COST EFFECTIVE GROUNDWATER
QUALITY SAMPLING NETWORK DESIGN
A Dissertation Presented
by
Graciela Herrera de Olivares
to
The Faculty of the Graduate College
of
The University of Vermont
In Partial Ful�llment of the Requirements
for the Degree of Doctor of Philosophy
Specializing in Mathematical Sciences
Emphasizing Applied Mathematics
May, 1998
Accepted by the Faculty of the Graduate College, The University of
Vermont, in partial ful�llment of the requirements for the degree of
Doctor of Philosophy, specializing in Mathematical Sciences,
emphasizing Applied Mathematics.
Dissertation Examination Committee:
Advisor
George F. Pinder, Ph.D.
Je�rey S. Buzas, Ph.D.
William D. Lakin, Ph.D.
Chairperson
Paul R. Bierman, Ph.D.
Interim Dean,
Andrew R. Bodman, Ph.D. Graduate College
Date: March 31, 1998
Abstract
Groundwater quality sampling networks are an aid in characterizing groundwatercontamination problems and in evaluating the performance of a remediation strategy.In this context the goal of a quality sampling network typically is to estimate con-taminant concentrations at some speci�ed locations in the aquifer. Often estimatingconcentrations of a contaminant plume in an e�cient way depends on both, the loca-tion of the sampling wells and the times when the contaminant samples are taken. Onthe other hand, performance costs of a sampling network can be a very large part ofoverall costs. Therefore, the design of a cost-e�ective groundwater-quality samplingnetwork can save much money.
In response to this need we have developed a methodology for the design of cost-e�ective groundwater-quality sampling networks in which sampling locations andsampling times are decision variables. The sampling networks obtained with thismethod are cost-e�ective in the sense that an e�cient use of the information pro-vided by each contaminant concentration sample leads to sampling programs thatwith a small numbers of samples can get accurate estimates.
As a �rst step to manage the data information in an e�cient way, we developed anestimation method that accounts for space-time correlations of the transport modelerror. The method is equivalent to a space-time kriging method in which the concen-tration mean and covariance matrix are obtained from a stochastic transport model.The method can accommodate several sources of variability. Taking advantage of cur-rent modeling practices, the estimation method uses a deterministic model developedfor a given groundwater quality problem and adds uncertainty to it.
Acknowledgements
Thank you Jos�e Luis for your patience and support during all these years. Thanks
also to my advisor Dr. George Pinder for the very rewarding experience of working
with him.
Being a member of the Research Center for Groundwater Remediation Design
(RCGRD) gave me the opportunity to interact and learn from other researchers and
students. I want to thank all its members. Also, thanks to my teachers.
I thank also the members of my dissertation examination committee for their
comments and suggestions.
ii
Dedication
To Jos�e Luis
To my parents
iii
Table of Contents
Acknowledgements : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : iiDedication : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : iiiList of Tables : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : viiList of Figures : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : viii1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1
1.1. Groundwater Contamination . . . . . . . . . . . . . . . . . . . . . . . 11.2. Groundwater Quality Sampling Networks . . . . . . . . . . . . . . . . 21.3. Dissertation Organization . . . . . . . . . . . . . . . . . . . . . . . . 4
2. Statistical De�nitions : : : : : : : : : : : : : : : : : : : : : : : : : : : : 52.1. Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2. Continuous Random Variables . . . . . . . . . . . . . . . . . . . . . . 62.3. Jointly Distributed Random Variables . . . . . . . . . . . . . . . . . . 82.4. Conditional Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . 112.5. Random Samples and Estimation . . . . . . . . . . . . . . . . . . . . 122.6. Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.7. Random Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3. Comprehensive Literature Review : : : : : : : : : : : : : : : : : : : : 183.1. Spatiotemporal Estimation Methods . . . . . . . . . . . . . . . . . . 18
3.1.1. Geostatistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.1.2. Stochastic Methods Based on PDEs . . . . . . . . . . . . . . . 22
3.2. Spatiotemporal Sampling Design . . . . . . . . . . . . . . . . . . . . 283.2.1. Sampling Network Design and Deterministic Modeling . . . . 283.2.2. Sampling Network Design and Stochastic Modeling . . . . . . 31
4. Estimation of Plumes in Motion in an Eulerian Framework. The
Role of the Model Error : : : : : : : : : : : : : : : : : : : : : : : : : : : : 384.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2. Spatiotemporal Estimation Methods . . . . . . . . . . . . . . . . . . 39
4.2.1. Geostatistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2.2. Stochastic Methods Based on PDEs . . . . . . . . . . . . . . . 42
4.3. Flow and Transport Equations . . . . . . . . . . . . . . . . . . . . . . 494.4. Stochastic Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.4.1. Hydraulic Conductivity Random Field . . . . . . . . . . . . . 504.4.2. Contaminant Concentration Random Fields . . . . . . . . . . 51
4.5. Model Error Time Correlations . . . . . . . . . . . . . . . . . . . . . 524.5.1. Model Error De�nition . . . . . . . . . . . . . . . . . . . . . . 52
iv
4.5.2. Model Error Statistical Properties . . . . . . . . . . . . . . . . 544.6. Consequences for the Estimation Process . . . . . . . . . . . . . . . . 61
4.6.1. Dynamic Kalman Filter . . . . . . . . . . . . . . . . . . . . . 614.6.2. Static Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . 634.6.3. Estimation Method Proposed in this Thesis . . . . . . . . . . 644.6.4. Estimation of Prior Moments by Stochastic Simulation . . . . 654.6.5. Conditional Variance . . . . . . . . . . . . . . . . . . . . . . . 664.6.6. Conditional Estimates . . . . . . . . . . . . . . . . . . . . . . 70
4.7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5. Cost E�ective Groundwater Quality Sampling Network Design : 785.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.2. Spatiotemporal Sampling Design . . . . . . . . . . . . . . . . . . . . 79
5.2.1. Sampling Network Design and Deterministic Modeling . . . . 805.2.2. Sampling Network Design and Stochastic Modeling . . . . . . 83
5.3. Sampling Design Methodology . . . . . . . . . . . . . . . . . . . . . . 895.3.1. Source Concentration Random Field . . . . . . . . . . . . . . 91
5.4. Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.5. Contaminant Transport Simulation . . . . . . . . . . . . . . . . . . . 935.6. Statistical Properties of the Hydraulic Conductivity and the Contam-
inant Concentration at the Source . . . . . . . . . . . . . . . . . . . . 945.7. Sampling Design Criteria . . . . . . . . . . . . . . . . . . . . . . . . . 955.8. Sampling Program. Test 1 . . . . . . . . . . . . . . . . . . . . . . . . 965.9. Plume Estimate Analysis. Test 1 . . . . . . . . . . . . . . . . . . . . 995.10. Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.10.1. Correlation Scale of Hydraulic Conductivity . . . . . . . . . . 1025.10.2. Correlation of the Contaminant Concentration at the Source . 1075.10.3. Variance of the Hydraulic Conductivity Field . . . . . . . . . 109
5.11. Residual Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.12. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119A. Formulas to Minimize the Estimate Variance . . . . . . . . . . . . . . 119References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6. Convergence Tests : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1256.1. Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.2. Test 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.2.1. Total Variance Analysis . . . . . . . . . . . . . . . . . . . . . 1266.2.2. Maximum Variance Analysis . . . . . . . . . . . . . . . . . . . 130
6.3. Correlation Scale of the Hydraulic Conductivity Field . . . . . . . . . 1336.3.1. Total Variance Analysis . . . . . . . . . . . . . . . . . . . . . 1336.3.2. Maximum Variance Analysis . . . . . . . . . . . . . . . . . . . 134
6.4. Time Correlation of the Concentration at the Source . . . . . . . . . 135
v
6.4.1. Total Variance Analysis . . . . . . . . . . . . . . . . . . . . . 1356.4.2. Maximum Variance Analysis . . . . . . . . . . . . . . . . . . . 135
6.5. Variance of the Hydraulic Conductivity Field . . . . . . . . . . . . . . 1376.5.1. Total Variance Analysis . . . . . . . . . . . . . . . . . . . . . 1376.5.2. Maximum Variance Analysis . . . . . . . . . . . . . . . . . . . 140
6.6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427. Conclusions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 146Comprehensive Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : 149Appendices : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 156Appendix A. Regression of a Contaminant Concentration Field Time-
Series : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 156
vi
List of Tables
3.1. Sampling network design for parameter estimation. . . . . . . . . . . 303.2. Sampling network design using geostatistical methods. . . . . . . . . 333.3. Sampling network designs using stochastic methods based on PDE's. 344.1. Input for the example problem. . . . . . . . . . . . . . . . . . . . . . 565.1. Sampling network design for parameter estimation. . . . . . . . . . . 825.2. Sampling network design using geostatistical methods. . . . . . . . . 855.3. Sampling network designs using stochastic methods based on PDE's. 865.4. Input for test 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.5. Parameter values for the six tests. . . . . . . . . . . . . . . . . . . . . 102A.1. Linear regression table. . . . . . . . . . . . . . . . . . . . . . . . . . . 157A.2. ANOVA table. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
vii
List of Figures
4.1. Graphical representation of the error correlation matrix. . . . . . . . 58
4.2. Three-dimensional representation of the model error correlation for thedi�erent nodes at all times with a) node 1 at the �rst time, b) node 2at the �rst time, c) node 3 at the �rst time, d) node 4 at the �rst time,and e) node 5 at the �rst time. . . . . . . . . . . . . . . . . . . . . . 59
4.3. Two-dimensional representation of the model error correlation for thedi�erent nodes at all times with a) node 1 at the �rst time, b node 2at the �rst time, c) node 3 at the �rst time, d) node 4 at the �rst time,and e) node 5 at the �rst time. . . . . . . . . . . . . . . . . . . . . . 60
4.4. Prior concentration estimate variances from the two models. . . . . . 67
4.5. Prior and posterior concentration estimate variances from model 1. . 68
4.6. Prior and posterior concentration estimate variances from model 2. . 69
4.7. Posterior concentration estimate variances from the two models . . . 69
4.8. Comparison between the prior concentration estimate and a concen-tration realization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.9. Comparison between the posterior concentration estimate from model1 and a concentration realization. . . . . . . . . . . . . . . . . . . . . 71
4.10. Comparison between the posterior concentration estimate from model2 and a concentration realization. . . . . . . . . . . . . . . . . . . . . 72
5.1. Flowchart for the proposed methodology. . . . . . . . . . . . . . . . . 91
5.2. a) Problem set up, Kalman �lter mesh, and boundary conditions for ow (h is in ft). b) Stochastic simulation mesh and boundary condi-tions for transport. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.3. Sampling program test 1, 39 samples. . . . . . . . . . . . . . . . . . . 97
5.4. Total variance vs. number of samples for tests 1, 2, and 3. Samples 1-10. 98
5.5. Maximum variance vs. number of samples for tests 1, 2, and 3. Samples1-10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.6. Comparison of the observed plume and the plume estimates (logarith-mic scale). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.7. Sampling program test 2, 34 samples. . . . . . . . . . . . . . . . . . . 103
5.8. Sampling program test 3, 27 samples. . . . . . . . . . . . . . . . . . . 104
5.9. Total variance vs. number of samples for tests 1, 2, and 3. Samples10-20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.10. Total variance vs. number of samples for tests 1, 2, and 3. Samples20-40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
viii
5.11. Maximum variance vs. number of samples for tests 1, 2, and 3. Samples10-20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.12. Maximum variance vs. number of samples for tests 1, 2, and 3. Samples20-40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.13. Sampling program test 4, 39 samples. . . . . . . . . . . . . . . . . . . 1085.14. Total variance vs. number of samples for tests 1 and 4. Samples 10-20. 109
5.15. Total variance vs. number of samples for tests 1 and 4. Samples 20-40. 1095.16. Maximum variance vs. number of samples for tests 1 and 4. Samples
10-20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.17. Maximum variance vs. number of samples for tests 1 and 4. Samples20-40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.18. Sampling program test 5, 42 samples. . . . . . . . . . . . . . . . . . . 1115.19. Sampling program test 6, 46 samples. . . . . . . . . . . . . . . . . . . 111
5.20. Total variance vs. number of samples for tests 1, 5, and 6. Samples 0-10.1125.21. Total variance vs. number of samples for tests 1, 5, and 6. Samples
10-20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.22. Total variance vs. number of samples for tests 1, 5, and 6. Samples20-40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.23. Maximum variance vs. number of samples for tests 1, 5, and 6. Samples1-10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.24. Maximum variance vs. number of samples for tests 1, 5, and 6. Samples10-20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.25. Maximum variance vs. number of samples for tests 1, 5, and 6. Samples20-40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.26. Total residual and con�dence interval for test 1. Total residual is shownwith a dotted line and the con�dence limits with a continuous line. a)First twenty samples. b) Last twenty samples . . . . . . . . . . . . . 116
5.27. Total residual and con�dence interval for test 2. Total residual is shownwith a dotted line and the con�dence limits with a continuous line. a)First twenty samples. b) Last twenty samples . . . . . . . . . . . . . 116
5.28. Total residual and con�dence interval for test 3. Total residual is shownwith a dotted line and the con�dence limits with a continuous line. a)First twenty samples. b) Last twenty samples . . . . . . . . . . . . . 117
5.29. Total residual and con�dence interval for test 4. Total residual is shownwith a dotted line and the con�dence limits with a continuous line. a)First twenty samples. b) Last twenty samples . . . . . . . . . . . . . 117
5.30. Total residual and con�dence interval for test 5. Total residual is shownwith a dotted line and the con�dence limits with a continuous line. a)First twenty samples. b) Last twenty samples . . . . . . . . . . . . . 117
5.31. Total residual and con�dence interval for test 6. Total residual is shownwith a dotted line and the con�dence limits with a continuous line. a)First twenty samples. b) Last twenty samples . . . . . . . . . . . . . 118
ix
6.1. Comparison of tests 1, 2, and 3. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plumerealizations. Prior estimate. . . . . . . . . . . . . . . . . . . . . . . . 127
6.2. Comparison of tests 1, 2, and 3. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plumerealizations. Posterior estimates, 10 samples. . . . . . . . . . . . . . . 128
6.3. Comparison of tests 1, 2, and 3. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plumerealizations. Posterior estimates, 20 samples. . . . . . . . . . . . . . . 129
6.4. Comparison of tests 1, 2, and 3. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plumerealizations. Posterior estimates, 30 samples. . . . . . . . . . . . . . . 129
6.5. Comparison of tests 1, 2, and 3. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plumerealizations. Posterior estimates, 40 samples. . . . . . . . . . . . . . . 129
6.6. Comparison of tests 1, 2, and 3. a) Maximum variance vs. numberof plume realizations. b) Relative maximum variance di�erences vs.number of plume realizations. Prior estimate. . . . . . . . . . . . . . 130
6.7. Comparison of tests 1, 2, and 3. a) Maximum variance vs. numberof plume realizations. b) Relative maximum variance di�erences vs.number of plume realizations. Posterior estimates, 10 samples. . . . . 131
6.8. Comparison of tests 1, 2, and 3. a) Maximum variance vs. numberof plume realizations. b) Relative maximum variance di�erences vs.number of plume realizations. Posterior estimates, 20 samples. . . . . 131
6.9. Comparison of tests 1, 2, and 3. a) Maximum variance vs. numberof plume realizations. b) Relative maximum variance di�erences vs.number of plume realizations. Posterior estimates, 30 samples. . . . . 132
6.10. Comparison of tests 1, 2, and 3. a) Maximum variance vs. numberof plume realizations. b) Relative maximum variance di�erences vs.number of plume realizations. Posterior estimates, 40 samples. . . . . 133
6.11. Comparison of tests 1 and 4. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plumerealizations. Prior estimate. . . . . . . . . . . . . . . . . . . . . . . . 135
6.12. Comparison of tests 1 and 4. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plumerealizations. Posterior estimates, 10 samples. . . . . . . . . . . . . . . 136
6.13. Comparison of tests 1 and 4. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plumerealizations. Posterior estimates, 20 samples. . . . . . . . . . . . . . . 136
6.14. Comparison of tests 1 and 4. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plumerealizations. Posterior estimates, 30 samples. . . . . . . . . . . . . . . 136
x
6.15. Comparison of tests 1 and 4. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plumerealizations. Posterior estimates, 40 samples. . . . . . . . . . . . . . . 137
6.16. Comparison of tests 1 and 4. a) Maximum variance vs. number ofplume realizations. b) Relative maximum variance di�erences vs. num-ber of plume realizations. Prior estimates. . . . . . . . . . . . . . . . 138
6.17. Comparison of tests 1 and 4. a) Maximum variance vs. number ofplume realizations. b) Relative maximum variance di�erences vs. num-ber of plume realizations. Posterior estimates, 10 samples. . . . . . . 138
6.18. Comparison of tests 1 and 4. a) Maximum variance vs. number ofplume realizations. b) Relative maximum variance di�erences vs. num-ber of plume realizations. Posterior estimates, 20 samples. . . . . . . 138
6.19. Comparison of tests 1 and 4. a) Maximum variance vs. number ofplume realizations. b) Relative maximum variance di�erences vs. num-ber of plume realizations. Posterior estimates, 30 samples. . . . . . . 139
6.20. Comparison of tests 1 and 4. a) Maximum variance vs. number ofplume realizations. b) Relative maximum variance di�erences vs. num-ber of plume realizations. Posterior estimates, 40 samples. . . . . . . 139
6.21. Comparison of tests 1, 5, and 6. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plumerealizations. Prior estimate. . . . . . . . . . . . . . . . . . . . . . . . 140
6.22. Comparison of tests 1, 5, and 6. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plumerealizations. Posterior estimates, 10 samples. . . . . . . . . . . . . . . 141
6.23. Comparison of tests 1, 5, and 6. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plumerealizations. Posterior estimates, 20 samples. . . . . . . . . . . . . . . 141
6.24. Comparison of tests 1, 5, and 6. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plumerealizations. Posterior estimates, 30 samples. . . . . . . . . . . . . . . 141
6.25. Comparison of tests 1, 5, and 6. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plumerealizations. Posterior estimates, 40 samples. . . . . . . . . . . . . . . 142
6.26. Comparison of tests 1, 5, and 6. a) Total maximum vs. number ofplume realizations. b) Relative maximum variance di�erences vs. num-ber of plume realizations. Prior estimates. . . . . . . . . . . . . . . . 143
6.27. Comparison of tests 1, 5, and 6. a) Maximum variance vs. numberof plume realizations. b) Relative maximum variance di�erences vs.number of plume realizations. Posterior estimates, 10 samples. . . . . 143
6.28. Comparison of tests 1, 5, and 6. a) Maximum variance vs. numberof plume realizations. b) Relative maximum variance di�erences vs.number of plume realizations. Posterior estimates, 20 samples. . . . . 143
xi
6.29. Comparison of tests 1, 5, and 6. a) Maximum variance vs. numberof plume realizations. b) Relative maximum variance di�erences vs.number of plume realizations. Posterior estimates, 30 samples. . . . . 144
6.30. Comparison of tests 1, 5, and 6. a) Maximum variance vs. numberof plume realizations. b) Relative maximum variance di�erences vs.number of plume realizations. Posterior estimates, 40 samples. . . . . 144
A.1. Chloride concentration vs. time at well 110. . . . . . . . . . . . . . . 157A.2. Regression for concentration logarithm vs. time logarithm at well 110. 158
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1
Chapter 1
Introduction
1.1 Groundwater Contamination
Groundwater contamination was recently recognized as an environmental problem.
For a long time it was thought that the geologic layers between the earth's surface and
the water table protected groundwater. It was not until the 1970's that the public in
the United States gave widespread recognition to this problem. During these years
information about several contaminant episodes was published in the popular press.
The most publicized of them is known as Love Canal. There, an emergency was
declared in Niagara Falls, New York, to protect the health of the inhabitants of the
zone.
After these episodes came into the attention of the public federal and state leg-
islation was approved to protect groundwater from contamination sources and to
regulate the clean up of contaminated groundwater. The interest in groundwater
quality monitoring has increased signi�cantly driven by this legislation.
The most important type of groundwater contamination of concern in the United
States today originates from hazardous chemicals. Those chemicals are used in a
large variety of production activities. Contaminant sources can be classi�ed as point
and nonpoint sources. These two terms describe the degree of localization of the
contaminant source. A point source is characterized by a small-scale identi�able
source, such as a leaking storage tank, disposal ponds, or a sanitary land�ll. This
type of source usually gives rise to a well-de�ned contaminant plume. We call a
2
contaminant problem a nonpoint source problem if it has a large scale and origi-
nates from contamination emanating from several small sources whose locations are
poorly de�ned. Some examples of nonpoint contaminant problems are herbicides
or pesticides used in farming, nitrates from household disposal systems, salt used
in highways during the winter, and acid rain. In these cases typically there are
not well-de�ned plumes but a large contaminated groundwater region with highly
variable concentrations. More information about groundwater contamination prob-
lems can be found in the books by Domenico and Schwartz [25] and by Kavanaugh
et al. [42].
1.2 Groundwater Quality Sampling Networks
Loaiciga et al. [49] divide the objectives of groundwater quality monitoring programs
into: ambient monitoring, detection monitoring, compliance monitoring, and research
monitoring. They describe these objectives in the following way. Ambient monitor-
ing establishes an understanding of characteristic regional groundwater variations
over time. Detection monitoring has the primary function of identifying the pres-
ence of targeted contaminants when their concentrations exceed background or es-
tablished levels. Compliance monitoring is enforced to verify the progress and success
of groundwater clean up and remediation works in disposal facilities. Research mon-
itoring consists of groundwater quality sampling tailored to meet speci�c research
goals.
Using this classi�cation, the groundwater quality sampling designs presented in
this dissertation are within the objectives of compliance monitoring. These are local
scale designs that could be used in the clean up of contaminated aquifers from point
sources. In remediation investigations groundwater quality sampling networks can
3
have di�erent functions. In the �rst phase of the investigation, when the problem
is being characterized, a sampling network can be used to obtain an initial estimate
of the contaminant plume. After a remediation technique is chosen and carried out,
a groundwater quality sampling network can be used to verify that the remediation
goals are being met.
It has been found that remediation e�orts frequently can take a long time [42]. The
number of places that have been under remediation works in the USA for many years
is large. Much money has being expended already and will be expended in the future
to pay for the monitoring at these sites and other sites that may require remediation
in the future. A methodology for the design of e�cient groundwater quality sampling
networks could save much money under these circumstances. In response to this need
we have developed a methodology for the design of cost-e�ective groundwater-quality
sampling networks.
Often groundwater contaminant plumes do not reach steady state in a short time.
Consequently, it is natural to expect that the estimation of groundwater contaminant
concentrations in an e�cient way will depend on both the location of the sampling
wells and the times when the water samples are taken. Therefore, in the sampling de-
signs that we are considering sampling locations and sampling times are both decision
variables.
On the other hand, the e�ciency of the groundwater sampling network depends
also on the e�ciency of the estimation method used to process the data obtained
from it. In situations like those described above, in which a contaminated site has
been under investigation for some time, often a deterministic mathematical model has
been calibrated for analyzing possible solutions for the problem. So, a second purpose
of this dissertation is to propose an estimation method that allows us to incorporate
uncertainty into such a model while being capable of handling the type of space-time
correlation that the contaminant concentrations of a plume in motion have.
4
1.3 Dissertation Organization
The present dissertation is divided into two parts. Two chapters are papers that
will be send for publication in a professional journal and the remaining chapters that
give information either de�ning some terms used in the papers or containing results
supporting the conclusions obtained in the two papers. The two papers contain their
own bibliography list and the bibliography for the rest of the chapters can be found
in the section called Comprehensive Bibliography.
The �rst three chapters are introductory chapters. The present chapter introduces
the topic of the thesis. Chapter 2 contains basic statistical de�nitions that are used
throughout the dissertation. Chapter 3 presents a comprehensive literature review
on the topics of spatiotemporal estimation methods and spatiotemporal sampling
network design.
Chapter 4, chapter 5, and chapter 6 contain all the results of the present work.
Chapter 4 is the �rst manuscript intended for publication included in this dissertation.
There we analyze some methods that have been used in the past to estimate ground-
water contaminant concentrations. Based on this analysis, we propose a method to
estimate contaminant concentrations of a plume in motion. In chapter 5, the second
manuscript intended for publication in the dissertation, we evaluate the estimation
method proposed in the preceding chapter in the context of groundwater quality
sampling network design using some hypothetical examples. Chapter 6 contains an
analysis of the number of plume realizations needed in the stochastic simulation for
the method to reach convergence for the examples of chapter 5.
Chapter 7 contains the conclusions of the dissertation.
5
Chapter 2
Statistical De�nitions
In this chapter we introduce basic statistical de�nitions and results that are used in
the rest of the thesis. All the statistical methods used in this dissertation are based
on these basic concepts. This chapter can be used for clarifying terms not familiar for
the reader. We do not pretend to give a detailed presentation of each topic mentioned,
instead we give references in which more information about these topics can be found.
2.1 Probability Theory
Two books that describe the concept of probability in a beautiful way are Gne-
denko [32] and Papoulis [55]. We base our discussion of the concept of probability on
the ideas presented in these books.
Gnedenko gives the following de�nition of probability: The theory of probability is
the mathematical discipline that studies the laws governing random phenomena.
But, what are random phenomena? If an event occurs each time the set of con-
ditions G are realized, we call it a deterministic event. If the event may or may not
occur each time the set of conditions G realized we call it a random event. The ran-
dom event A is of interest if, when we repeat the realization of conditions G several
times, the occurrences of A do not deviate much from an average value. This average
value is used to characterize the random event A.
6
The objective of the theory is to study those averages. To this end, probabilities
are associated with the events of interest. A typical example of the way probabilities
can be assigned to events is the tossing of a coin. If a coin is ipped several times
and the frequency with which tails are obtained is registered, after a large number
of ips a frequency close to one half would be obtained. Then we can say that the
probability that a tail will be obtained when a coin is tossed is equal to 0:5. If we
de�ne the event A as the occurrence of a tail, we write
P (A) = 0:5:
Papoulis distinguishes three steps in any probabilistic investigation:
1. We �nd the probability P (A) of certain events A (for example using frequencies
as in the coin tossing experiment).
