Contents -- Introduction -- 1. Basic concepts of the probability theory -- 1.1 The Definition of Probability -- 1.1.1 Set of elementary events -- 1.1.2 Probability model with a finite number of outcomes -- 1.1.3 Relative-frequency definition of probability -- 1.1.4 Classical definition of probability -- 1.1.5 Geometrical definition of probability -- 1.1.6 Exercises -- 1.2 Basic Properties of Probability -- 1.2.1 Addition of probabilities -- 1.2.2 Nonindependent and independent events -- 1.2.3 The Bayes formula and complete probability -- 1.2.4 Exercises -- 1.3 Distribution Functions -- 1.3.1 Random variables -- 1.3.2 Distribution function -- 1.3.3 The density function -- 1.3.4 The distribution and density of function of one random argument -- 1.3.5 Random vectors -- 1.3.6 Marginal and conditional distributions -- 1.3.7 The distributive law of two random variables -- 1.3.8 Exercises -- 1.4 The Numerical Characteristics of Probability Distributions -- 1.4.1 Mathematical expectation -- 1.4.2 Variance and correlation coefficients -- 1.4.3 Quantiles -- 1.4.4 Characteristics of a density function -- 1.4.5 Exercises -- 1.5 Characteristic and Generating Functions -- 1.5.1 Moment generating function -- 1.5.2 Probability generator -- 1.5.3 Semi-invariants or cumulants -- 1.5.4 Exercises -- 1.6 The Limit Theorems -- 1.6.1 Convergence in probability -- 1.6.2 Chebyshev inequality -- 1.6.3 The law of averages (Chebyshev's theorem) -- 1.6.4 Generalised Chebyshev's theorem -- 1.6.5 Markov's theorem -- 1.6.6 Bernoulli theorem -- 1.6.7 Poisson theorem -- 1.6.8 The central limit theorem -- 1.6.9 Exercises -- 1.7 Discrete Distribution Functions -- 1.7.1 Binomial distribution -- 1.7.2 Poisson distribution -- 1.7.3 Geometrical distribution -- 1.7.4 Exercises -- 1.8 Continuous Distributions -- 1.8.1 Univariate normal distribution.

1.8.2 Multivariate normal distribution -- 1.8.3 Uniform distribution -- 1.8.4 2 distribution -- 1.8.5 Student's distribution (t-distribution) -- 1.8.6 Fisher distribution and Z-distribution -- 1.8.7 Triangular distribution -- 1.8.8 Beta distribution -- 1.8.9 Exponential distribution -- 1.8.10 Laplace distribution -- 1.8.11 Cauchy distribution -- 1.8.12 Logarithmic normal distribution -- 1.8.13 Significance of the normal distribution -- 1.8.14 Confidence intervals -- 1.8.15 Exercises -- 1.9 Information and Entropy -- 1.9.1 Entropy of the set of discrete states of system -- 1.9.2 Entropy of the complex system -- 1.9.3 Shannon information (discrete case) -- 1.9.4 Entropy and information for systems with a continuous set of states -- 1.9.5 Fisher information -- 1.9.6 Exercises -- 1.10 Random Functions and its Properties -- 1.10.1 Properties of random functions -- 1.10.2 Properties of the correlation function -- 1.10.3 Action of the linear operator on a random function -- 1.10.4 Cross correlation function -- 1.10.5 Wiener-Khinchin theorem and power spectrum -- 1.10.6 Definition of estimations of the characteristics of random variables -- 2. Elements of mathematical statistics -- 2.1 The Basic Concepts of the Decision Theory -- 2.1.1 Distribution class of the statistical problem -- 2.1.2 The structure of the decision space and the loss function -- 2.1.3 Decision rule -- 2.1.4 Sufficient statistic -- 2.2 Estimate Properties -- 2.2.1 Consistency -- 2.2.2 Bias -- 2.2.3 Rao-Cramer inequality. Efficiency -- 2.2.4 Sufficiency -- 2.2.5 Asymptotic normality -- 2.2.6 Robustness -- 3. Models of measurement data -- 3.1 Additive Model -- 3.2 Models of the Quantitative Interpretation -- 3.3 Regression Model -- 3.4 The Models of Qualitative Interpretation -- 3.5 The Models of Qualitative-Quantitative Interpretation -- 3.6 Random Components of Model and its Properties.

