Spatiotemporal random fields: theory and applications

Front Cover -- Spatiotemporal Random Fields -- Spatiotemporal Random FieldsTheory and Applications -- Copyright -- Dedication -- Contents -- Preface -- I - SPACE, TIME, SPACE-TIME, RANDOMNESS, AND PROBABILITY -- 1. INTRODUCTION -- 2. SPACE-TIME CONTINUUM AND KOLMOGOROV PROBABILITY SPACE -- 2.1 SPACE-TIME ARGUMENTS: POINTS, LAGS, SEPARATIONS, AND METRICS -- 2.2 TRANSFORMATIONS AND INVARIANCE IN SPACE-TIME -- 2.3 SPACE-TIME INTERPRETATIONS -- 2.4 FUNCTIONS OF SPACE-TIME ARGUMENTS -- 3. RANDOM VARIABLES IN SPACE-TIME -- 3.1 KOLMOGOROV'S PROBABILITY THEORY -- 3.2 USEFUL INEQUALITIES -- 3.3 CONVERGENCE OF RANDOM VARIABLE SEQUENCES -- II - SPATIOTEMPORAL RANDOM FIELDS -- 1. INTRODUCTION -- 1.1 THE SPACE-TIME COMPONENT -- 1.2 THE RANDOMNESS COMPONENT -- 2. CHARACTERIZATION OF SCALAR SPATIOTEMPORAL RANDOM FIELDS -- 2.1 PROBABILISTIC STRUCTURE -- 2.2 THE CHARACTERISTIC FUNCTION -- 2.3 SPATIOTEMPORAL VARIABILITY FUNCTIONS: COMPLETE (OR FULL) AND PARTIAL -- 2.4 ANALYSIS IN THE SPECTRAL DOMAIN -- 2.5 DATA-INDEPENDENT SPATIOTEMPORAL VARIABILITY FUNCTION -- 2.6 SOME NOTICEABLE SPECIAL CASES OF THE SPATIOTEMPORAL RANDOM FIELD THEORY -- 2.7 SPACE-TIME SEPARABILITY -- 3. PHYSICAL INSIGHT BEHIND THE RANDOM FIELD CONCEPT -- 3.1 RANDOM FIELD REALIZATIONS -- 3.2 PROBABLE VERSUS ACTUAL -- 3.3 PROBABILITY AND THE OBSERVATION EFFECT -- 3.4 SELF-CONSISTENCY AND PHYSICAL FIDELITY -- 4. GEOMETRY OF SPATIOTEMPORAL RANDOM FIELDS -- 5. VECTOR SPATIOTEMPORAL RANDOM FIELDS -- 6. COMPLEX SPATIOTEMPORAL RANDOM FIELDS -- 7. CLASSIFICATIONS OF THE SPATIOTEMPORAL RANDOM FIELD MODEL -- 7.1 FIRST CLASSIFICATION: DISCRETE VERSUS CONTINUOUS ARGUMENTS -- 7.2 SECOND CLASSIFICATION: SCALAR VERSUS VECTOR RANDOM FIELDS AND ARGUMENTS -- 7.3 THIRD CLASSIFICATION: PROBABILITY LAW SHAPES -- 7.4 FOURTH CLASSIFICATION: SPACE-TIME VARIABILITY.

