Statistical Methods of Geophysical Data Processing

This textbook contains a consideration of the wide field of problems connected with statistical methods of processing of observed data, with the main examples and considered models related to geophysics and seismic exploration. This textbook will be particularly helpful to students and professionals from various fields of physics, connected with an estimation of the parameters of the physical objects by experimental data. The reader can also find many important topics, which are the basis for statistical methods of estimation and inverse problem solutions. Contents......Page 12 Introduction......Page 6 1.1.1 Set of elementary events......Page 19 1.1.2 Probability model with a finite number of outcomes......Page 22 1.1.4 Classical definition of probability......Page 23 1.1.6 Exercises......Page 24 1.2.1 Addition of probabilities......Page 25 1.2.2 Nonindependent and independent events......Page 27 1.2.3 The Bayes formula and complete probability......Page 28 1.3.1 Random variables......Page 29 1.3.2 Distribution function......Page 30 1.3.3 The density function......Page 32 1.3.4 The distribution and density of function of one random argument......Page 33 1.3.5 Random vectors......Page 34 1.3.6 Marginal and conditional distributions......Page 35 1.3.7 The distributive law of two random variables......Page 36 1.3.8 Exercises......Page 39 1.4.1 Mathematical expectation......Page 41 1.4.2 Variance and correlation coefficients......Page 42 1.4.3 Quantiles......Page 44 1.4.4 Characteristics of a density function......Page 45 1.4.5 Exercises......Page 47 1.5 Characteristic and Generating Functions......Page 48 1.5.1 Moment generating function......Page 49 1.5.2 Probability generator......Page 50 1.5.4 Exercises......Page 51 1.6.2 Chebyshev inequality......Page 52 1.6.3 The law of averages (Chebyshev's theorem)......Page 53 1.6.5 Markov's theorem......Page 54 1.6.8 The central limit theorem......Page 55 1.7 Discrete Distribution Functions......Page 57 1.7.1 Binomial distribution......Page 58 1.7.2 Poisson distribution......Page 59 1.7.3 Geometrical distribution......Page 60 1.7.4 Exercises......Page 61 1.8.1 Univariate normal distribution......Page 63 1.8.2 Multivariate normal distribution......Page 65 1.8.4 2 distribution......Page 69 1.8.5 Student's distribution (t-distribution)......Page 70 1.8.6 Fisher distribution and Z-distribution......Page 71 1.8.9 Exponential distribution......Page 73 1.8.10 Laplace distribution......Page 74 1.8.12 Logarithmic normal distribution......Page 75 1.8.13 Significance of the normal distribution......Page 76 1.8.14 Confidence intervals......Page 77 1.8.15 Exercises......Page 78 1.9.1 Entropy of the set of discrete states of system......Page 80 1.9.2 Entropy of the complex system......Page 81 1.9.3 Shannon information (discrete case)......Page 82 1.9.4 Entropy and information for systems with a continuous set of states......Page 84 1.9.5 Fisher information......Page 87 1.10 Random Functions and its Properties......Page 90 1.10.1 Properties of random functions......Page 92 1.10.2 Properties of the correlation function......Page 94 1.10.4 Cross correlation function......Page 95 1.10.5 Wiener–Khinchin theorem and power spectrum......Page 96 1.10.6 Definition of estimations of the characteristics of random variables......Page 99 2.1 The Basic Concepts of the Decision Theory......Page 101 2.1.2 The structure of the decision space and the loss function......Page 103 2.1.3 Decision rule......Page 106 2.1.4 Sufficient statistic......Page 108 2.2 Estimate Properties......Page 109 2.2.2 Bias......Page 110 2.2.3 Rao–Cramer inequality. Efficiency......Page 112 2.2.5 Asymptotic normality......Page 115 2.2.6 Robustness......Page 116 3.1 Additive Model......Page 117 3.2 