3.7 Model with Random Parameters -- 3.8 A Priori Information -- 4. The functional relationships of sounding signal elds and parameters of the medium -- 4.1 Seismology and Seismic Prospecting -- 4.2 Acoustics of the Ocean -- 4.3 Wave Electromagnetic Fields in Geoelectrics and Ionospheric Sounding -- 4.4 Atmospheric Sounding -- 5. Ray theory of wave eld propagation -- 5.1 Basis of the Ray Theory -- 5.2 Ray Method for the Scalar Wave Equation -- 5.3 Shortwave Asymptotic Form of the Solution of the One-Dimensional Helmholtz Equation (WKB Approximation) -- 5.4 The Elements of Elastic Wave Ray Theory . -- 5.5 The Ray Description of Almost-Stratified Medium -- 5.6 Surface Wave in Vertically Inhomogeneous Medium -- 5.7 Ray Approximation of Electromagnetic Fields -- 5.8 Statement of Problem of the Ray Kinematic Tomography -- 6. Methods for parameter estimation of geophysical objects -- 6.1 The Method of Moments -- 6.2 Maximum Likelihood Method -- 6.3 The Newton-Le Cam Method -- 6.4 The Least Squares Method -- 6.5 LSM

7.1 Test of Parametric Hypothesis -- 7.2 Criterion of a Posteriori Probability Ratio -- 7.3 The Signal Resolution Problem -- 7.4 Information Criterion for the Choice of the Model -- 7.5 The Method of the Separation of Interfering Signals -- 8. Algorithms of approximation of geophysical data -- 8.1 The Algorithm of Univariate Approximation by Cubic Splines -- 8.2 Periodic and Parametric Spline Functions -- 8.3 Application of the Spline Functions for Histogram Smoothing -- 8.4 Algorithms for Approximation of Seismic Horizon Subject to Borehole Observations -- 8.4.1 The Markovian type of correlation along the beds and no correlation between beds -- 8.4.2 Markovian correlation between the beds and no correlation along bed -- 8.4.3 Conformance inspection of seismic observation to borehole data concerning bed depth -- 8.4.4 Incorporation of random nature of depth measurement using borehole data -- 8.4.5 Application of a posteriori probability method to approximation of seismic horizon -- 8.4.6 Case of uncorrelated components of random vector -- 8.4.7 Approximation of parameters of approximation horizon by the orthogonal polynomials -- 8.4.8 Numerical examples of application of approximation algorithms -- 8.5 Algorithm of Approximation of Formation Velocity with the Use of Areal Observations with Borehole Data -- 9. Elements of applied functional analysis for problem of estimation of the parameters of geophysical objects -- 9.1 Elements of Applied Functional Analysis -- 9.2 Ill-Posed Problems -- 9.3 Statistical Estimation in the Terms of the Functional Analysis -- 9.4 Elements of the Mathematical Design of Experiment -- 10. Construction and interpretation of tomographic functionals -- 10.1 Construction of the Model of Measurements -- 10.2 Tomographic Functional -- 10.3 Examples of Construction and Interpretation of Tomographic Functionals.

10.3.1 Scalar wave equation -- 10.3.2 The Lame equation in an isotropic in nite medium -- 10.3.3 The transport equation of the stationary sounding signal -- 10.3.4 The diffusion equation -- 10.4 Ray Tomographic Functional in the Dynamic and Kinematic Interpretation of the Remote Sounding Data -- 10.5 Construction of Incident and Reversed Wave Fields in Layered Reference Medium -- 11. Tomography methods of recovering the image of medium -- 11.1 Elements of Linear Tomography -- 11.1.1 Change of variables -- 11.1.2 Differentiation of generalized function -- 11.2 Connection of Radon Inversion with Diffraction Tomography -- 11.3 Construction of Algorithms of Reconstruction Tomography -- 11.4 Errors of Recovery, Resolving Length and Backus and Gilbert Method -- 11.5 Back Projection in Diffraction Tomography -- 11.6 Regularization Problems in 3-D Ray Tomography -- 11.7 Construction of Green Function for Some Type of Sounding Signals -- 11.7.1 Green function for the wave equation -- 11.7.2 Green function for "Poisson equation" -- 11.7.3 Green function for Lame equation in uniform isotropic in nite medium -- 11.7.4 Green function for diffusion equation -- 11.7.5 Green function for operator equation of the second genus -- 11.8 Examples of the Recovery of the Local Inhomogeneity Parameters by the Diffraction Tomography Method -- 11.8.1 An estimation of the resolution -- 11.8.2 An estimation of the recovery accuracy of inhomogeneities parameters -- 12. Methods of transforms and analysis of the geophysical data -- 12.1 Fourier Transform -- 12.1.1 Fourier series -- 12.1.2 Fourier integral -- 12.2 Laplace Transform -- 12.3 Z-Transform -- 12.4 Radon Transform for Seismogram Processing -- 12.5 Gilbert Transform and Analytical Signal -- 12.6 Cepstral Transform -- 12.7 Bispectral Analysis -- 12.8 Kalman Filtering.

12.9 Multifactor Analysis and Processing of Time Series.