7.5 FIFTH CLASSIFICATION: SPATIOTEMPORAL RANDOM FIELD MEMORY VERSUS INDEPENDENCE -- 8. CLOSING COMMENTS -- 8.1 THE METHODOLOGICAL IMPORTANCE OF SPACE-TIME -- 8.2 A CONCEPTUAL MEETING POINT FOR MODELERS AND EXPERIMENTALISTS -- 8.3 THERE IS NO ASSUMPTIONLESS MODELING -- III - SPACE-TIME METRICS -- 1. BASIC NOTIONS -- 1.1 FORMAL AND PHYSICAL ASPECTS OF SPACE-TIME METRIC DETERMINATION -- 1.2 SPACE-TIME METRIC FORMS -- 1.3 DERIVED SPACE-TIME METRICS -- 1.4 SPACE-TIME METRIC DIFFERENTIALS -- 1.5 SPECIFYING SPACE-TIME RELATIONSHIPS IN THE COVARIANCE FUNCTION -- 2. COVARIANCE DIFFERENTIAL FORMULAS -- 3. SPACE-TIME METRIC DETERMINATION FROM PHYSICAL CONSIDERATIONS -- 4. EXAMPLES -- 5. CONCERNING THE ZETA COEFFICIENTS -- 6. CLOSING COMMENTS -- IV - SPACE-TIME CORRELATION THEORY -- 1. FOCUSING ON SPACE-TIME VARIABILITY FUNCTIONS -- 1.1 BASICS OF SPACE-TIME CORRELATION THEORY -- 1.2 PHYSICAL INVESTIGATIONS BASED ON SPACE-TIME CORRELATION THEORY -- 2. SPACE-TIME VARIABILITY FUNCTIONS IN TERMS OF SCALAR SPACE-TIME STATISTICS -- 2.1 LOCALITY: ONE-POINT SPACE-TIME VARIABILITY FUNCTIONS -- 2.2 NONLOCALITY: OMNIDIRECTIONAL TWO-POINT SPACE-TIME VARIABILITY FUNCTIONS -- 2.3 NONLOCALITY: DIRECTION-SPECIFIC SPACE-TIME VARIABILITY FUNCTIONS -- 2.4 PHYSICAL CONSIDERATIONS AND ASSUMPTIONS OF SPACE-TIME VARIABILITY FUNCTIONS -- 2.5 FORMAL AND PHYSICAL COVARIANCE PERMISSIBILITY -- 3. BASIC PROPERTIES OF COVARIANCE FUNCTIONS -- 4. CROSS-SPACE-TIME VARIABILITY FUNCTIONS -- 5. CORRELATION OF GAUSSIAN AND RELATED SPATIOTEMPORAL RANDOM FIELDS -- 5.1 GENERAL CONSIDERATIONS -- 5.2 GAUSSIAN PROPERTIES -- 6. CORRELATION THEORY OF COMPLEX SPATIOTEMPORAL RANDOM FIELDS -- 6.1 BASIC NOTIONS -- 6.2 OTHER TYPES OF COMPLEX COVARIANCE FUNCTIONS -- 6.3 GAUSSIAN COMPLEX SPATIOTEMPORAL RANDOM FIELDS.

6.4 COMPLEX-VALUED VERSUS REAL-VALUED COVARIANCE FUNCTIONS OF SPACE-TIME HOMOSTATIONARY RANDOM FIELDS -- 6.5 SOME METHODOLOGICAL CONSIDERATIONS -- V - TRANSFORMATIONS OF SPATIOTEMPORAL RANDOM FIELDS -- 1. INTRODUCTION -- 2. FOURIER TRANSFORMATION -- 2.1 CHARACTERISTIC FUNCTIONS -- 2.2 HARMONIZABLE RANDOM FIELDS AND COVARIANCE FUNCTIONS -- 2.3 TRANSFER FUNCTION AND EVOLUTIONARY MEAN POWER -- 2.4 FOURIER TRANSFORM OF VECTOR SPATIOTEMPORAL RANDOM FIELDS -- 3. SPACE TRANSFORMATION -- 3.1 BASIC NOTIONS -- 3.2 SPACE TRANSFORMATION OF SPATIOTEMPORAL RANDOM FIELDS -- 3.3 SPACE TRANSFORMATION FOR SPATIOTEMPORAL VARIABILITY FUNCTIONS -- 3.4 SPACE TRANSFORMATION IN THE SIMULATION OF SPATIOTEMPORAL RANDOM FIELDS -- 3.5 SPACE TRANSFORMATION IN THE SOLUTION OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATION -- 4. THE TRAVELING TRANSFORMATION -- 4.1 BASIC NOTIONS -- 4.2 DETERMINATION OF THE TRAVELING VECTOR -- 4.3 TRAVELING TRANSFORMATION IN SPATIOTEMPORAL RANDOM FIELD