IINNVVEERRSSEE PPRROOBBLLEEMMSS IINN GGEEOOPPHHYYSSIICCSS GGEEOOSS 556677 A Set of Lecture Notes by Professors Randall M. Richardson and George Zandt Department of Geosciences University of Arizona Tucson, Arizona 85721 Revised and Updated Summer 2003 Geosciences 567: PREFACE (RMR/GZ) TABLE OF CONTENTS PREFACE ........................................................................................................................................v CHAPTER 1: INTRODUCTION ................................................................................................. 1 1.1 Inverse Theory: What It Is and What It Does ........................................................ 1 1.2 Useful Definitions ................................................................................................... 2 1.3 Possible Goals of an Inverse Analysis .................................................................... 3 1.4 Nomenclature .......................................................................................................... 4 1.5 Examples of Forward Problems .............................................................................. 7 1.5.1 Example 1: Fitting a Straight Line ........................................................... 7 1.5.2 Example 2: Fitting a Parabola .................................................................. 8 1.5.3 Example 3: Acoustic Tomography .......................................................... 9 1.5.4 Example 4: Seismic Tomography .......................................................... 10 1.5.5 Example 5: Convolution ....................................................................... 10 1.6 Final Comments .................................................................................................... 11 CHAPTER 2: REVIEW OF LINEAR ALGEBRA AND STATISTICS...................................... 12 2.1 Introduction ........................................................................................................... 12 2.2 Matrices and Linear Transformations..................................................................... 12 2.2.1 Review of Matrix Manipulations ............................................................ 12 2.2.2 Matrix Transformations ......................................................................... 15 2.2.3 Matrices and Vector Spaces ................................................................... 19 2.2 Probability and Statistics........................................................................................ 20 2.3.1 Introduction ............................................................................................ 20 2.3.2 Definitions, Part 1 ................................................................................... 20 2.3.3 Some Comments on Applications to Inverse Theory .............................. 23 2.3.4 Definitions, Part 2 .................................................................................. 24 CHAPTER 3: INVERSE METHODS BASED ON LENGTH ................................................... 28 3.1 Introduction ........................................................................................................... 28 3.2 Data Error and Model Parameter Vectors .............................................................. 28 3.3 Measures of Length ............................................................................................... 28 3.4 Minimizing the Misfit: Least Squares ................................................................... 30 3.4.1 Least Squares Problem for a Straight Line ............................................. 30 3.4.2 Derivation of the General Least Squares Solution .................................. 33 3.4.3 Two Examples of Least Squares Problems ............................................ 35 3.4.4 Four-Parameter Tomography Problem ................................................... 37 3.5 Determinancy of Least Squares Problems.............................................................. 38 3.5.1 Introduction ............................................................................................ 38 3.5.2 Even-Determined Problems: M = N ...................................................... 39 3.5.3 Overdetermined Problems: Typically, N > M ......................................... 39 3.5.4 Underdetermined Problems: Typically M > N ....................................... 39 3.6 Minimum Length Solution..................................................................................... 40 3.6.1 Background Information ........................................................................ 40 3.6.2 Lagrange Multipliers .............................................................................. 41 3.6.3 Application to the Purely Underdetermined Problem ............................. 44 i Geosciences 567: PREFACE (RMR/GZ) 3.6.4 Comparison of Least Squares and Minimum Length Solutions.............. 46 3.6.5 Example of Minimum Length Problem ................................................. 46 3.7 Weighted Measures of Length............................................................................... 47 3.7.1 Introduction ............................................................................................ 47 3.7.2 Weighted Least Squares.......................................................................... 47 3.7.3 Weighted Minimum Length ................................................................... 50 3.7.4 Weighted Damped Least Squares ........................................................... 52 3.8 A Priori Information and Constraints..................................................................... 53 3.8.1 Introduction ............................................................................................ 53 3.8.2 A First Approach to Including Constraints ............................................. 54 3.8.3 A Second Approach to Including Constraints ........................................ 56 3.8.4 Example From Seismic Receiver Functions ........................................... 59 3.9 Variance of the Model Parameters.......................................................................... 60 3.9.1 Introduction ............................................................................................60 3.9.2 Application to Least Squares .................................................................. 60 3.9.3 Application to the Minimum Length Problem ........................................ 61 3.9.4 Geometrical Interpretation of Variance ................................................... 61 CHAPTER 4: LINEARIZATION OF NONLINEAR PROBLEMS............................................ 65 4.1 Introduction ........................................................................................................... 65 4.2 Linearization of Nonlinear Problems ..................................................................... 65 4.3 General Procedure for Nonlinear Problems .......................................................... 68 4.4 Three Examples ..................................................................................................... 68 4.4.1 A Linear Example ................................................................................... 68 4.4.2 A Nonlinear Example ............................................................................. 70 4.4.3 Nonlinear Straight-Line Example ........................................................... 75 4.5 Creeping vs Jumping (Shaw and Orcutt, 1985) .................................................... 79 CHAPTER 5: THE EIGENVALUE PROBLEM ........................................................................ 82 5.1 Introduction ........................................................................................................... 82 5.2 The Eigenvalue Problem for Square (M × M) Matrix A ........................................ 82 5.2.1 Background ............................................................................................ 82 5.2.2 How Many Eigenvalues, Eigenvectors? .................................................. 83 5.2.3 The Eigenvalue Problem in Matrix Notation .......................................... 84 5.2.4 Summarizing the Eigenvalue Problem for A .......................................... 87 5.3 Geometrical Interpretation of the Eigenvalue Problem for Symmetric A ............... 87 5.3.1 Introduction ............................................................................................ 87 5.3.2 Geometrical Interpretation ...................................................................... 88 5.3.3 Coordinate System Rotation ................................................................... 92 5.3.4 Summarizing Points ............................................................................... 93 5.4 Decomposition Theorem for Square A .................................................................. 94 5.4.1 The Eigenvalue Problem for AT ............................................................. 94 5.4.2 Eigenvectors for AT ................................................................................ 94 5.4.3 Decomposition Theorem for Square Matrices ........................................ 95 5.4.4 Finding the Inverse A–1 for the M × M Matrix A ................................ 101 5.4.5 What Happens When There Are Zero Eigenvalues? ............................ 102 5.4.6 Some Notes on the Properties of SP and RP ........................................ 105 5.5 Eigenvector Structure of mLS .............................................................................