Statistical Methods of Geophysical Data Processing

The following sections are included:

• Introduction

• Contents

The following sections are included:

• The Definition of Probability
  • Set of elementary events
  • Probability model with a finite number of outcomes
  • Relative-frequency definition of probability
  • Classical definition of probability
  • Geometrical definition of probability
  • Exercises

• Basic Properties of Probability
  • Addition of probabilities
  • Nonindependent and independent events
  • The Bayes formula and complete probability
  • Exercises

• Distribution Functions
  • Random variables
  • Distribution function
  • The density function
  • The distribution and density of function of one random argument
  • Random vectors
  • Marginal and conditional distributions
  • The distributive law of two random variables
  • Exercises

• The Numerical Characteristics of Probability Distributions
  • Mathematical expectation
  • Variance and correlation coefficients
  • Quantiles
  • Characteristics of a density function
  • Exercises

• Characteristic and Generating Functions
  • Moment generating function
  • Probability generator
  • Semi-invariants or cumulants
  • Exercises

• The Limit Theorems
  • Convergence in probability
  • Chebyshev inequality
  • The law of averages (Chebyshev's theorem)
  • Generalised Chebyshev's theorem
  • Markov's theorem
  • Bernoulli theorem
  • Poisson theorem
  • The central limit theorem
  • Exercises

• Discrete Distribution Functions
  • Binomial distribution
  • Poisson distribution
  • Geometrical distribution
  • Exercises

• Continuous Distributions
  • Univariate normal distribution
  • Multivariate normal distribution
  • Uniform distribution
  • χ²–distribution
  • Student's distribution (t-distribution)
  • Fisher distribution and Z-distribution
  • Triangular distribution
  • Beta distribution
  • Exponential distribution
  • Laplace distribution
  • Cauchy distribution
  • Logarithmic normal distribution
  • Significance of the normal distribution
  • Confidence intervals
  • Exercises

• Information and Entropy
  • Entropy of the set of discrete states of system
  • Entropy of the complex system
  • Shannon information (discrete case)
  • Entropy and information for systems with a continuous set of states
  • Fisher information
  • Exercises

• Random Functions and its Properties
  • Properties of random functions
  • Properties of the correlation function
  • Action of the linear operator on a random function
  • Cross correlation function
  • Wiener–Khinchin theorem and power spectrum
  • Definition of estimations of the characteristics of random variables

The following sections are included:

• The Basic Concepts of the Decision Theory
  • Distribution class of the statistical problem
  • The structure of the decision space and the loss function
  • Decision rule
  • Sufficient statistic

• Estimate Properties
  • Consistency
  • Bias
  • Rao–Cramer inequality. Efficiency
  • Sufficiency
  • Asymptotic normality
  • Robustness

The following sections are included:

• Additive Model

• Models of the Quantitative Interpretation

• Regression Model

• The Models of Qualitative Interpretation

• The Models of Qualitative-Quantitative Interpretation

• Random Components of Model and its Properties

• Model with Random Parameters

• A Priori Information

The following sections are included:

• Seismology and Seismic Prospecting

• Acoustics of the Ocean

• Wave Electromagnetic Fields in Geoelectrics and Ionospheric Sounding

• Atmospheric Sounding

The following sections are included:

• Basis of the Ray Theory

• Ray Method for the Scalar Wave Equation

• Shortwave Asymptotic Form of the Solution of the One-Dimensional Helmholtz Equation (WKB Approximation)

• The Elements of Elastic Wave Ray Theory

• The Ray Description of Almost-Stratified Medium

• Surface Wave in Vertically Inhomogeneous Medium

• Ray Approximation of Electromagnetic Fields

• Statement of Problem of the Ray Kinematic Tomography

The following sections are included:

• The Method of Moments

• Maximum Likelihood Method

• The Newton–Le Cam Method

• The Least Squares Method

• LSM — Nonlinear Model

• LSM — Orthogonal Polynomials (Chebyshev Polynomials)

• LSM — Case of Linear Constraints

• Linear Estimation — Case of Nonstationary Model

• Bayes' Criterion and Method of Statistical Regularization

• Method of Maximum a Posteriori Probability

• The Recursion Algorithm of MAP

• Singular Analysis and Least Squares Method
  • Resolution matrix

• The Method of Least Modulus

• Robust Methods of Estimation
  • Reparametrization algorithm
  • Huber robust method
  • Andrews robust method

• Interval Estimation

• The Method of Backus and Gilbert of the Linear Inverse Problem Solution

• Genetic Algorithm
  • Coding
  • Selection
  • Crossover
  • Mutation
  • Choice

The following sections are included:

• Test of Parametric Hypothesis

• Criterion of a Posteriori Probability Ratio

• The Signal Resolution Problem

• Information Criterion for the Choice of the Model

• The Method of the Separation of Interfering Signals

The following sections are included:

