mmc2.geofisica.unam.mxmmc2.geofisica.unam.mx/cursos/geoest/Articulos/Geostatistics/Cos… · Ac kno wledgemen ts Thank y ou Jos e Luis for our patience and supp ort during all these

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



5.9. Total variance vs. number of samples for tests 1, 2, and 3. Samples10-20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105


viii

5.11. Maximum variance vs. number of samples for tests 1, 2, and 3. Samples10-20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106


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


5.23. Maximum variance vs. number of samples for tests 1, 5, and 6. Samples1-10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113



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






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




6.6. 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




6.11. 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



x


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




6.21. 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





6.26. 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



xi



A.1. Chloride concentration vs. time at well 110. . . . . . . . . . . . . . . 157A.2. Regression for concentration logarithm vs. time logarithm at well 110. 158

xii

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.

References

[1] Ababou, R., Three-Dimensional Flow in Random Porous Media, Vol. 2, MIT,Cambridge, (1988).

[2] ASCE Task Committee on Geostatistical Techniques in Geohydrology: Geor-gakakos, A. P., P. K. Kitanidis, H. A. Loaiciga, R. A. Olea, S. R. Yates, and S.Rouhani (chairman), Review of geostatistics in geohydrology. I: Basic concepts,J. Hydr. Engrg., 116 (1990), 612{632.

[3] Bear, J., Dynamics of Fluids in Porous Media, Dover, New York, (1988).

[4] Bilonick, R. A., The space-time distribution of sulfate deposition in the north-eastern United States, Atmospheric Environmental, 19 (1985), 1829{1845.

[5] Bras R. L., and I. Rodr��guez-Iturbe, Random Functions and Hydrology, Addison-Wesley, New York, (1985).

[6] Christakos, G., On certain classes of spatiotemporal random �elds with applica-tions to space-time data processing, IEEE Transactions on Systems, Man, and

Cybernetics, 21 (1991), 861{875.

[7] Christakos, G., Random Field Models in Earth Sciences, Academic Press, SanDiego, (1992).

[8] Christakos, G., and V .R. Raghu, Modeling and estimation of space/time waterquality parameters, CMR News, U. of North Carolina at Chapel Hill 2 (1994),14{22.

74

[9] Cressie, N. A., Statistics for Spatial Data, Wiley and Sons, New York, (1993).

[10] Dagan, G., Stochastic modeling of groundwater ow by unconditional and con-ditional probabilities, 2. The solute transport., Water Resour. Res., 18 (1982),835{848.

[11] Dagan, G., Solute transport in heterogeneous porous formations, J. Fluid Mech.,

145 (1984), 151{177.

[12] Dagan, G., Flow and Transport in Porous Formations, Springer Verlag, NewYork, (1989).

[13] Dagan, G., I. Butera, and E. Grella, Impact of concentration measurements uponestimation of ow and transport parameters: The Lagrangian approach, Water

Resour. Res., 32 (1996), 297{306.

[14] Dagan, G., and S. P. Neuman, Nonasymptotic behavior of a common Eulerianapproximation for transport in random velocity �elds, Water Resour. Res., 27

(1991), 3249{3256.

[15] Kavanaugh, M. C., et al., Alternatives for Ground Water Cleanup, NationalAcademy Press, Washington, D.C., (1994).

[16] de Marsily, G., and J.-P. Delhomme, Geostatistics: from mining to hydrology,in: Proc. 1989 Nat. Con. on Hydraulics Engineering, ed. by Ports, M. A., ASCE,New York (1989), pp. 659{666.

[17] Egbert, G. D., and Lettenmaier D. P., Stochastic modeling of the space-timestructure of atmospheric chemical deposition, Water Resour. Res., 22 (1986),165{179.

[18] Ezzedine, S., and Y. Rubin, A geostatistical approach to the conditional estima-tion of spatially distributed solute concentration and notes on the use of tracerdata in the inverse problem, Water Resour. Res., 32 (1996), 853{861.

[19] Gelhar, L. W., A. L. Gutjahr, and R. L. Na�, Stochastic analysis of macrodis-persion in a strati�ed aquifer, Water Resour. Res., 15 (1979), 1387{1740.

