Inversion de datos geofisicos
Páginas: 72 (17828 palabras)
Publicado: 18 de agosto de 2010
Probabilistic Approach to Inverse Problems∗
Klaus Mosegaard† & Albert Tarantola‡ November 16, 2002
Abstract In ‘inverse problems’ data from indirect measurements are used to estimate unknown parameters of physical systems. Uncertain data, (possibly vague) prior information on model parameters, and a physical theory relating the model parameters to the observations are the fundamental elementsof any inverse problem. Using concepts from probability theory, a consistent formulation of inverse problems can be made, and, while the most general solution of the inverse problem requires extensive use of Monte Carlo methods, special hypotheses (e.g., Gaussian uncertainties) allow, in some cases, an analytical solution to part of the problem (e.g., using the method of least squares).
∗ Thistext has been published as a chapter of the International Handbook of Earthquake & Engineering Seismology (Part A), Academic Press, 2002, pages 237–265. It is here complete, with its appendixes. † Niels Bohr Institute; Juliane Maries Vej 30; 2100 Copenhagen OE; Denmark; mailto:klaus@gfy.ku.dk ‡ Institut de Physique du Globe; 4, place Jussieu; 75005 Paris; France; mailto:tarantola@ipgp.jussieu.fr1
Contents
1 Introduction 1.1 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Brief Historical Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Elements of Probability 2.1 Volume . . . . . . . . . . . . . . . . . . 2.2 Probability . . . . . . . . . . . . . . . . 2.3 HomogeneousProbability Distributions 2.4 Conjunction of Probabilities . . . . . . . 2.5 Conditional Probability Density . . . . . 2.6 Marginal Probability Density . . . . . . 2.7 Independence and Bayes Theorem . . . 3 Monte Carlo Methods 3.1 Random Walks . . . . . . . . . 3.2 The Metropolis Rule . . . . . . 3.3 The Cascaded Metropolis Rule 3.4 Initiating a Random Walk . . . 3.5 Convergence Issues . . . . . .. 4 4 5 6 6 7 7 10 11 13 13 13 15 15 16 16 17 17 17 18 18 20 20 20 21 23 23 24 26 26 26 27 27 27 28 28 29 30 30 32 32 32 34 34 35 35
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4 Probabilistic Formulation of Inverse Problems 4.1 Model Parameters and Observable Parameters . . . 4.2 Prior Information on Model Parameters . . . . . . . 4.3 Measurements and Experimental Uncertainties . . . 4.4 Joint ‘Prior’ Probability Distribution in the (M, D) 4.5 Physical Laws as Mathematical Functions . . . . . . 4.5.1 Physical Laws . . . . . . . . . . . . . . . . .. 4.5.2 Inverse Problems . . . . . . . . . . . . . . . . 4.6 Physical Laws as Probabilistic Correlations . . . . . 4.6.1 Physical Laws . . . . . . . . . . . . . . . . . . 4.6.2 Inverse Problems . . . . . . . . . . . . . . . .
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Klaus Mosegaard† & Albert Tarantola‡ November 16, 2002
Abstract In ‘inverse problems’ data from indirect measurements are used to estimate unknown parameters of physical systems. Uncertain data, (possibly vague) prior information on model parameters, and a physical theory relating the model parameters to the observations are the fundamental elementsof any inverse problem. Using concepts from probability theory, a consistent formulation of inverse problems can be made, and, while the most general solution of the inverse problem requires extensive use of Monte Carlo methods, special hypotheses (e.g., Gaussian uncertainties) allow, in some cases, an analytical solution to part of the problem (e.g., using the method of least squares).
∗ Thistext has been published as a chapter of the International Handbook of Earthquake & Engineering Seismology (Part A), Academic Press, 2002, pages 237–265. It is here complete, with its appendixes. † Niels Bohr Institute; Juliane Maries Vej 30; 2100 Copenhagen OE; Denmark; mailto:klaus@gfy.ku.dk ‡ Institut de Physique du Globe; 4, place Jussieu; 75005 Paris; France; mailto:tarantola@ipgp.jussieu.fr1
Contents
1 Introduction 1.1 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Brief Historical Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Elements of Probability 2.1 Volume . . . . . . . . . . . . . . . . . . 2.2 Probability . . . . . . . . . . . . . . . . 2.3 HomogeneousProbability Distributions 2.4 Conjunction of Probabilities . . . . . . . 2.5 Conditional Probability Density . . . . . 2.6 Marginal Probability Density . . . . . . 2.7 Independence and Bayes Theorem . . . 3 Monte Carlo Methods 3.1 Random Walks . . . . . . . . . 3.2 The Metropolis Rule . . . . . . 3.3 The Cascaded Metropolis Rule 3.4 Initiating a Random Walk . . . 3.5 Convergence Issues . . . . . .. 4 4 5 6 6 7 7 10 11 13 13 13 15 15 16 16 17 17 17 18 18 20 20 20 21 23 23 24 26 26 26 27 27 27 28 28 29 30 30 32 32 32 34 34 35 35
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4 Probabilistic Formulation of Inverse Problems 4.1 Model Parameters and Observable Parameters . . . 4.2 Prior Information on Model Parameters . . . . . . . 4.3 Measurements and Experimental Uncertainties . . . 4.4 Joint ‘Prior’ Probability Distribution in the (M, D) 4.5 Physical Laws as Mathematical Functions . . . . . . 4.5.1 Physical Laws . . . . . . . . . . . . . . . . .. 4.5.2 Inverse Problems . . . . . . . . . . . . . . . . 4.6 Physical Laws as Probabilistic Correlations . . . . . 4.6.1 Physical Laws . . . . . . . . . . . . . . . . . . 4.6.2 Inverse Problems . . . . . . . . . . . . . . . .
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