Models of the Quantitative Interpretation......Page 119 3.3 Regression Model......Page 120 3.4 The Models of Qualitative Interpretation......Page 122 3.5 The Models of Qualitative-Quantitative Interpretation......Page 123 3.6 Random Components of Model and its Properties......Page 124 3.8 A Priori Information......Page 129 4.1 Seismology and Seismic Prospecting......Page 133 4.2 Acoustics of the Ocean......Page 141 4.3 Wave Electromagnetic Fields in Geoelectrics and Ionospheric Sounding......Page 144 4.4 Atmospheric Sounding......Page 149 5.1 Basis of the Ray Theory......Page 153 5.2 Ray Method for the Scalar Wave Equation......Page 156 5.3 Shortwave Asymptotic Form of the Solution of the One-Dimensional Helmholtz Equation (WKB Approximation)......Page 162 5.4 The Elements of Elastic Wave Ray Theory .......Page 164 5.5 The Ray Description of Almost-Stratified Medium......Page 165 5.6 Surface Wave in Vertically Inhomogeneous Medium......Page 170 5.7 Ray Approximation of Electromagnetic Fields......Page 173 5.8 Statement of Problem of the Ray Kinematic Tomography......Page 177 6.1 The Method of Moments......Page 181 6.2 Maximum Likelihood Method......Page 182 6.3 The Newton–Le Cam Method......Page 183 6.4 The Least Squares Method......Page 185 6.5 LSM | Nonlinear Model......Page 187 6.6 LSM | Orthogonal Polynomials (Chebyshev Polynomials)......Page 188 6.7 LSM | Case of Linear Constraints......Page 190 6.8 Linear Estimation | Case of Nonstationary Model......Page 192 6.9 Bayes' Criterion and Method of Statistical Regularization......Page 193 6.10 Method of Maximum a Posteriori Probability......Page 194 6.11 The Recursion Algorithm of MAP......Page 196 6.12 Singular Analysis and Least Squares Method......Page 197 6.13 The Method of Least Modulus......Page 202 6.14 Robust Methods of Estimation......Page 204 6.14.1 Reparametrization algorithm......Page 205 6.14.2 Huber robust method......Page 206 6.14.3 Andrews robust method......Page 207 6.15 Interval Estimation......Page 208 6.16 The Method of Backus and Gilbert of the Linear Inverse Problem Solution......Page 211 6.17.1 Coding......Page 214 6.17.4 Mutation......Page 215 6.17.5 Choice......Page 217 7.1 Test of Parametric Hypothesis......Page 219 7.2 Criterion of a Posteriori Probability Ratio......Page 220 7.3 The Signal Resolution Problem......Page 224 7.4 Information Criterion for the Choice of the Model......Page 226 7.5 The Method of the Separation of Interfering Signals......Page 229 8. Algorithms of approximation of geophysical data......Page 16 8.1 The Algorithm of Univariate Approximation by Cubic Splines......Page 235 8.2 Periodic and Parametric Spline Functions......Page 241 8.3 Application of the Spline Functions for Histogram Smoothing......Page 244 8.4 Algorithms for Approximation of Seismic Horizon Subject to Borehole Observations......Page 245 8.4.1 The Markovian type of correlation along the beds and no correlation between beds......Page 247 8.4.3 Conformance inspection of seismic observation to borehole data concerning bed depth......Page 248 8.4.4 Incorporation of random nature of depth measurement using borehole data......Page 250 8.4.5 Application of a posteriori probability method to approximation of seismic horizon......Page 252 8.4.6 Case of uncorrelated components of random vector......Page 253 8.4.7 Approximation of parameters of approximation horizon by the orthogonal polynomials......Page 255 8.4.8 Numerical examples of application of approximation algorithms......Page 256 8.5 Algorithm of Approximation of Formation Velocity with the Use of Areal Observations with Borehole Data......Page 259 9.1 Elements of Applied Functional Analysis......Page 265 9.2 Ill-Posed Problems......Page 282 9.3 Statistical Estimation in the Terms of the Functional Analysis......Page 