[20] Gelhar, L. W., Stochastic Subsurface Hydrology, Prentice Hall, Englewood Cli�s,New Jersey, (1993).

[21] Graham, W. D., and D. McLaughlin, Stochastic Analysis of Nonstationary Sub-surface Solute Transport: 1. Unconditional Moments, Water Resour. Res., 25

(1989), 215{232.

75

[22] Graham, W. D., and D. McLaughlin, Stochastic Analysis of Nonstationary Sub-surface Solute Transport: 2. Conditional Moments, Water Resour. Res., 25

(1989), 2331{2355.

[23] Graham, W. D., and D. McLaughlin, A stochastic model of solute transport ingroundwater: application to the Borden, Ontario, tracer test, Water Resour.

Res., 27 (1991), 1345{1359.

[24] Isaaks, E. H., and R. Mohan Srivastava, Applied Geostatistics, Oxford UniversityPress, New York, (1989).

[25] Jazwinski, A. H., Stochastic Processes and Filtering Theory (1970), AcademicPress, San Diego.

[26] Jinno, K., A. Kawamura, T. Ueda, and H. Yoshinaga, Prediction of the concen-tration distribution of groundwater pollutants, in: Groundwater Management:

Quantity and Quality (Proceedings of the Benidorm Symposium, October 1989).IAHS publ. no. 188, (1989), pp. 131{142.

[27] Journel, A. G., and C. J. Huijbregts, Mining Geostatistics, Academic Press,London, (1978).

[28] Li, S.-G., and D. McLaughlin, A nonstationary spectral method for solvingstochastic groundwater problems: unconditional analysis, Water Resour. Res.,

27 (1991), 1589{1605.

[29] Loaiciga, H. A., An optimization approach for groundwater quality monitoringnetwork design, Water Resour. Res., 25 (1989), 1771{1782.

[30] McLaughlin, D., L. B. Reid, S. G. Li, and J. Hyman, A stochastic method forcharacterizing ground-water contamination, Ground Water, 31 (1993), 237{249.

[31] Neuman, S. P., Eulerian-Lagrangian theory of transport in space-time nonsta-tionary velocity �elds: Exact nonlocal formalism by conditional moments andweak approximation, Water Resour. Res., 29 (1993), 633{645.

[32] Neuman, S.P., Levin, O., Orr, S., Paleologos, E., Zhang, D., and Zhang, Y.-K., Nonlocal representations of subsurface ow and transport by conditionalmoments. In Computational Stochastic Mechanics (chapter 20), edited by Cheng,A.H.-D., Yang, and C.Y., (1993), pp. 451{473.

[33] Ripley, B. D., Stochastic Simulation, John Wiley, New York, (1987).

[34] Rodr��guez-Iturbe, I., and Mej��a J. M., The design of rainfall networks in timeand space, Water Resour. Res., 10 (1974), 713{728.

76

[35] Rouhani, S., and T. J. Hall, Space-time kriging of groundwater data, in: Geo-statistics, Vol. 2, ed. by Amstrong, M., Kluwer Academic, Dordrecht (1989),pp. 639{650.

[36] Rouhani, S., and H. Wackernagel, Multivariate geostatistical approach to space-time data analysis, Water Resour. Res., 26 (1990), 585{591.

[37] Rubin, Y., Prediction of tracer plume migration in disordered porous media bythe method of conditional probabilities, Water Resour. Res., 27 (1991), 1291{1308.

[38] Shafer, J. M., M. D. Varljen, and H. Allen Werhmann, Identifying temporalchange in the spatial correlation of regional groundwater contamination, in: Proc.1989 Nat. Conf. on Hydraulics Engineering, ed. by Ports, M.A., ASCE, New York(1989), pp. 398{403.

[39] Smith, L., and R. Allan Freeze, Stochastic analysis of steady state groundwater ow in a bounded domain 1. One-dimensional simulations, Water Resour. Res.,

15 (1979), 521{528.

