Introduction.- Preliminary Remarks.- Why Random Fields?.- Notation and
Definitions.- Noise and Errors.- Spatial Data and Basic Processing
Procedures.- A Personal Selection of Relevant Books.- Trend Models and
Estimation.- Empirical Trend Estimation.- Regression Analysis.- Global
Trend Models.- Local Trend Models.- Trend Estimation based on Physical
Information.- Trend Based on the Laplace Equation.- Basic Notions of
Random Fields.- Introduction.- Single-Point Description.- Stationarity
and Statistical Homogeneity.- Variogram versus Covariance.-
Permissibility of Covariance Functions.- Permissibility of Variogram
Functions.- Additional Topics of Random Field Modeling.- Ergodicity.-
Statistical Isotropy.- Anisotropy.- Anisotropic Spectral Densities.-
Multipoint Description of Random Fields.- Geometric Properties of Random
Fields.- Local Properties.- Covariance Hessian Identity and Geometric
Anisotropy.- Spectral Moments.- Length Scales of Random Fields.- Fractal
Dimension.- Long-Range Dependence.- Intrinsic Random Fields.- Fractional
Brownian Motion.- Classification of Random Fields.- Gaussian Random
Fields.- Multivariate Normal Distribution.- Field Integral Formulation.-
Useful Properties of Gaussian Random Fields.- Perturbation Theory for
Non-Gaussian Probability Densities.- Non-stationary Covariance
Functions.- Further Reading.- Random Fields based on Local
Interactions.- Spartan Spatial Random Fields.- Two-point Functions and
Realizations.- Statistical and Geometric Properties.- Bessel-Lommel
Covariance Functions.- Lattice Representations of Spartan Random
Fields.- Introduction to Gauss-Markov Random Fields.- From Spartan
Random Fields to Gauss-Markov Random Fields.- Lattice Spectral Density.-
SSRF Lattice Moments.- SSRF Inverse Covariance Operator on Lattices.-
Spartan Random Fields and Langevin Equations.- Introduction to
Stochastic Differential Equations.- Classical Harmonic Oscillator.-
Stochastic Partial Differential Equations.- Spartan Random Fields and
Stochastic Partial Differential Equations.- Covariance and Green's
functions.- Whittle-Matérn Stochastic Partial Differential Equation.-
Diversion in Time Series.- Spatial Prediction Fundamentals.- General
Principles of Linear Prediction.- Deterministic Interpolation.-
Stochastic Methods.- Simple Kriging.- Ordinary Kriging.- Properties of
the Kriging Predictor.- Topics Related to the Application of Kriging.-
Evaluating Model Performance.- More on Spatial Prediction.- Linear
Generalizations of Kriging.- Nonlinear Extensions of Kriging.-
Connections with Gaussian Process Regression.- Bayesian Kriging.-
Continuum Formulation of Linear Prediction.- The "Local-Interaction"
Approach.- Big Spatial Data.- Basic Concepts and Methods of Estimation.-
Estimator Properties.- Estimating the Mean with Ordinary Kriging.-
Variogram Estimation.- Maximum Likelihood Estimation.- Cross
Validation.- More on Estimation.- The Method of Normalized
Correlations.- The Method of Maximum Entropy.- Stochastic Local
Interactions.- Measuring Ergodicity.- Beyond the Gaussian Models.-
Trans-Gaussian Random Fields.- Gaussian Anamorphosis.- Tukey g-h Random
Fields.- Transformations based on Kappa Exponentials.- Hermite
Polynomials.- Multivariate Student's t-distribution.- Copula Models.-
The Replica Method.- Binary Random Fields.- The Indicator Random Field.-
Ising Model.- Generalized Linear Models.- Simulations.- Introduction.-
Covariance Matrix Factorization.- Spectral Simulation Methods.-
Fast-Fourier-Transform Simulation.- Randomized Spectral Sampling.-
Conditional Simulation based on Polarization Method.- Conditional
Simulation based on Covariance Matrix Factorization.- Monte Carlo
Methods.- Sequential Simulation of Random Fields.- Simulated Annealing.-
Karhunen-Loève Expansion.- Karhunen-Loève Expansion of Spartan Random
Fields.- Epilogue.- A Jacobi's Transformation Theorems.- B Tables of
SSRF Properties.- C Linear Algebra Facts.- D Kolmogorov-Smirnov Test.-
Glossary.- References.- Index
