Global Optimization Methods in Geophysical Inversion (eBook)

Front Cover 1
Global Optimization Methods in Geophysical Inversion 4
Copyright Page 5
Contents 10
Preface 6
Chapter 1. Preliminary Statistics 14
1.1. Random variables 14
1.2. Random numbers 15
1.3. Probability 16
1.4. Probability distribution, distribution function and density function 18
1.5. Joint and marginal probability distributions 22
1.6. Mathematical expectation, moments, variances, and covariances 23
1.7. Conditional probability 26
1.8. Monte Carlo integration 28
1.9. Importance sampling 29
1.10. Stochastic processes 31
1.11. Markov chains 33
1.12. Homogeneous, inhomogeneous, irreducible and aperiodic Markov chains 37
1.13. The limiting probability 38
Chapter 2. Direct, Linear and Iterative-linear Inverse Methods 40
2.1. Direct inversion methods 41
2.2. Model based inversion methods 47
2.3. Linear/linearized inverse methods 50
2.4. Iterative linear methods for quasi-linear problems 63
2.5. Bayesian formulation 66
2.6. Solution using probabilistic formulation 73
2.7. Summary 77
Chapter 3. Monte Carlo Methods 80
3.1. Enumerative or grid search techniques 80
3.2. Monte Carlo inversion 84
3.3. Hybrid Monte Carlo–linear inversion 89
3.4. Directed Monte Carlo methods 90
Chapter 4. Simulated Annealing Methods 92
4.1. Metropolis algorithm 96
4.2. Heat bath algorithm 104
4.3. Simulated annealing without rejected moves 115
4.4. Fast simulated annealing 116
4.5. Very fast simulated reannealing 119
4.6. Mean 123
4.7. Using SA in geophysical inversion 135
4.8. Summary 136
Chapter 5. Genetic Algorithms 138
5.1. A classical GA 138
5.2. Schemata and the fundamental theorem of genetic algorithms 150
5.3. Problems 152
5.4. Combining elements of SA into a new GA 153
5.5. A mathematical model of a GA 157
5.6. Multimodal fitness functions, genetic drift 165
5.7. Uncertainty estimates 168
5.8. Evolutionary programming 168
5.9. Summary 170
Chapter 6. Geophysical Applications of SA and G A 172
6.1. 1-D Seismic waveform inversion 174
6.2. Pre-stack migration velocity estimation 205
6.3. Inversion of resistivity sounding data for 1-D earth models 225
6.4. Inversion of resistivity profiling data for 2-D earth models 229
6.5. Inversion of magnetotelluric sounding data for 1-D earth models 238
6.6. Stochastic reservoir modeling 240
6.7. Seismic deconvolution by mean field annealing and Hopfield network 247
Chapter 7. Uncertainty Estimation 258
7.1. Methods of Numerical Integration 259
7.2. Simulated annealing: The Gibbs’ sampler 263
7.3. Genetic algorithm: The parallel Gibbs’ sampler 266
7.4. Numerical examples 267
7.5. Summary 276
References 282
Subject Index 292