[40] Smith, L., and F. W. Schwartz, Mass transport, 1. A stochastic analysis ofmacroscopic dispersion, Water Resour. Res., 16 (1980), 303{313.

[41] Smith, L., and F. W. Schwartz, Mass transport, 2. Analysis of uncertainty inprediction, Water Resour. Res., 17 (1981), 351{369.

[42] Smith, L., and F. W. Schwartz, Mass transport, 3. Role of hydraulic conductivitydata in prediction, Water Resour. Res., 17 (1981), 1463{1479.

[43] Solow, A. R., and S. M. Gorelick, Estimating monthly stream ow values bycokriging, Mathematical Geology, 18 (1986), 785{809.

[44] Stein, M., A simple model for spatio-temporal processes, Water Resour. Res.,

22 (1986), 2107{2110.

[45] Tang, D. H., and G. F. Pinder, Simulation of groundwater ow and mass trans-port under uncertainty, Advances in Water Resources, 1 (1977), 25{30.

[46] Tang, D. H., and G. F. Pinder, Analysis of mass transport with uncertain pa-rameters, Water Resources Research, 15 (1979), 1147{1155.

[47] Van Genuchten, M. Th., and W. J. Alves, Analytical Solutions of the One-

Dimensional Convective-Dispersive Solute Transport Equation, Tech. Bull. 1661,U.S. Dept. of Agric., Washington, D.C., (1982).

[48] Yu, Y.-S., M. Heidari, and W. Guang-Te, Optimal estimation of contaminanttransport in ground water, Water Resources Bulletin, 25 (1989), 295{300.

77

[49] Zeger, S. L., Exploring an ozone spatial time series in the frequency domain, J.of the Am. Stat. Assoc., 80 (1985), 323{331.

[50] Zhang, D., and Neuman S. P., Eulerian-Lagrangian analysis of transport condi-tioned on hydraulic data 1. Analytical-numerical approach, Water Resour. Res.,

31 (1995), 39{51.

[51] Zhang, D., and S. P. Neuman, Eulerian-Lagrangian analysis of transport con-ditioned on hydraulic data 2. E�ects of log transmissivity and hydraulic headmeasurements, Water Resour. Res., 31 (1995), 53{63.

[52] Zhang, D., and S. P. Neuman, Eulerian-Lagrangian analysis of transport condi-tioned on hydraulic data 3. Spatial moments, Water Resour. Res., 31 (1995),65{75.

[53] Zhang, D., and S. P. Neuman, Eulerian-Lagrangian analysis of transport con-ditioned on hydraulic data 4. Uncertain initial plume state and non-Gaussianvelocities, Water Resour. Res., 31 (1995), 77{88.

[54] Zou, S., and A. Parr, Optimal estimation of two-dimensional contaminant trans-port, Ground Water, 33 (1995), 319{325.

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

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

1231231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

1231231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

1231231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

1231231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

1231231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

1231231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

1231231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

1231231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

1231231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

1231231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

123123123123

123123123123

123123123123

1234123412341234

123123123123

123123123123

1234123412341234

1231231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

1231231231234

12341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

123123123

123123123

123412341234

123123123

123123123

123412341234

123123123

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


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


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


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


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


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


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


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


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


12 14 16 18 20samples

5000

10000

15000

20000

25000

sT2

Test 6

Test 5

Test 1


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


2 4 6 8 10samples

10000

20000

30000

40000

50000

60000

sM2

Test 6

Test 5

Test 1


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


25 30 35 40samples

50

100

150

200

250

300

350

sM2

Test 6

Test 5

Test 1


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)


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)


5 10 15 20smpls

-2000

-1000

1000

2000

25 30 35 40smpls

-100

-50

50

100

(a) (b)


5 10 15 20smpls

-2000

-1000

1000

2000

25 30 35 40smpls

-150

-100

-50

50

100

150

(a) (b)


118

5 10 15 20smpls

-2000

-1000

1000

2000

25 30 35 40smpls

-200

-100

100

200

(a) (b)


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.

References


[2] Andricevic, R., Coupled withdrawal and sampling designs for groundwater sup-ply models, Water Resour. Res., 29 (1993), 5{16.