290 9.4 Elements of the Mathematical Design of Experiment......Page 307 10.1 Construction of the Model of Measurements......Page 311 10.2 Tomographic Functional......Page 315 10.3.1 Scalar wave equation......Page 316 10.3.2 The Lame equation in an isotropic in nite medium......Page 317 10.3.3 The transport equation of the stationary sounding signal......Page 324 10.3.4 The diffusion equation......Page 326 10.4 Ray Tomographic Functional in the Dynamic and Kinematic Interpretation of the Remote Sounding Data......Page 327 10.5 Construction of Incident and Reversed Wave Fields in Layered Reference Medium......Page 330 11.1 Elements of Linear Tomography......Page 333 11.1.1 Change of variables......Page 334 11.1.2 Differentiation of generalized function......Page 335 11.2 Connection of Radon Inversion with Diffraction Tomography......Page 340 11.3 Construction of Algorithms of Reconstruction Tomography......Page 345 11.4 Errors of Recovery, Resolving Length and Backus and Gilbert Method......Page 352 11.5 Back Projection in Diffraction Tomography......Page 358 11.6 Regularization Problems in 3-D Ray Tomography......Page 365 11.7.1 Green function for the wave equation......Page 369 11.7.2 Green function for "Poisson equation"......Page 371 11.7.3 Green function for Lame equation in uniform isotropic in nite medium......Page 372 11.7.4 Green function for diffusion equation......Page 376 11.7.5 Green function for operator equation of the second genus......Page 377 11.8.1 An estimation of the resolution......Page 378 11.8.2 An estimation of the recovery accuracy of inhomogeneities parameters......Page 379 12.1.1 Fourier series......Page 389 12.1.2 Fourier integral......Page 390 12.2 Laplace Transform......Page 392 12.3 Z-Transform......Page 393 12.4 Radon Transform for Seismogram Processing......Page 394 12.5 Gilbert Transform and Analytical Signal......Page 396 12.6 Cepstral Transform......Page 398 12.7 Bispectral Analysis......Page 399 12.8 Kalman Filtering......Page 400 12.9 Multifactor Analysis and Processing of Time Series......Page 402 12.10 Wiener Filter......Page 407 12.11 Predictive-Error Filter and Maximum Entropy Principle......Page 408 A.1.1 Examples of numerical simulation of random values......Page 415 A.1.2 Construction of histogram......Page 416 A.1.3.2 Binomial distribution......Page 417 A.1.3.4 2-distribution......Page 419 A.1.3.8 Geometrical distribution......Page 420 A.1.3.15 Exercises......Page 421 A.1.4.1 Kolmogorov criterion......Page 422 A.1.6 Time series......Page 423 A.2.2 Signals and their spectral characteristics......Page 424 A.2.3 Multifactor analysis......Page 425 A.2.3.1 Exercises......Page 427 A.2.4 Cepstral transform......Page 429 A.2.4.1 Exercises......Page 430 A.3.1.1 Gravitational attraction of the sphere......Page 431 A.3.3 Computer simulation of the seismic field......Page 432 A.3.4 Deconvolution by the Wiener filter......Page 433 A.3.4.1 Exercise......Page 434 A.3.5 Quantitative interpretation......Page 435 A.3.5.1 Exercises......Page 437 A.3.6 Qualitative interpretation......Page 438 A.3.7 Diffraction tomography......Page 439 A.3.7.1 Exercises......Page 441 Appendix B Tables......Page 443 Bibliography......Page 445 Index......Page 451 "This textbook contains a consideration of the wide field of problems connected with statistical methods of processing of observed data, with the main examples and considered models related to geophysics and seismic exploration. This textbook wil be particularly helpful to students and professionals from various fields of physics, connected with an estimation of the parameters of the physical objects by experimental data. The reader can also find many important topics, which are the basis for statistical methods of estimation and inverse problem solutions."--Jacket