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TABLE OF CONTENTS
Chapter I: FUNDAMENTAL PRINCIPLES OF STOCHASTIC ENVIRONMENTAL HEALTH
MODELLING 1
1. On the method of environmental health science 1
2. Fundamental principles of the stochastic mode of thinking 7
3. The notion of spatiotemporal continuity 16
3.1 Philosophical aspects of space and time 17
3.2 The space-time continuum concept 18
3.3 Natural vs. health processes 19
3.4 Space-time coordinates 20
4. The field concept in environmental health science 22
5. Knowledge bases in the space-time domain. 26
6. Spatiotemporal scales 27
6.1 Spatiotemporal scales for natural processes 28
6.2 Spatiotemporal scales for health processes 31
6.3 The hierarchy of scales 32
7. An overview of the environmental health paradigm 34
Chapter II: ENVIRONMENTAL EXPOSURE FIELDS AND THEIR HEALTH EFFECTS 37
1. The holistic environmental exposure -health effect perspective 37
2. Studying environmental exposure fields 42
2.1 Pollution sources 42
2.2 Environmental pollutants 43
2.3 Characterization of the environment 44
2.4 Environmental compartments 45
2.5 Environmental media 46
2.6 Natural fields 47
2.7 Evolution mechanisms 49
3. Human Exposure Concepts 53
3.1 Basic notions 53
3.2 Mathematical expressions for exposure 59
3.3 Mathematical expressions for biomarkers 64
3.4 Exposure as a practical biomarker estimator 68
4. Health Effect Concepts 69
4.1 The health impact pathway and the various kinds of health effects
4.2 Mathematical models for health effect indicators 71
4.3 Mathematical models for individual-based and population-based
dose and Exposure-response curves 73
4.4 Spatiotemporal coordinates in human exposure analysis 77
Chapter III: SPATIOTEMPORAL RANDOM FIELDS IN EXPOSURE ANALYSIS AND
ASSESSMENT 81
1. A theatrical introduction to the random field concept 81
2. The spatiotemporal random field 83
2.1 Basic mathematical formulation 84
2.2 Characterization in terms of ensemble functions 87
3. S/TRF classifications 90
4. Space-time metrics 95
5. Spatiotemporal correlation models 97
5.1 General properties 97
5.2 Criteria of permissibility 99
5.3 Useful tools for constructing spatiotemporal correlation
Functions 101
6. Separable space-time covariance models 102
6.1 Separability in the space-time domain 102
6.2 Classes of spatial covariance models 103
7. Natural models and covariance functions 109
8. Regression S/TRF models 114
8.1 Mathematical formulation 114
8.2 Statistical inference issues 116
9. Wave representation of S/TRF 118
Chapter IV: MODELLING EXPOSURE HETEROGENEITIES 121
1. A class of spatiotemporal random fields 121
1.1 The central idea and definitions 121
1.2 Determination of the Q-operator in practice and its physical
significance 125
1.3 An S/TRF decomposition 127
2. Spatiotemporal polynomial notation 127
3. Orders of spatiotemporal continuity v/u 129
4. Continuum representations of S/TRF-v/u 132
5. Discrete representations: spatiotemporal increments of orders v/u 135
6. Spectral Representations 139
7. Generalized Spatiotemporal random fields 141
8. Permissibility criteria 147
8.1 Analysis in the continuous spatiotemporal domain
The generalized spatiotemporal covariance 148
8.2 Some examples in the discrete space-time domain 152
8.3 Permissibility conditions for polynomial covariances 154
8.4 Permissibility conditions for vector S/TRF 157
9. Generalized covariance models derived from FDE's 160
10. Space-time separability 163
11. New generalized spatiotemporal covariance models 164
11.1 The Polynomial model 164
11.2 The Exponential model 165
11.3 The Gaussian model 169
Chapter V: SPATIOTEMPORAL MAPPING OF ENVIRONMENTAL HEALTH PROCESSES-
THE BME APPROACH 171
1. About spatiotemporal maps 171
2. Space-time mapping Fundamentals 173