• The Algorithm of Univariate Approximation by Cubic Splines

• Periodic and Parametric Spline Functions

• Application of the Spline Functions for Histogram Smoothing

• Algorithms for Approximation of Seismic Horizon Subject to Borehole Observations
  • The Markovian type of correlation along the beds and no correlation between beds
  • Markovian correlation between the beds and no correlation along bed
  • Conformance inspection of seismic observation to borehole data concerning bed depth
  • Incorporation of random nature of depth measurement using borehole data
  • Application of a posteriori probability method to approximation of seismic horizon
  • Case of uncorrelated components of random vector
  • Approximation of parameters of approximation horizon by the orthogonal polynomials
  • Numerical examples of application of approximation algorithms

• Algorithm of Approximation of Formation Velocity with the Use of Areal Observations with Borehole Data

The following sections are included:

• Elements of Applied Functional Analysis

• Ill-Posed Problems

• Statistical Estimation in the Terms of the Functional Analysis

• Elements of the Mathematical Design of Experiment

The following sections are included:

• Construction of the Model of Measurements

• Tomographic Functional

• Examples of Construction and Interpretation of Tomographic Functionals
  • Scalar wave equation
  • The Lame equation in an isotropic infinite medium
  • The transport equation of the stationary sounding signal
  • The diffusion equation

• Ray Tomographic Functional in the Dynamic and Kinematic Interpretation of the Remote Sounding Data

• Construction of Incident and Reversed Wave Fields in Layered Reference Medium

The following sections are included:

• Elements of Linear Tomography
  • Change of variables
  • Differentiation of generalized function

• Connection of Radon Inversion with Diffraction Tomography

• Construction of Algorithms of Reconstruction Tomography

• Errors of Recovery, Resolving Length and Backus and Gilbert Method

• Back Projection in Diffraction Tomography

• Regularization Problems in 3-D Ray Tomography

• Construction of Green Function for Some Type of Sounding Signals
  • Green function for the wave equation
  • Green function for "Poisson equation"
  • Green function for Lame equation in uniform isotropic infinite medium
  • Green function for diffusion equation
  • Green function for operator equation of the second genus

• Examples of the Recovery of the Local Inhomogeneity Parameters by the Diffraction Tomography Method
  • An estimation of the resolution
  • An estimation of the recovery accuracy of inhomogeneities parameters

The following sections are included:

• Fourier Transform
  • Fourier series
  • Fourier integral

• Laplace Transform

• Z-Transform

• Radon Transform for Seismogram Processing

• Gilbert Transform and Analytical Signal

• Cepstral Transform

• Bispectral Analysis

• Kalman Filtering

• Multifactor Analysis and Processing of Time Series
  • • I. Evolvent of 1-D time series to multidimensional one
    • II. Analysis of principal components: Singular decomposition of the sample correlation matrix
    • III. Selection of the principal components
    • IV. Recovery of 1-D time series
    • Selection of parameters

• Wiener Filter

• Predictive-Error Filter and Maximum Entropy Principle

The following sections are included:

• Appendix A Computer exercises
  • Statistical Methods
    • Examples of numerical simulation of random values
      • Exercises
    • Construction of histogram
      • Exercises
    • Description of a random variable
      • Beta distribution
      • Binomial distribution
      • Cauchy distribution
      • χ²-distribution
      • Exponential distribution
      • Fisher distribution
      • Univariate normal (Gaussian) distribution
      • Geometrical distribution
      • Laplace distribution
      • Logarithmic normal distribution
      • Poisson distribution
      • Student distribution
      • Uniform distribution
      • Two-dimensional normal distribution
      • Exercises
    • Computer simulation of random values
      • Kolmogorov criterion
      • Exercises
    • Confidence intervals
      • Exercises
    • Time series
      • Correlated and uncorrelated time series
      • Autocorrelation and crosscorrelation functions
      • Exercises
  • Transforms
    • Fourier transform
    • Signals and their spectral characteristics
      • Exercises
    • Multifactor analysis
      • Exercises
    • Cepstral transform
      • Exercises
  • Direct and Inverse Problem Solution
    • Computer simulation of gravitational attraction
      • Gravitational attraction of the sphere
      • Gravitational attraction of the uniform cylinder
    • Computer simulation of magnetic field
      • Magnetic induction of a dipole
      • Magnetic induction of the horizontal cylinder
    • Computer simulation of the seismic field
      • Exercises
    • Deconvolution by the Wiener filter
      • Exercise
    • Quantitative interpretation
      • Exercises
    • Qualitative interpretation
      • Exercises
    • Diffraction tomography
      • Exercises

• Appendix B Tables

• Bibliography

• Index