2. We assume that probabilities satisfy certain axioms, and by deductive reasoning
we determine from the probabilities P (A) of certain events A the probabilities
P (B) of other events B.
3. We make a physical prediction using the numbers P (B) determined in the
previous step.
The theory of probability deals with step 2. Steps 1 and 3 are the subject of statistics.
Here we will not explain the axioms on which the theory of probability is built,
these axioms are included in many standard probability books including Casella and
Berger [11], Gnedenko [32] and Papoulis [55].
2.2 Continuous Random Variables
With every random variableX, we associate a function called the distribution function
of X. The distribution function of the random variable X, denoted by FX(x), is
7
de�ned by
FX(x) = P (X � x);
for all x.
This function determines the properties of a random variable. Using FX(x) we can
compute the probability that X has to take values on any subset of the real line.
A random variableX is called a continuous random variable if there exists a density
function fX such that
FX(x) =Z x
�1fX(�)d�; �1 � x � 1:
A continuous random variable can take values in a continuous interval of the real line.
When a stochastic approach is used to model hydrology problems, most of the time
we use continuous random variables to represent hydrologic variables. This is because
variables like hydraulic conductivity, hydraulic heads and groundwater concentrations
can take any value in an interval of the real line, in this case any positive value. In
the following de�nitions we assume that all random variables are continuous.
The expected value or mean of a random variable X, denoted by EfXg, is
EfXg =Z 1
�1xfX(x)dx
if the integral or sum exists. The variance of a random variable X is VarX =
EfX � EfXgg2. The positive square root of the variance is called the standard
deviation of the random variable. The standard deviation is a measure of the degree
of spread of the distribution function of X about its mean.
Frequently the mean of a random variable is used to describe the central tendency
of a process. Its standard deviation or its variance is used as a measure of how much
the outcomes of the variable can di�er from this central tendency. Therefore, random
phenomena often are described in terms of their mean and their variance.
8
Gaussian variables are very important in statistics in general, and in hydrology
in particular. This is because many problems can be modeled using this type of
variable. A random variable is Gaussian, or equivalently, normally distributed if its
density function is given by
fX(x) =1p2��X
exp
"�1
2
�x�mX
�X
�2#;
where mX and �X are constant parameters. It is easy to show that mX = EfXgand that �2X = VarX. The two parameters mX and �2X characterize the Gaussian
density. We write X � N(mX ; �2X) to denote a random variable X that is Gaussian
with mean mX and variance �2X .
In this dissertation we model hydraulic conductivities as lognormal random vari-
ables. We say that Y is a lognormal random variable if Y = exp(X), where X is
normally distributed. If X � N(mX ; �2X), then the density function of Y is
fY (y) =1p
2��Xyexp
24�1
2
log y �mX
�X
!235 :
2.3 Jointly Distributed Random Variables
When solving hydrology problems usually we want to simultaneously estimate several
variables that have some kind of relationship one to the other. For example we could
be interested in estimating hydraulic conductivity at several locations in an aquifer.
If we model conductivity at each location as separate variables we would be dealing
with variables that have some relationship between them. For this kind of problem we
de�ne the concept of jointly distributed random variables. The continuous random
variables X1; : : : ; Xn are said to be jointly distributed if they are de�ned on the same
probability space.
9
For notational convenience we introduce the random vector X:
X =
0BBBBB@
X1
...
Xn
1CCCCCA ;
and the corresponding vector of values,
x =
0BBBBB@
x1...
xn
1CCCCCA :
The joint distribution function of X is
FX(x) = P (X1 � x1; : : : ; Xn � xn);
where P (X1 � x1; : : : ; Xn � xn) is the probability of the set fX1 � x1g\ : : :\fXn �xng. The joint density function of X, fX, satis�es
FX(x) =Z x1
�1� � �
Z xn
�1fX(�1; : : : ; �n)d�1; : : : ; d�n:
If we are interested only in the subset X1; : : : ; Xm of the variables X1; : : : ; Xn
(m < n), we can obtain their density function from that of the original set. Let
X = (X1; : : : ; Xn), Y = (X1; : : : ; Xm) and Z = (Xm+1; Xm+2; : : : ; Xn). The joint
density function of X can be written as fX(x) = fY;Z(y; z). The marginal density
function of the vector Z is de�ned as
fZ(z) =ZfY;Z(y; z)dy;
where dz is the volume element of <m and the integral is over <m.
A measure of the degree of relationship between random variables is given by the
covariance. The covariance of two variables Xk and Xl is de�ned by
Cov(Xk; Xl) = Ef(Xk � EfXkg)(Xl � EfXlg)g: (2.1)
10
Note that
Cov(Xk; Xk) = Var(Xk):
If the two variables Xk and Xl represent di�erent physical entities, sometimes 2.1 is
called the cross-covariance of the two variables.
If the covariance is normalized using the variance of each variable we obtain what
is known as the correlation coe�cient of Xk and Xl:
�(Xk; Xl) =Cov(Xk; Xl)
Var(Xk)Var(Xl):
If no relation exists between two random variables we call them independent. Two
jointly distributed random variables X1 and X2 are said to be independent if
Eff1(X1)f2(X2)g = Eff1(X1)gEff2(X2)g;
for all �xed functions f1, f2, provided these expectations exist. In a similar way, we
say that X1; : : : ; Xn are mutually independent if
Eff1(X1) � � �fn(Xn)g = Eff1(X1)g � � �Effn(Xn)g;
for all �xed functions f1; : : : ; fn. Two jointly distributed random variables are said
to be uncorrelated if their second moments are �nite and if
Cov(Xk; Xl) = 0:
When we put together a group of variables in a random vector, we can still talk
about its mean and its covariance. We generalize the de�nitions of these terms to
random vectors.
If X is a random vector, its expectation or mean vector is
EfXg =
0BBBBB@
EfX1g...
EfXng
1CCCCCA :
11
The covariance matrix of the random vector X is
PX = Ef(X� EfXg)(X� EfXg)T):
The de�nition of Gaussian variables can also be generalized to include jointly
distributed variables. The joint density function of n jointly normally distributed
random variables X1; : : : ; Xn is:
fX(x) =1
(2�)n=2jPxj1=2 exp��1
2(x� Efxg)TP�1
x(x� Efxg)
�:
We use
X � N(EfXg; PX)
to mean that the random vector X is Gaussian with mean vector EfXg and co-
variance matrix PX, where PX is a positive de�nite matrix. If the jointly normally
distributed random variables EfX1g; : : : ; EfXng are pairwise uncorrelated, then theyare independent.
2.4 Conditional Probabilities
Conditional probabilities will play an important role in this thesis. We will use them
to obtain estimates of the concentrations in a contaminant plume from a mathematical
model and concentration data.
Given two events A and B, we de�ne the conditional probability function P (AjB)of event A given event B by
P (AjB) = P (A \ B)
P (B); (P (B) > 0):
The conditional density function fXjY(xjy) of X given Y = y is de�ned by
fXjY(xjy) = fX;Y(x;y)
fY(y)=
fX;Y(x;y)RfX;Y(x;y)dx
:
12
The conditional expectation of the random vector X given the random vector Y is
de�ned by
EfXjYg =ZxfXjY(xjy)dx:
The conditional covariance matrix is
PXjY = Ef(X� EfXjYg)(X� EfXjYg)TjYg:
2.5 Random Samples and Estimation
Usually when a problem is under investigation, a series of experiments can be per-
formed so that the outcome of one experiment does not in uence the outcome of any
other experiment. Then the outcomes of the experiments can be analyzed using the
concept of a random sample.
The random variables X1; : : : ; Xn are called a random sample of size n from the
population f(x) if X1; : : : ; Xn are mutually independent random variables with the
same probability density function f(x).
If we are interested in estimating the mean value of the outcomes of the experiments
describe above, we can use the sample mean. The sample mean, denoted X, is the
arithmetic average of the values in a random sample,
X =1
n
nXi=1
Xi:
In a similar way we can estimate the variance of the outcomes using the sample
variance. The sample variance is de�ned by
S2X =
1
n� 1
nXi=1
�Xi �X
�2:
The standard deviation is estimated by the sample standard deviation. The sample
standard deviation, SX , is the positive square root of S2X .
13
These estimates have the following property. If X1; : : : ; Xn is a random sample
from a population with mean mX and variance �2X , then
EfXg = mX
and
EfS2Xg = �2X :
We say that the sample mean is an unbiased estimator of mX and the sample variance
is an unbiased estimator of �2X .
It is also possible to estimate the covariance and the correlation of jointly dis-
tributed variables using random samples. Let X1; : : : ; Xn and Y1; : : : ; Yn be random
samples from a joint distribution fX;Y with covariance Cov(X; Y ). Then the covari-
ance samplePn
1 (Xi�X)(Yi�Y )=(n�1) is an unbiased estimate of Cov(X; Y ). While
the correlation samplePn
1 (Xi � X)(Yi � Y )=(SXSY ) is an unbiased estimate of the
correlation coe�cient �(X; Y ).
2.6 Stochastic Processes
In this dissertation we use two di�erent representations for groundwater contaminant
concentrations. The �rst one is a stochastic process and the second a random �eld.
In this section we de�ne the �rst one and in the next section we explain what we
understand by random �elds.
A vector stochastic process fXt; t 2 Tg is a family of random vectors indexed by
the set of real numbers T . An observation of the stochastic process fXt; t 2 Tgis called a realization of the process. The collection of all possible realizations is
called the ensemble of the stochastic process. A stochastic process is characterized
by specifying the joint density function f(X1; : : : ;Xn) for all �nite sets fti 2 Tg.
14
We say that the stochastic process has a discrete state space if the random vectors
Xt are discrete. If they are continuous we say that the stochastic process has a
continuous state space. If the parameter set T is discrete we say that the stochastic
process is a discrete parameter process; if it is continuous, we say that the stochastic
process is a continuous parameter process.
White Gaussian random sequences are used often to represent random errors. A
white Gaussian random sequence fXn; n = 1; 2; : : :g is a sequence for which all the
X's are mutually independent and each is normally distributed. The probability law
of a white Gaussian random sequence is completely determined by the mean vectors
EfXng and the covariance matrices Pn = Ef(Xn�EfXng)(Xn�EfXng)Tg, n � 1.
This is because the X's are mutually independent, i.e.,
Ef(Xn � EfXng)(Xm � EfXmg)gT) = 0; if n 6= m:
A time series is a sequence of observations ordered in time. Time series are an-
alyzed using stochastic processes that emphasize the correlation between variables
depending on their distance on time. Some common methods used to model and
estimate a time series are regression analysis, ARIMA methods and spectral analysis
methods [2].
2.7 Random Fields
The book of Christakos [13] explains the concept of random �elds and its applications
in earth sciences. We follow his terminology in this section. A spatial random �eld
fX(s); s 2 � <ng is a family of random variables (or random vectors) with param-
eter s from <n. We say that it is a continuous parameter random �eld or discrete
parameter random �eld according to whether the parameter s is continuous or dis-
15
crete. As with stochastic processes, the complete characterization of a random �eld
can be obtained from its joint probability density functions to an arbitrary order.
The mean of the random �eld X(s) is a function of s:
mX(s) = EfX(s)g:
If the random �eld is scalar, the mean is a number, if it is a vector random �eld the
mean is a vector.
The covariance function of a scalar random �eld X(s) is
cX(s; s0) = Ef(X(s)�mX(s))(X(s0)�mX(s
0))g;
and its correlation function is,
�(s; s0) =cX(s; s
0)
�X(s)�X(s0):
If the random �eld is a vector random �eld, its covariance matrix is
cX(s; s0) = Ef(X(s)�mX(s))(X(s0)�mX(s
0))gT;
The random �eld X(s) is said to be stationary in the wide sense (or homogeneous)
if it has �nite second moments, its mean is a constant, and its covariance depends
only on the distance between the two vectors. This is,
mX(s) = m
and
cX(s; s0) = cX(s� s0):
The random �eld is isotropic in the wide sense if it has a constant mean and its
covariance depends only on the length of the di�erence between the two vectors but
not on the direction of this di�erence vector. This is,
cX(s; s0) = cX(js� s0j):
16
The variogram is de�ned as the variance of the increment X(s+ r)�X(s). For a
stationary random �eld the variogram depends on the di�erence vector r but not on
the position vector s, i.e.
2 (r) = Var(X(s+ r)�X(s)):
The function (r) is called the semivariogram and it satis�es
(r) = cX(0)� cX(r); for all r:
For isotropic spatial random �elds two measures of the extent of spatial correlations
commonly used are the correlation scale and the correlation range. The correlation
scale �X is
�X =1
�2X
Z 1
0cX(r)dr:
The value of the covariance is approximately 50% of the variance value at the distance
r = �X . The range � is the distance for which the value of the correlation is 5% of
the value of the variance.
In chapters 4 and 5 we represent the hydraulic conductivity by a spatial random
�eld, stationary and isotropic and we use an exponential semivariogram to character-
ize it. We call a semivariogram an exponential semivariogram if it has the following
form:
(r) = �2X
�1� exp
��r
a
��;
where a is a constant. For this semivariogram �X = a and � = 3a.
We call X(s; t) a spatiotemporal random �eld if s 2 , where is a subset of the
Euclidean space <n and t 2 T , where T is a subset of the positive real line. A random
�eld X(s) is called an ergodic random �eld if the mean and/or covariance of X(s) are
the same as those of any single realization of the �eld. If the condition is satis�ed,
we talk about ergodicity in the mean and/or the covariance, respectively. One of the
17
representations that we use in chapter 5 for groundwater contaminant concentrations
is a nonergodic spatiotemporal random �eld.
We use a Kalman �lter in this dissertation to condition groundwater contaminant
concentration estimates on groundwater contaminant concentration data. We intro-
duce the formulas of the �lter in chapter 4. A complete derivation of the �lter can be
found in the book of Jazwinski [39]. Two useful books to learn about stochastic meth-
ods in hydrologic problems were written by Dagan [20] and by Gelhar [30]. The last
book contains examples on the application of perturbation methods to derive closed
form stochastic equations describing statistical moments of some hydrologic variables.
Perturbation methods have been used to obtain partial di�erential equations that de-
scribe the mean and variance of groundwater contaminant concentrations.
18
Chapter 3
Comprehensive Literature Review
In this section we present a comprehensive literature review. This review is divided
into two parts, the �rst presents work done in the area of spatiotemporal estimation
methods while the second presents work done in the topic of spatiotemporal ground-
water quality sampling network design. These reviews are taken from chapters 4 and
5 and included here to give unity to the thesis.
3.1 Spatiotemporal Estimation Methods
Contaminant concentration can be modeled as a random �eld that has correlations in
both time and space. Estimation methods often used in hydrology to estimate con-
taminant concentration are time series and geostatistical methods. These methods
consider variables that are either time correlated or space correlated. Practice has
shown that these methods are not adequate when working with variables which have
both time and space correlation [13]. In recent years, some researchers have used
space-time estimation methods to estimate di�erent hydrologic variables. In what
follows we give a brief summary of works in which either a generalization of tradi-
tional methodologies is used to process spatiotemporal data or stochastic methods
based on partial di�erential equations are applied to obtain groundwater concentra-
tion estimates conditioned on data.
19
3.1.1 Geostatistics
Rouhani et al. [5] describe geostatistics as a collection of techniques for the solution
of estimation problems involving spatial variables. Of the geostatistical methods the
most popular is kriging. Kriging methods were originally developed in the mining in-
dustry. Due to the kind of estimation problems that are of interest in that �eld these
methods are speci�cally designed to model spatial variability [5, 13, 17, 23, 37, 41].
Therefore, these methods use functions that describe only the spatial correlation
structure of the variables involved. Two functions commonly used for this purpose
are the variogram and the covariance functions. The function that represents the cor-
relation structure must be estimated from data before applying any kriging method.
Linear kriging methods are the most commonly used. These methods generate es-
timators by weighting the measurements with coe�cients obtained from the mini-
mization of the mean square error, subject to unbiased conditions [5, 37]. Stationary
and nonstationary variables can be estimated. Some common kriging methods are
simple kriging, ordinary kriging, and universal kriging. The �rst two are used to
solve stationary problems, and the last is used to solve nonstationary problems. The
estimation of groundwater contaminant concentration is typically a nonstationary
problem.
Some e�orts to analyze spatiotemporal data in areas di�erent from groundwater
hydrology were made by Bilonick [8], Egbert [26], Solow and Gorelick [69], and
Zeger [84]. All these works use kriging as the estimation method. Christakos [13]
presents a review of Earth Sciences works that deal with the analysis of spatiotem-
poral data. Research done to extend geostatistical methods to account for time
correlation in hydrology estimation problems are presented next.
A simple model for spatiotemporal processes was introduced by Stein [71]. In
this model the spatiotemporal random �eld is represented as the sum of a function
20
that depends only on space, a function that depends only on time, and a random
error depending on both space and time. It was assumed that the random error
spatial semivariogram was the same at all times and that errors at di�erent times
were independent from each other. Due to its simplicity, the method is easy to use.
Simultaneously, the lack of space-time cross terms in the process trend description
makes the method inapplicable in many problems of interest.
Rouhani and coworkers used two di�erent approaches to represent time variability
by kriging methods. In the �rst one [61], Rouhani and Hall extended universal kriging
to the time domain. They represented the space-time phenomena as a realization of
a random function in n + 1 dimensions, n spatial and one time dimension. The
drift of the random variable was represented by a polynomial with no space-time
cross terms, while the covariance was expressed as the sum of a spatial covariance
function and a temporal covariance function. They used this geostatistical method
to estimate a piezometric surface. The authors found that their method produced a
better spatial map for a given time using all the available data than using only the
data available for that time. The authors faced some challenges due to ill-conditioned
kriging matrices resulting from scale di�erences between spatial and temporal changes
in the data.
In the second work [62], Rouhani and Wackernagel combined time series and krig-
ing. They studied the space-time structure of a hydrological data set emphasizing the
temporal domain but accounting for space-time correlation. To do this, the authors
modeled the observed values at each measurement site as separate, but correlated,
time-series. Each time series was a random function composed of a sum of random
processes, each related to a speci�c temporal scale. This methodology was applied
to analyze monthly piezometric data in a basin south of Paris, France. Limitations
of this approach become evident. It is possible to forecast and estimate missing data
with this method but not to estimate data at an unsampled site. In addition, the
21
number of variograms and cross variograms that need to be estimated: m(m+ 1) for
m observation wells, is very large.
Rodr��guez-Iturbe and Mej��a [58] developed a methodology for the design of precip-
itation networks in time and space. They worked out sampling programs to estimate
two variables; the long-term mean areal rainfall value, and the mean area rainfall
value of a storm event. The authors modeled rainfall as a process with space-time
correlation, and expressed the space-time covariance function as the product of a
temporal covariance function and a spatial covariance function.
Christakos and Raghu [14] applied a method developed by Christakos in a pre-
vious work [12] to the study of groundwater quality data using space-time random
�elds. This method is based on spatiotemporal random �elds, and can be applied
to data with space nonhomogeneous and time nonstationary correlation structures.
It provides estimates that are optimal in the mean-square-error sense. Due to the
data characteristics, the authors modeled spatiotemporal continuity at a local scale
and represented the covariance parameters as space-time distributed variables. They
decompose the covariance matrices into a space homogeneous/time stationary part
plus a polynomial in time and space. The objective in this work was to estimate the
covariance parameters based on concentration data and then to estimate or predict
concentrations at locations and times with no data. In the case study considered, the
water quality data consisted of spring sodium ion (Na+) concentration measurements
from the Dyle river basin upstream.
Research analyzing the identi�cation of temporal change in the spatial correlation
of groundwater contamination was presented by Shafer et al. [64]. They use the
jackknife method to estimate the semivariogram of groundwater quality data and
its approximate con�dence limits. Their approach was demonstrated using nitrate-
nitrogen concentrations in samples of shallow groundwater.
22
3.1.2 Stochastic Methods Based on PDEs
A second way to proceed is to model the contaminant transport problem using a
stochastic equation. Here the random sources are addressed explicitly in the stochastic
equation. Equation parameters, boundary and initial conditions, and an extra term
called the model error can be random variables. Random measurement errors can be
considered as well. The books by Dagan [20] and by Gelhar [30] explain this approach
in some detail.
During the last twenty years a great deal of development on the theoretical as-
pects of stochastic modeling of groundwater contamination has occurred. A driving
force for these e�orts has been the attempt to develop a theory capable of reproduc-
ing the kind of contaminant dispersion that has been observed in groundwater �eld
problems and that is not captured by traditional mathematical models. Commonly,
two di�erent approaches have been taken, the Lagrangian approach and the Eulerian
approach. The �rst one regards the solute body as a collection of particles and the
transport process is represented in terms of its random trajectories. In this framework
the ensemble spatial moments are generally analyzed. Some early works using this
viewpoint are those of Smith and Schwartz from a numerical perspective [66{68] and
those of Dagan from an analytical one [18, 19].
In the Eulerian approach a solute transport equation in Eulerian coordinates is
postulated and some of its parameters, commonly the velocity, are regarded as ran-
dom. From this equation the ensemble moments of interest are obtained using some
approximations, for example the contaminant concentration ensemble mean and vari-
ance or some ensemble spatial moments. Some pioneering works using this approach
were done by Gelhar [31] using spectral methods and by Tang and Pinder [73, 74]
using perturbation methods.
Some authors have analyzed the e�ects of conditioning the estimates obtained
23
using these approaches with hydraulic conductivity data and/or hydraulic head data.
For the Lagrangian approach some examples are the works of Dagan [18], Ezzedine
and Rubin [27], Rubin [63], and Smith and Schwartz [67]. For the Eulerian approach
the work of Graham and McLaughlin [34].
A di�erent approach was adopted by Neuman and coworkers. Neuman developed
a Eulerian-Lagrangian theory on transport conditioned on hydraulic data [53]. The
theory yields a transport equation, with the dispersive ux given exactly in terms of
conditional Lagrangian kernels. Also, an approximate expression is obtained for the
space-time concentration covariance function. Numerical developments to implement
Neuman's theory and his analyses of conditioning di�erent contaminant estimates on
log-transmissivity and/or head data are presented in a series of four papers by Zhang
and Neuman [85{88].
More relevant for our study are those works that have conditioned transport esti-
mates with contaminant concentration data. We present �rst the works that develop
estimation methods in a Eulerian framework.
In a series of two papers Graham and McLaughlin developed a stochastic de-
scription of transient solute plumes [33, 34]. In the �rst paper they derived, using
perturbation techniques, a system of coupled partial di�erential equations that de-
scribes propagation of the ensemble mean concentration, the velocity concentration
cross covariance, and concentration covariance from the conservative solute transport
equation. The only source of uncertainty that they addressed was steady-state ve-
locity variability. They compared the contaminant concentration mean and variance
from their model with the same estimates obtained from stochastic simulation and
the comparison was favorable.
In the second part of the work they develop the equations necessary to use an
extended Kalman �lter to condition these moments with measurements of hydraulic
conductivity, hydraulic head, and solute concentration. The method works sequen-
24
tially: prior moments are obtained, samples are taken from regions with predicted
high uncertainty, then moments are conditioned on new data and a new set of samples
is chosen using the predicted variance. In a later work [35] the authors applied this
methodology to a �eld problem (a tracer test). Only concentration data was used to
condition the velocity and contaminant concentration means. They concluded that
the conditional prediction uncertainty is underestimated by the stochastic model.
The authors suggest that this may be due to the lack of uncertainty in the initial and
boundary conditions of their model, or to the non-Gaussian behavior that they found
the conditional concentration residuals had.
A similar approach for estimating a groundwater contaminant plume was used by
McLaughlin et al. in a �eld application [52]. In this work the concentration covariance
matrix and the cross-covariance matrices between concentration and the hydrological
variables of interest are obtained from a nonstationary spectral method developed
by Li and McLaughlin [47]. Conditioning is done by a Bayesian method equivalent
to the static Kalman �lter explained in section 4.6.2. The method can be applied to
transient problems but in this application solute transport was modeled as stationary.
The sources of uncertainty considered were hydraulic conductivity and contaminant
concentration at the source. The contaminant concentration estimate was conditioned
on hydraulic conductivity data, head data and contaminant concentration data. It
was found that predicted errors were generally higher than expected, especially near
the edges of the contaminant plume. The authors argue that this may be due to the
in uence of recharge variability.
Dagan and Neuman [22] showed that the approximation used by Graham and
McLaughlin to derive the system of equations is inconsistent because terms neglected
are of the same order as terms retained. It is not clear in which cases this inconsistency
may give rise to serious errors.
Yu et al. [83] apply a linear Kalman �lter to estimate groundwater solute con-
25
centration. Concentration is modeled using a two-dimensional advective dispersive
transport equation. The idea of the work is to improve the concentration estimate
obtained from a numerical solution of the transport equation conditioning it with
data through a Kalman �lter. The authors give an example in which the Galerkin
�nite element method is used to discretize the equation. The model error considered
is due to numerical approximations. A problem with known solution is solved. The
data for the estimation procedure are obtained from the analytical solution. The
model error covariance matrix was determined from the di�erences between the �nite
element model values and the analytical solution. A comparison between the analyt-
ical solution, the numerical solution and the Kalman �lter procedure solution showed
that signi�cant improvement can be accomplished by using the suggested algorithm.
Zou and Parr [89] used a methodology very similar to the one explained above.
They give an example in which the explicit �nite di�erence method is used to ob-
tain the state equation of the system. In this example measurement and numerical
models errors are the uncertainty sources; the authors argue that parameter uncer-
tainty can be accounted for indirectly by the model error term. A problem with
known solution is solved. The "data" for the estimation procedure are obtained
from a second numerical solution, in this case from the MOC method. The model
error covariance matrix was determined from the di�erences between the �nite dif-
ference model values and the analytical solutions. The measurement error covari-
ance matrix was obtained from the di�erence between the MOC model values and
the analytical solutions. A comparison between the analytical solution, the numer-
ical solution, and the Kalman �lter solution showed that signi�cant improvement
can be accomplished by combining the two numerical solutions through a Kalman
�lter.
