Thurber
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Preface
This textbook evolved from a course in geophysical inverse methods taught dur-
ing the past decade at New Mexico Tech, first by Rick Aster and, for the last
five years, jointly between Rick Aster and Brian Borchers. The audience for
the course has included a broad range of first– or second–year graduate stu-
dents (and occasionally advanced undergraduates) from geophysics, hydrology,
mathematics, astronomy, and other disciplines. Cliff Thurber joined this c ol-
laboration during the past three years and has taught a similar course at the
University of Wisconsin.
Our principal goal for this text is to promote fundamental understanding of
parameter estimation and inverse problem philosophy and methodology, specifi-
cally regarding such key issues as uncertainty, ill–posedness, regularization, bias,
and resolution. We emphasize theoretical points with illustrative examples, and
MATLAB codes that implement these examples are provided on a companion
CD. Throughout the examples and exercises, a CD icon indicates that there is
additional material on the CD. Exercises include a mix of programming and
theoretical problems.
This book has necessarily had to distill a tremendous body of mathematics
and science going back to (at least) Newton and Gauss. We hope that it will find
a broad audience of students and professionals interested in the general problem
of estimating physical models from data. Because this is an introductory text
surveying a very broad field, we have not been able to go into great depth.
However, each chapter has a “notes and further reading” section to help guide
the reader to further exploration of specific topics. Where appropriate, we have
also directly referenced research contributions to the field.
Some advanced topics have been deliberately omitted from the book be -
cause of space limitations and/or because we expect that many readers would
not be sufficiently familiar with the required mathematics. For example, read-
ers with a strong mathematical background may be surprised that we consider
only inverse problems with discrete data and discretized models. By doing this
we avoid the much of the technical complexity of functional analysis. Some
advanced applications and topics that we have omitted include inverse scatter-
ing problems, seismic diffraction tomography, wavelets, data assimilation, and
expectation maximization (EM) methods.
We expect that readers of this book will have prior familiarity with cal-
culus, differential equations, linear algebra, probability, and statistics at the
iii
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iv PREFACE
undergraduate level. In our experience, many students are in need of at least
a review of these topics, and we typically spend the first two to three weeks of
the course reviewing this material from Appendices A, B, and C.
Chapters 1 through 5 form the heart of the book, and should be covered
in sequence. Chapters 6, 7, and 8 are independent of each other, but depend
strongly on the material in Chapters 1 through 5. As such, they may be covered
in any order. Chapters 9 and 10 are independent of Chapters 6 through 8, but
are most appropriately covered in sequence. Chapter 11 is independent of the
specifics of Chapters 6 through 10, and provides an alternate view on, and
summary of, key statistical and inverse theory issues.
If significant time is allotted for review of linear algebra, vector calculus,
probability, and statistics in the appendices, there will probably not be time
to cover the entire book in one semester. However, it should be possible for
teachers to cover the majority of the material by selectively using material in
the chapters following Chapter 5.
We esp e cially wish to acknowledge our own professors and mentors in this
field, including Kei Aki, Robert Parker, and Peter Shearer. We thank our
many colleagues, including our own students, who provided sustained encour-
agement and feedback, particularly James Beck, Elena Resmerita, Charlotte
Rowe, Tyson Strand, and Suzan van der Lee. Stuart Anderson, Greg Beroza,
Ken Creager, Ken Dueker, Eliza Michalopoulou, Paul Segall, Anne Sheehan,
and Kristy Tiampo deserve special mention for their classroom testing of early
versions of this text. Robert Nowack, Gary Pavlis, Randall Richardson, and
Steve Roecker provided thorough reviews that substantially improved the final
manuscript. We offer special thanks to Per Christian Hansen of the Technical
University of Denmark for collaboration in the incorporation of his Regulariza-
tion Tools, which we highly recommend as an adjunct to this text. We also thank
the editorial staff at Academic Press, especially Frank Cynar and Jennifer Hele,
for essential advice and direction. Suzanne Borchers and Susan Delap provided
valuable pro ofreading and graphics expertise. Brian Borchers was a visiting
fellow at the I nstitute for Pure and Applied Mathematics (IPAM) at UCLA,
and Rick Aster was partially supported by the New Mexico Tech Geophysical
Research Center during the preparation of the text.
Rick Aster, Brian Borchers, and Cliff Thurber
June, 2004
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Contents
Preface iii
1 Introduction 1
1.1 Classification of Inverse Problems . . . . . . . . . . . . . . . . . . 1
1.2 Examples of Parameter Estimation Problems . . . . . . . . . . . 4
1.3 Examples of Inverse Problems . . . . . . . . . . . . . . . . . . . . 8
1.4 Why Inverse Problems are Hard . . . . . . . . . . . . . . . . . . 12
1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.6 Notes and Further Reading . . . . . . . . . . . . . . . . . . . . . 16
2 Linear Regression 17
2.1 Introduction to Linear Regression . . . . . . . . . . . . . . . . . . 17
2.2 Statistical Aspects of Least Squares . . . . . . . . . . . . . . . . 19
2.3 Unknown Measurement Standard Deviations . . . . . . . . . . . 28
2.4 L
1
Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5 Monte Carlo Error Propagation . . . . . . . . . . . . . . . . . . . 37
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.7 Notes and Further Reading . . . . . . . . . . . . . . . . . . . . . 42
3 Discretizing Continuous Inverse Problems 43
3.1 Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Quadrature Methods . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Expansion in Terms of Represe nters . . . . . . . . . . . . . . . . 49
3.4 Expansion in Terms of Orthonormal Basis Functions . . . . . . . 50
3.5 The Method of Backus and Gilbert . . . . . . . . . . . . . . . . . 51
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.7 Notes and Further Reading . . . . . . . . . . . . . . . . . . . . . 56
4 Rank Deficiency and Ill–Conditioning 59
4.1 The SVD and the Generalized Inverse . . . . . . . . . . . . . . . 59
4.2 Covariance and Re solution of the Generalized Inverse Solution . 65
4.3 Instability of the Generalized Inverse Solution . . . . . . . . . . . 68
4.4 An Example of a Rank Deficient Problem . . . . . . . . . . . . . 71
4.5 Discrete Ill-Posed Problems . . . . . . . . . . . . . . . . . . . . . 78
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