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Inverse Problems in Geophysics

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TOC

1. Introduction
2. Solving finite linear systems of equations
2.1. LINEAR MODEL ESTIMATION
2.2. LEAST-SQUARES ESTIMATION
2.3. MINIMUM NORM ESTIMATION
2.4. MIXED DETERMINED PROBLEMS
2.5. THE CONSISTENCY PROBLEM FOR THE LEAST-SQUARES SOLUTION
2.6. THE CONSISTENCY PROBLEM FOR THE MINIMUM-NORM SOLUTION
2.7. THE NEED FOR A MORE GENERAL REGULARIZATION
2.8. THE TRANSFORMATION RULES FOR THE WEIGHT MATRICES

3. Linear inverse problems with continuous models
3.1. CONTINUOUS MODELS AND BASIS FUNCTIONS
3.2. SPECTRAL LEAKAGE, THE PROBLEM
3.3. SPECTRAL LEAKAGE, THE CURE
3.4. SPECTRAL LEAKAGE AND GLOBAL TOMOGRAPHY

4. The single scattering approximation and linearized waveform inversion
4.1. THE BORN APPROXIMATION
4.2. INVERSION AND MIGRATION
4.3. THE BORN APPROXIMATION FOR TRANSMISSION DATA
4.4. SURFACE WAVE INVERSION OF THE STRUCTURE UNDER NORTH-AMERICA

5. Rayleigh’s principle and perturbed eigenfrequencies
5.1. RAYLEIGH-SCHR¨ODINGER PERTURBATION THEORY
5.2. THE PHASE VELOCITY PERTURBATION OF LOVE WAVES

6. Fermat’s theorem and seismic tomography
6.1. FERMAT’S THEOREM, THE EIKONAL EQUATION AND SEISMIC TOMOGRAPHY

7. Nonlinearity and ill-posedness
7.1. EXAMPLE 1, NON-LINEARITY AND THE INVERSE PROBLEM FOR OF THE SCHR¨ODINGER EQUATION
7.2. EXAMPLE 2, NON-LINEARITY AND SEISMIC TOMOGRAPHY

8. Model appraisal for nonlinear inverse problems
8.1. NONLINEAR BACKUS-GILBERT THEORY
8.2. GENERATION OF POPULATIONS OF MODELS THAT FIT THE DATA
8.3. USING DIFFERENT INVERSION METHODS

9. Epilogue

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