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The book's goal is to describe in sufficient detail the fundamentals of several optimization methods with application to geophysical inversion such that students, researchers and practitioners will be able to design practical algorithms to solve their specific geophysical inversion problems.

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One of the major goals of geophysical inversion is to find earth models that explain the geophysical observations.

Both local and global optimization methods are used in the estimation of material properties from geophysical data.

Contents

Preface

Chapter 1. Preliminary Statistics

1.1. Random variables

1.2. Random numbers

1.3. Probability

1.4. Probability distribution. distribution function and density function

1.4.1. Examples of distribution and density functions

1.5. Joint and marginal probability distributions

1.6. Mathematical expectation. moments. variances. and covariances

1.7. Conditional probability

1.8. Monte Carlo integration

1.9. Importance sampling

1.10. Stochastic processes

1.11. Markov chains

1.12. Homogeneous. inhomogeneous. irreducible and aperiodic Markov chains

1.13. The limiting probability

Chapter 2. Direct. Linear and Iterative-linear Inverse Methods

2.1. Direct inversion methods

2.2. Model based inversion methods

2.3. Linear/linearized inverse methods

2.3.1. Solution of linear inverse problems

2.3.2. Stability and uniqueness-singular value decomposition analysis

2.3.3. Methods of constraining the solution

2.4. Iterative linear methods for quasi-linear problems

2.5. Bayesian formulation

2.6. Solution using probabilistic formulation

2.6.1. Linear case

2.6.3. Case of weak non-linearity

2.6.3. Quasi-linear case

2.7. Summary

Chapter 3. Monte Carlo Methods

3.1. Enumerative or grid search techniques

3.2. Monte Carlo inversion

3.3. Hybrid Monte Carlo-linear inversion

3.4. Directed Monte Carlo methods

Chapter 4. Simulated Annealing Methods

4.1. Metropolis algorithm

4.1.1. Mathematical model and asymptotic convergence

4.2. Heat bath algorithm

4.2.1. Mathematical model and asymptotic convergence

4.3. Simulated annealing without rejected moves

4.4. Fast simulated annealing

4.5. Very fast simulated reannealing

4.6. Mean

4.6.1. Neurons

4.6.2. Hopfield neural networks

4.6.3. Avoiding local minimum

4.6.4. Mean field theory

4.7. Using SA in geophysical inversion

4.7.1 . Bayesian formulation

4.8. Summary

Chapter 5. Genetic Algorithms

5.1. A classical GA

5.1.1. Coding

5.1.2. Selection

5.1.3. Crossover

5.1.4. Mutation

5.2. Schemata and the fundamental theorem of genetic algorithms

5.3. Problems

5.4. Combining elements of SA into a new GA

5.5. A mathematical model of a GA

5.6. Multimodal fitness functions, genetic drift

5.7. Uncertainty estimates

5.8. Evolutionary programming

5.9. Summary

Chapter 6. Geophysical Applications of SA and G A

6.1. 1 -D Seismic waveform inversion

6.1.1. Application of heat bath SA

6.1.2. Application of GA

6.1.3. Real data examples

6.1.4. Hybrid GNLI

6.2. Pre-stack migration velocity estimation

6.2.1. 1-D earth structure

6.2.2. 2-D earth structure

6.3. Inversion of resistivity sounding data for I-D earth models

6.3.1. Exact parameterization

6.3.2. Over parameterization with smoothing

6.4. Inversion of resistivity profiling data for 2-D earth models

6.4.1. Inversion of synthetic data

6.4.2. Inversion of field data

6.5. Inversion of magnetotelluric sounding data for 1-D earth models

6.6. Stochastic reservoir modeling

6.7. Seismic deconvolution by mean field annealing and Hopfield network

Chapter 7. Uncertainty Estimation

7.1. Methods of Numerical Integration

7.1.1. Grid search or enumeration

7.1.2. Monte Carlo integration

7.1.3. Importance sampling

7.1.4. Multiple MAP estimation

7.2. Simulated annealing: The Gibbs’ sampler

7.3. Genetic algorithm: The parallel Gibbs’ samp

7.4. Numerical examples

7.4.1. Inversion of noisy synthetic vertical electric sounding data

7.5. Summary

References

Subject Index

with TOC BookMarkLinks

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