ESTIMATION: THE SPACE-TIME PROJECTION TECHNIQUE -- 5. CLOSING COMMENTS -- VI - GEOMETRICAL PROPERTIES OF SPATIOTEMPORAL RANDOM FIELDS -- 1. INTRODUCTION -- 2. STOCHASTIC CONVERGENCE -- 3. STOCHASTIC CONTINUITY -- 3.1 BASIC TYPES OF STOCHASTIC CONTINUITY -- 3.2 EQUIVALENCE, MODIFICATION, AND SEPARABILITY44TO BE DISTINGUISHED FROM COVARIANCE SEPARABILITY. -- 4. STOCHASTIC DIFFERENTIATION -- 4.1 BASIC NOTATION AND DEFINITIONS -- 4.2 COVARIANCES OF RANDOM FIELD DERIVATIVES -- 4.3 MEAN SQUARELY DIFFERENTIABILITY CONDITIONS -- 4.4 ALMOST SURELY DIFFERENTIABILITY CONDITIONS -- 5. THE CENTRAL LIMIT THEOREM -- 6. STOCHASTIC INTEGRATION -- VII - AUXILIARY HYPOTHESES OF SPATIOTEMPORAL VARIATION -- 1. INTRODUCTION -- 1.1 HYPOTHESIS 1: HOMOSTATIONARITY -- 1.2 HYPOTHESIS 2: ISOSTATIONARITY -- 1.3 HYPOTHESIS 3: HETEROGENEITY -- 1.4 HYPOTHESIS 4: ERGODICITY -- 1.5 HYPOTHESIS 5: SEPARABILITY -- 1.6 HYPOTHESIS 6: SYMMETRY.

1.7 HYPOTHESIS 7: LOCATIONAL DIVERGENCE -- 2. SPACE-TIME HOMOSTATIONARITY -- 2.1 OMNIDIRECTIONAL SPATIOTEMPORAL VARIABILITY FUNCTIONS -- 2.2 DIRECTION-SPECIFIC SPATIOTEMPORAL COVARIANCE FUNCTION -- 2.3 ANISOTROPIC FEATURES -- 2.4 SPATIOTEMPORAL VARIOGRAM AND STRUCTURE FUNCTIONS: OMNIDIRECTIONAL AND DIRECTION SPECIFIC -- 3. SPECTRAL REPRESENTATIONS OF SPACE-TIME HOMOSTATIONARITY -- 3.1 SPECTRAL FUNCTIONS OF SPACE-TIME HOMOSTATIONARY RANDOM FIELDS -- 3.2 PROPERTIES OF THE SPECTRAL DENSITY FUNCTION -- 3.3 PARTIAL SPECTRAL REPRESENTATIONS -- 3.4 MORE ON DISPERSION RELATIONS -- 3.5 SEPARABILITY ASPECTS -- 4. THE GEOMETRY OF SPACE-TIME HOMOSTATIONARITY -- 4.1 DIFFERENTIATION FORMULAS: PHYSICAL AND SPECTRAL DOMAINS -- 4.2 STOCHASTIC CONTINUITY AND DIFFERENTIABILITY -- 4.3 SPATIOTEMPORAL RANDOM FIELD INTEGRABILITY -- 5. SPECTRAL MOMENTS AND LINEAR RANDOM FIELD TRANSFORMATIONS -- VIII - ISOSTATIONARY SCALAR SPATIOTEMPORAL RANDOM FIELDS -- 1. INTRODUCTION -- 1.1 BASIC CONSIDERATIONS -- 1.2 POWER-LAW CORRELATIONS -- 1.3 PHYSICAL CONSIDERATIONS OF VARIOGRAM FUNCTIONS -- 2. RELATIONSHIPS BETWEEN COVARIANCE DERIVATIVES AND SPACE-TIME ISOSTATIONARITY -- 3. HIGHER-ORDER SPATIOTEMPORAL VARIOGRAM AND STRUCTURE FUNCTIONS -- 4. SEPARABLE CLASSES OF SPACE-TIME ISOSTATIONARY COVARIANCE MODELS -- 5. A SURVEY OF SPACE-TIME COVARIANCE MODELS -- 6. SCALES OF SPATIOTEMPORAL DEPENDENCE AND THE UNCERTAINTY PRINCIPLE -- 6.1 SCALES FOR SPATIOTEMPORAL RANDOM FIELDS WITH RESTRICTED SPACE-TIME VARIABILITY -- 6.2 RELATIONSHIPS BETWEEN PHYSICAL AND SPECTRAL DOMAINS: THE UNCERTAINTY PRINCIPLE -- 7. ON THE ERGODICITY HYPOTHESES OF SPATIOTEMPORAL RANDOM FIELDS -- IX - VECTOR AND MULTIVARIATE RANDOM FIELDS -- 1. INTRODUCTION -- 2. HOMOSTATIONARY AND HOMOSTATIONARILY CONNECTED CROSS-SPATIOTEMPORAL VARIABILITY FUNCTIONS AND CROSS-SPECTRAL DENSITY FUNCTIONS.

