Geophysical Data Analysis (eBook)

Front Cover 1
Geophysical Data Analysis: Discrete Inverse Theory 4
Copyright Page 5
Contents 6
Preface 12
Introduction 14
CHAPTER 1. DESCRIBING INVERSE PROBLEMS 20
1.1 Formulating Inverse Problems 20
1.2 The Linear Inverse Problem 22
1.3 Examples of Formulating Inverse Problems 23
1.4 Solutions to Inverse Problems 30
CHAPTER 2. SOME COMMENTS ON PROBABILITY THEORY 34
2.1 Noise and Random Variables 34
2.2 Correlated Data 37
2.3 Functions of Random Variables 40
2.4 Gaussian Distributions 42
2.5 Testing the Assumption of Gaussian Statistics 44
2.6 Confidence Intervals 46
CHAPTER 3. SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 1: THE LENGTH METHOD 48
3.1 The Lengths of Estimates 48
3.2 Measures of Length 49
3.3 Least Squares for a Straight Line 52
3.4 The Least Squares Solution of the Linear Inverse Problem 53
3.5 Some Examples 55
3.6 The Existence of the Least Squares Solution 58
3.7 The Purely Underdetermined Problem 61
3.8 Mixed–Determined Problems 63
3.9 Weighted Measures of Length as a Type of A Priori Information 65
3.10 Other Types of A Priori Information 68
3.11 The Variance of the Model Parameter Estimates 71
3.12 Variance and Prediction Error of the Least Squares Solution 71
CHAPTER 4. SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 2: GENERALIZED INVERSES 74
4.1 Solutions versus Operators 74
4.2 The Data Resolution Matrix 75
4.3 The Model Resolution Matrix 77
4.4 The Unit Covariance Matrix 78
4.5 Resolution and Covariance of Some Generalized Inverses 79
4.6 Measures of Goodness of Resolution and Covariance 80
4.7 Generalized Inverses with Good Resolution and Covariance 81
4.8 Sidelobes and the Backus-Gilbert Spread Function 84
4.9 The Backus-Gilbert Generalized Inverse for the Underdetermined Problem 86
4.10 Including the Covariance Size 88
4.11 The Trade-off of Resolution and Variance 89
CHAPTER 5. SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 3: MAXIMUM LIKELIHOOD METHODS 92
5.1 The Mean of a Group of Measurements 92
5.2 Maximum Likelihood Solution of the Linear Inverse Problem 95
5.3 A Priori Distributions 96
5.4 Maximum Likelihood for an Exact Theory 100
5.5 Inexact Theories 102
5.6 The Simple Gaussian Case with a Linear Theory 104
5.7 The General Linear, Gaussian Case 105
5.8 Equivalence of the Three Viewpoints 108
5.9 The F Test of Error Improvement Significance 109
5.10 Derivation of the Formulas of Section 5.7 110
CHAPTER 6. NONUNIQUENESS AND LOCALIZED AVERAGES 114
6.1 Null Vectors and Nonuniqueness 114
6.2 Null Vectors of a Simple Inverse Problem 115
6.3 Localized Averages of Model Parameters 116
6.4 Relationship to the Resolution Matrix 117
6.5 Averages versus Estimates 118
6.6 Nonunique Averaging Vectors and A Priori Information 119
CHAPTER 7. APPLICATIONS OF VECTOR SPACES 122
7.1 Model and Data Spaces 122
7.2 Householder Transformations 124
7.3 Designing Householder Transformations 128
7.4 Transformations That Do Not Preserve Length 130
7.5 The Solution of the Mixed – Determined Problem 131
7.6 Singular-Value Decomposition and the Natural Generalized Inverse 132
7.7 Derivation of the Singular-Value Decomposition 137
7.8 Simplifying Linear Equality and Inequality Constraints 138
7.9 Inequality Constraints 139
CHAPTER 8. LINEAR INVERSE PROBLEMS AND NON-GAUSSIAN DISTRIBUTIONS 146
8.1 L1 Norms and Exponential Distributions 146
8.2 Maximum Likelihood Estimate of the Mean of an Exponential Distribution 148
8.3 The General Linear Problem 150
8.4 Solving L1 Norm Problems 151
8.5 The L8 Norm 154
CHAPTER 9. NONLINEAR INVERSE PROBLEMS 156
9.1 Parameterizations 156
9.2 Linearizing Parameterizations 160
9.3 The Nonlinear Inverse Problem with Gaussian Data 160
9.4 Special Cases 166
9.5 Convergence and Nonuniqueness of Nonlinear L2 Problems 166
9.6 Non-Gaussian Distributions 169
9.7 Maximum Entropy Methods 173
CHAPTER 10. FACTOR ANALYSIS 174
10.1 The Factor Analysis Problem 174
10.2 Normalization and Physicality Constraints 178
10.3 Q-Mode and R-Mode Factor Analysis 180
10.4 Empirical Orthogonal Function Analysis 180
CHAPTER 11. CONTINUOUS INVERSE THEORY AND TOMOGRAPHY 184
11.1 The Backus – Gilbert Inverse Problem 184
11.2 Resolution and Variance Trade-off 186
11.3 Approximating Continuous Inverse Problems as Discrete Problems 187
11.4 Tomography and Continuous Inverse Theory 189
11.5 Tomography and the Radon Transform 190
11.6 The Fourier Slice Theorem 191
11.7 Backprojection 192
CHAPTER 12. SAMPLE INVERSE PROBLEMS 196
12.1 An image Enhancement Problem 196
12.2 Digital Filter Design 200
12.3 Adjustment of Crossover Errors 203
12.4 An Acoustic Tomography Problem 207
12.5 Temperature Distribution in an Igneous Intrusion 211
12.6 L1, L2, and L8 Fitting of a Straight Line 215
12.7 Finding the Mean of a Set of Unit Vectors 220
12.8 Gaussian Curve Fitting 223
12.9 Earthquake Location 226
12.10 Vibrational Problems 230
CHAPTER 13. NUMERICAL ALGORITHMS 234
13.1 Solving Even-Determined Problems 235
13.2 Inverting a Square Matrix 242
13.3 Solving Underdetermined and Overdetermined Problems 244
13.4 L2 Problems with Inequality Constraints 253
13.5 Finding the Eigenvalues and Eigenvectors of a Real Symmetric Matrix 264
13.6 The Singular-Value Decomposition of a Matrix 267
13.7 The Simplex Method and the Linear Programming Problem 269
CHAPTER 14. APPLICATIONS OF INVERSE THEORY TO GEOPHYSICS 274
14.1 Earthquake Location and the Determination of the Velocity Structure of the Earth from Travel Time Data 274
14.2 Velocity Structure from Free Oscillations and Seismic Surface Waves 278
14.3 Seismic Attenuation 280
14.4 Signal Correlation 280
14.5 Tectonic Plate Motions 281
14.6 Gravity and Geomagnetism 282
14.7 Electromagnetic Induction and the Magnetotelluric Method 283
14.8 Ocean Circulation 284
APPENDIX A: Implementing Constraints with Lagrange Multipliers 286
APPENDIX B: L2 Inverse Theory with Complex Quantities 288
REFERENCES 290
INDEX 294
INTERNATIONAL GEOPHYSICS SERIES 300