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Global Optimization Methods in Geophysical Inversion

1st Edition - July 24, 1995

Authors: M.K. Sen, P.L. Stoffa

eBook ISBN:

9 7 8 - 0 - 0 8 - 0 5 3 2 5 6 - 1

One of the major goals of geophysical inversion is to find earth models that explain the geophysical observations. Thus the branch of mathematics known as optimization has found… Read more

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One of the major goals of geophysical inversion is to find earth models that explain the geophysical observations. Thus the branch of mathematics known as optimization has found significant use in many geophysical applications.

Both local and global optimization methods are used in the estimation of material properties from geophysical data. As the title of the book suggests, the aim of this book is to describe the application of several recently developed global optimization methods to geophysical problems.

• The well known linear and gradient based optimization methods have been summarized in order to explain their advantages and limitations

• The theory of simulated annealing and genetic algorithms have been described in sufficient detail for the readers to understand the underlying fundamental principles upon which these algorithms are based

• The algorithms have been described using simple flow charts (the algorithms are general and can be applied to a wide variety of problems

Students, researchers and practitioners will be able to design practical algorithms to solve their specific geophysical inversion problems. The book is virtually self-contained so that there are no prerequisites, except for a fundamental mathematical background that includes a basic understanding of linear algebra and calculus.

Preface. 1. Preliminary Statistics. Random variables. Random numbers. Probability. Probability distribution, distribution function and density function. Joint and marginal probability distributions. Mathematical expectation, moments, variances, and covariances. Conditional probability. Monte Carlo integration. Importance sampling. Stochastic processes. Markov chains. Homogeneous, inhomogeneous, irreducible and aperiodic Markov chains. The limiting probability. 2. Direct, Linear and Iterative-linear Inverse Methods. Direct inversion methods. Model based inversion methods. Linear/linearized inverse methods. Iterative linear methods for quasi-linear problems. Bayesian formulation. Solution using probabilistic formulation. 3. Monte Carlo Methods. Enumerative or grid search techniques. Monte Carlo inversion. Hybrid Monte Carlo-linear inversion. Directed Monte Carlo methods. 4. Simulated Annealing Methods. Metropolis algorithm. Heat bath algorithm. Simulated annealing without rejected moves. Fast simulated annealing. Very fast simulated reannealing. Mean field annealing. Using SA in geophysical inversion. 5. Genetic Algorithms. A classical GA. Schemata and the fundamental theorem of genetic algorithms. Problems. Combining elements of SA into a new GA. A mathematical model of a GA. Multimodal fitness functions, genetic drift, GA with sharing, and repeat (parallel) GA. Uncertainty estimates. Evolutionary programming - a variant of GA. 6. Geophysical Applications of SA and GA. 1-D Seismic waveform inversion. Pre-stack migration velocity estimation. Inversion of resistivity sounding data for 1-D earth models. Inversion of resistivity profiling data for 2-D earth models. Inversion of magnetotelluric sounding data for 1-D earth models. Stochastic reservoir modeling. Seismic deconvolution by mean field annealing and Hopfield network. 7. Uncertainty Estimation. Methods of numerical integration. Simulated annealing: The Gibbs' sampler. Genetic algorithm: The parallel Gibbs' sampler. Numerical examples. References. Subject Index.