[3] Andricevic, R., and E. Foufola-Georgiou, A transfer function approach to sam-pling network design for groundwater contamination, Water Resour. Res., 27

(1991), 2759{2769.

[4] Babu, D., G. F. Pinder, A. Niemi, D. P. Ahlfeld, and S. A. Stotho�, Chemi-cal transport by three dimensional groundwater ows, Rep. 84-WR-3, PrincetonUniversity, Department of Civil Engineering. Princeton, N.J., (1993).

122

[5] Bog�ardi, I., and A. B�ardossy, Multicriterion network design using geostatistics,Water Resour. Res., 21 (1985), 199{208.

[6] Cleveland, T. G., and W. W-G. Yeh, Sampling network design for transportparameter identi�cation, J. Water Resour. Plng. and Mgmt., 116 (1990), 764{783.

[7] Cleveland, T. G., and W. W-G. Yeh, Optimal con�guration and scheduling ofground-water tracer test, J. Water Resour. Plng. and Mgmt., 117 (1991), 37{51.

[8] Deutsch, C. V., and A. G. Journel, GSLIB, Geostatistical Software Library and

User's Guide, Oxford University Press, New York, (1992).

[9] Freeze, R. A., J. Massman, Smith L., T. Sperling, and B. James, Hydrogeologicaldecision analysis: 1. A framework, Ground Water, 28 (1990), 738{766.

[10] Freeze, R. A., T. Sperling, B. James, J. Massman, and L. Smith, Hydrogeologicaldecision analysis: 4. The concept of data worth and its use in the developmentof site investigation strategies, Ground Water, 30 (1992), 574{588.


[12] Graham, W. D., and D. McLaughlin, Stochastic Analysis of Nonstationary Sub-surface Solute Transport: 2. Conditional Moments, Water Resour. Res., 25

(1989), 2331{2355.

[13] Graham, W. D., and D. McLaughlin, Stochastic Analysis of Nonstationary Sub-surface Solute Transport: 1. Unconditional Moments, Water Resour. Res., 25

(1989), 215{232.


Res., 27 (1991), 1345{1359.

[15] Hoel, P. G., S. Port, and C. Stone, Introduction to Stochastic Processes, WavelandPress, Inc., Illinois, (1972).

[16] James, B. R., and S. M. Gorelick, When enough is enough: The worth of moni-toring data in aquifer remediation design, Water Resour. Res., 30 (1994), 3499{3513.


123

[18] Knopman, D. S., and C. I. Voss, Behavior of sensitivities in the one-dimensionaladvection-dispersion equation: implications for parameter estimation and sam-pling design, Water Resour. Res., 23 (1987), 253{ 272.

[19] Knopman, D. S., and C. I. Voss, Further comments on sensitivities, parameterestimation, and sampling design in one-dimensional analysis of solute transportin porous media, Water Resour. Res., 24 (1988), 225{238.

[20] Knopman, D. S., and C. I. Voss, Discrimination among one-dimensional modelsof solute transport in porous media: implications for sampling design, Water

Resour. Res., 24 (1988), 1859{1876.

[21] Knopman, D. S., C. I. Voss, and S.P. Garabedian, Sampling design for ground-water solute transport: tests methods and analysis of Cape Cod tracer tastedata, Water Resour. Res., 27 (1991), 925{949.


[23] Loaiciga, H. A., R. J. Charbeneau, L. G. Everett, G. E. Fogg, and B. F. Hobbs,Review of ground-water quality monitoring network design, J. Hydr. Engrg., 118(1992), 11{37.

[24] McGrath, W. A., Sampling Network Design to Delineate Groundwater Contam-

inant Plumes. Ph. D. Thesis, University of Vermont, Burlington, (1997).

[25] McGrath, W. A., and G. F. Pinder, Delineating a contaminant plume boundary,in: Computational Methods in Water Resources X, ed. by Peters, A. et al., KluwerAcademic, (1994), pp. 317{324.


[27] Pettyjohn, W. A., Cause and e�ect of cyclic changes in ground-water quality,Ground-Water Monitoring Review, 2 (1982), 43{49.


