Environmental Spatiotemporal Mapping and Groundwater Flow Modeling
using the BME and ST methods

A dissertation from Marc Serre

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COVER PAGES (pdf  postcript)

TABLE OF CONTENTS

Chapter

I. INTRODUCTION (pdf  postcript) 1

II. INTRODUCTION TO SPACE/TIME RANDOM FIELD MODELLING IN

THE LIGHT OF UNCERTAIN PHYSICAL KNOWLEDGE (pdf  postcript) 4

2.1. Introduction To Space/Time Random Fields 5

2.1.1. The Space/Time Random Field 5

2.1.2. A Review of Some Space-Time Covariance Function 10

2.1.3. Choosing a Covariance Model from Experimental Data 13

2.1.4. Stochastic Simulation of S/TRF 14

2.2. Defining the Physical Knowledge Bases Available in Spatiotemporal

Mapping 17

2.3. Stochastic Simulation of S/TRF in the Light of Uncertain Physical

Knowledge 20

2.4. A Review of Some Kriging Methods. 24

2.4.1. Simple Kriging 25

2.4.2. Indicator Kriging 27

2.4.3. Simple Kriging With Measurement Error 29

III. THE BASIC EQUATIONS OF THE BME MAPPING METHOD (pdf  postcript) 33

3.1. A Rapid Tour of the BME Approach 33

3.2. Different Forms of the General Knowledge Operator 35

3.2.1. A General Solution of the Entropy Maximization Problem 36

3.2.2. The Gaussian Form: When Only the Mean and Covariance

Function Are Known 37

3.2.3. Going Beyond The Gaussian Form: Incorporating Physical

Laws 41

3.3. BME Estimates 48

3.4. Uncertainty Assessment 49

IV. COMPUTATIONAL BME FOR SOFT DATA OF INTERVAL TYPE (pdf  postcript) 51

4.1. The Processing Operator for Specificatory Knowledge of Interval Type 51

4.2. A Proposed Formulation of the Posterior PDF for Efficient

Computation 54

4.3. The BME Mode Estimate 56

4.4. Moments of the BME Posterior PDF 57

4.4.1. The BME Mean Estimate 57

4.4.2. Variance Of The BME Posterior PDF 58

4.4.3. Skewness of the Posterior PDF 59

4.5. Numerical Implementation 60

4.5.1. Implementation Considerations 60

4.5.2. Numerical Work for the BME Estimation Method 61

4.6. BMEintEst Version 1.0, a Program for BME Estimation Using Interval

Soft Data 63

4.7. Simulated Comparison with Kriging Methods 66

4.7.1. Comparison Between BME and Indicator Kriging 67

4.7.2. Comparison Between BME and Simple Kriging Methods 71

4.8. The Lyon Case Study 77

V. COMPUTATIONAL BME FOR SOFT DATA OF PROBABILISTIC TYPE (pdf  postcript) 83

5.1. The Processing Operator For Specificatory Knowledge Of Probabilistic

Type 83

5.2. A Proposed Formulation of the Posterior Pdf For Efficient Computation 85

5.3. BME Mode Estimate 86

5.4. Moments of the BME Posterior PDF 87

5.5. Numerical Implementation 88

5.6. Simulated Case Studies 89

5.6.1. A Simulation Based Comparison Between BME and Kriging

Methods 90

5.6.2. Map Based Comparison Between Bme And Kriging Methods 98

5.7. The Equus Beds Case Study 106

VI. VECTOR BME (pdf  postcript) 119

6.1. What are Vector S/TRF's in BME Mapping? 119

6.2. The General Knowledge Operator 120

6.3. The Specificatory Knowledge Operator 122

6.4. A Proposed Formulation of the BME Posterior PDF for Efficient

Computation 124

6.5. Moments of the BME Posterior PDF 125

6.6. Numerical Implementation 126

6.7. Applications of Computational Vector BME in Environmental Sciences 127

6.8. The Mortality/Temperature Association Case Study for North Carolina 129

VII. MULTI-POINT BME ESTIMATION AND BME CONFIDENCE SETS (pdf  postcript) 146

7.1. The Multi-Point Mapping Approach 146

7.2. A proposed Formulation of the Posterior PDF for Efficient Computation 148

7.3. Multi-point BME Mode Estimate 150

7.4. Moments of the multi-point BME posterior pdf 151

7.5. Uncertainty Assessment: Going Beyond the Confidence Intervals- The

BME Confidence Sets 155

7.6. Simulated Case Studies 157

7.6.1 Multi-point Mapping versus single-point Mapping: Test Case 1 158

7.6.2 Multi-point Mapping Versus Single-Point Mapping: Test Case 2 161

7.6.3 BME Confidence Sets in Action 165

VIII. BME Studies Of Stochastic Differential Equations Representing Physical Law (pdf  postcript) 170

8.1. Introduction 171

8.2. Incorporating Physical Laws in BME mapping 172

8.3. A Numerical Example for the Incorporation of a Physical Law in the

General Knowledge 173

8.3.1. The physical law and the BME framework 174

8.3.2. Numerical results 176

8.4. BME Mapping of Unidimentional Ground water Flow 181

8.4.1. A Formulation of the One-Dimensional Flow Problem 181

8.4.2. Using Darcy Law to Obtain Hydraulic Head Moments 183

8.4.3. The BME Prior PDF 184

8.4.4. The BME Posterior PDF 186

8.4.5. Simulated Case Study 187

IX. CONCLUSION (pdf  postcript) 195

8.1. Suggestions for Future Work 199

8.1.1. Space Transformation Theory for Bounded Flow Domain 199

8.1.2. BME Mapping of Data with Known Distribution 200

8.1.3. BME Mapping Using Statistical Moments of High Orders 200

8.1.4. BME Mapping Using Physical Laws 200

APPENDICES (pdf  postcript)

APPENDIX A: PROBABILITY SPACE AND RANDOM VARIABLES 202

APPENDIX B: SOLUTION OF THE ENTROPY MAXIMIZATION PROBLEM 205

APPENDIX C: INVERSE OF PARTITIONED MATRICES 207

C.1. Properties of the Inverse of Partitioned Matrices 207

C.2. Property of the First Schur Complement 207

C.3. Property of the Second Schur Complement 208

C.4. Properties of the Determinant 209

APPENDIX D: PROPERTIES OF MULTIVARIATE NORMAL

DISTRIBUTIONS 210

D.1. The Multivariate Normal Distribution: 210

D.2. Normalization Property 211

D.3. Centered Second Order Moment 211

D.4. Conditional Distribution. 212

D.5. Marginal Distribution 212

D.6. Approximation of the Integral over a Hyper Rectangle of the Subset

of aRandom Vector which is Multivariate Normal 215

D.7. Useful Integrals 218

D.8. Moments of the univariate Gaussian pdf 219

APPENDIX E: THE THEORY OF SPACE TRANSFORMATIONS AND

THEIR NUMERICAL IMPLEMENTATION 220

E.2. A Brief Review of the Theory of Space Transformations 220

E.3. Numerical Implementations of the Space Transformations 226

APPENDIX F: PROOF OF CONFIDENCE SETS 228

REFERENCES (pdf  postcript) 229