A method based on the extended Kalman �lter was developed by Jinno et al. [40]
to predict the contaminant concentration of groundwater pollutants. A stochastic
26
one-dimensional convection-dispersion equation is used to model the transport prob-
lem. The only random term included in the equation is the model error and it is
represented as a white Gaussian random sequence. This partial di�erential equation
is transformed into an ordinary di�erential equation expanding the concentration and
the model error using Fourier series. A discrete system is obtained by approximating
the time derivative using �nite di�erences. The Fourier coe�cients for the concen-
tration expansion and the transport equation parameters are estimated using an ex-
tended Kalman �lter. Two synthetic examples were presented. In these examples the
data used to obtain the estimates were contaminant concentrations measurements,
and measurements on the parameters of the equation. Each observation contained
some measurement error. The e�ects on the accuracy of the estimates of the sampling
frequency was analyzed.
A work that deserves special attention is that of Loaiciga [48] because his approach
is similar to ours. This author combines kriging with a stochastic transport equation
to estimate pollutant concentrations. To estimate the concentration at a spatiotem-
poral point in which a sample is not available the author uses a linear estimate
employing contaminant concentration data at all spatiotemporal points. Assuming
that parameters are known Loaiciga derives the elements of the concentration covari-
ance matrix from the advection-dispersion equation governing mass transport. The
only source of uncertainty that he considers when obtaining the covariance matrix
is the model error. The concentration mean at each point must be known to sat-
isfy the unbiased condition. These values are again calculated from the transport
equation.
A deterministic equation that describes the mean solute concentration is obtained
by perturbing the solute concentration and the seepage velocity. In section 4.5 it will
be shown that when one assumes the transport model errors at di�erent times are
independent, as Loaiciga does in his work, important concentration correlations are
27
disregarded. When proceeding in this way information contained in a contaminant
sample about contaminant concentration at other space-time locations is overlooked.
For this reason we think that Loaiciga's method suggests that more samples than
are actually necessary are required to obtain a concentration estimate with a given
degree of certainty.
Using the Lagrangian approach Dagan et al. [21] study the impact of concentra-
tion measurements upon estimation of ow and transport parameters. The authors
analyze a simple case in which a solute body of constant concentration is injected
into an aquifer and the objective is to obtain the statistical distribution of the solute
concentration. The e�ect of pore-scale dispersion is neglected, such that concentra-
tion stays constant and the volume of the solute body is preserved. An expression is
obtained for the conditional mean and the conditional variance of a generic variable
when conditioned on concentration data. The conditional mean obtained is the same
as the one obtained by cokriging. But, in contrast to the conditional variance in
cokriging, the variance obtained in this paper di�ers depending on the value of the
measured concentration. The conditional mean and variance of log-transmissivity
and of the plume centriod are analized. Contaminant concentration conditioned on
contaminant concentration data is not analyzed.
A work by Neuman et al. [54] contains an example of the application of Neuman's
theory in the estimation of vertically averaged concentration of an inorganic solute
from a tracer experiment using concentration data. In this example the tracer con-
centration is estimated at a given time using a subset of the data available at that
time and it is compared with the actual plume determined on the basis of all avail-
able data. As was mentioned before, an expression for a spatiotemporal contaminant
concentration covariance matrix is obtained from Neuman's theory. We are not aware
of any paper where this covariance matrix has been used to estimate concentrations
of a moving contaminant plume using concentration data sampled at di�erent times.
28
3.2 Spatiotemporal Sampling Design
There are many works in which the problem of groundwater quality sampling network
design is analyzed assuming either that the sampling times have been preselected
or that the contaminant concentration has reached steady state. In these works all
sampling decisions involve only space but not time [4,9,28,29,38,51,52,59,60,75,78,79].
Loaiciga et al. [49] and McGrath [50] present an extensive review of works dealing
with these kinds of sampling designs. In what follows we review works in which
sampling network designs use decision variables that depend on space and time.
3.2.1 Sampling Network Design and Deterministic Modeling
When the transport equation is used to describe the evolution of a contaminant
plume in a deterministic framework, the plume behavior is completely determined
by initial conditions, boundary conditions and the equation parameter values. Using
the transport equation to model a speci�c problem requires that these conditions
and values be chosen using site information. Initial and boundary conditions can be
�gured out from historical information and the hydrogeological characteristics of the
site under investigation. Frequently the velocity parameter of the transport equation
is obtained from the ow equation and other parameters from a model calibration
process. A second way to obtain these parameters is using solute concentration data
when solving what is called the inverse problem [72, 82]. Once the parameters are
speci�ed contaminant plume predictions are obtained solving the equation.
Spatiotemporal sampling network design for parameter estimation of a determin-
istic model has been a subject of recent research. Three papers that propose this kind
of network design are those of Knopman et al. [46], and Cleveland and Yeh [15, 16]
(see Table 3.1). In these works parameter estimation is done within a stochastic
framework but the parameters are assumed to be deterministic.
29
Knopman and Voss [43] analyzed the spatiotemporal behavior of sensitivities for
parameters of one-dimensional advection-dispersion equations when parameters are
estimated from a regression model. The use of an equation with a closed form solu-
tion allowed them to calculate sensitivities from exact derivatives. They found that
sampling at points in space and time with high sensitivity to a parameter yield ac-
curate estimations for that parameter, but designs that minimize the variance of one
parameter may not minimize the variance of other parameters. Therefore, they sug-
gest applying a multiobjective approach when optimal sampling designs are proposed.
This analysis was extended to parameters associated with �rst order chemical-decay,
boundary conditions, initial conditions, and multilayer transport [44].
In a later paper their results were the basis for developing a multiobjective sampling
design for parameter estimation and model discrimination [45]. Model discrimination
implies working with more than one transport model when �tting the data; the au-
thors obtained parameter estimation for all the models simultaneously. They used
a composite D-optimal objective function with the idea of maximizing information
for each set of parameters; they measure information by a function of the sensitivity
matrices. Knopman et al. [46] tested the design using bromide concentration data
collected during the Cape Cod, Massachusetts, natural gradient test. Designs consist
of the downstream distances of rows of fully screened wells oriented perpendicular to
the groundwater ow direction and the timing of sampling to be carried out on each
row. Characteristics of this paper are summarized in Table 3.1, it was chosen as a
representative element of this set of works.
Cleveland and Yeh [15, 16] (see Table 3.1) use a maximal information criterion
to select between di�erent designs. Information is measured by a weighted sum of
squared sensitivities, this criterion was chosen after the Knopman and Voss results.
The authors develop the sampling methodology under the assumption that once sam-
pling has begun at a site it continues until the end of the experiment. The examples
30
Table 3.1: Sampling network design for parameter estimation. KD9101 [46],CT9001 [15], CT9101 [16], v velocity, K conductivity, T transmissivity, S storage co-e�cient, R retardation factor, �L longitudinal dispersivity, �T transverse dispersivity,ne e�ective porosity, � decay parameter, Ca input source strength, C0 dimensionlessinitial concentration, c concentration.
KD9101 CT9001 CT9101
Objective
function
D-optimal Weighted
information
matrix trace
Weighted
information
matrix trace
Dimensions
transport eq.
1 2 2
Aquiferlayers
1 or 2 1 1
Sampling
dimensions
2 2 1
Parameters
estimated
v, �L, Ca,
C0, �
K, S, ne, �L,
�T , R
T , S, ne, �L,
�T , R
Estimation
method
Gauss-Newton
nonlinear
regression
Least-squares Least-squares
Kind of data c c c
presented assume that prior estimates of the parameters are available, the authors
suggest that a sequential approach design can be used to update estimates. In the �rst
work two dimensions are considered, one in the direction of the ow (horizontal) and
the second is depth (vertical); possible sampling locations vary in those directions. In
the second work the transport equation that describes the tracer concentration does
not consider changes in the vertical direction. The total experimental duration is
divided into several stages and a decision is made at the beginning of each stage. The
addition of only one sampling location at a time is considered. Sampling locations
31
are selected on the line that joins injection and extraction points.
3.2.2 Sampling Network Design and Stochastic Modeling
A second option to model a pollutant plume is within a stochastic framework. Two
types of stochastic methods used in hydrology are geostatistical methods and methods
based on partial di�erential equations. Estimates of contaminant concentrations can
be obtained through these methods using contaminant concentration measurements
or measurements of other variables correlated with contaminant concentrations, as
are hydraulic heads and hydraulic conductivities. When a stochastic model is used, on
top of obtaining an estimate for the groundwater pollutant concentrations we get the
uncertainty associated with the estimate. Next we summarize some works that deal
with the problem of spatiotemporal sampling network design for di�erent hydrologic
variables within a stochastic context.
Geostatistical estimation
In chapter 4 we described some works that propose extensions of geostatistical meth-
ods, created to deal exclusively with space variability, to include time variability; here
we are interested in describing the applications of these methodologies in hydrologic
sampling design problems.
Rodr��guez-Iturbe and Mej��a [58] developed a methodology for the design of pre-
cipitation networks in time and space. They worked out sampling programs for two
variables; the long-term mean areal rainfall value, and the mean area rainfall value
of a storm event. For both variables they analyzed two di�erent sampling programs:
simple random sampling, where each station is located with a uniform probability dis-
tribution over the whole space; strati�ed random sampling, where the area is divided
into many non overlapping subareas and k sampling points are chosen randomly in
32
each subarea. The authors estimated the rainfall process using a generalized geo-
statistical method explained in chapter 4. They discuss trade o�s of time sampling
versus space sampling and conclude that in the design of rainfall networks it is im-
portant to consider spatial correlation and time correlation. A summary of the paper
is given in Table 3.2.
Loaiciga [48] (see Table 3.2) combines some elements of kriging with the transport
equation to estimate pollutant concentrations. For the details on the estimation
method see chapter 4. He proposes a spatiotemporal groundwater sampling network
design that involves two steps: parameter estimation, and network optimization.
For network optimization, the objective used by Loaiciga is to choose where and
when to sample to minimize the variance of the concentration estimate error subject
to budget constraints and unbiasedness. The determination of an optimal sampling
plan is posed as a mixed integer programming problem. The author applies the
methodology to �nd the optimal sampling program of a chloride plume distribution.
In this application a design of a sampling network that selects sampling locations and
sampling times was demonstrated. The objective was to minimize the variance of
concentration estimation error along the cells bordering a river that is in the region
at a given time. Surprisingly, the optimized sampling plan yielded a solution such
that each chosen sampling location had to be sampled during the entire sampling
period. Loaiciga attributed this result to the quasi-steady nature of the contaminant
plume. Our results from chapter 4 support the idea that this uninterrupted sampling
schedule may be a consequence of the time-uncorrelated model errors used by Loaiciga
when deriving the contaminant concentration covariance matrix.
Stochastic methods based on PDE's
When a stochastic transport equation is used in the modeling of a contaminant plume,
transport parameters, boundary and initial conditions can be random variables. So, in
33
Table 3.2: Sampling network design using geostatistical methods. RI7401 [58],LH8901 [48], c solute concentration, rf rainfall.
RI7401 LH89101
Objective
function
Error variance Error variance
Sampling
dimensions
2 2
Estimationmethod
Kriging Kriging
Kind of data rf c
Covariance
function
Product
factorization
Obtained
from transport
equation
stochastic modeling sources of uncertainty could be parameter variability, boundary or
initial conditions variability and measurement errors. The model does not describe a
single plume but a set of possible plumes. The characteristics of each possible plume
depend on the probability characteristics of the parameters and of the initial and
boundary conditions. When using this kind of description usually only the �rst two
moments of the pollutant concentration are estimated. If concentration, hydraulic
heads and/or hydraulic conductivity data are available, parameters and pollutant
concentration moments can be estimated using the equation and the data.
Three works are presented here in which sampling networks are designed for the
estimation of hydrologic variables using stochastic methods based on partial di�eren-
tial equations (see Table 3.3). Andricevic [3] and Yangxiao et al. [81] work with the
ow equation while Graham and McLauglin [34] work with the transport equation.
In Table 3.3 some di�erences can be appreciated in the way the coe�cients are rep-
resented and the data is used in these three works. An explanation of the di�erences
between Kalman �lters and extended Kalman �lters can be found in Jazwinski [39].
34
Table 3.3: Sampling network designs using stochastic methods based on PDEs.AR9301 [3], GW8902 [34], YZ9101 [81], v velocity, K conductivity, T transmissivity,Sy speci�c yield, S storage coe�cient, bc boundary conditions, f external uxes, cconcentration, h hydraulic head, w model error, Q model error covariance, Efg ex-pected value, Phh hydraulic head covariance, Pcc solute concentration covariance, Pvvvelocity covariance, Pcv concentration-velocity cross covariance, PhT hydraulic head-transmissivity cross covariance, Phf hydraulic head-external uxes cross covariance.
AR9301 GW8902 YZ9101
Objective To formulate
coupled
withdrawal and
sampling designs
for groundwater
supply models
To develop a
stochastic
description of
transient solute
plumes
To monitor
spatiotemporal
changes of
groundwater
head, caused by
groundwaterabstraction
Equation
dimensions
2 2 2
Aquifer
layers
1 1 2
Sampling
dimensions
2 2 2
Variables
estimated
Efhg,Phh, PhT , Phf
Efcg, Efvg,EflnKg,Pcc, Pvv , Pcv
Q, Ra,
S, Sy, T
Estimation
method
Kalman �lter Extended
Kalman �lter
Kalman �lter
Random
coe�cients
bc, f , T v w
Kind of data h c, h, K h
35
Andricevic proposes a coupled formulation of withdrawal and sampling designs for
groundwater supply models. He employs a sequential approach: the withdrawal de-
sign is conditioned on collected measurements, while the hydraulic head response on
the withdrawal design guides the future development of the sampling network. The
author describes the withdrawal design as a discrete time optimal control problem,
and he solves it by a loop stochastic control method. A random penalty-type additive
cost function is used as the objective function. The cost function is decomposed into
the deterministic and stochastic parts. A Bayesian framework is use for the mini-
mization of the deterministic part of this objective function. The sampling design's
objective is minimizing the uncertainty in the objective function of the groundwater
withdrawal program and to reduce the uncertainty in the measured variable.
The sampling criterion is expressed mathematically as the sensitivity of the ob-
jective function stochastic part of the withdrawal design to the uncertainty in the
hydraulic head distribution multiplied by the variance of the hydraulic head. The
ow equation is employed to predict output uncertainty in hydraulic heads through
�rst and second moment analysis, and the Kalman �lter algorithm is used to condi-
tion these moments with data. The algorithm looks for the best locations to measure
hydraulic heads to minimize the sampling criterion. In the sequential design, mea-
surements are used to update the covariance matrix of the estimation error hydraulic
head, which in turn changes the objective function for the withdrawal design. Reduc-
tion in hydraulic conductivity, external uxes, and boundary condition uncertainties
due to the hydraulic head measurements are considered when the hydraulic head
covariance is updated.
Yangxiao et al. [81] combined the parameter estimation procedure proposed in
a previous work [76] with a network design problem. The objective of the sampling
design is to monitor spatiotemporal changes of groundwater heads, caused by ground-
water abstraction. The only uncertain term considered in the ow equation is the
36
model error. The authors propose to use a Kalman �lter to estimate ow equation
parameters (called deterministic parameters) and some parameters associated with
the model error covariance matrix (called stochastic parameters). The calibration is
performed for a period in which the statistical stationary conditions are met and in
which all the matrices required in the Kalman �lter algorithm are assumed �xed (they
do not change with time). The method estimates the parameters sequentially: �rst an
estimate of the covariance model error is proposed and the �rst calibration round of
deterministic parameters is performed, then these parameters are kept constant and
the �rst round of stochastic parameters is obtained. These two steps are repeated
several times until a preestablished error criterion for both sets of parameters is met.
Two �eld examples were analyzed. In both the sampling frequencies were kept
�xed and the network densities were minimized under the constraint of a given thresh-
old value for the standard deviation of the estimation error. Several alternatives were
analyzed and the best one was chosen by inspection. The authors discuss the relative
importance of spatial network density and sampling frequency relating them with the
response time of the system. They found that if the system reacts fast, the spatial
optimization of the network is important. If the system reacts slowly, both temporal
optimization and spatial optimization are important.
The model errors considered by Yangxiao et al. in this work are uncorrelated on
time. It is unknown how important are the time correlations of the model errors for
the ow equation. It could be expected that these correlations are not as important as
is shown in chapter 4 are for the transport equation because the ow solution usually
reaches steady state in a short period of time. This does not happen often with the
transport solutions.
Graham and McLaughlin developed a stochastic description of transient solute
plumes in a series of two papers [33,34] (see Table 3.3). Their work has consequences
for sampling network design because they make this description site speci�c combin-
37
ing stochastic equations and data. The method works sequentially: prior moments
are obtained, samples are taken from regions with predicted high uncertainty, then
moments are conditioned on new data, and a new set of samples is chosen using the
predicted variance. The number of samples chosen at each round is decided arbitrar-
ily. In a later work [35] the authors applied this methodology to a �eld problem (a
tracer test). Their main interest was to evaluate the performance of the stochastic
model and they did not provide a sampling design analysis.
In contrast with Graham and McLaughlin's approach in our method we chose
sampling locations and its sampling schedule for a period of time, in our method
there is not need to collect samples after a sampling desicion is made to keep going as
is needed in Graham and McLaughlin method. This makes possible to decide as part
of the process the number of samples to be taken at each time instead of deciding
this number arbitrarily. Also, we decide where to sample and when to sample based
on the reduction of the concentration estimate variance at all locations at all times
which does not necessarily coincides with the locations with greatest variance.
38
Chapter 4
Estimation of Plumes in Motion in an Eulerian
Framework. The Role of the Model Error
4.1 Introduction
It has been recognized that groundwater remediation frequently can take a long
time [15]. The number of places that have been under remediation in the USA for
many years is large. Much money has being expended already and will be expended
in the future to pay the monitoring work at these sites and other sites that may
require remediation in the future. A methodology for the design of e�cient ground-
water quality sampling networks could save a considerable amount of money in these
circumstances. In response to this need we have developed a methodology for the
design of cost-e�ective groundwater-quality sampling networks.
Often groundwater contaminant plumes do not reach steady state in a short time.
Consequently, it is natural to expect that the estimation of groundwater contaminant
concentrations in an e�cient way will depend on both the location of the sampling
wells and the times when the water samples are taken. Therefore, in the sampling de-
signs that we are considering sampling locations and sampling times are both decision
variables.
On the other hand, the e�ciency of the groundwater sampling network depends
also on the e�ciency of the estimation method used to process the data obtained
from it. As a �rst step to obtain cost-e�ective groundwater quality sampling designs
we analyze in this chapter some estimation methods that have been used in the
39
past within an Eulerian framework. We de�ne what we understand by the transport
equation model error and we study some of its statistical characteristics. Based on
our �ndings, we propose a method to estimate contaminant concentrations of a plume
in motion. In the next chapter we present some hypothetical examples to evaluate
the method in the context of groundwater quality sampling network design.
4.2 Spatiotemporal Estimation Methods
Contaminant concentration can be modeled as a random �eld that has correlations
in both time and space. Estimation methods often used in hydrology to estimate
contaminant concentration are time series and geostatistical methods. These meth-
ods consider variables that are either time correlated or space correlated. Practice
has shown that these methods are not adequate when working with variables which
have both time and space correlation [7]. In recent years, some researchers have used
space-time estimation methods to estimate di�erent hydrologic variables. In what
follows we give a brief summary of works in which either a generalization of tradi-
tional methodologies is used to process spatiotemporal data or stochastic methods
based on partial di�erential equations are applied to obtain groundwater concentra-
tion estimates conditioned on data.
4.2.1 Geostatistics
Rouhani et al. [2] describe geostatistics as a collection of techniques for the solution
of estimation problems involving spatial variables. Of the geostatistical methods the
most popular is kriging. Kriging methods were originally developed in the mining in-
dustry. Due to the kind of estimation problems that are of interest in that �eld these
40
methods are speci�cally designed to model spatial variability [2,7,9,16,24,27]. There-
fore, these methods use functions that describe only the spatial correlation structure
of the variables involved. Two functions commonly used for this purpose are the
variogram and the covariance functions. The function that represents the correlation
structure must be estimated from data before applying any kriging method. Linear
kriging methods are the most commonly used. These methods generate estimators by
weighting the measurements with coe�cients obtained from the minimization of the
mean square error, subject to unbiased conditions [2,24]. Stationary and nonstation-
ary variables can be estimated. Some common kriging methods are simple kriging,
ordinary kriging, and universal kriging. The �rst two are used to solve stationary
problems, and the last is used to solve nonstationary problems. The estimation of
groundwater contaminant concentration is typically a nonstationary problem.
Some e�orts to analyze spatiotemporal data in areas di�erent from groundwater
hydrology were made by Bilonick [4], Egbert [17], Solow and Gorelick [43], and
Zeger [49]. All these works use kriging as the estimation method. Christakos [7]
presents a review of Earth Sciences works that deal with the analysis of spatiotem-
poral data. Research done to extend geostatistical methods to account for time
correlation in hydrology estimation problems are presented next.
A simple model for spatiotemporal processes was introduced by Stein [44]. In
this model the spatiotemporal random �eld is represented as the sum of a function
that depends only on space, a function that depends only on time, and a random
error depending on both space and time. It was assumed that the random error
spatial semivariogram was the same at all times and that errors at di�erent times
were independent from each other. Due to its simplicity, the method is easy to use.
Simultaneously, the lack of space-time cross terms in the process trend description
makes the method inapplicable in many problems of interest.
Rouhani and coworkers used two di�erent approaches to represent time variability
41
by kriging methods. In the �rst one [35], Rouhani and Hall extended universal kriging
to the time domain. They represented the space-time phenomena as a realization of a
random function in n+1 dimensions, n spatial and one time dimension. The drift of
the random variable was represented by a polynomial with no space-time cross terms,
while the covariance was expressed as the sum of a spatial covariance function and
a temporal covariance function. They used this geostatistical method to estimate a
piezometric surface. The authors found that their method produced a better spatial
map for a given time using all the available data than using only the data available for
that time. The authors faced some challenges due to ill-conditioned kriging matrices
resulting from scale di�erences between spatial and temporal changes in the data.
In the second work [36], Rouhani and Wackernagel combined time series and krig-
ing. They studied the space-time structure of a hydrological data set emphasizing the
temporal domain but accounting for space-time correlation. To do this, the authors
modeled the observed values at each measurement site as separate, but correlated,
time-series. Each time series was represented as a random function composed of a
sum of random processes, each related to a speci�c temporal scale. This methodology
was applied to analyze monthly piezometric data in a basin south of Paris, France.
Limitations of this approach become evident. It is possible to forecast and estimate
missing data with this method but not to estimate data at an unsampled site. In
addition, the number of variograms and cross variograms that need to be estimated:
m(m + 1) for m observation wells, is very large.
Rodr��guez-Iturbe and Mej��a [34] developed a methodology for the design of precip-
itation networks in time and space. They worked out sampling programs to estimate
two variables; the long-term mean areal rainfall value, and the mean area rainfall
value of a storm event. The authors modeled rainfall as a process with space-time
correlation, and expressed the space-time covariance function as the product of a
temporal covariance function and a spatial covariance function.
42
Christakos and Raghu [8] applied a method developed by Christakos in a pre-
vious work [6] to the study of groundwater quality data using space-time random
�elds. This method is based on spatiotemporal random �elds, and can be applied
to data with space nonhomogeneous and time nonstationary correlation structures.
It provides estimates that are optimal in the mean-square-error sense. Due to the
data characteristics, the authors modeled spatiotemporal continuity at a local scale
and represented the covariance parameters as space-time distributed variables. They
decompose the covariance matrices into a space homogeneous/time stationary part
plus a polynomial in time and space. The objective in this work was to estimate the
covariance parameters based on concentration data and then to estimate or predict
concentrations at locations and times with no data. In the case study considered, the
water quality data consisted of spring sodium ion (Na+) concentration measurements
from the Dyle river basin upstream.
Research analyzing the identi�cation of temporal change in the spatial correlation
of groundwater contamination was presented by Shafer et al. [38]. They use the
jackknife method to estimate the semivariogram of groundwater quality data and
its approximate con�dence limits. Their approach was demonstrated using nitrate-
nitrogen concentrations in samples of shallow groundwater.
4.2.2 Stochastic Methods Based on PDEs
A second way to proceed is to model the contaminant transport problem using a
stochastic equation. Here the random sources are addressed explicitly in the stochastic
equation. Equation parameters, boundary and initial conditions, and an extra term
called the model error can be random variables. Random measurement errors can be
considered as well. The books by Dagan [12] and by Gelhar [20] explain this approach
in some detail.
43
During the last twenty years a great deal of development on the theoretical as-
pects of stochastic modeling of groundwater contamination has occurred. A driving
force for these e�orts has been the attempt to develop a theory capable of repro-
ducing the kind of contaminant dispersion that has been observed in groundwater
�eld problems and that is not captured by traditional mathematical models. Com-
monly, two di�erent approaches have been taken, the Lagrangian approach and the
Eulerian approach. The �rst one regards the solute body as a collection of particles
and the transport process is represented in terms of its random trajectories. In this
framework the ensemble spatial moments are generally analyzed. Some early works
using this viewpoint are those of Smith and Schwartz from a numerical perspective
( [40], [41], [42]) and those of Dagan from an analytical one ( [10], [11]).
In the Eulerian approach a solute transport equation in Eulerian coordinates is
postulated and some of its parameters, commonly the velocity, are regarded as ran-
dom. From this equation the ensemble moments of interest are obtained using some
approximations, for example the contaminant concentration ensemble mean and vari-
ance or some ensemble spatial moments. Some pioneering works using this approach
were done by Gelhar [19] using spectral methods and by Tang and Pinder [45, 46]
using perturbation methods.
Some authors have analyzed the e�ects of conditioning the estimates obtained
using these approaches with hydraulic conductivity data and/or hydraulic head data.
For the Lagrangian approach some examples are the works of Dagan [10], Ezzedine
and Rubin [18], Rubin [37], and Smith and Schwartz [41]. For the Eulerian approach
the work of Graham and McLaughlin [22].