2.1 The basic epistemic framework of the BME approach 173
2.2 Other mapping approaches 175
2.3 Operational concepts or scale 176
3. Mathematical formulation of the BME approach 177
4. Some analytical results 185
4.1 Multi-point BME 185
4.2 Single-point BME 186
5. Other Formulations of the BME approach 189
5.1 Functional BME 189
5.2 Vector BME 191
6. BME VS. MMSE estimates 192
Chapter VI: SPATIOTEMPORAL MMSE MAPPING 193
1. Introduction 193
2. Mathematical Formulation 194
3. Linear MMSE space-time mapping 195
3.1 Mathematical formulation 195
3.2 Recursive formulations 199
3.3 More properties 199
4. Homogeneous-Stationary S/TRF: Space-time Kriging forms 201
4.1 Biased MMSE mapping 201
4.2 Unbiased MMSE mapping 202
5. Nonhomogeneous-nonstationary S/TRF: The case of S/TRF- v/u 206
6. Vector MMSE space-time mapping 212
7. Implementation issues - Applications 214
7.1 Practical map generation 214
7.2 Trend determination 211
8. Regression ~/TRF 214
9. BME vs. MMSE estimators 226
10. BME vs. Kriging estimators 276
Chapter VII: STOCHASTIC PARTIAL DIFFERENTIAL EQUATION MODELLING OF
FLOW AND TRANSPORT 231
1. Introduction 231
2. Perturbation expansions 235
3. The diagrammatic approach 240
3.1 The diagrammatic language-Physical space 241
3.2 Diagrams in frequency space 243
4. Self-consistent equations 247
4.1 The basic concepts 247
4.2 The asymptotic limit 251
5. Diagrammatic methods in subsurface hydrology 252
6. Renormalization group analysis-Change of scale 253
6.1 The basic concepts 254
6.2 Diagrammatic approximations for the self-energy 258
6.3 Frequency space RNG treatment of transport 259
6.4 Perturbative RNG schemes 261
6.5 4G analysis or subsurface pollution transport 263
7. The space transformation technique 265
7.1 Basic definitions 265
7.2 Some properties of ST's 267
7.3 Numerical implementations of ST 270
7.4 Applications of STs in SPDE solving 272
7.5 Differential Geometric approach to multiphase flow 275
7.6 Boundary effects 276
Chapter VIII: STOCHASTIC PHYSIOLOGICALLY-BASED POLLUTOKINETIC
MODELLING 279
1. Introduction 289
2. Compartmental analysis of pollutokinetics 280
3. One-compartment stochastic pollutokinetics 284
4. Multi-compartment stochastic pollutokinetics 290
5. Stochastic pollutokinetics and health effects 292
6. Some generalizations 297
Chapter IX: STOCHASTIC EXPOSURE AND HEALTH INDICATORS 299
1. Introduction 299
2. Stochastic exposure indicators 302
2.1 The exposure S/TRF-pair 303
2.2 The stochastic exposure indicator concept 305
2.3 One-point stochastic exposure indicators 306
2.4 Two-point stochastic exposure indicators 309
2.5 Uncertainty in the exposure threshold level 313
2.6 Other properties of exposure indicators 315
2.7 Ergodicity and sample averaging of exposure indicators 318
3. Stochastic cell-based health effect indicators 322
3.1 Cellular structures 322
3.2 The threshold postulate 323
3.3 Relationships between the basic cell indicators 329
3.4 Derived cell indicators 331
3.5 Scale and modelling effects 331
3.6 The multistage postulate 335
4. Stochastic formulation of traditional population health effect
indicators 340
4.1 Two fundamental health effect indicators 341
4.2 Person-time rates 346
4.3 Rate ratios 349
4.4 Spatiotemporal variation and mapping of population health
indicators - The case of the North Carolina breast cancer
incidence distribution 349
5. correlation analysis 355
5.1 Population health damage indicators 355
5.2 Linear PER model 359
5.3 Uniform population density 360
5.4 Stochastic trends and uncertainty assessment in exposure-damage
models 361
5.5 Tame delayed PER relationships 363
6. Spatiotemporal maps of health damage indicators - The ozone case 363
7. Analysis of disease - Exposure association 369
Bibliography 375
Index 391