2.1 BASIC NOTIONS AND INTERPRETATIONS -- 2.2 GEOMETRY OF VECTOR SPATIOTEMPORAL RANDOM FIELDS -- 3. SOME SPECIAL CASES OF COVARIANCE FUNCTIONS -- 4. SOLENOIDAL AND POTENTIAL VECTOR SPATIOTEMPORAL RANDOM FIELDS -- 5. PARTIAL CROSS-COVARIANCE AND CROSS-SPECTRAL FUNCTIONS -- 6. HIGHER-ORDER CROSS-SPATIOTEMPORAL VARIABILITY FUNCTIONS -- 7. ISOSTATIONARY VECTOR SPATIOTEMPORAL RANDOM FIELDS -- 7.1 DIRECT (LAG-BASED) SPACE-TIME ISOSTATIONARITY -- 7.2 COMPOSITE LAG-FIELD-BASED SPACE-TIME ISOSTATIONARITY -- 7.3 LINKS WITH SOLENOIDAL AND POTENTIAL SPATIOTEMPORAL RANDOM FIELDS -- 8. EFFECTIVE DISTANCES AND PERIODS -- X - SPECIAL CLASSES OF SPATIOTEMPORAL RANDOM FIELDS -- 1. INTRODUCTION -- 2. FROZEN SPATIOTEMPORAL RANDOM FIELDS AND TAYLOR'S HYPOTHESIS -- 2.1 BASIC NOTIONS -- 2.2 SPECTRAL DOMAIN ANALYSIS -- 2.3 DIFFERENTIAL EQUATION REPRESENTATIONS -- 2.4 EXTENSIONS OF THE FROZEN RANDOM FIELD MODEL -- 2.4.1 Spectral Domain Decomposition -- 2.4.2 Nonstationary Representation -- 2.4.3 Elliptic Representation -- 2.4.4 Random Velocity -- 2.4.5 Heterogeneous Representation -- 2.5 INTEGRALS OF FROZEN SPATIOTEMPORAL RANDOM FIELDS -- 2.6 VECTOR FROZEN SPATIOTEMPORAL RANDOM FIELDS -- 3. PLANE-WAVE SPATIOTEMPORAL RANDOM FIELDS -- 4. LOGNORMAL SPATIOTEMPORAL RANDOM FIELDS -- 5. SPHERICAL SPATIOTEMPORAL RANDOM FIELDS -- 6. LAGRANGIAN SPATIOTEMPORAL RANDOM FIELDS -- XI - CONSTRUCTION OF SPATIOTEMPORAL PROBABILITY LAWS -- 1. INTRODUCTION -- 2. DIRECT PROBABILITY DENSITY MODEL CONSTRUCTION TECHNIQUES -- 2.1 THE INDEPENDENCY TECHNIQUES -- 2.2 THE SPHERICAL SYMMETRY TECHNIQUE -- 2.3 THE TRANSFORMATION TECHNIQUE -- 3. FACTORA-BASED PROBABILITY DENSITY MODEL CONSTRUCTION TECHNIQUES -- 4. COPULA-BASED PROBABILITY DENSITY MODEL CONSTRUCTION TECHNIQUES -- 5. STOCHASTIC DIFFERENTIAL EQUATION-BASED PROBABILITY DENSITY MODEL CONSTRUCTION TECHNIQUES.

5.1 THE TRANSFORMATION OF VARIABLES APPROACH.