A di�erent approach was adopted by Neuman and coworkers. Neuman developed
a Eulerian-Lagrangian theory on transport conditioned on hydraulic data [31]. The
theory yields a transport equation, with the dispersive ux given exactly in terms of
conditional Lagrangian kernels. Also, an approximate expression is obtained for the
44
space-time concentration covariance function. Numerical developments to implement
Neuman's theory and his analyses of conditioning di�erent contaminant estimates on
log-transmissivity and/or head data are presented in a series of four papers by Zhang
and Neuman ( [50], [51], [52], and [53]).
More relevant for our study are those works that have conditioned transport esti-
mates with contaminant concentration data. We present �rst the works that develop
estimation methods in a Eulerian framework.
In a series of two papers Graham and McLaughlin developed a stochastic de-
scription of transient solute plumes [21, 22]. In the �rst paper they derived, using
perturbation techniques, a system of coupled partial di�erential equations that de-
scribes propagation of the ensemble mean concentration, the velocity concentration
cross covariance, and concentration covariance from the conservative solute transport
equation. The only source of uncertainty that they addressed was steady-state ve-
locity variability. They compared the contaminant concentration mean and variance
from their model with the same estimates obtained from stochastic simulation and
the comparison was favorable.
In the second part of the work they develop the equations necessary to use an
extended Kalman �lter to condition these moments with measurements of hydraulic
conductivity, hydraulic head, and solute concentration. The method works sequen-
tially: prior moments are obtained, samples are taken from regions with predicted
high uncertainty, then moments are conditioned on new data and a new set of samples
is chosen using the predicted variance. In a later work [23] the authors applied this
methodology to a �eld problem (a tracer test). Only concentration data was used to
condition the velocity and contaminant concentration means. They concluded that
the conditional prediction uncertainty is underestimated by the stochastic model.
The authors suggest that this may be due to the lack of uncertainty in the initial and
boundary conditions of their model, or to the non-Gaussian behavior that they found
45
the conditional concentration residuals had.
A similar approach for estimating a groundwater contaminant plume was used by
McLaughlin et al. in a �eld application [30]. In this work the concentration covariance
matrix and the cross-covariance matrices between concentration and the hydrological
variables of interest are obtained from a nonstationary spectral method developed
by Li and McLaughlin [28]. Conditioning is done by a Bayesian method equivalent
to the static Kalman �lter explained in section 4.6.2. The method can be applied to
transient problems but in this application solute transport was modeled as stationary.
The sources of uncertainty considered were hydraulic conductivity and contaminant
concentration at the source. The contaminant concentration estimate was conditioned
on hydraulic conductivity data, head data and contaminant concentration data. It
was found that predicted errors were generally higher than expected, especially near
the edges of the contaminant plume. The authors argue that this may be due to the
in uence of recharge variability.
Dagan and Neuman [14] showed that the approximation used by Graham and
McLaughlin to derive the system of equations is inconsistent because terms neglected
are of the same order as terms retained. It is not clear in which cases this inconsistency
may give rise to serious errors.
Yu et al. [48] apply a linear Kalman �lter to estimate groundwater solute con-
centration. Concentration is modeled using a two-dimensional advective dispersive
transport equation. The idea of the work is to improve the concentration estimate
obtained from a numerical solution of the transport equation conditioning it with
data through a Kalman �lter. The authors give an example in which the Galerkin
�nite element method is used to discretize the equation. The model error considered
is due to numerical approximations. A problem with known solution is solved. The
data for the estimation procedure are obtained from the analytical solution. The
model error covariance matrix was determined from the di�erences between the �nite
46
element model values and the analytical solution. A comparison between the analyt-
ical solution, the numerical solution and the Kalman �lter procedure solution showed
that signi�cant improvement can be accomplished by using the suggested algorithm.
Zou and Parr [54] used a methodology very similar to the one explained above.
They give an example in which the explicit �nite di�erence method is used to obtain
the state equation of the system. In this example measurement and numerical models
errors are the uncertainty sources; the authors argue that parameter uncertainty
can be accounted for indirectly by the model error term. A problem with known
solution is solved. The "data" for the estimation procedure are obtained from a
second numerical solution, in this case from the MOC method. The model error
covariance matrix was determined from the di�erences between the �nite di�erence
model values and the analytical solutions. The measurement error covariance matrix
was obtained from the di�erence between the MOC model values and the analytical
solutions. A comparison between the analytical solution, the numerical solution, and
the Kalman �lter solution showed that signi�cant improvement can be accomplished
by combining the two numerical solutions through a Kalman �lter.
A method based on the extended Kalman �lter was developed by Jinno et al. [26]
to predict the contaminant concentration of groundwater pollutants. A stochastic
one-dimensional convection-dispersion equation is used to model the transport prob-
lem. The only random term included in the equation is the model error and it is
represented as a white Gaussian random sequence. This partial di�erential equation
is transformed into an ordinary di�erential equation expanding the concentration and
the model error using Fourier series. A discrete system is obtained by approximating
the time derivative using �nite di�erences. The Fourier coe�cients for the concen-
tration expansion and the transport equation parameters are estimated using an ex-
tended Kalman �lter. Two synthetic examples were presented. In these examples the
data used to obtain the estimates were contaminant concentrations measurements,
47
and measurements on the parameters of the equation. Each observation contained
some measurement error. The e�ects on the accuracy of the estimates of the sampling
frequency was analyzed.
A work that deserves special attention is that of Loaiciga [29] because his approach
is similar to ours. This author combines kriging with a stochastic transport equation
to estimate pollutant concentrations. To estimate the concentration at a spatiotem-
poral point in which a sample is not available the author uses a linear estimate
employing contaminant concentration data at all spatiotemporal points. Assuming
that parameters are known Loaiciga derives the elements of the concentration covari-
ance matrix from the advection-dispersion equation governing mass transport. The
only source of uncertainty that he considers when obtaining the covariance matrix is
the model error. The concentration mean at each point must be known to satisfy the
unbiased condition. These values are again calculated from the transport equation.
A deterministic equation that describes the mean solute concentration is obtained
by perturbing the solute concentration and the seepage velocity. In section 4.5 it will
be shown that when one assumes the transport model errors at di�erent times are
independent, as Loaiciga does in his work when calculating the covariance matrix,
important concentration correlations are disregarded. When proceeding in this way
information contained in a contaminant sample about contaminant concentration
at other space-time locations is overlooked. For this reason we think that Loaiciga's
method suggests that more samples than are actually necessary are required to obtain
a concentration estimate with a given degree of certainty.
Using the Lagrangian approach Dagan et al. [13] study the impact of concentra-
tion measurements upon estimation of ow and transport parameters. The authors
analyze a simple case in which a solute body of constant concentration is injected
into an aquifer and the objective is to obtain the statistical distribution of the solute
concentration. The e�ect of pore-scale dispersion is neglected, such that concentra-
48
tion stays constant and the volume of the solute body is preserved. An expression is
obtained for the conditional mean and the conditional variance of a generic variable
when conditioned on concentration data. The conditional mean obtained is the same
as the one obtained by cokriging. But, in contrast to the conditional variance in
cokriging, the variance obtained in this paper di�ers depending on the value of the
measured concentration. The conditional mean and variance of log-transmissivity
and of the plume centroid are analyzed. Contaminant concentration conditioned on
contaminant concentration data is not analyzed.
A work by Neuman et al. [32] contains an example of the application of Neuman's
theory in the estimation of vertically averaged concentration of an inorganic solute
from a tracer experiment using concentration data. In this example the tracer con-
centration is estimated at a given time using a subset of the data available at that
time and it is compared with the actual plume determined on the basis of all avail-
able data. As was mentioned before, an expression for a spatiotemporal contaminant
concentration covariance matrix is obtained from Neuman's theory. We are not aware
of any paper where this covariance matrix has been used to estimate concentrations
of a moving contaminant plume using concentration data sampled at di�erent times.
Our purpose is to use an estimation method that allows us to incorporate uncer-
tainty to models that originally have been developed under a deterministic viewpoint.
The estimation methods used by Loaiciga [29], Yu et al. [48], and Zou and Parr [54]
have the desired characteristic. In these works the transport model with determinis-
tic coe�cients is used and uncertainty is accounted for through an additive random
term, here called the model error. These model errors at di�erent times are assumed
to be independent and normally distributed. Zou and Parr [54] suggested that the
uncertainty of hydraulic properties can be taken into account by the model error and
Loaiciga implicitly is making the same assumption. We think that to obtain realistic
groundwater quality sampling network designs it is fundamental to account for the
49
uncertainty in groundwater velocity. To our knowledge it has not been analyzed any-
where whether stochastic transport models with deterministic parameters and model
errors uncorrelated on time can capture the statistical properties that contaminant
concentration �elds have that are obtained from models with random velocity pa-
rameters. We investigate this problem in what follows as a �rst step to proposing an
estimation method appropriate for our purposes.
4.3 Flow and Transport Equations
A model used often in groundwater quality problems is the steady state ow equa-
tion and the conservative convection-dispersion transport equation, coupled through
Darcy's Law:
r � (K � rh) = 0; (4.1)
@c
@t�r � (D � rc� V c) = 0; (4.2)
V = �K�rh; (4.3)
where h is hydraulic head, K is hydraulic conductivity, c is solute concentration, D
is hydrodynamic dispersion, � is e�ective porosity, and V is pore velocity.
Equation (4.1), describes the ow of water through the aquifer when it has reached
steady state. The hydraulic conductivity tensor, K, is a parameter that characterizes
the capacity of the porous medium to transmit water. The transport equation (4.2)
describes the changes in the contaminant concentration through time for a conserva-
tive solute. Darcy's Law (4.3) gives a rule to calculate the pore velocity of groundwater
in the aquifer using the heads from the ow equation and the hydraulic conductivity.
It is recognized that the most uncertain parameter in this set of equations is hy-
draulic conductivity. Hydraulic conductivity is a highly variable property, it can vary
50
several orders of magnitude within a few meters, and it is measured indirectly. This
makes its estimation extremely di�cult. Since conductivity is used in Darcy's Law
to calculate velocity, errors in hydraulic conductivity can produce important errors
in contaminant concentration results. Consequently, many researchers have adopted
a stochastic point of view in analyzing and estimating contaminant concentrations.
In this approach which will be followed herein hydraulic conductivities are modeled
by a spatially correlated random �eld.
4.4 Stochastic Simulation
We use stochastic simulation to analyze the statistical characteristics of the model
error and to calculate some inputs for the estimation method that we propose. Here
we introduce the basic concepts in stochastic simulation, also called Monte Carlo
simulation. The text by Ripley [33] is a good introductory book on this subject.
Stochastic simulation involves using a model to analyze and predict the behavior of
a real process that contains elements that can be interpreted as random. As is common
in statistics, the analysis and estimation of a random process require sampling some
of its realizations. Stochastic simulation involves obtaining realizations of the solution
of a stochastic model and analyzing the statistical properties of these realizations.
4.4.1 Hydraulic Conductivity Random Field
We model hydraulic conductivity as a spatial random �eld with a lognormal proba-
bility distribution. To characterize the random �eld it is su�cient to �nd its mean
and covariance structure. This can be done through a geostatistical analysis of �eld
data. Given the hydraulic conductivity mean and correlation structure, it is possible
to generate a set of conductivity realizations with the given statistical structure.
51
We assume that the conductivity �eld is homogeneous, stationary and isotropic.
That is, the conductivity mean and its variance have a constant value at all locations
and the covariance of the conductivity at two locations depends only on the distance
between them, i.e.,
Cov(K(r1); K(r2)) = F (jr1 � r2j);
where r1, and r2 are two position vectors, jrj denotes the norm of the vector r, and
F is a scalar function.
Instead of working with the continuous conductivity random �eld, we approximate
it with a discrete random �eld. This �eld is related with a numerical mesh, where
each node on the mesh has a random conductivity variable associated with it. As with
standard numerical methods, as we re�ne the mesh, we obtain a better approximation
to the continuous �eld. Ababou [1] shows that for a random variable with correlation
scale � locally integrated over a length �x,
Y =1
�x
Z �x
0Y (x0)dx0;
there is a reduction in variance and an increase in the correlation scale in comparison
with the original variable Y (x). He suggested that discretization e�ects may be
avoided when the ratio � = �=�x of correlation scale to discretization scale is larger
than 1 + �2Y , where �2Y is the variance of Y , i.e.
�
�x� 1 + �2Y : (4.4)
4.4.2 Contaminant Concentration Random Fields
As a consequence of modeling hydraulic conductivity by a random �eld, the velocity
and contaminant concentration, that are functions of the conductivity, become ran-
dom �elds as well. When stochastic simulation is used to analyze the characteristics
of the contaminant concentration �eld we proceed as follows. Hydraulic conductivity
52
realizations are obtained. The ow equation is solved numerically using each realiza-
tion. The values obtained determine a velocity �eld that is, in turn, used to solve the
contaminant transport equation and produce a realization of the plume. From the
plume realizations we can obtain the desired quantities for the analysis, for example
concentration means and concentration variances at a group of locations.
4.5 Model Error Time Correlations
In this section we analyze the statistical properties of the model error when it has
been de�ned while accounting for velocity uncertainty. We show, through a one-
dimensional example that the model errors can be correlated on time. Finally we
show the consequences of considering time uncorrelated model errors when condi-
tioning contaminant concentration estimates with contaminant concentration data
from groundwater samples.
4.5.1 Model Error De�nition
In an e�ort to model the groundwater quality problem using stochastic concepts some
authors, including Yu et al. [48], Zou and Parr [54], Jinno et al. [26] and Loaiciga [29],
have used an equation of the form
cn+1 = �cn + bn + �nwn+1;
where the vector cn is a discrete representation of concentrations at time tn, each
vector wn represents the model error at that time, bn is a known vector, and � and
�n are known matrices. The vector sequence fwn+1; n = 0; 1; : : :g is a zero-mean
white Gaussian sequence.
53
In what follows we propose a way to relate the model error sequence fwn+1; n =
0; 1; : : :g with the concentration random �eld obtained from the conservative transport
equation with random velocity. We do this for the continuous transport equation and
then we obtain the model errors wn discretizing this equation.
We demonstrate the concept that we are proposing using a one-dimensional trans-
port equation, but the same idea applies when using two-dimensional and three-
dimensional equations.
The one-dimensional version of equation (4.2) is
@c
@t� @
@x
D@c
@x� V c
!= 0: (4.5)
Here we are assuming that D and V are random coe�cients. We rewrite the velocity
and dispersion coe�cients as their mean plus a deviation from their mean, V =
�V + V 0; D = �D +D0. Substituting in equation (4.5.1) we obtain
@c
@t� @
@x
�D@c
@x� �V c
!= v(x; t); (4.6)
where the continuous model error is given by
v(x; t) =@
@x
D0 @c
@x� V 0c
!: (4.7)
Note that the model error de�ned in this way does not necessarily have zero mean.
We interpret the model error v as an error due to the omission of random coe�-
cients in the equation. This is the term that we would have to add to equation (4.6) to
obtain from it the solutions of equation (4.5). If the velocity mean and the dispersion
mean are known it is possible to obtain realizations of v from equation (4.6) by sub-
stituting for c concentration values of realizations of the contaminant concentration
obtained from equation (4.5).
The equation obtained after discretizing the transport equation is
Acn+1 = Bcn + an + vn+1; (4.8)
54
or, solving for cn+1,
cn+1 = �cn + bn + wn+1; (4.9)
where the vector bn has been de�ned so that the expected value of the model error
wn+1 is zero, and � = A�1B. The form and value of the coe�cients appearing in
equations (4.8) and (4.9) depend upon the particular numerical technique used for
approximating the ow and transport equations.
To this point we have de�ned wn+1 in such a way that the concentration random
�eld obtained from equation (4.9) has the same characteristics as the concentration
random �eld obtained from the transport equation with random coe�cients. Now we
want to analyze the statistical characteristics of this model error. To do this we use
stochastic simulation.
Realizations of the model error wn+1 are obtained in the following way. Velocity
�eld realizations are obtained from the steady state ow equation with random con-
ductivities coupled with Darcy's Law. Dispersion coe�cient realizations are related
to velocity realizations through the formula D = �V , where � is a constant known
dispersivity coe�cient. The mean velocity and the mean dispersion coe�cients are
calculated from these realizations. For each velocity realization a concentration real-
ization is obtained as a solution of equation (4.5) and it is substituted into equation
wn+1 = cn+1 � �cn + bn: (4.10)
Recall that the matrix � is a function of the mean velocity and mean dispersion.
4.5.2 Model Error Statistical Properties
A set of tests are performed using the one-dimension transport equation with bound-
ary conditions:
55
c(0; t) = 1; and@c
@x(1; t) = 0 (4.11)
and initial conditions
c(x; 0) = 0: (4.12)
Realizations for log-hydraulic conductivities are obtained from an ARIMA method
from Bras and Rodr��guez Iturbe [5]. The domain is subdivided into equally sized in-
tervals and for each simulation a single conductivity value is assigned to each of them.
If h(0) = h0, and h(L) = hL are the boundary conditions for the ow equation, and
Ki is the hydraulic conductivity value at the i-th subinterval, the velocity obtained
when solving equations (4.1) and (4.3) is [3, 39]
V = � hL � h0��x
Pi 1=Ki
: (4.13)
We use this expression to calculate velocity realizations. Boundary conditions for
ow are set to h0 = 55 ft, and hL = 0 ft. A 1320 ft length domain is used and it is
subdivided into twenty subintervals (�x = 66 ft). A period of two years is simulated.
Values for the required input parameters are summarized in table 4.1. The conduc-
tivity �eld is homogeneous, isotropic, lognormally distributed, and with correlation,
�logK(x) =e
�1�
jxj�logK
��2logK � 1
e�2logK � 1
if jxj < �logK; (4.14)
where �logK is the correlation scale of the random �eld and �2logK is the log-conduc-
tivity variance. The log-conductivity mean is set equal to 3:055, the log-conductivity
variance equal to 0:7, and the correlation scale equal to 264 ft. The value for the
dispersivity coe�cient is 33 ft and porosity is 25 %.
The transport equation is solved analytically for each velocity realization. The
formula used to obtain the concentration solution neglects boundary e�ects [47], it is:
c(x; t) =1
2
"Erfc
x� V t
2pDt
!+ exp
�V x
D
�Erfc
x + V t
2pDt
!#:
56
Table 4.1: Input for the example problem.
�logK (ft) �2logK � (ft) � �x (ft) �t (days)
264 0.7 33 0.25 66 15.21
A �nite di�erence scheme is used to obtain realizations of the model error. This
is, the matrix � in equation (4.10) is obtained from the central �nite di�erence dis-
cretization of the transport equation. Forty eight time steps are used for the two years
(�t = 15:21 days). The values for the log-conductivity variance, the log-conductivity
correlation scale, and the spatial discretization interval length satisfy the criteria pro-
posed by Ababou (4.4). On the other hand, Gelhar [20] presents a table of data on
variance and correlation scale of saturated log-hydraulic conductivity from sites of
several dimensions and our parameters are in agreement with those values.
For simplicity, we analyze only the correlations of the errors on a subgrid of the
original grid and for a subset of the times used in the numerical discretization. The
subgrid is regular, with �ve elements on it. Its nodes are denoted by xi; i = 1; : : : ; 6,
where xi = i4�x = i264 feet. Only six times are considered, ti = i8�t = i121:7 days,
i = 1; : : : ; 6.
Model error correlations are determined from a set of 1000 stochastic simulations.
The correlation between the errors at two space-time locations (xi; tj) and (xp; tq) is
de�ned as
�w(xi; tj; xp; tq) =Efw(xi; tj)w(xp; tq)g�w(xi; tj)�w(xp; tq)
:
To simplify notation we use �ij;pq to denote the correlation above and wij to denote
w(xi; tj). We calculate these correlations from the output of the stochastic simulation
using the sample correlation:
�ij;pq =
PNk=1w
kijw
kpqPN
k=1(wkij)
2PN
k=1(wkpq)
2;
where wkij is the k-th model error realization.
57
A graphical representation of the model error correlations is shown in �gure 4.1.
We are representing the fourth order tensor �ij;pq with a two dimensional matrix.
In the vertical and horizontal axes we have the same coordinates. The �rst �ve
coordinates seen along the vertical axes are the �ve locations at the �rst time, the
second �ve are the same �ve locations at the second time and so on. Then the �rst �ve
by �ve square on the left lower corner contains the correlations between the model
errors at the �rst time at all locations, this is, �ij;pq; p = 1; q = 1; i; j = 1; : : : ; 5.
In the same way, each �ve by �ve square in the diagonal contains the correlations
between the model errors at a �xed time for all locations. Each o� diagonal square
contains the model error correlations at di�erent times, for example, the second �ve
by �ve square in the �rst �ve by �ve row is �ij;pq; p = 1; q = 2; i; j = 1; : : : ; 5. All the
correlations are positive with values between 0 and 1, a dark square indicates that
the correlation is close to zero, a light square indicates that it is close to one. If the
model errors were uncorrelated on time, all the �ve by �ve blocks o� the diagonal
would be black. Clearly this is not the case.
Figure 4.2 shows a di�erent representation of a subset of the same results. It
contains �ve di�erent graphs of correlation model errors. In each graph a node and
a time are �xed. That is, the graphs show the values of �i0;j0;p;q; p = 1; : : : ; 5; q =
1; : : : ; 6. The time �xed is t1 for all the graphs and the �xed node changes in each
graph. On top of each graph the node and time �xed are shown. One axis shows the
times t1 to t6 and the other the nodes x1 to x5. The vertical axes shows the value of
the model error correlations. In these graphs the changes in correlations in time and
space can be observed.
In graph 4.2a we can observe that the model error w1;1 has strong correlations
with the model errors at some other nodes and times. It is noticeable that the cor-
relation peaks are located along one of the diagonals of the domain. In �gure 4.3
the same correlations are shown but �xing one node at a time. The letters for cor-
58
x5,t1 x5,t2 x5,t3 x5,t4 x5,t5 x5,t6
x1,t1x2,t1x3,t1x4,t1x5,t1
x5,t2
x5,t3
x5,t4
x5,t5
x5,t6
Figure 4.1: Graphical representation of the error correlation matrix. A black squareindicates weak correlation and a light square strong correlation. For an explanationsee the text.
responding graphs are the same. If the model errors were uncorrelated on time the
only correlation values di�erent from zero in each one of these graphs would be at
time t1, on the vertical axes. In the graphs 4.3a and 4.3b clearly this is not the case.
The correlations of the errors w3;1, and w4;1 with errors at other times are not as
strong (�gures 4.2c, 4.3c, 4.2d, and 4.3d), and for the model error at the last node,
x5 (�gures 4.2e, and 4.3e) the correlations with errors at times di�erent from t1 are
practically zero.
A Shapiro-Wilks test is applied to the model error realizations at a set of spa-
tiotemporal points to measure normality. Most of the model errors tested are not
normally distributed, their distributions skew to the left or to the right depending on
their spatiotemporal location.
In this example it is shown that model errors can have strong time correlations.
One may ask what are the consequences on the estimation process when these corre-
lations are neglected. We analyze this in the next section.
59
(a) (b)8x1, t1<
1
23
45
6
t
1
2
3
4
5
x
0.20.40.60.81
r
23
45
6
t
8x2, t1<
1
23
45
6
t
1
2
3
4
5
x
0.25
0.5
0.75
1
r
23
45
6
t
(c) (d)8x3, t1<
1
23
45
6
t
1
2
3
4
5
x
0.25
0.5
0.75
1
r
23
45
6
t
8x4, t1<
1
23
45
6
t
1
2
3
4
5
x
00.250.5
0.75
1
r
23
45
6
t
(e)8x5, t1<
1
23
45
6
t
1
2
3
4
5
x
00.250.5
0.75
1
r
23
45
6
t
Figure 4.2: Three-dimensional representation of the model error correlation for thedi�erent nodes at all times with a) node 1 at the �rst time, b) node 2 at the �rsttime, c) node 3 at the �rst time, d) node 4 at the �rst time, and e) node 5 at the �rsttime.
60
2 3 4 5 6t
0.2
0.4
0.6
0.8
1r
2 3 4 5 6t
0.2
0.4
0.6
0.8
1r
(a) (b)
2 3 4 5 6t
0.2
0.4
0.6
0.8
1r
2 3 4 5 6t
0.2
0.4
0.6
0.8
1r
(c) (d)
2 3 4 5 6t
0.2
0.4
0.6
0.8
1r
Node 5
Node 4
Node 3
Node 2
Node 1
(e)
Figure 4.3: Two-dimensional representation of the model error correlation for thedi�erent nodes at all times with a) node 1 at the �rst time, b node 2 at the �rst time,c) node 3 at the �rst time, d) node 4 at the �rst time, and e) node 5 at the �rst time.
61
4.6 Consequences for the Estimation Process
To give an idea of the consequences of modeling groundwater quality problems using
models with time uncorrelated errors, we compare some statistical properties of the
concentration estimates obtained from a model with those characteristics, here called
model 1, with those obtained from a model with time correlated errors, here called
model 2. To do this we have to use an estimation method that allows us to combine
the estimates that we get from the stochastic simulation with data. For model 1 Yu et
al. [48] and Zou and Parr [54] used a Kalman �lter, we can use the same method. For
model 2 we can use a geostatistical method like Loaiciga did [29], or a static Kalman
�lter. We use the second option. In what follows the principles of the Kalman �lter
are explained and the formulas that we are using are presented.
4.6.1 Dynamic Kalman Filter
The Kalman �lter obtains linear minimum-variance unbiased-estimates for the state
of a system from noisy data. It also establishes a way to update these estimates when
a new measurement becomes available with no need to refer to old data. The �lter is
based on two equations: the state equation and the measurement equation. The �rst
describes the evolution of the system state over time and the second relates the state
with data. For a complete derivation of the Kalman �lter equations see [25].
Consider a discrete system whose state at time tn is denoted by xn,
xn+1 = �xn + bn + �nwn+1; (4.15)
where the vectors xn and bn are (N � 1) column vectors, � and �n are N � N
matrices. The vector bn and the matrices � and �n are deterministic and known, and
each vector wn represents the model error at the given time. The vector sequence
fwn+1; n = 0; 1; : : :g is a white Gaussian sequence with covariance Qn+1. Let fzn; n =
62
1; 2; : : :g be a sequence of measurements of the corresponding system states xn. Each
vector zn is the set of l measurements at time tn,
zn =
0BBBBBBBBB@
z1n
z2n...
zln
1CCCCCCCCCA;
where l � N .
Let these samples be related with the state through the linear measurement equa-
tions,
zn = Hnxn + vn:
Hn is the sampling matrix at time tn; it describes the linear combinations of state
variables which give rise to zn. The dimension of the measurement matrix is l�N , with
l corresponding to the number of measurements at time n and N to the dimension of
the state. The set fvn; n = 1; 2; : : :g represents the measurement error and it is a white
Gaussian sequence, with mean zero and covariance matrixRn. The measurement error
sequence fvng and the state xn are independent. The distribution of x0 is known.
It can be shown that the minimum-variance unbiased estimate for the state variable
at time tn given the measurements z1; : : : ; zk, denoted xkn, is the expected value of the
state xk given the data, this is, xkn = Efxnjz1; z2; : : : ; zkg. Note that in this notation
the subscript identi�es the time in which the state is estimated and the superscript
the number of measurements that are used to obtain the estimate. The covariance
matrix of the error of this estimate is
P kn = Ef(xn � xkn)(xn � xkn)
Tg;
where T denotes transpose.
63
The idea of the Kalman �lter is that given the estimate of the state at time tn
using the n data available, xnn = Efxnjz1; z2; : : : ; zng, it is possible to predict (or
forecast) the state at the next time using this estimate and the system equation. For
this purpose the next equations are used;
xnn+1 = �xnn + bn (4.16)
P nn+1 = �P n
n�T + �nQn+1�
T
n: (4.17)
When the next measurement zn+1 becomes available the last prediction can be up-
dated using the following equations,
xn+1n+1 = xnn+1 +Kn+1(zn+1 �Hn+1xnn+1); (4.18)
P n+1n+1 = P n
n+1 �Kn+1Hn+1Pnn+1; (4.19)
where
Kn+1 = P nn+1H
T
n+1fHn+1Pnn+1H
T
n+1 +Rn+1g�1 (4.20)
is the Kalman gain. These formulas are used sequentially, starting from a given prior
estimate x00 for the state at time t0 and its covariance matrix P 00 .
4.6.2 Static Kalman Filter
To account for the space time correlations that the transport model error has we
cannot use the dynamic Kalman �lter because in its derivation it is assumed that the
model errors are not time-correlated. However, we can use the static version of the
�lter. We will explain �rst the formulas used in the �lter and then we will explain
how to use this �lter in our problem.
In the static �lter the state variable does not change with time. The state equa-
tion (4.15) is replaced by
xn+1 = xn;
64
what leads to P nn+1 = P n
n . This means that equations (4.16) and (4.17) are not
necessary when working with static variables. For this case there are a couple of
changes. Each one of the measurements fzn; n = 1; 2; : : :g are taken from a �xed
state denoted by x, and the superscripts are not related with time any more but they
indicate the order in which the samples are taken. In the Kalman �lter formulas
( 4.18, 4.19, and 4.20) all the subscripts related with time can be dropped. The
formulas that we will use in this work are written below.
Given a prior estimate of the system state x0 with known distribution and its
covariance matrix P 0, the minimum variance linear estimate for the state can be
obtained sequentially through the following formulas:
xn+1 = xn +Kn+1 (zn+1 �Hn+1xn) ; (4.21)
P n+1 = P n �Kn+1Hn+1Pn; (4.22)
where
Kn+1 = P nHT
n+1
�Hn+1P
nHT
n+1 +Rn+1
��1; (4.23)
the state estimate given n data is
xn = Efxjz1; z2; : : : ; zng;
and the error covariance matrix is
P n = Ef(x� xn)(x� xn)Tg: (4.24)
4.6.3 Estimation Method Proposed in this Thesis
How can we use the static Kalman �lter to estimate concentrations on moving con-
taminant plumes? It is possible to calculate the space-time concentration covariance
matrix from the two models we have presented. It is not di�cult to prove that for
65
model 2 this covariance matrix is identical to the one that would be obtained from
the stochastic simulation. This is a direct consequence of the de�nition of the model
error. If we de�ne the state variable as the vector of concentrations at all locations
and all times, the concentration space-time covariance matrix from the stochastic
simulation would be an estimate of that of the state vector. Then, to condition the
concentration estimate from the stochastic simulation using concentration data, we
can use in the Kalman �lter the space-time mean concentration from the simulation as
the prior estimate, and its covariance matrix as the prior estimate covariance matrix.
In the Kalman �lter equations( 4.21), (4.22), and (4.23) we substitute x with C,
the contaminant concentration vector that contains concentrations at all positions
and all times:
C = (c1;1; c2;1; : : : ; cM;1; c1;2; c2;2; : : : ; cM;2; : : : ; cM;T );
where ci;l is the concentration at location xi at time tl, M is total number of space
points and T is the total number of times. The corresponding covariance matrix is,
P =
0BBBBBBBBB@
P1;1;1;1 P1;1;2;1 � � � P1;1;M;1 P1;1;1;2 � � � P1;1;M;T
P2;1;1;1 P2;1;2;1 � � � P2;1;M;1 P2;1;1;2 � � � P2;1;M;T
......
......
...
PM;T ;1;1 PM;T ;2;1 � � � PM;T ;M;1 PM;T ;1;2 � � � PM;T ;M;T
1CCCCCCCCCA:
4.6.4 Estimation of Prior Moments by Stochastic Simulation
As mentioned before, we use stochastic simulation to obtain the prior plume mean
and its covariance matrix. If ckil denotes the k-th concentration realization at location
xi at time tl, then the concentrations on the mean plume obtained with N realizations
would be,
cil =1
N
NXk=1
ckil:
66
The element (i; l; p; s) of the corresponding covariance matrix would be,
Cov(cil; cps) = Covil;ps =1
N � 1
NXk=1
(ckil � cil)(ckps � cps):
We de�ne C0 as the vector that contains cik for all locations (xi), and all times tk,
and P 0 as the matrix of the covariances, Covil;ps, associated with this vector.
The estimation method that we are proposing can be thought of as a space-time
kriging method in which the concentration expected mean and its covariance matrix
are obtained from the stochastic transport equation. So, the concept is very similar
to Loaiciga's [29], the main di�erence being the way in which we are addressing the
contaminant concentration correlations.
4.6.5 Conditional Variance
We start by comparing the estimate variance predicted by the two models when no
sample is �ltered. For model 1 (the model with time-uncorrelated errors) the prior
variance of the vector of concentration estimates cn are the elements in the diagonal
of the covariance matrix
P 0n+1 = �P 0
n�T +Qn+1 (4.25)
where Qn+1 is the covariance matrix of the model error vector wn+1 with itself. The
covariance matrices Qn, n = 1; : : : are obtained from the stochastic simulation. We
can calculate the estimate error covariance matrix step by step in time, starting with
P0 = 0. The prior covariance matrix P0 is zero because c0 = 0 is assumed to be
known with certainty. For model 2 (the model with time-correlated errors) the prior
variance for all times is obtained from the stochastic simulation.
It is pertinent to note that to calculate the estimate error variance at the locations
we are interested in using equation (4.25) it is necessary to calculate the variance at all
locations on the discretized interval (in this case 20 locations) for all the forty-eight
67
2 3 4 5x
0.034
0.068
0.102
sw2
Time 4
2 3 4 5x
0.034
0.068
0.102
sw2
Time 5
2 3 4 5x
0.034
0.068
0.102
sw2
Time 6
2 3 4 5x
0.034
0.068
0.102
sw2
Time 1
2 3 4 5x
0.034
0.068
0.102
sw2
Time 2
2 3 4 5x
0.034
0.068
0.102
sw2
Time 3
Figure 4.4: Prior concentration estimate variances from the two models. Stars - timeuncorrelated model errors, diamonds - time correlated model errors.
numerical steps. In contrast for model 2, using the space-time covariance matrix,
we can calculate the covariance matrix only at those locations of interest (in this
case 36 spatiotemporal locations) and use it to calculate the estimate variances only
at the locations of interest. At �rst glance it may be thought that calculating the
spatiotemporal covariance matrix is a drawback when, in fact, it allows us to separate
the estimation process from the discretization used for the numerical problem. Of
course, if the concentration estimates are wanted at all the locations on the numerical
mesh, at all the numerical time steps, storing the spatiotemporal covariance matrix
could be a limiting factor when working with big problems.
A comparison between the prior estimate variances obtained from the two models
is shown in �gure 4.4. The variances are shown for the six times t1; t2; : : : ; t6. The
variance obtained from the model with time uncorrelated errors is shown in the plot
with stars and the variance from the model with correlated errors is shown in the plot
with diamonds. The model with uncorrelated errors underestimates the concentration
variances.
If we use the Kalman �lter to estimate concentrations when concentration data
68
2 3 4 5x
0.002
0.004
0.006
0.008
sw2
Time 4
2 3 4 5x
0.002
0.004
0.006
0.008
sw2
Time 5
2 3 4 5x
0.002
0.004
0.006
0.008
sw2
Time 6
2 3 4 5x
0.002
0.004
0.006
0.008
sw2
Time 1
2 3 4 5x
0.002
0.004
0.006
0.008
sw2
Time 2
2 3 4 5x
0.002
0.004
0.006
0.008
sw2
Time 3
Figure 4.5: Prior and posterior concentration estimate variances from model 1 whena sample at (x1; t1) is �ltered. Stars - prior variances, diamonds - posterior variances.
is available, the variance of the estimate error does not depend on the value of the
data. So, we can compare the variance predicted by the two models based only on
the location and time at which the sample �ltered is taken.
The variance predicted by model 1 when a measurement at location x1 and time t1
is �ltered is shown in �gure 4.5 together with the variance predicted by the same model
for the prior estimate. The posterior variance at the sampling location is zero because
we are assuming that the sampling error is zero. It can be seen that the information
obtained by �ltering the sample travels from time t1 to times t2 and t3 but it has been
almost lost at the last two times where the prior variance and the posterior variance
are almost identical. In contrast when model 2 is used when �ltering a measurement
form the same space-time location the information is conserved along the six times
(�gure 4.6). Figure 4.7 shows a comparison between the posterior variances predicted
by the two models when a sample from location (x1; t1) is �ltered.
The same comparison is done when a sample at the space-time location (x2; t1)
is �ltered. Again it was found that the information obtained from the sample is
preserved longer for model 2 that for model 1.
69
2 3 4 5x
0.034
0.068
0.102
sw2
Time 4
2 3 4 5x
0.034
0.068
0.102
sw2
Time 5
2 3 4 5x
0.034
0.068
0.102
sw2
Time 6
2 3 4 5x
0.034
0.068
0.102
sw2
Time 1
2 3 4 5x
0.034
0.068
0.102
sw2
Time 2
2 3 4 5x
0.034
0.068
0.102
sw2
Time 3
Figure 4.6: Prior and posterior concentration estimate variances from model 2 whena sample at (x1; t1) is �ltered. Stars - prior variances, diamonds - posterior variances.
2 3 4 5x
0.01
0.02
0.03
0.04
sw2
Time 4
2 3 4 5x
0.01
0.02
0.03
0.04
sw2
Time 5
2 3 4 5x
0.01
0.02
0.03
0.04
sw2
Time 6
2 3 4 5x
0.01
0.02
0.03
0.04
sw2
Time 1
2 3 4 5x
0.01
0.02
0.03
0.04
sw2
Time 2
2 3 4 5x
0.01
0.02
0.03
0.04
sw2
Time 3
Figure 4.7: Posterior concentration estimate variances from the two models when asample at (x1; t1) is �ltered. Stars - model 1, diamonds - model 2.
70
4.6.6 Conditional Estimates
To show the di�erences that the conditioned contaminant concentration estimates
obtained from the two models can have, we obtain these estimates when a given
contaminant concentration curve is observed. To do this we choose one of the con-
taminant concentration realizations obtained in the stochastic simulation and we take
the contaminant concentration samples equal to the value of this concentration real-
ization.
The prior concentration estimate for both methods can be calculated recursively
by the formula
c0n+1 = �c0n + bn;
with c00 = 0. We point out again that c0n is equal to the mean concentration at
time tn obtained from the stochastic simulations, this is because of the way bn was
de�ned.
The realization chosen and the prior estimate are shown in �gure 4.8. The concen-
tration realization is the stared curve and the mean concentration is represented by
the plot with diamonds. The concentration is equal to one at the �rst node, x0, at all
times for both curves because that is the value of the boundary condition. The prior
estimate curve has a shape similar to that of the realization curve but it is di�erent
to it at several points. The idea is that this initial estimate should get closer to the
realization when samples from the realization concentrations are �ltered. We will
show the e�ect of �ltering one sample in this example.
The concentration estimate obtained from model 1 when �ltering a sample from
(x1; t1) is shown in �gure 4.9. The concentration is equal for the two curves at location
(x1; t1), this is because the sample is taken a that location. There are some small
changes in the stared curve at time 2 and time 3, but for the last three times the
curve remains essentially unaltered.
71
1 2 3 4 5x
0.20.40.60.8
1
c Time 4
1 2 3 4 5x
0.20.40.60.8
1
c Time 5
1 2 3 4 5x
0.20.40.60.8
1
c Time 6
1 2 3 4 5x
0.20.40.60.8
1
c Time 1
1 2 3 4 5x
0.20.40.60.8
1
c Time 2
1 2 3 4 5x
0.20.40.60.8
1
c Time 3
Figure 4.8: Comparison between the prior concentration estimate and a concentrationrealization. Stars - prior concentration estimate, diamond - concentration realization.
1 2 3 4 5x
0.20.40.60.8
1
c Time 4
1 2 3 4 5x
0.20.40.60.8
1
c Time 5
1 2 3 4 5x
0.20.40.60.8
1
c Time 6
1 2 3 4 5x
0.20.40.60.8
1
c Time 1
1 2 3 4 5x
0.20.40.60.8
1
c Time 2
1 2 3 4 5x
0.20.40.60.8
1
c Time 3
Figure 4.9: Comparison between the posterior concentration estimate from model 1and a concentration realization. Stars - posterior concentration estimate from model1, diamond - concentration realization.
72
1 2 3 4 5x
0.20.40.60.8
1
c Time 4
1 2 3 4 5x
0.20.40.60.8
1
c Time 5
1 2 3 4 5x
0.20.40.60.8
1
c Time 6
1 2 3 4 5x
0.20.40.60.8
1
c Time 1
1 2 3 4 5x
0.20.40.60.8
1
c Time 2
1 2 3 4 5x
0.20.40.60.8
1
c Time 3
Figure 4.10: Comparison between the posterior concentration estimate from model 2and a concentration realization. Stars - posterior concentration estimate from model2, diamond - concentration realization.
When the sample from location (x1; t1) is �ltered using model 2 the concentration
estimate gets much closer to the concentration realization values (�gure 4.10). At time
1, the estimate from the two models is the same but as time passes the information
provided by the sample is a lot larger for model 2 than for model 1. The errors at
the last three times get very small for the estimate from model 2.
The norm of the estimate error is
jjejj =sX
i;k
(c(xi; tk)� c(xi; tk))2;
where c is the estimate and c is the realization concentration. Comparing this norm
before and after conditioning we have that for the prior estimate jjejj = 0:512, for the
estimate from model 1 jjejj = 0:459, and for the estimate from model 2 jjejj = 0:344.
This means that for this particular example using model 1 the error is reduced 11%
and using model 2 it is reduced 33%.
73
4.7 Conclusions
From this analysis we conclude that disregarding the time correlations of the model er-
rors can lead to estimation methods that need many more samples to obtain the same
degree of certainty as a model incorporating model error time correlations. Also, the
variance as a measure of error in the concentration estimates can be misleading when
model error correlations are not accounted for. For the example shown here the con-
centration estimates obtained from the model with time-correlated errors with only
one sample are very good even when the model errors are not normally distributed.
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78
Chapter 5
Cost E�ective Groundwater Quality Sampling
Network Design
5.1 Introduction
Groundwater quality sampling networks are an aid in characterizing groundwater
contamination problems and in evaluating the performance of a remediation strategy.
In this context the goal of a quality sampling network typically is to estimate con-
taminant concentrations at some speci�ed locations in the aquifer. Often estimating
concentrations of a contaminant plume in an e�cient way depends on both the loca-
tion of the sampling wells and the times when the contaminant samples are taken. On
the other hand, performance costs of a sampling network can be a very large part of
overall costs. Therefore, the design of a cost-e�ective water-quality sampling network
can save much money. In response to this need we have developed a methodology for
the design of a cost-e�ective water-quality sampling network. In the design sampling
locations and sampling times are decision variables.
In chapter 4, we analyzed some statistical characteristics of groundwater contam-
inant concentrations that turned out to be important when estimating this variable
in an Eulerian framework. Based on our �ndings, we proposed a method to estimate
contaminant concentrations of a plume in motion. In this chapter we present some
hypothetical examples to evaluate the method in the context of groundwater sampling
network design.
In chapter 4 we developed a linear estimation method that can accommodate sev-
79
eral sources of variability. When the estimation method is linear, the error variance
estimate does not depend on the value of the concentration estimate. This allows
the comparison of the error variance of the contaminant concentration estimate pro-
duced by several possible sampling strategies with no need of knowing any sampled
contaminant concentration. When sampling decisions involve several sampling times,
linearity of the estimation method is very important. If the method is not linear, the
estimate error variance depends on the contaminant concentration data. In this case,
approximations have to be made using the data available when the sampling decision
is made.
One possibility is to use a sequential approach: sampling locations are selected
only for one sampling time, then the data is gathered at the given time and are used
to update the error variance estimate and make a new sampling decision. In this
approach the number of sampling locations to be selected at each time has to be
decided in advance. In contrast, when the estimation error is linear, as in the method
that we propose, the number of sampling locations at each time does not have to be
stipulated.
We assume that hydraulic conductivity has already been estimated. This assump-
tion makes sense in situations in which a site has already been investigated for some
time and bothxs contaminant concentration data and hydraulic conductivity data
are available. Our purpose is to apply this methodology in cases in which a quality
sampling network may already exist and we want to make it more e�cient.
5.2 Spatiotemporal Sampling Design
There are many works in which the problem of groundwater quality sampling network
design is analyzed assuming either that the sampling times have been preselected or
80
that the contaminant concentration has reached steady state. In these works all
sampling decisions involve only space but not time [3, 5, 9, 10, 16, 25, 26, 29, 30, 33,
35, 36]. Loaiciga et al. [23] and McGrath [24] present an extensive review of works
dealing with these kinds of sampling designs. In what follows we review works in
which sampling network designs use decision variables that depend on space and
time.
5.2.1 Sampling Network Design and Deterministic Modeling
When the transport equation is used to describe the evolution of a contaminant
plume in a deterministic framework, the plume behavior is completely determined
by initial conditions, boundary conditions and the equation parameter values. Using
the transport equation to model a speci�c problem requires that these conditions
and values be chosen using site information. Initial and boundary conditions can be
�gured out from historical information and the hydrogeological characteristics of the
site under investigation. Frequently the velocity parameter of the transport equation
is obtained from the ow equation and other parameters from a model calibration
process. A second way to obtain these parameters is using solute concentration data
when solving what is called the inverse problem [32, 38]. Once the parameters are
speci�ed contaminant plume predictions are obtained solving the equation.
Spatiotemporal sampling network design for parameter estimation of a determin-
istic model has been a subject of recent research. Three papers that propose this
kind of network design are those of Knopman et al. [21], and Cleveland and Yeh [6,7]
(see Table 5.1). In these works parameter estimation is done within a stochastic
framework but the model is assumed to be deterministic.
Knopman and Voss [18] analyzed the spatiotemporal behavior of sensitivities for
parameters of one-dimensional advection-dispersion equations when parameters are
81
estimated from a regression model. The use of an equation with a closed form solu-
tion allowed them to calculate sensitivities from exact derivatives. They found that
sampling at points in space and time with high sensitivity to a parameter yield ac-
curate estimations for that parameter, but designs that minimize the variance of one
parameter may not minimize the variance of other parameters. Therefore, they sug-
gest applying a multiobjective approach when optimal sampling designs are proposed.
This analysis was extended to parameters associated with �rst order chemical-decay,
boundary conditions, initial conditions, and multilayer transport [19].
In a later paper their results were the basis for developing a multiobjective sampling
design for parameter estimation and model discrimination [20]. Model discrimination
implies working with more than one transport model when �tting the data; the au-
thors obtained parameter estimation for all the models simultaneously. They used
a composite D-optimal objective function with the idea of maximizing information
for each set of parameters; they measure information by a function of the sensitivity
matrices. Knopman et al. [21] tested the design using bromide concentration data
collected during the Cape Cod, Massachusetts, natural gradient test. Designs consist
of the downstream distances of rows of fully screened wells oriented perpendicular to
the groundwater ow direction and the timing of sampling to be carried out on each
row. Characteristics of this paper are summarized in Table 5.1, it was chosen as a
representative element of this set of works.
Cleveland and Yeh [6,7] (see Table 5.1) use a maximal information criterion to se-
lect between di�erent designs. Information is measured by a weighted sum of squared
sensitivities, this criterion was chosen after the Knopman and Voss results. The au-
thors develop the sampling methodology under the assumption that once sampling has
begun at a site it continues until the end of the experiment. The examples presented
assume that prior estimates of the parameters are available, the authors suggest that
a sequential approach design can be used to update estimates. In the �rst work two
82
Table 5.1: Sampling network design for parameter estimation. KD9101 [21],CT9001 [6], CT9101 [7], v velocity, K conductivity, T transmissivity, S storage coef-�cient, R retardation factor, �L longitudinal dispersivity, �T transverse dispersivity,ne e�ective porosity, � decay parameter, Ca input source strength, C0 dimensionlessinitial concentration, c concentration.
KD9101 CT9001 CT9101
Objective
function
D-optimal Weighted
information
matrix trace
Weighted
information
matrix trace
Dimensions
transport eq.
1 2 2
Aquifer
layers
1 or 2 1 1
Sampling
dimensions
2 2 1
Parameters
estimated
v, �L, Ca,
C0, �
K, S, ne, �L,
�T , R
T , S, ne, �L,
�T , R
Estimation
method
Gauss-Newton
nonlinear
regression
Least-squares Least-squares
Kind of data c c c
dimensions are considered, one in the direction of the ow (horizontal) and the sec-
ond is depth (vertical); possible sampling locations vary in those directions. In the
second work the transport equation that describes the tracer concentration does not
consider changes in the vertical direction. The total experimental duration is divided
into several stages and a decision is made at the beginning of each stage. The addition
of only one sampling location at a time is considered. Sampling locations are selected
on the line that joins injection and extraction points.
83
5.2.2 Sampling Network Design and Stochastic Modeling
A second option to model a pollutant plume is within a stochastic framework. Two
types of stochastic methods used in hydrology are geostatistical methods and methods
based on partial di�erential equations. Estimates of contaminant concentrations can
be obtained through these methods using contaminant concentration measurements
or measurements of other variables correlated with contaminant concentrations, as
are hydraulic heads and hydraulic conductivities. When a stochastic model is used, on
top of obtaining an estimate for the groundwater pollutant concentrations we get the
uncertainty associated with the estimate. Next we summarize some works that deal
with the problem of spatiotemporal sampling network design for di�erent hydrologic
variables within a stochastic context.
Geostatistical estimation
In chapter 4 we described some works that propose extensions of geostatistical meth-
ods, created to deal exclusively with space variability, to include time variability; here
we are interested in describing the applications of these methodologies in hydrologic
sampling design problems.
Rodr��guez-Iturbe and Mej��a [28] developed a methodology for the design of pre-
cipitation networks in time and space. They worked out sampling programs for two
variables; the long-term mean areal rainfall value, and the mean area rainfall value
of a storm event. For both variables they analyzed two di�erent sampling programs:
simple random sampling, where each station is located with a uniform probability dis-
tribution over the whole space; strati�ed random sampling, where the area is divided
into many non overlapping subareas and k sampling points are chosen randomly in
each subarea. The authors estimated the rainfall process using a generalized geo-
statistical method explained in chapter 4. They discuss trade o�s of time sampling
84
versus space sampling and conclude that in the design of rainfall networks it is im-
portant to consider spatial correlation and time correlation. A summary of the paper
is given in Table 5.2.
Loaiciga [22] (see Table 5.2) combines some elements of kriging with the transport
equation to estimate pollutant concentrations. For the details on the estimation
method see chapter 4. He proposes a spatiotemporal groundwater sampling network
design that involves two steps: parameter estimation, and network optimization.
For network optimization, the objective used by Loaiciga is to choose where and
when to sample to minimize the variance of the concentration estimate error subject
to budget constraints and unbiasedness. The determination of an optimal sampling
plan is posed as a mixed integer programming problem. The author applies the
methodology to �nd the optimal sampling program of a chloride plume distribution.
In this application a design of a sampling network that selects sampling locations and
sampling times was demonstrated. The objective was to minimize the variance of
concentration estimation error along the cells bordering a river that is in the region
at a given time. Surprisingly, the optimized sampling plan yielded a solution such
that each chosen sampling location had to be sampled during the entire sampling
period. Loaiciga attributed this result to the quasi-steady nature of the contaminant
plume. Our results from chapter 4 support the idea that this uninterrupted sampling
schedule may be a consequence of the time-uncorrelated model errors used by Loaiciga
when deriving the contaminant concentration covariance matrix.
Stochastic methods based on PDEs
When a stochastic transport equation is used in the modeling of a contaminant plume,
transport parameters, boundary and initial conditions can be random variables. So, in
stochastic modeling sources of uncertainty could be parameter variability, boundary or
85
Table 5.2: Sampling network design using geostatistical methods. RI7401 [28],LH8901 [22], c solute concentration, rf rainfall.
RI7401 LH89101
Objective
function
Error variance Error variance
Sampling
dimensions
2 2
Estimationmethod
Kriging Kriging
Kind of data rf c
Covariance
function
Product
factorization
Obtained
from transport
equation
initial conditions variability and measurement errors. The model does not describe a
single plume but a set of possible plumes. The characteristics of each possible plume
depend on the probability characteristics of the parameters and of the initial and
boundary conditions. When using this kind of description usually only the �rst two
moments of the pollutant concentration are estimated. If concentration, hydraulic
heads and/or hydraulic conductivity data are available, parameters and pollutant
concentration moments can be estimated using the equation and the data.
Three works are presented here in which sampling networks are designed for the
estimation of hydrologic variables using stochastic methods based on partial di�er-
ential equations (see Table 5.3). Andricevic [2] and Yangxiao et al. [37] work with
the ow equation while Graham and McLauglin [12] work with the transport equa-
tion. In Table 5.3 some di�erences can be appreciated in the way the coe�cients are
represented and the data is used in these three works. An explanation of the di�er-
ences between Kalman �lters and extended Kalman �lters can be found in Jazwin-
ski [17].
86
Table 5.3: Sampling network designs using stochastic methods based on PDEs.AR9301 [2], GW8902 [12], YZ9101 [37], v velocity, K conductivity, T transmissivity,Sy speci�c yield, S storage coe�cient, bc boundary conditions, f external uxes, cconcentration, h hydraulic head, w model error, Q model error covariance, Efg ex-pected value, Phh hydraulic head covariance, Pcc solute concentration covariance, Pvvvelocity covariance, Pcv concentration-velocity cross covariance, PhT hydraulic head-transmissivity cross covariance, Phf hydraulic head-external uxes cross covariance.
AR9301 GW8902 YZ9101
Objective To formulate
coupled
withdrawal and
sampling designs
for groundwater
supply models
To develop a
stochastic
description of
transient solute
plumes
To monitor
spatiotemporal
changes of
groundwater
head, caused by
groundwater
abstraction
Equation
dimensions
2 2 2
Aquifer
layers
1 1 2
Sampling
dimensions
2 2 2
Variables
estimated
Efhg,Phh, PhT , Phf
Efcg, Efvg,EflnKg,Pcc, Pvv , Pcv
Q, Ra,
S, Sy, T
Estimation
method
Kalman �lter Extended
Kalman �lter
Kalman �lter
Randomcoe�cients
bc, f , T v w
Kind of data h c, h, K h
87
Andricevic proposes a coupled formulation of withdrawal and sampling designs for
groundwater supply models. He employs a sequential approach: the withdrawal de-
sign is conditioned on collected measurements, while the hydraulic head response on
the withdrawal design guides the future development of the sampling network. The
author describes the withdrawal design as a discrete time optimal control problem,
and he solves it by a loop stochastic control method. A random penalty-type additive
cost function is used as the objective function. The cost function is decomposed into
the deterministic and stochastic parts. A Bayesian framework is use for the mini-
mization of the deterministic part of this objective function. The sampling design's
objective is minimizing the uncertainty in the objective function of the groundwater
withdrawal program and to reduce the uncertainty in the measured variable.
The sampling criterion is expressed mathematically as the sensitivity of the ob-
jective function stochastic part of the withdrawal design to the uncertainty in the
hydraulic head distribution multiplied by the variance of the hydraulic head. The
ow equation is employed to predict output uncertainty in hydraulic heads through
�rst and second moment analysis, and the Kalman �lter algorithm is used to condi-
tion these moments with data. The algorithm looks for the best locations to measure
hydraulic heads to minimize the sampling criterion. In the sequential design, mea-
surements are used to update the covariance matrix of the estimation error hydraulic
head, which in turn changes the objective function for the withdrawal design. Reduc-
tion in hydraulic conductivity, external uxes, and boundary condition uncertainties
due to the hydraulic head measurements are considered when the hydraulic head
covariance is updated.
Yangxiao et al. [37] combined the parameter estimation procedure proposed in
a previous work [34] with a network design problem. The objective of the sampling
design is to monitor spatiotemporal changes of groundwater heads, caused by ground-
water abstraction. The only uncertain term considered in the ow equation is the
88
model error. The authors propose to use a Kalman �lter to estimate ow equation
parameters (called deterministic parameters) and some parameters associated with
the model error covariance matrix (called stochastic parameters). The calibration is
performed for a period in which the statistical stationary conditions are met and in
which all the matrices required in the Kalman �lter algorithm are assumed �xed (they
do not change with time). The method estimates the parameters sequentially: �rst an
estimate of the covariance model error is proposed and the �rst calibration round of
deterministic parameters is performed, then these parameters are kept constant and
the �rst round of stochastic parameters is obtained. These two steps are repeated
several times until a preestablished error criterion for both sets of parameters is met.
Two �eld examples were analyzed. In both the sampling frequencies were kept
�xed and the network densities were minimized under the constraint of a given thresh-
old value for the standard deviation of the estimation error. Several alternatives were
analyzed and the best one was chosen by inspection. The authors discuss the relative
importance of spatial network density and sampling frequency relating them with the
response time of the system. They found that if the system reacts fast, the spatial
optimization of the network is important. If the system reacts slowly, both temporal
optimization and spatial optimization are important.
The model errors considered by Yangxiao et al. in this work are uncorrelated on
time. It is unknown how important are the time correlations of the model errors for
the ow equation. It could be expected that these correlations are not as important as
is shown in chapter 4 are for the transport equation because the ow solution usually
reaches steady state in a short period of time. This does not happen often with the
transport solutions.
Graham and McLaughlin developed a stochastic description of transient solute
plumes in a series of two papers [12,13] (see Table 5.3). Their work has consequences
for sampling network design because they make this description site speci�c combin-
89
ing stochastic equations and data. The method works sequentially: prior moments
are obtained, samples are taken from regions with predicted high uncertainty, then
moments are conditioned on new data, and a new set of samples is chosen using the
predicted variance. The number of samples chosen at each round is decided arbitrar-
ily. In a later work [14] the authors applied this methodology to a �eld problem (a
tracer test). Their main interest was to evaluate the performance of the stochastic
model and they did not provide a sampling design analysis.
In contrast with Graham and McLaughlin's approach in our method we chose
sampling locations and its sampling schedule for a period of time, in our method
there is not need to collect samples after a sampling desicion is made to keep going as
is needed in Graham and McLaughlin method. This makes possible to decide as part
of the process the number of samples to be taken at each time instead of deciding
this number arbitrarily. Also, we decide where to sample and when to sample based
on the reduction of the concentration estimate variance at all locations at all times
which does not necessarily coincides with the locations with greatest variance.
5.3 Sampling Design Methodology
We now evaluate the estimation method proposed in the previous chapter in the
context of groundwater quality sampling network design. The hypothetical examples
presented in this chapter are two-dimensional.
Following the notation used in the one-dimensional example of chapter 4, we denote
the vector containing contaminant concentrations at all locations and all times as C.
We order them by placing together the concentration for all locations at a �xed time,
and within a given time we put �rst the concentrations corresponding to the �rst row
of the estimation mesh, then those which correspond to the second row and so on.
90
We de�ne a row of the mesh by the nodes on the mesh with the same y coordinate.
The concentrations are ordered by increasing time:
C = (c111; c121; : : : ; cNM1; c112; c122; : : : ; cNM2; : : : ; cNMT );
where cijl is the concentration at location (xi; yj) at time tl. The corresponding
covariance matrix follows the same order. For example, if we work with a rectangular
mesh with four nodes, (x1; y1), (x1; y2), (x2; y1), (x2; y2), and we want to estimate the
concentration on these nodes at two times, t1; t2, then we would have eight concentra-
tion values associated with eight space-time positions. The concentration covariance
matrix would be,
P =
0BBBBBBBBB@
P1;1;1;1;1;1 P1;1;1;1;2;1 P1;1;1;2;1;1 P1;1;1;2;2;1 P1;1;1;1;1;2 � � � P1;1;1;2;2;2
P1;2;1;1;1;1 P1;2;1;1;2;1 P1;2;1;2;1;1 P1;2;1;2;2;1 P1;2;1;1;1;2 � � � P1;2;1;2;2;2
......
......
.... . .
...
P2;2;2;1;1;1 P2;2;2;1;2;1 P2;2;2;2;1;1 P2;2;2;2;2;1 P2;2;2;1;1;2 � � � P2;2;2;2;2;2
1CCCCCCCCCA;
where Pijl;pqs is the covariance of cijl and cpqs.
The way in which the estimation method developed in the last chapter is coupled
with the sampling design algorithm is illustrated in �gure 5.1 with a owchart. As
was explained in that chapter, in the estimation method a stochastic system of equa-
tions is used to model the problem. An initial (also referred as prior in this work)
contaminant concentration estimate and its covariance matrix are obtained from this
system through stochastic simulation. The prior estimate is conditioned on data using
a Kalman �lter. The variance of the posterior estimates obtained from the Kalman
�lter, does not depend on the value of the contaminant concentration in the samples
�ltered, it only depends on the spatiotemporal location of the sample. Therefore, it is
possible to evaluate the quality of the estimates produced by sampling networks with
di�erent sampling schedules and to choose the one that satis�es a level of certainty
with the smallest number of samples.
91
Selection of sampligprogram
Stochastic simulation:Prior estimate
Kalman filter:Posterior estimate
Figure 5.1: Flowchart for the proposed methodology.
5.3.1 Source Concentration Random Field
In the examples presented in chapter 4 the only source of uncertainty considered was
hydraulic conductivity. We think that a second important source of uncertainty when
modeling contaminant plumes is the contaminant concentration at the source.
Often a trend can be observed in a time-series of groundwater contaminant con-
centrations sampled from a single well. However large deviations from the trend also
can be noticed. When comparing time-series from di�erent wells located close to each
other, the curves described by each contaminant concentration series often are similar:
the deviations from the contaminant concentration trend at one sampling well seem
to be related to the deviations from the contaminant trend at other wells. We think
that the basic behavior of each time-series is related with the behavior of the contam-
inant concentration at the source. If the volume of the leaking contaminant source
is constant, the amount of contaminant leaked to the aquifer at a given time may
depend on di�erent factors. An obvious one is the in�ltration at the source, which in
turn strongly depends on rainfall events. This can explain the peaks present at each
92
contaminant series and the similitude observed at di�erent wells. Pettyjohn [27] stud-
ied the causes of groundwater quality uctuations in shallow and sur�cial aquifers,
his interpretation concurs with ours.
In the examples presented here, we model the contaminant concentration at the
source as a random variable. To do this, we incorporate in the stochastic simulation a
random term that is added to a deterministic function that models the contaminant
concentration at the source.
5.4 Example Problems
A synthetic problem is presented to evaluate the contaminant concentration estimates
produced by the method that we propose. In �gure 5.2a a contaminant source is
located on the left hand side of a square region with 0.5 miles side length. On
the right-hand side a river is present. We want to choose a contaminant-sampling
program to obtain an estimate of contaminant concentrations of the moving plume
during a two-year period. The concentrations will be estimated on the nodes of what
we call the Kalman �lter mesh, it is shown in �gure 5.2a. Six estimation times are
considered during this two-year period, that is, every 121.66 days an estimate for the
plume is obtained. The sampling program consists of a sampling schedule for wells
located on the nodes of the Kalman �lter mesh. The possible sampling times coincide
with estimation times. Well locations and sampling times must be selected. In this
example, the criterion to evaluate the estimates is a function of the estimate variance
but other criteria can be used.
The Kalman �lter mesh is playing the role of two meshes that in general could be
di�erent. These are the estimation mesh and the sampling mesh. The �rst one would
include all the nodes on which contaminant concentration estimates are wanted, and
the second one would include all the nodes that are possible sampling locations.
93
Riv
er
Contaminantsource
∂∂
h
y= 0
∂∂
h
y= 0
h = 0
h = 50
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c
∂∂
c
y= 0
∂∂
c
y= 0
∂∂
cx
= 0
(a) (b)
Figure 5.2: a) Problem set up, Kalman �lter mesh, and boundary conditions for ow (h is in ft). b) Stochastic simulation mesh and boundary conditions for trans-port.
5.5 Contaminant Transport Simulation
A second mesh, which we call the stochastic simulation mesh, is required for the
numerical solution of the transport equation. We divided the domain into 40 �40 equally sized elements. The stochastic simulation mesh is shown in �gure 5.2b.
Note that the Kalman �lter mesh is a submesh of the stochastic simulation mesh.
Boundary conditions for ow and transport are included in �gures 5.2a and 5.2b,
respectively. Concentrations are in ppm and hydraulic heads are in feet. Forty-
eight time-steps are used to simulate the two-year period. The contaminant source is
active during all of this period. The Princeton Transport Code (PTC) [4], a three-
dimensional numerical simulator, is used in two dimensional mode to solve these
equations.
94
5.6 Statistical Properties of the Hydraulic Conductivity and
the Contaminant Concentration at the Source
We assume for the purposes of this example that the conductivity �eld is lognormally
distributed, homogeneous, stationary, and isotropic. The mean value of the �eld is
3.055 and the semivariogram that represents the log-conductivity spatial correlation
structure is
K(h) = �2logK
"1� exp
� h
�logK
!#;
where �2logK is the variance of logK, and �logK is the correlation scale. The correlation
scale is the separation at which the correlation decreases to the e�1 level [11].
Concentrations at the source are modeled as identically distributed independent
random functions. On each node, the concentration is represented as a time series,
with
c(t) = exp(�14 + 3t + e(t));
where, e(t) is a zero-mean random perturbation, normally distributed and with vari-
ance 0.1948. While the variance was obtained from the analysis of a time series of
�eld data, the form of the exponent is hypothetical. For each source node, at each
simulation time-step, a di�erent random perturbation is used. The time correlation
of the random perturbations is modeled with the variogram
c(t) = 0:1948�1� exp
�� t
�c
��;
where �c is the concentration correlation at the source.
We use a method called sequential Gaussian simulation (SGS) from the GSLIB
package [8] to obtain hydraulic conductivity realizations. This method was chosen
because of its e�ciency.
A set of tests was done to check the convergence of the method described above
with respect to the number of realizations. The analysis of the results is presented
95
in appendix 6. It was concluded that for all the tests done in this work 3,000 plume
realizations are enough to obtain convergence.
5.7 Sampling Design Criteria
As was mentioned erlier, we use a function of the estimate variance as a measure of the
estimate error. The value of this function at a given space-time location depends on
the samples taken erlier. One sampling location at a time is chosen. The one selected
is that which reduces the function of the estimate variance the most, given previous
sampling decisions. The variances of the concentration estimates are obtained from
the Kalman �lter. The variances after taking n samples are the elements on the
diagonal of the covariance matrix P n. Sampling stops when the value of the function
of the estimate variances is less than a preestablished value. In the examples following
we use each new sample to estimate the concentration and their variances in the
whole two-year period considered. Then, a sample taken at a given time, contributes
to estimating concentrations at both past times and future times.
The function that we use in these examples to evaluate the quality of a given
estimate is the total variance, �2T . The total variance of the estimate obtained con-
ditioning on n samples is the sum of these estimate variances over all locations and
times. For these examples this is,
�2T(n) =Xi;j;l
�2ijl(n);
where �2ijl(n) is the variance of the estimate at the i; j location on the Kalman �lter
mesh and at the l-th estimation time. As was mentioned before, the variances of
the estimate obtained, conditioed with n samples, are the diagonal elements of the
covariance matrix P n. The formula used to minimize the total variance is explained in
appendix A. This formula is equivalent to that obtained by Rouhani for kriging [29].
96
Table 5.4: Input for test 1.
�logK (ft) �2logK �c (days) �2c �L (ft) �T (ft) � �x (ft) �y (ft) �t (days)
264 0.7 11 0.195 33 3.3 0.25 66 66 15.21
In the following examples the criterion employed to stop the sampling program
is obtaining an estimate with a total variance less than or equal to a given value.
This value is determined using the expected number of nodes, nd, that exhibit a
concentration larger than a given threshold. In these examples the threshold is 1
ppm. Thus, the number nd is calculated by counting the number of nodes with
concentration larger than 1 ppm in the prior estimate. We calculate the average
variance of the estimates using �2T /nd and we impose the condition that this average
variance be less than 1 ppm2.
5.8 Sampling Program. Test 1
All the examples analyzed here assume samples with no error. Parameter values for
this test are shown in table 5.4. The sampling program chosen by the algorithm when
minimizing the total variance is shown in �gure 5.3. The �gure includes six squares
that represent the Kalman �lter mesh at the six sampling times. The numbers indicate
the order in which the samples are chosen. Number 1, for example, indicates that the
�rst sample chosen is at the contaminant source at the sixth sampling time: (x1; y4; t6).
There are thirty-nine samples in the program. This is the number of samples necessary
to obtain a total estimate variance less than or equal to 52 ppm2 (there are 52 nodes
with expected concentration values larger than 1 ppm, i.e. nd = 52).
Twelve of the �rst twenty sampling locations are chosen at the source. The other
eight samples are at the two central rows of the sampling mesh, either at the �fth
97
t=4
7
8
29
30
31
32
33
t=5
3
45
6
11
12 15
16
21
22
23
28
35
t=6
1
2 19
20
2527
34 36
37
38
t=1
17
18
t=2
13
14
39
t=3
9
10
24
26
Figure 5.3: Sampling program test 1, 39 samples.
or sixth times. This tendency to �rst place the samples at the contaminant source
is due to the large concentration variance at those locations. Checking the estimate
variance values at the di�erent space-time locations before and after taking a sample
from the source, we note that the variances remain essentially unaltered except at
the place where the sample is taken. This means that sampling at the source does
not give signi�cant information about the concentration at other locations.
This can be con�rmed by comparing the plot of the total variance against the
number of samples (�g. 5.4) with the corresponding plot for the maximum variance
against number of samples (�g. 5.5). The total variance of the prior estimate is about
310,000 ppm2, and the maximum variance about 60,000 ppm, when the �rst sample is
taken at the source location (x1; y3; t6) the total variance is reduced to about 250,000
ppm2. In other words, the location of the maximum variance is chosen �rst, and
consequently about 92% of the reduction in the total variance is due to the reduction
of the concentration variance at that location. In comparison, when the third sample
is chosen at location (x2; y3; t5) the variance at that locations accounts for 32% of the
reduction in the total variance.
98
2 4 6 8 10samples
50000
100000
150000
200000
250000
300000
sT2
Test 3
Test 2
Test 1
Figure 5.4: Total variance vs. number of samples for tests 1, 2, and 3. Samples 1-10.
2 4 6 8 10samples
10000
20000
30000
40000
50000
60000
sM2
Test 3
Test 2
Test 1
Figure 5.5: Maximum variance vs. number of samples for tests 1, 2, and 3. Samples1-10.
All of the �rst twenty samples are chosen on the third and fourth rows of the
sampling mesh. This suggests that to reduce the total variance of the concentration
estimate it is important to obtain �rst the central tendency of the plume. The last
nineteen samples appear to de�ne the spreading of the plume. Fourteen of them
are located where the prior plume has its boundaries. These boundaries are shown
in �g. 5.6a. The number of locations selected at a given time increases with time,
reaches a peak at t = 5, with thirteen samples, and decreases to ten samples at t = 6.
The sampling time that gives the most information is then t = 5. It is interesting to
note that this is not the time at which the expected plume has the largest variances;
this time is t = 6.
99
5.9 Plume Estimate Analysis. Test 1
To get a feeling for the quality of the plume estimates that are obtained by the
proposed method we compare these estimates with a preselected plume from which
samples are hypothetically taken. The preselected or "observed" plume is chosen
arbitrarily from the set of realizations. Figure 5.6a shows the prior estimate and
the observed plume. The observed plume is shown in white contours and the prior
estimate in black contours. The prior estimate is the plume calculated from the
stochastic simulation. A logarithmic scale is used. As can be observed, the prior
estimate estimates the extent of the plume well but not the spreading. It has a
symmetric shape, as expected, but the observed plume leans toward the upper part
of the region. Also, at the last three illustrated times the observed plume has a
bifurcation in the middle.
Figure 5.6b compares the plume estimate, obtained using the �rst ten samples
chosen, with the observed plume. The plume estimate is shown in black contours.
Sampled locations are marked with black bullets. The e�ect of combining the prior
estimates with these ten samples, through the Kalman �lter, is the elongation and
spreading of the plume estimate. The general shape of the plume remains the same.
Now, slightly larger values are obtained in the upper part of the domain than in the
lower part. The spreading of the plume is captured better, but the length of the
plume is overestimated.
When ten more samples are used (�gure 5.6c), the estimate captures better the
shape of the observed plume, especially at the last two times, where six samples
have been added. Note that the plume shape is changed substantially for the fourth
estimation time although no samples have been added there.
When a program of thirty samples is used (�gure 5.6d), the shape of the plume is
captured better at the second, third, and forth estimation times, but the spreading
100
t=1 t=2 t=3
t=4 t=5 t=6
t=1 t=2 t=3
t=4 t=5 t=6
t=1 t=2 t=3
t=4 t=5 t=6
t=1 t=2 t=3
t=4 t=5 t=6
(a) (b)
t=1 t=2 t=3
t=4 t=5 t=6
t=1 t=2 t=3
t=4 t=5 t=6
t=1 t=2 t=3
t=4 t=5 t=6
t=1 t=2 t=3
t=4 t=5 t=6
(c) (d)
t=1 t=2 t=3
t=4 t=5 t=6
t=1 t=2 t=3
t=4 t=5 t=6
(e)
Figure 5.6: Comparison of the observed plume and the plume estimates (logarithmicscale). The observed plume is in white contours and the estimates are in blackcontours. Black dots indicate sampling locations. a) Prior estimate. Plume estimatefor a sampling program of: b) 10 samples, c) 20 samples, d) 30 samples, and e) 39samples.
101
at the �rst time is overestimated. The width of the lower plume �nger is corrected
(four samples are added in that zone) but the upper �nger is overestimated. At the
last time the direction in which the observed plume is leaning is captured better.
For a program of thirty-nine samples (�gure 5.6e) the estimate gets very close to the
reference plume.
5.10 Sensitivity Analysis
In this section we analyze the e�ects of changing di�erent statistical parameters on
the sampling program, the total variance, and the maximum variance of the esti-
mates. Di�erent values of the correlation scale of the logK, the variance of this
variable, and the time-correlation scale of the concentration at the source are consid-
ered. Table 5.5 contains the values used for these parameters for the di�erent test
cases. The coe�cients of the transport equation are kept the same as for test 1 (see
table 5.4). Gelhar [11] presents a table of data on variance and correlation scale of
saturated log-hydraulic conductivity from several sites of di�erent dimensions. Our
parameters agree with those values. On the other hand, the discretization used, with
�x = �y = 66 ft, and the log-hydraulic conductivity variances and correlation scales
used in these examples satisfy the criteria
� � 1 + �2logK
proposed by Ababou [1] with exception of test 3. In this test � = 1:3 and 1+�2logK =
1:7. Since the di�erence between the two numbers is small we believe that we are
not introducing a large error by using these parameters. Also, the results of this test
seem to be congruent with the results of the other tests.
102
Table 5.5: Parameter values for the six tests.
Test 1 Test 2 Test 3 Test 4 Test 5 Test 6
�logK(ft) 264 176 88 264 264 264
�2logK 0.7 0.7 0.7 0.7 1.0 1.3
�c(days) 11 11 11 33 11 11
5.10.1 Correlation Scale of Hydraulic Conductivity
We analyze the results for tests 1, 2, and 3 to evaluate the e�ects of changing the
correlation scale of logK on the sampling programs. One e�ect is the modi�cation
of the concentration-estimate variances. We will discuss this when analyzing the
behavior of the total variance for each test. For our present discussion it is enough to
say that plumes obtained from a highly correlated hydraulic conductivity �eld have
larger variances than those obtained from a �eld with weaker correlations. Also, the
expected spreading of the plumes for test 1 is larger than that for test 3. For that
reason the expected plume has more locations with expected concentrations larger
than 1 ppm for test 1 than for test 3, and more samples are needed to obtain a given
degree of certainty for the estimates of test 3 than for those of test 1. The number
of nodes with concentration larger than 1, i.e. nd is 39, 34, and 27 for test 1, test 2,
and test 3, respectively. As was explained in page 96, we stop the sampling process
for each of these tests when the total variance gets smaller than its corresponding
nd value. For test 2, 34 samples are needed to reach that value and for test 3, 27
samples.
The sampling programs chosen for the corresponding tests are shown in �gures 5.3,
5.7, and 5.8. The �rst twenty sampling locations are almost the same for all three
tests, but they are chosen in a slightly di�erent order. The only location that di�ers
is (x4; y4; t6) that is chosen in test 3 instead of (x4; y4; t5), which is chosen in the other
103
t=4
7
8
2730
32
t=5
3
45
6
11
12 17
18
23
25
29
31
t=6
1
2 19
20
24
26
28 33
34
t=1
15
16
t=2
13
14
t=3
9
10
21
22
Figure 5.7: Sampling program test 2, 34 samples.
two tests. For test 1 and test 2, the only di�erences in the selection order is for the
15th, 16th, 17th, and 18th samples. Between the �rst twenty sampling locations for
test 2 and test 3 the di�erences are the locations chosen for the 3rd, 4th, 5th, 6th,
and 18th samples. In the three tests these twenty sampling locations contain all the
possible samples at the contaminant source, that is, twelve of the twenty locations
are chosen at the source. All of these samples are taken on the third and fourth rows
of the sampling mesh. Considering the number of samples taken at each sampling
time, for the three tests, using these twenty samples the sampling time that gives the
most information is t = 5.
The last samples exhibit greater di�erences in the order in which they are chosen.
These di�erences are related with the expected spreading of the plume. The sampling
programs for test 2 and test 3 have the same sampling locations at the �rst three
sampling times. The sampling program for test 1 contains only one extra location
during these periods: (x2; y3; t2). The expected spreading of the mean (or prior) plume
has an in uence on the number of samples selected along the non-central rows and
along the last two columns. The sampling program for test 1 contains ten sampling
104
t=4
7
8
2325
26
t=5
3
4 5
6 11
12 17
t=6
1
2
18
19
20 24
27
t=1
15
16
t=2
13
14
t=3
9
10
21
22
Figure 5.8: Sampling program test 3, 27 samples.
locations on these rows and columns. In comparison, the sampling program for test
2 has only six samples along them and test 3 does not have any. For all of the tests,
only spatial locations along the central rows of the mesh are taken more than once.
Since changes in the log-hydraulic conductivity correlation scale means modi�ca-
tion of the concentration-estimates variances, the total variance is altered as well.
Figures 5.4, 5.9, and 5.10, show a comparison of the total variance versus number of
samples used to obtain the estimates. For comparison we use 40 samples in each one
of the tests. As can be observed in these �gures, when the log-conductivity correlation
scale increases, so does the total variance of the estimates. Thus, the estimates of test
1, for which logK has the largest correlation scale, have the largest total variance,
and the estimates of test 3, for which logK has the smallest correlation scale, have
the smallest total variance.
This can be explained by the behavior of the plumes in the di�erent conductiv-
ity �elds. When the �eld is highly correlated, the plume realizations tend to have
irregular shapes, like the shape of the observed plume of test 1 (�g. 5.6). This means
that for a given location the possible concentration values vary more than when the
105
12 14 16 18 20samples
5000
10000
15000
sT2
Test 3
Test 2
Test 1
Figure 5.9: Total variance vs. number of samples for tests 1, 2, and 3. Samples 10-20.
25 30 35 40samples
200
400
600
800
1000
sT2
Test 3
Test 2
Test 1
Figure 5.10: Total variance vs. number of samples for tests 1, 2, and 3. Samples20-40.
hydraulic conductivity �eld is less correlated, in which case plumes tend to have more
homogeneous shapes.
As more samples are taken, the proportional di�erences between the estimate total-
variances for the three tests increase. For example, for the prior estimates (0 samples)
the total variance of the estimate of test 1 is 316,904 ppm2, and that for the estimate
of test 3 is 267,448 ppm2; for the plume estimates wherein 20 samples are used, the
corresponding total variances are 1,018 ppm2 and 214 ppm2, respectively. This gives
a proportion of 311 for test 1 and of 1245 for test 3. This shows that the amount
of information obtained per sample decreases when the log-conductivity correlation
scale increases. The result is congruent with a decrease in the concentration �eld
correlation when the log-conductivity correlation scale increases.
106
12 14 16 18 20samples
500
1000
1500
2000
2500
sM2
Test 3
Test 2
Test 1
Figure 5.11: Maximum variance vs. number of samples for tests 1, 2, and 3. Samples10-20.
In contrast the maximum variances of the estimates have values that are very
similar to each other when less than 20 samples are used (�gs. 5.5, and 5.11). It is
natural to have the same maximum variance for these tests because, for the prior
estimate, the maximum variance at each time occurs at the source locations. Since
the contaminant concentration realizations at these nodes are the same for the three
tests, at the source nodes we expect to have the same prior variances.
As was discussed above for test 1, the �rst sample is selected at one of the space-
time locations where the estimate has maximum variance. When the sample is taken
at that location the concentration variance goes to zero there and the variance of
the estimate at the neighboring location, (x1; y4; t6), is reduced. For that reason the
estimate maximum variance is reduced (originally these two points had the same
variance). The second sample is taken again at the location where the estimate has
maximum variance and the maximum variance drops down at that point on the curve.
The third sampling location is the same for tests 1 and 2 but not for test 3. For this
last test the location sampled is at the source but this is not the case for the other
two tests. In any event, the maximum variance remains the same for the three tests.
This is because even when a location with maximum estimate variance is selected
for test 3, there is a second location with the same estimate variance, this location
107
25 30 35 40samples
20
40
60
80
sM2
Test 3
Test 2
Test 1
Figure 5.12: Maximum variance vs. number of samples for tests 1, 2, and 3. Samples20-40.
is (x1; y3; t5). When that location is sampled next, the maximum estimate variance
for test 3 is reduced but not for tests 1 and 2. From this discussion it is clear that
a at portion of the plot indicates that either the maximum has not been sampled
or there is a second point where the maximum is attained. More di�erences between
maximum variances can be observed when a maximum is attained in the interior of
the region, for example when 10 samples have been taken. When more than twenty
locations have been sampled (�g. 5.12), and therefore all the possible samples at the
source have been taken, there are more di�erences in the three plots.
5.10.2 Correlation of the Contaminant Concentration at the
Source
Now we analyze the e�ect of the correlation-scale of the contaminant concentration at
its source. The two tests conducted for this section, test 1 and test 4, have 52 nodes
with expected concentration larger than 1 ppm2, and we use sampling programs with
39 samples for both. The �rst twenty one sampling locations selected are identical for
tests 1 and 4 (�gs. 5.3 and 5.13). After this, the locations selected are the same with
the exception of three samples, but the order in which they are chosen is di�erent. The
locations that di�er are as follows: for test 1, (x3; y4; t4), (x5; y3; t4), and (x5; y2; t6) are
108
t=4
7
8
2831
32
t=5
3
45
6
11
12 15
16
21
24
25
29
37
t=6
1
2 19
20
2627
30 33
34
38
39
t=1
17
18
t=2
13
14 35
36
t=3
9
10
22
23
Figure 5.13: Sampling program test 4, 39 samples.
selected but not for test 4. For test 4, locations (x3; y3; t6), (x4; y3; t6), and (x4; y4; t6)
are included instead. So, when concentration correlations at the source are larger,
more weight is placed on the central part of the plume at the last sampling time and
in test 1 these samples are taken either on the boundary of the prior plume or on the
central portion of the plume at time t = 4.
The total variances for tests 1 and 4 are very similar. For the ten �rst samples
they are identical, so we do not show the plots. There are some small di�erences for
the variances obtained between the 10th and the 40th samples (�gures 5.14 and 5.15).
It is surprising that the estimate variances for test 1 get smaller than those for test 4.
One could expect the contrary because the correlation at the source is larger for the
last test. One reason for obtaining this similitude in the total variance could be due
to the long sampling steps that are used. Both tests have a time-correlation scale at
the source smaller than the sampling steps (that are equal to about four months).
The maximum variances for these two tests are very similar as well. They are
almost identical for the �rst ten samples (for this reason we do not show the plots).
There are some small di�erences at the 13th, 18th and 19th samples (�gure 5.14).
109
12 14 16 18 20samples
5000
10000
15000
sT2
Test 4
Test 1
Figure 5.14: Total variance vs. number of samples for tests 1 and 4. Samples 10-20.
25 30 35 40samples
200
400
600
800
1000
sT2
Test 4
Test 1
Figure 5.15: Total variance vs. number of samples for tests 1 and 4. Samples 20-40.
Even when the sampling locations are the same, the estimate variances are a little
di�erent. The maximum variances of the last twenty samples have more di�erences
(�gure 5.15). From the shape of the plots it seems that the sampling program for
test 1 contains the locations with maximum estimate variance more often than the
one for test 4 (a at portion of the plot indicates that the maximum has not been
sampled).
5.10.3 Variance of the Hydraulic Conductivity Field
A comparison of the results for tests 1, 5, and 6 show the e�ect of changes in the
variance of the hydraulic conductivity �eld on the sampling program and the plume
estimates. The prior plume obtained from a hydraulic conductivity �eld with a large
110
12 14 16 18 20samples
500
1000
1500
2000
2500
sM2
Test 4
Test 1
Figure 5.16: Maximum variance vs. number of samples for tests 1 and 4. Samples10-20.
25 30 35 40samples
20
40
60
80
sM2
Test 4
Test 1
Figure 5.17: Maximum variance vs. number of samples for tests 1 and 4. Samples20-40.
variance has larger variances than a plume obtained from a hydraulic conductivity
�eld with a smaller variance. So the expected spreading of the plumes for test 1 is
smaller than that for test 6. The number of nodes with concentration larger than 1,
i.e. nd, are 60, and 67 for test 5 and test 6 respectively. For test 5, 42 samples are
used and for test 6, 27 samples.
The sampling programs selected for these three tests are shown in �gures 5.3, 5.18,
and 5.19. As in all the previous tests, the �rst twenty sampling locations contain
all the possible samples at the contaminant source. The most signi�cant di�erences
between the sampling programs are the number of samples taken at extreme locations
(rows 2 and 5, and columns 5 and 6): for test 1 we have ten samples, for test 5
twelve, and for test 6 �fteen. Again a heavy weight is given to the central portion of
111
t=4
7
8
38
39
40
t=5
3
45
6
9
10 15
16
19
21
23
25
2731
33
36
t=6
1
2
20
22
24 30
32
34
35
37
41
t=1
17
18
t=2
13
14 42
t=3
11
12
26
28
29
Figure 5.18: Sampling program test 5, 42 samples.
t=4
17
8 38
40
41
t=5
35
6
9
10 14
15
17
21
22
23
24
25
33
35
36
t=6
2
4 18
26
27
29
31
32
34
39
42
44
46
t=1
19
20
t=2
13
16
37
45
t=3
11
12 28
30
43
Figure 5.19: Sampling program test 6, 46 samples.
the sampling mesh and sampling time t = 5 is the one from which more samples are
taken in each test.
Figures 5.20, 5.21, and 5.22, show a comparison of the total variance versus
number of samples used to obtain the estimates. We use 40 samples in each one of
the tests. As can be observed in these �gures, when the log-conductivity variance
increases so does the total variance of the estimates. So, the estimates of test 1, for
112
2 4 6 8 10samples
50000
100000
150000
200000
250000
300000
350000
sT2
Test 6
Test 5
Test 1
Figure 5.20: Total variance vs. number of samples for tests 1, 5, and 6. Samples 0-10.
12 14 16 18 20samples
5000
10000
15000
20000
25000
sT2
Test 6
Test 5
Test 1
Figure 5.21: Total variance vs. number of samples for tests 1, 5, and 6. Samples10-20.
which the variance of logK has the largest value, have the largest total variances.
As more samples are taken, the proportional di�erences between the estimates total-
variances for the three tests increase. That is, the amount of information obtained
per sample decreases when the log-conductivity variance increases.
The maximum variance of the estimates for tests 1, 5, and 6 are shown in �g-
ures 5.23, 5.24, and, 5.25. Again, for the �rst eighteen samples the maximum variance
plots are very similar for the three tests because most of the time the maximum is at-
tained at source locations. After all the space-time source locations are sampled more
di�erences can be observed in the graphs. Test 6 has the largest maximal estimate
variances, test 5 the second larger, and test 1 the smallest.
113
12 14 16 18 20samples
5000
10000
15000
20000
25000
sT2
Test 6
Test 5
Test 1
Figure 5.22: Total variance vs. number of samples for tests 1, 5, and 6. Samples20-40.
2 4 6 8 10samples
10000
20000
30000
40000
50000
60000
sM2
Test 6
Test 5
Test 1
Figure 5.23: Maximum variance vs. number of samples for tests 1, 5, and 6. Samples1-10.
5.11 Residual Analysis
In his classic book on �ltering theory, Jazwinski [17] suggests using the predicted
residuals to evaluate the performance of the �lter. The n+1-th residual is de�ned as
the estimate error given the previous n data z1; z2; : : : ; zn. In practice the residuals
can be calculated only at locations where data is available but in our examples, when
we analyze the estimates for a previously selected \observed" plume, we can calculate
the residuals at all locations each time a knew sample is added. The vector of residuals
that we use is
rn = Z � Cn;
114
12 14 16 18 20samples
1000
2000
3000
4000
5000
6000
sM2
Test 6
Test 5
Test 1
Figure 5.24: Maximum variance vs. number of samples for tests 1, 5, and 6. Samples10-20.
25 30 35 40samples
50
100
150
200
250
300
350
sM2
Test 6
Test 5
Test 1
Figure 5.25: Maximum variance vs. number of samples for tests 1, 5, and 6. Samples20-40.
where Z are the contaminant concentrations of the reference plume at all the space-
time locations and Cn is the concentration estimate obtained from the Kalman �lter
conditioning with the n data z1; z2; : : : ; zn.
By de�nition the covariance matrix of the residuals is P n (eq. 4.24) and the vari-
ances of these residuals are the values on the diagonal of this matrix. Therefore,
assuming that the contaminant concentration at each space-time location is normaly
distributed we can obtain con�dence intervals for the residuals using the correspond-
ing variances predicted by the Kalman �lter.
For the examples presented in this work to analyse the plume estimates it would be
almost impossible to make a separate analysis for the residuals of the concentration
115
estimates at each node and at each time. Instead we analyze the addition of the
components of the residual vector. To this end we de�ne the total residual by
tot res(n) =Xijl
(zijl � cnijl); (5.1)
where cnijl is the Kalman �lter estimate at location xi;j at time tl when n samples are
used, and zijl is the concentration of the \observed" plume at the same location. The
indices i; j correspond to the Kalman mesh nodes and l ranges over the six sampling
times.
Since each residual rnijl = zijl � cnijl is assumed normaly distributed so is the total
residual tot res(n). The mean of the residuals is zero because cn is an unbiased
estimate, for this reason Eftot res(n)g = 0.
By de�nition
tot res(n) =Xijl
rnijl =NXs=1
rns ;
where s is the position in the space-time vector C (see page 5.3 for the de�nition of
this vector) of the concentration cijk. It is not di�cult to prove that [15]
Var(tot res(n)) =NX
s1=1
NXs2=1
P ns1;s2: (5.2)
Note that by de�nition P ns;t = Cov(rns ; r
nt ). We denote by �2TR(n) the variance of the
total residual.
We approximate the 95% con�dence interval for the total residual with the interval
(tot res(n)� 2�TR(n); tot res(n) + 2�TR(n)):
In �gures 5.26 to 5.31, we show the total residual obtained using a number of samples
versus the number of samples and the con�dence interval limits. For simplicity we
show the total residual value for forty samples in all the tests. The total residual is
shown with a dotted line and the con�dence interval limits with continuous lines. For
each one of these �gures we expect to have about 95% of the total residual values
116
5 10 15 20smpls
-2000
-1000
1000
2000
25 30 35 40smpls
-100
-50
50
100
(a) (b)
Figure 5.26: Total residual and con�dence interval for test 1. Total residual is shownwith a dotted line and the con�dence limits with a continuous line. a) First twentysamples. b) Last twenty samples
5 10 15 20smpls
-1500
-1000
-500
500
1000
1500
25 30 35 40smpls
-75
-50
-25
25
50
75
(a) (b)
Figure 5.27: Total residual and con�dence interval for test 2. Total residual is shownwith a dotted line and the con�dence limits with a continuous line. a) First twentysamples. b) Last twenty samples
inside the con�dence interval. As can be observed, that proportion is satis�ed in each
one of the tests.
5.12 Conclusions
We developed a linear estimation method that can accommodate several sources of
variability and that can be used in the design of groundwater quality sampling net-
works in which sampling locations and sampling times are decision variables. Taking
advantage of current modeling practices, the estimation method uses a determinis-
tic model developed for a given groundwater quality problem. There is no need to
117
5 10 15 20smpls
-1000
-500
500
1000
25 30 35 40smpls
-40
-20
20
40
(a) (b)
Figure 5.28: Total residual and con�dence interval for test 3. Total residual is shownwith a dotted line and the con�dence limits with a continuous line. a) First twentysamples. b) Last twenty samples
5 10 15 20smpls
-2000
-1000
1000
2000
25 30 35 40smpls
-100
-50
50
100
(a) (b)
Figure 5.29: Total residual and con�dence interval for test 4. Total residual is shownwith a dotted line and the con�dence limits with a continuous line. a) First twentysamples. b) Last twenty samples
5 10 15 20smpls
-2000
-1000
1000
2000
25 30 35 40smpls
-150
-100
-50
50
100
150
(a) (b)
Figure 5.30: Total residual and con�dence interval for test 5. Total residual is shownwith a dotted line and the con�dence limits with a continuous line. a) First twentysamples. b) Last twenty samples
118
5 10 15 20smpls
-2000
-1000
1000
2000
25 30 35 40smpls
-200
-100
100
200
(a) (b)
Figure 5.31: Total residual and con�dence interval for test 6. Total residual is shownwith a dotted line and the con�dence limits with a continuous line. a) First twentysamples. b) Last twenty samples
make use of specialized stochastic modeling software, the only software requirement
to apply the method is to have a random �eld simulator for variables with correlation
in two or three dimensions (like some of the programs in the GSLIB package [8]).
A disadvantage of the method is the large number of stochastic plume realizations
necessary to reach convergence. In these simple examples we used 3000 simulations.
The synthetic examples presented show that this method can obtain estimates with
small uncertainty for a contaminant plume in motion with a small number of water
samples. It was demonstrated that the use of the total variance of the estimates,
together with the maximum variance of the estimates provides a tool to analyze the
results with no need to analyze statistical characteristics of the estimates at each
node.
As particular conclusions we have:
� There is a tendency to �rst place the samples at the contaminant source. This
tendency is due to the large concentration variance at those locations.
� Sampling at the source does not give signi�cant information about the concen-
tration at other locations.
� There is a tendency to sample �rst on the third and fourth rows of the sampling
119
mesh. This suggests that to reduce the total variance of the concentration
estimate it is important to obtain �rst the central tendency of the plume.
� The sample locations chosen last de�ne the boundary of the plume.
� The number of samples needed to obtain a certain degree of certainty increases
with the hydraulic conductivity correlation scale and also with its variance.
� Samples have to be distributed in a wider area when the hydraulic conductivity
correlation scale is large and when the variance is large.
� For the tests presented in this work the e�ect of changes on contaminant time-
correlation at the source on the sampling program and the predicted estimate
variance are insigni�cant. An analysis of the e�ects of changes in the sampling
frequency is lacking.
A Formulas to Minimize the Estimate Variance
As is mentioned in the text, according to our strategy each new concentration sample
selected is the one that reduces the most the total variance �2T . We present in this
appendix the formulas used to choose a new sample.
To simplify the notation we denote a space-time location (xi; yj; tl) by �s, with a
single index and the variance of the concentration estimate at this location using k
samples, P ks;s. The index s indicates the position at which the variable cijk is located
in the space-time vector C de�ned in page 90. The covariance of cl and cs given k
samples is denoted by P kl;s. Let N be the total number of spaciotemporal locations of
interest.
We choose one sampling space-time location at a time and the sampling locations
is chosen from the nodes of the Kalman �lter mesh and from the six sampling times.
120
In this case the sampling matrix H is a vector. If the k-th sampling location is �j
then the corresponding sampling vector is
Hk = (0; 0; : : : ; 1; 0; : : : ; 0) ;
where the number 1 located at the j-th position. The sampling error covariance
associated with sampling at the j-th location is a number that we denot by rj.
From the Kalman �lter formulas 4.22 and 4.23, we obtain the e�ect on the variance
of the concentration estimate at location i of sampling at location j. The product
P kHT
k+1 is
P kHT
k+1 =�P k1;j; : : : ; P
kN;j
�T;
and �Hk+1P
kHT
k+1 +Rk+1
��1=
1
P kjj + rj
:
Substituting in equation 4.23 we get
Kk+1 =1
P kjj + rj
�P k1;j; : : : ; P
kN;j
�T:
Note that P k+1 in equation 4.22 can be written as
P k+1 = (I �Kk+1Hk+1)Pk;
where I is the N by N identity matrix. The matrix I �Kk+1Hk+1 is
I �Kk+1Hk+1 =
0BBBBBBBBBBBBBBBBBBB@
1 0 : : : 0 � P k1;jP kjj+rj
0 : : : 0
0 1 0 : : : � P k2;jP kjj+rj
.... . .
......
...
1� (P kjj )2
P kjj+rj
.... . . 0
0 : : : 0 � P kN;j
P kjj+rj
0 : : : 1
1CCCCCCCCCCCCCCCCCCCA
:
121
The diagonal elements of the matrix (I �Kk+1Hk+1)Pk are
P k+1ii = P k
ii �(P k
i;j)2
rj + P kjj
:
Therefore, the change on the estimate variance P kii at location �i due to a sample
from location �j is due to the term(P ki;j)
2
P kjj+rj
.
The e�ect of taking a new sample at location �j on the total variance of the
contaminant plume estimate is given by
�2T (k + 1) =Xi
P k+1ii (5.3)
=Xi
P kii �
1
rj + P kjj
Xi
�P ki;j
�2: (5.4)
The �rst sum in the last equality is the total variance given k samples. Therefore,
�2T (k + 1) = �2T (k)�1
rj + P kjj
Xi
�P ki;j)�2:
The second sum on the right-hand side of the last equality is always less than or
equal to the total variance given k samples, then �2T (k+ 1) is minimum ifP
i
�(P ki;j)
2
ri+P kii
�
is maximum. Each new sample is chosen by inspection using this formula.
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125
Chapter 6
Convergence Tests
In this chapter we determine the number of plume realizations necessary to obtain
an accurate concentration estimate. To this end we analyze the behavior of the
contaminant-estimate total variance, �2T , and of its maximum variance, �2M , as func-
tions of the number of plume realizations.
6.1 Methodology
We made two di�erent kinds of plots. The �rst kind presents the prior-estimate total
variances (or the maximum variances) as a function of the number of realizations. The
second kind shows the total variances (or the maximum variances) of the posterior
estimates obtained using 10, 20, 30, and 40 samples, versus the number of plume
realizations. The �rst kind of plot helps us in analyzing the convergence of the prior-
estimate total variance obtained from the stochastic simulation. The second kind of
plot is an aid for testing the convergence of the sampling design procedure proposed
in this work. This is, the convergence of the combination of the stochastic simulation
and the selection of the sampling program using the Kalman �lter.
It is important to note that the selection of the sampling program depends not
only on the prior estimate variance, but also on the covariance matrix of this estimate
and on the formulas used to select the sampling program. Thus, the convergence of
the prior estimate variance is not enough for the procedure to convergence.
126
The underlying idea of these tests is that if the estimate total variance pre-
dicted by the Kalman �lter when a given number of samples are �ltered does not
change much when adding more realizations, we consider that the procedure has
converged.
To obtain a relative measure of the change of the total variance when adding a
number of realizations, we de�ne the relative di�erences of the total variances:
��2T�2T
=�2T (�nr (n + 1))� �2T (�nr n)
�2T (�nr n); (6.1)
where �2T (N) is the total variance of the estimate obtained when N realizations
are used, n�nr is the current number of realizations and �nr is a �xed increment
in the number of realizations. In an analogous way we de�ne the relative di�er-
ences of the maximum variances. In the following examples the increment used is
�nr = 200. We analyze �rst the convergence of test 1. Later, we analyze the
e�ects of changing the log-conductivity correlation scale (tests 1, 2 and 3) on con-
vergence, the e�ects of changing the time-correlation of concentrations at the con-
taminant source (tests 1 and 2), and the e�ects of changing the log-conductivity
variance (tests 1, 4, and 5). The parameters de�ning each test are shown in ta-
ble 5.5.
6.2 Test 1
6.2.1 Total Variance Analysis
In �gure 6.1a the plot of the prior-estimate total-variance against number of realiza-
tions for tests 1, 2, and 3 are shown. Here we analyze the behavior of the plots for
test 1, in the next section we compare the plots for the three tests. Note that when
the number of realizations is between four hundred and sixteen hundred the total
127
1000 2000 3000 4000p.r.
270000
280000
290000
300000
310000
320000
sT2
Test 3
Test 2
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.01
0.02
0.03
0.04
D sT2 ê sT
2
Test 3
Test 2
Test 1
(a) (b)
Figure 6.1: Comparison of tests 1, 2, and 3. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plume realizations.Prior estimate.
variance increases with the number of realizations. After that, the total variance di-
minishes and after thirty-two hundred realizations it changes very little. We explain
this behavior in the following way. Di�erent plume realizations may have contami-
nant concentration values that di�er much at a given position. When few realizations
are used to obtain the estimate, the di�erence between the concentration realizations
and the mean concentration is large at several locations. This produces an increase
in the variance. After a certain number of realizations have been produced, the mean
concentration changes very little at each location and the concentration deviations
from the mean get smaller. At that point the total variance diminishes because of
the averaging process. After this, equilibrium is reached.
The relative di�erences of the prior estimate total-variances against number of
simulations are shown in �gure 6.1b. Equation 6.1 de�nes the values represented
by the vertical axis, the number of realizations in the horizontal axis corresponds
to n�nr, the smaller number used in the equation to calculate the di�erence. As
expected, the relative di�erences tend to get smaller when more realizations are used
to obtain the estimates. A relative di�erence less than .01 is obtained for the �rst
time when the number of realizations gets to one thousand. After thirty-eight hundred
128
1000 2000 3000 4000p.r.
12000
14000
16000
18000
sT2
Test 3
Test 2
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.02
0.04
0.06
0.08
DsT2 ê sT
2
Test 3
Test 2
Test 1
(a) (b)
Figure 6.2: Comparison of tests 1, 2, and 3. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plume realizations.Posterior estimates, 10 samples.
realizations the di�erence is less than 0.0008.
The total variances predicted by the Kalman �lter for the estimates obtained using
ten samples are shown in �gure 6.2a. The total variance increases initially until it
reaches a peak at twelve hundred realizations. After this, the total-variance stabilizes.
Note that the number of realizations necessary to reach stability is the same as for
the prior estimate total-variance. The corresponding relative di�erences (�gure 6.2b)
have values that oscillate between 0.0 and 0.01 when the number of realizations is
twelve hundred or more.
The total variance of the posterior estimates when more than 10 samples are used
(�gs. 6.3a, 6.4a, and 6.5a) are increasing functions of the number of realizations. Each
curve oscillates initially and gets a smooth shape at a di�erent number of realizations.
The curves for the sampling programs with twenty and thirty samples get this smooth
shape after twelve hundred realizations. The curve for the sampling program of forty
samples gets it after a thousand realizations. Their corresponding relative di�erences
(�gs. 6.3b, 6.4b, 6.5b) show that when more samples are taken, a larger number of
realizations is needed to obtain the same degree of relative change.
129
1000 2000 3000 4000p.r.
400
600
800
1000
sT2
Test 3
Test 2
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.02
0.04
0.06
0.08
0.1
0.12
DsT2 ê sT
2
Test 3
Test 2
Test 1
(a) (b)
Figure 6.3: Comparison of tests 1, 2, and 3. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plume realizations.Posterior estimates, 20 samples.
1000 2000 3000 4000p.r.
25
50
75
100
125
150
sT2
Test 3
Test 2
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.1
0.2
0.3
0.4
0.5
DsT2 ê sT
2
Test 3
Test 2
Test 1
(a) (b)
Figure 6.4: Comparison of tests 1, 2, and 3. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plume realizations.Posterior estimates, 30 samples.
1000 2000 3000 4000p.r.
10
20
30
40sT
2
Test 3
Test 2
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.1
0.2
0.3
0.4
0.5
0.6
DsT2 ê sT
2
Test 3
Test 2
Test 1
(a) (b)
Figure 6.5: Comparison of tests 1, 2, and 3. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plume realizations.Posterior estimates, 40 samples.
130
500 1000 1500 2000 2500 3000 3500 4000p.r.
58000
59000
60000
61000
62000
63000
sM2
Test 3
Test 2
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.01
0.02
0.03
0.04
0.05
0.06DsM
2 ê sM2
Test 3
Test 2
Test 1
(a) (b)
Figure 6.6: Comparison of tests 1, 2, and 3. a) Maximum variance vs. number ofplume realizations. b) Relative maximum variance di�erences vs. number of plumerealizations. Prior estimate.
6.2.2 Maximum Variance Analysis
The plot of the maximum variance of the prior estimate (�g. 6.6a) oscillates initially
and when the number of realizations is larger than eighteen hundred it gets a de-
scending tendency. The relative di�erences of the maximum variances, denoted here
by��2M�2M
have a general descending tendency when the number of plume realizations
increases. All the values after two thousand realizations are smaller than 0.01, the
only exception is the maximum variance di�erence at thirty-six realizations.
The maximum variance of the estimates obtained using 10 samples have a large
drop at one thousand realizations (�g. 6.7a). We think that this indicates that a
change in the sampling program is realized at this point. In the analysis done in
chapter 4 we noticed that these sudden changes in the maximum variance were ob-
served when samples were taken at points where the maximum was attained. Often,
this characteristic was present in the maximum variance curves during the �rst twenty
samples, when the concentrations at the contaminant source were sampled. We think
that the jump observed in the plot presently analyzed, indicates that a sampling at
the location where the maximum estimate variance is attained for the estimates ob-
tained with less than one thousand realizations, is included in the sampling program
131
500 1000 1500 2000 2500 3000 3500 4000p.r.
2500
3000
3500
4000
4500
5000
sM2
Test 3
Test 2
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.1
0.2
0.3
0.4
DsM2 ê sM
2
Test 3
Test 2
Test 1
(a) (b)
Figure 6.7: Comparison of tests 1, 2, and 3. a) Maximum variance vs. number ofplume realizations. b) Relative maximum variance di�erences vs. number of plumerealizations. Posterior estimates, 10 samples.
500 1000 1500 2000 2500 3000 3500 4000p.r.
40
60
80
100
120
140
160
sM2
Test 3
Test 2
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.2
0.4
0.6
0.8
DsM2 ê sM
2
Test 3
Test 2
Test 1
(a) (b)
Figure 6.8: Comparison of tests 1, 2, and 3. a) Maximum variance vs. number ofplume realizations. b) Relative maximum variance di�erences vs. number of plumerealizations. Posterior estimates, 20 samples.
for the �rst time at one thousand realizations. The plot of the relative di�erences
of the maximum variances (�g. 6.7b) has a peak at 1000 realizations, re ecting the
sudden change in values of the maximum estimate variances.
Similar behavior is observed in the plot of the maximum variance of the estimates
obtained using 20 samples (�g. 6.8a). Here, it seems that the sampling program
changes and goes back to its original state several times, resulting in a step like
plot. Again, we can relate the peaks in the plot of the maximum-variances relative
di�erences (�g. 6.8b) with these sampling program changes.
132
500 1000 1500 2000 2500 3000 3500 4000p.r.
4
6
8
10
12
14
sM2
Test 3
Test 2
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.2
0.4
0.6
0.8
DsM2 ê sM
2
Test 3
Test 2
Test 1
(a) (b)
Figure 6.9: Comparison of tests 1, 2, and 3. a) Maximum variance vs. number ofplume realizations. b) Relative maximum variance di�erences vs. number of plumerealizations. Posterior estimates, 30 samples.
For the sampling programs with thirty samples, the maximum-variance curve is
a lot smoother (�g. 6.9a). In chapter 4 we observed that for all the tests (using
three thousand plume realizations) the sampling program with 30 samples contained
the samples at all the possible sampling space-time locations at the source. So, we
think that the sampling programs of thirty samples most probably include all these
samples. The maximum variance values oscillate initially but after the realizations
reach twelve hundred these values stabilize. This indicates that more than twelve
hundred realizations are required for the methodology proposed in this thesis to con-
verge if a program of thirty samples is selected. The plot of the corresponding relative
di�erences (�g. 6.9b) has smaller peaks than the corresponding plot for a sampling
program with twenty samples (�g. 6.8b).
When forty samples are selected, the maximum estimate variance plot (�g. 6.10a)
behaves similarly. The plot oscillates initially, and at the point of two thousand
realizations the values do not change much anymore. Small value jumps can be
noticed, indicating changes in the sampling schedule that do not change the result-
ing maximum variance by much. As in the previous case, this may indicate that
more than two thousand realizations are required if a program of forty samples is
133
500 1000 1500 2000 2500 3000 3500 4000p.r.
1
2
3
4
5
6
sM2
Test 3
Test 2
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.2
0.4
0.6
0.8
1
DsM2 ê sM
2
Test 3
Test 2
Test 1
(a) (b)
Figure 6.10: Comparison of tests 1, 2, and 3. a) Maximum variance vs. number ofplume realizations. b) Relative maximum variance di�erences vs. number of plumerealizations. Posterior estimates, 40 samples.
selected. The plot of the corresponding relative di�erences (�g. 6.10b) re ects this
behavior.
6.3 Correlation Scale of the Hydraulic Conductivity Field
Tests 1, 2, and 3 di�er only in the hydraulic conductivity correlation scale. Next we
analyze the e�ects of changing the correlation scale of hydraulic conductivity on the
convergence of the procedure developed comparing the results for these three tests.
6.3.1 Total Variance Analysis
The shapes of the plots of the prior estimate total-variances are very similar, they
di�er mainly in the scale (�g. 6.1a). We believe that this similitude is due to the
use of a common seed for the three tests. The corresponding relative di�erences have
almost identical values (�g. 6.1b).
The total variances of the posterior estimates (�gs. 6.2a, 6.3a, 6.4a, and 6.5a) di�er
more as the number of samples increases. The di�erences between these curves can
134
be appreciated better in the total-variance relative di�erences plots. The resulting
values of the relative di�erences when using 10 samples (�g. 6.2b), are still similar for
the three tests but the graphs have some peaks at di�erent ordinates. The di�erences
get greater when 20 samples are used (�g. 6.3b). The plots for test 1 and test 2 have
large peaks but the one for test 3 is a lot smoother. When 30 samples are used in
the estimation process (�g. 6.4b), the total variance di�erences for the three tests get
similar values again. For the posterior estimates obtained with 40 samples (�g. 6.1b)
the relative total variance di�erences have again values that are similar but the plot
for test 1 has larger peaks between 500 and 2000 realizations and afterwards it gets
very at.
6.3.2 Maximum Variance Analysis
The maximum variances of the prior estimates for the three tests are identical
(�gs 6.6a). When ten samples are included in the sampling program the curves
remain similar (�gs. 6.7a). The plot for test 1 is the most di�erent, having large
maximum variance values for the �rst one thousand realizations. The plots get very
di�erent when more samples are taken (�gs. 6.8a, 6.9a, and 6.10a). The maximum
variances of the posterior estimates for test 1 are the most di�erent. Even when the
maximum variance values di�er greatly for some of these plots, the relative di�erences
of the maximum variances are similar (�gs. 6.6b, 6.7b, 6.8b, 6.9b, and 6.10b). The
only exception to this are the plots for the posterior estimates with twenty samples.
These results indicate that the rate of convergence of the method is relatively
insensitive to changes in the hydraulic conductivity correlation scale.
135
1000 2000 3000 4000p.r.
305000
310000
315000
320000
sT2
Test 4
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.01
0.02
0.03
0.04
DsT2 ê sT
2
Test 4
Test 1
(a) (b)
Figure 6.11: Comparison of tests 1 and 4. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plume realizations.Prior estimate.
6.4 Time Correlation of the Concentration at the Source
The e�ect of changes on the correlation scale of the contaminant concentration at the
source on the convergence rate are minor.
6.4.1 Total Variance Analysis
Comparison between results of test 1 and test 4 (�gs. 6.11a, 6.12a, 6.13a, 6.14a, and
6.15a) show that the total variances of the concentration estimates are very similar
before and after taking samples. The total variances are consistently slightly larger
for the estimates of test 4, which has a correlation scale at the source larger than
test 1, with exception of the total variance for the prior estimate (�g. 6.11a) which
is larger for test 1 when a small number of plume realizations is used. The relative
di�erences of the total variance are also very similar, especially when the number of
realizations are large (�gs. 6.11b, 6.12b, 6.13b, 6.14b, and 6.15b).
6.4.2 Maximum Variance Analysis
The maximum variances are also similar for these two tests. The maximum variances
of the prior estimates are larger for test 1 than for test 4 (�g. 6.16a, 6.17a, 6.18a,
136
1000 2000 3000 4000p.r.
17500
18000
18500
19000
19500
sT2
Test 4
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.02
0.04
0.06
0.08
DsT2 ê sT
2
Test 4
Test 1
(a) (b)
Figure 6.12: Comparison of tests 1 and 4. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plume realizations.Posterior estimates, 10 samples.
1000 2000 3000 4000p.r.
900
1000
1100
1200sT
2
Test 4
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.02
0.04
0.06
0.08
0.1
0.12
0.14DsT
2 ê sT2
Test 4
Test 1
(a) (b)
Figure 6.13: Comparison of tests 1 and 4. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plume realizations.Posterior estimates, 20 samples.
1000 2000 3000 4000p.r.100
120
140
160
180
200sT
2
Test 4
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.05
0.1
0.15
0.2
0.25
0.3
DsT2 ê sT
2
Test 4
Test 1
(a) (b)
Figure 6.14: Comparison of tests 1 and 4. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plume realizations.Posterior estimates, 30 samples.
137
1000 2000 3000 4000p.r.
20
25
30
35
40
sT2
Test 4
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.1
0.2
0.3
0.4
0.5
0.6
DsT2 ê sT
2
Test 4
Test 1
(a) (b)
Figure 6.15: Comparison of tests 1 and 4. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plume realizations.Posterior estimates, 40 samples.
6.19a, and 6.20a). When samples are added to the estimate, the maximum variances
decrease faster for test 1 than for test 4. The maximum variances of the estimates for
test 1 are smaller than those of the estimates for test 4 when 30 or more samples are
added (�g. 6.19a). The posterior estimates when 40 samples are used in the Kalman
�lter are again very similar for both tests, a smaller maximum variance is predicted for
one test or the other depending on the number of plume realizations used to obtain the
prior estimate. The relative di�erences of the maximum variances have values of the
same order of magnitude when the number of plume realizations is large (�gs. 6.16b,
6.17b, 6.18b, 6.19b, and 6.20b). Thus, the number of realizations required to obtain
a certain value of relative-di�erence total-variance or relative-di�erence maximum-
variance is very similar for the two tests.
6.5 Variance of the Hydraulic Conductivity Field
6.5.1 Total Variance Analysis
When the log-conductivity variance increases, the total variance for the prior esti-
mates increase (�gs. 6.21a, 6.22a, 6.23a, 6.24a, and 6.25a). The same behavior is
138
500 1000 1500 2000 2500 3000 3500 4000p.r.
58000
59000
60000
61000
62000
63000
sM2
Test 4
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.01
0.02
0.03
0.04
0.05
0.06
DsM2 ê sM
2
Test 4
Test 1
(a) (b)
Figure 6.16: Comparison of tests 1 and 4. a) Maximum variance vs. number ofplume realizations. b) Relative maximum variance di�erences vs. number of plumerealizations. Prior estimates.
500 1000 1500 2000 2500 3000 3500 4000p.r.
3000
3500
4000
4500
5000
5500
sM2
Test 4
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.1
0.2
0.3
0.4
DsM2 ê sM
2
Test 4
Test 1
(a) (b)
Figure 6.17: Comparison of tests 1 and 4. a) Maximum variance vs. number ofplume realizations. b) Relative maximum variance di�erences vs. number of plumerealizations. Posterior estimates, 10 samples.
500 1000 1500 2000 2500 3000 3500 4000p.r.
100
120
140
160
sM2
Test 4
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.2
0.4
0.6
0.8
DsM2 ê sM
2
Test 4
Test 1
(a) (b)
Figure 6.18: Comparison of tests 1 and 4. a) Maximum variance vs. number ofplume realizations. b) Relative maximum variance di�erences vs. number of plumerealizations. Posterior estimates, 20 samples.
139
500 1000 1500 2000 2500 3000 3500 4000p.r.
14
16
18
20
22
24
sM2
Test 4
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.05
0.1
0.15
0.2
0.25
DsM2 ê sM
2
Test 4
Test 1
(a) (b)
Figure 6.19: Comparison of tests 1 and 4. a) Maximum variance vs. number ofplume realizations. b) Relative maximum variance di�erences vs. number of plumerealizations. Posterior estimates, 30 samples.
500 1000 1500 2000 2500 3000 3500 4000p.r.
3
4
5
6
7sM
2
Test 4
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.2
0.4
0.6
0.8
1
DsM2 ê sM
2
Test 4
Test 1
(a) (b)
Figure 6.20: Comparison of tests 1 and 4. a) Maximum variance vs. number ofplume realizations. b) Relative maximum variance di�erences vs. number of plumerealizations. Posterior estimates, 40 samples.
140
1000 2000 3000 4000p.r.
320000
340000
360000
380000sT
2
Test 6
Test 5
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.01
0.02
0.03
0.04
0.05
DsT2 ê sT
2
Test 6
Test 5
Test 1
(a) (b)
Figure 6.21: Comparison of tests 1, 5, and 6. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plume realizations.Prior estimate.
observed after samples have been added: all the posterior estimates for test 1 have
the smallest total variance, then those for test 5 and the estimates of test 6 have the
largest total variance. As before, the relative di�erences (�gs. 6.21b, 6.22b, 6.23b,
6.24b, and 6.25b) di�er more when more samples are taken, and for a given number
of samples the dissimilarities tend to diminish when the number of plume realizations
increases.
The total-variance relative-di�erences get smaller with a smaller number of real-
izations for test 1 than for tests 5 and 6. The order of magnitude of the di�erences is
very similar for tests 5 and 6 for all the posterior estimates with exception of those
obtained when 20 samples are taken. There the di�erences for test 6 can have values
of about 0.1 even after 2500 realizations.
6.5.2 Maximum Variance Analysis
The maximum variance is identical for the prior estimates of the three tests. After
10 samples are added (�g. 6.27a) the maximum variances of the estimates of test
1 drop to about 3000 ppm2 but the maximum variances of the estimates of tests 5
and 6 remain almost identical. When more than 10 samples are taken (�gs. 6.28a,
141
1000 2000 3000 4000p.r.
18000
20000
22000
24000
26000
sT2
Test 6
Test 5
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.02
0.04
0.06
0.08
DsT2 ê sT
2
Test 6
Test 5
Test 1
(a) (b)
Figure 6.22: Comparison of tests 1, 5, and 6. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plume realizations.Posterior estimates, 10 samples.
1000 2000 3000 4000p.r.1000
1500
2000
2500
3000
3500
sT2
Test 6
Test 5
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.05
0.1
0.15
0.2
0.25
DsT2 ê sT
2
Test 6
Test 5
Test 1
(a) (b)
Figure 6.23: Comparison of tests 1, 5, and 6. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plume realizations.Posterior estimates, 20 samples.
1000 2000 3000 4000p.r.
100
200
300
400
500
600
700
sT2
Test 6
Test 5
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.05
0.1
0.15
0.2
0.25
0.3
DsT2 ê sT
2
Test 6
Test 5
Test 1
(a) (b)
Figure 6.24: Comparison of tests 1, 5, and 6. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plume realizations.Posterior estimates, 30 samples.
142
1000 2000 3000 4000p.r.
20
40
60
80
100
120
140
sT2
Test 6
Test 5
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.1
0.2
0.3
0.4
0.5
0.6
0.7
DsT2 ê sT
2
Test 6
Test 5
Test 1
(a) (b)
Figure 6.25: Comparison of tests 1, 5, and 6. a) Total variance vs. number of plumerealizations. b) Relative total variance di�erences vs. number of plume realizations.Posterior estimates, 40 samples.
6.29a, and 6.30a) the maximum variance of the estimates of each one of the tests
get a di�erent value. Again, the maximum variance increases when the value of the
log-conductivity variance increases. In �gure 6.28a the plot for test 6 has a shape
similar to a step function.
We explained above that we believe that this behavior of the maximum variance
is due to changes in the sampling program involving sampling locations at the con-
taminant source. The relative di�erences of these variances (�gs. 6.26b, 6.27b, 6.28b,
6.29b, and 6.30b) have the largest di�erences for the three tests when 20 samples are
taken.
6.6 Conclusions
From this analysis we conclude that the total variance is a measure suitable for
the convergence analysis of the methodology that we are proposing. The maximum
variance can help in distinguishing estimate variance changes due to sampling program
changes from changes due to the averaging process itself. From the results of the total
variance we conclude that for the tests presented in this thesis 3000 realizations were
143
500 1000 1500 2000 2500 3000 3500 4000p.r.
58000
59000
60000
61000
62000
63000
sM2
Test 6
Test 5
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.01
0.02
0.03
0.04
0.05
0.06DsM
2 ê sM2
Test 6
Test 5
Test 1
(a) (b)
Figure 6.26: Comparison of tests 1, 5, and 6. a) Total maximum vs. number ofplume realizations. b) Relative maximum variance di�erences vs. number of plumerealizations. Prior estimates.
500 1000 1500 2000 2500 3000 3500 4000p.r.
3500
4000
4500
5000
5500
6000
sM2
Test 6
Test 5
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.1
0.2
0.3
0.4
DsM2 ê sM
2
Test 6
Test 5
Test 1
(a) (b)
Figure 6.27: Comparison of tests 1, 5, and 6. a) Maximum variance vs. number ofplume realizations. b) Relative maximum variance di�erences vs. number of plumerealizations. Posterior estimates, 10 samples.
500 1000 1500 2000 2500 3000 3500 4000p.r.100
200
300
400
500
600
sM2
Test 6
Test 5
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.2
0.4
0.6
0.8
1
DsM2 ê sM
2
Test 6
Test 5
Test 1
(a) (b)
Figure 6.28: Comparison of tests 1, 5, and 6. a) Maximum variance vs. number ofplume realizations. b) Relative maximum variance di�erences vs. number of plumerealizations. Posterior estimates, 20 samples.
144
500 1000 1500 2000 2500 3000 3500 4000p.r.
20
30
40
50
60
sM2
Test 6
Test 5
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.2
0.4
0.6
0.8
1
DsM2 ê sM
2
Test 6
Test 5
Test 1
(a) (b)
Figure 6.29: Comparison of tests 1, 5, and 6. a) Maximum variance vs. number ofplume realizations. b) Relative maximum variance di�erences vs. number of plumerealizations. Posterior estimates, 30 samples.
500 1000 1500 2000 2500 3000 3500 4000p.r.
5
10
15
20
sM2
Test 6
Test 5
Test 1
500 1000 1500 2000 2500 3000 3500p.r.
0.2
0.4
0.6
0.8
1
DsM2 ê sM
2
Test 6
Test 5
Test 1
(a) (b)
Figure 6.30: Comparison of tests 1, 5, and 6. a) Maximum variance vs. number ofplume realizations. b) Relative maximum variance di�erences vs. number of plumerealizations. Posterior estimates, 40 samples.
145
enough to obtain convergence. A second conclusion is that for the tests presented in
this thesis changes in hydraulic conductivity correlation scale, hydraulic conductivity
variance, and contaminant concentration time-correlation at the contaminant source,
do not a�ect the rate of convergence. On the other hand, the number of samples
to be included in the sampling design can play an important role in the number of
realizations needed for the method to converge.
146
Chapter 7
Conclusions
In this dissertation, we developed a linear estimation method that can accommodate
several sources of variability and that can be used in the design of groundwater qual-
ity sampling networks in which sampling locations and sampling times are decision
variables.
The �rst step in the development of this estimation method was to test numerically
the statistical characteristics of the model error. It was proved that the model errors
at di�erent times are correlated and that they are not normally distributed. In the
example shown, the reduction in estimate uncertainty resulting from the addition of
data when model-error time-correlations are considered is much greater than when
they are not. From the analysis we concluded that disregarding the time correlations
of the model errors can lead to estimation methods that need many more samples to
obtain the same degree of certainty than those needed by a model addressing model
error time correlations. Also, the variance as a measure of error in the concentration
estimates can be misleading when model error correlations are not accounted for.
We developed an estimation method that accounts for space-time correlations. The
method is equivalent to a space-time kriging method in which the concentration mean
and covariance matrix are obtained from a stochastic transport model. The method
can accommodate several sources of variability. Taking advantage of current modeling
practices, the estimation method uses a deterministic model developed for a given
groundwater quality problem and adds uncertainty to it. To apply the method there
147
is no need to make use of specialized stochastic modeling software. The only software
requirement is a random simulator for variables with correlation in two or three
dimensions (like some of the programs in the GSLIB package [8]). A disadvantage of
the method is the large number of stochastic plume realizations necessary to reach
convergence.
A procedure to choose sampling locations and sampling times, minimizing the
total spatiotemporal variance step by step, was applied. This procedure is a general-
ization to space and time of the variance reduction approach used by Rouhani [59].
The synthetic two-dimensional examples presented show that this method can obtain
estimates with small uncertainty for a contaminant plume in motion with few water
samples. It was demonstrated that the use of the total variance of the estimates in
combination with the maximum variance of the estimates provides a tool for analyz-
ing the results with no need to analyze statistical characteristics of the estimates at
each node.
In these examples two sources of uncertainty were considered: hydraulic conduc-
tivities and concentrations at the contaminant source. Some particular conclusions
drawn from these examples are:
� There is a tendency to �rst place the samples at the contaminant source. This
tendency is due to the large concentration variance at those locations.
� Sampling at the source does not give signi�cant information about the concen-
tration at other locations.
� There is a tendency to sample �rst on the third and fourth rows of the sampling
mesh. This suggests that to reduce the total variance of the concentration
estimate it is important to obtain �rst the central tendency of the plume.
� The sample locations chosen last de�ne the boundary of the plume.
148
� The number of samples needed to obtain a certain degree of certainty increases
with the hydraulic conductivity correlation scale and with its variance.
� Samples have to be distributed in a wider area when the hydraulic conductivity
correlation scale is large and when the variance is large.
� For the tests presented in this work the e�ect of changes on contaminant time-
correlation at the source on the sampling program and the predicted estimate
variance are insigni�cant. An analysis of the e�ects of changes in the sampling
mesh and in the sampling frequency is needed.
149
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156
Appendix A
Regression of a Contaminant Concentration FieldTime-Series
A time series of �eld data was used to obtain the variability of the contaminantconcentration at its source. In the present appendix, the analysis of the data ispresented.
The data that is analyzed is from the CIBA-GEIGY site located at Toms River,New Jersey. It consists of a series of chloride concentration data sampled from well110, from October of 1976 to July of 1980. This well was chosen because it is locatedclose to one of the chloride sources at the plant and there were 24 samples available.This number was considered large enough to obtain a meaningful linear regressionfor the data. A plot of the concentration series against time is shown in �gure A.1.As can be observed in the plot, chloride was not sampled at the well with a regularfrequency. On the average a sample was taken every three months but there aremonths in which more than one sample was taken. It was found that a better linearregression was obtained in a log-log scale.
Using the computational package \Linear Regression" from the program Math-ematica version 3.0 [80], the results shown in table A.1 were obtained. This tableincludes the parameters estimated, their standard errors, and t-statistics for testingwhether each parameter is zero. The p-values are calculated by comparing the ob-tained statistics to the t distribution with n � p degrees of freedom, where n is thesample size and p is the number of parameters estimated. The regression equationobtained is
c(t) = �3:72518t7 + 128:343t6 � 1681:75t5 + 9166:96t4
�234161t2 + 1:01605� 106t� 1:41475� 106;
where, c denotes concentration and t denotes time. The t-statistics values obtainedindicate that the coe�cients are di�erent from zero using a 1% level of signi�cance.The p values indicate also that the parameters are di�erent from zero and that all ofthem have a similar signi�cance.
A table for the analysis of variance A.2 provides a comparison of the given modelto a smaller one including only a constant term. The table includes the degreesof freedom, the sum of squares and the mean squares due to the model (in therow labeled model) and due to the residuals (in the row labeled error). The F -test compares the two models by the ratio of their mean squares. If the value of F is
157
400 600 800 1000 1200 1400 1600Time HdaysL
0
200
400
600
800
Chl
orid
eco
ncen
trat
ion
Hppm
L
Figure A.1: Chloride concentration vs. time at well 110.
Table A.1: Linear regression table.
Estimate S.E. TStat. PValue
1 �1:41475106 258426 -5.47447 0.0000509385
x 1:01605106 185279 5.48389 0.0000500069
x2 -234160.66 42624.9 -5.49352 0.0000490730
x4 9166.96 1662.76 5.51311 0.0000472286
x5 -1681.75 304.504 -5.52292 0.0000463317
x6 "128.343 23.1974 5.53267 0.0000454570
x7 -3.72518 0.672133 -5.54232 0.0000446100
large, the null hypothesis supporting the smaller model is rejected. The mean squareregression error is 0.194805. This is the variance used in the stochastic simulations ofthe concentration at the contaminant source. The F -test value provides evidence toreject the smaller model at a signi�cant level of 1%. Figure A.2 shows the logarithmof the raw data, the �tted curve, and the 95% con�dence intervals for the predictedvalues of observations.
158
Table A.2: ANOVA table.
D.F. Sum of Sq. Mean Sq. FRatio Pvalue
Model 6 13.0101 2.16836 32.6228 4:1632710�8
Error 16 1.06348 0.0664675
Total 22 14.0736
5.75 6 6.25 6.5 6.75 7 7.25Log t
4
5
6
7
Log c
Figure A.2: Regression for concentration logarithm vs. time logarithm at well 110.