[29] Rouhani, S., Variance reduction analysis, Water Resour. Res., 21 (1985), 837{846.

[30] Rouhani, S., and T. J. Hall, Geostatistical schemes for groundwater sampling,J. Hydrology, 103 (1988), 85{102.

[31] Steel, R. G. D., and J. H. Torrie, Principles and Procedures of Statistics,McGraw-Hill, New York, (1960).

124

[32] Sun, N., Inverse Problems in Groundwater Modeling, Kluwer Academic, Dor-drecht, The Netherlands, (1994).

[33] Tucciarelli, T., and G. Pinder, Optimal data acquisition strategy for the devel-opment of a transport model for groundwater remediation, Water Resour. Res.,

27, (1991), 577{588.

[34] Van Geer, F., C. B. Te Stroet, and Z. Yangxiao, Using Kalman �ltering toimprove and quantify the uncertainty of numerical groundwater simulations. 1.The role of system noise and its calibration., Water Resour. Res., 27 (1991),1987{1994.

[35] Wagner, B. J., Sampling design methods for groundwater modeling under un-certainty, Water Resour. Res., 31 (1995), 2581{2591.

[36] Woldt, W., and I. Bogardi, Ground water monitoring network design using mul-tiple criteria decision making and geostatistics, Water Resour. Bull., 28 (1992),45{62.

[37] Yangxiao, Z., C. B. Te Stroet, and F. Van Geer, Using Kalman �ltering toimprove and quantify the uncertainty of numerical groundwater simulations. 2.Application to monitoring network design.,Water Resour. Res., 27 (1991), 1995{2006.

[38] Yeh, W. W.-G., Review of parameter identi�cation procedures in groundwaterhydrology: the inverse problem, Water Resour. Res., 22 (1986), 95{108.

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)


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)


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)


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)


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)


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)


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.


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.


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.


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).


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)


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)


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)


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


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)


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)


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)


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.


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)


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)


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)


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)


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)


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)


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)


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)


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

Comprehensive Bibliography


[2] Anderson, T. W., The Statistical Analysis of Time Series, John Wiley, NewYork, (1994).

[3] Andricevic, R., Coupled withdrawal and sampling designs for groundwater sup-ply models, Water Resour. Res., 29 (1993), 5{16.

[4] Andricevic, R., and E. Foufola-Georgiou, A transfer function approach to sam-pling network design for groundwater contamination, Water Resour. Res., 27

(1991), 2759{2769.

[5] ASCE Task Committee on Geostatistical Techniques in Geohydrology: Geor-gakakos, A. P., P. K. Kitanidis, H. A. Loaiciga, R. A. Olea, S. R. Yates, and S.Rouhani (chairman), Review of geostatistics in geohydrology. I: Basic concepts,J. Hydr. Engrg., 116 (1990), 612{632.

[6] Babu, D., G. F. Pinder, A. Niemi, D. P. Ahlfeld, and S. A. Stotho�, Chemicaltransport by three dimensional groundwater ows, Rep. 84-WR-3, University ofVermont, Department of Civil Engineering. Burlington, VT, (1993).

[7] Bear, J., Dynamics of Fluids in Porous Media, Dover, New York, (1988).

[8] Bilonick, R. A., The space-time distribution of sulfate deposition in the north-eastern United States, Atmospheric Environmental, 19 (1985), 1829{1845.

[9] Bog�ardi, I., and A. B�ardossy, Multicriterion network design using geostatistics,Water Resour. Res., 21 (1985), 199{208.

[10] Bras R. L., and I. Rodr��guez-Iturbe, Random Functions and Hydrology, Addison-Wesley, New York, (1985).

[11] Casella, G., and R. L. Berger, Statistical Inference, Wadsworth & Brooks, Paci�cGrove, California, (1990).

[12] Christakos, G., On certain classes of spatiotemporal random �elds with applica-tions to space-time data processing, IEEE Transactions on Systems, Man, and

Cybernetics, 21 (1991), 861{875.

150

[13] Christakos, G., Random Field Models in Earth Sciences, Academic Press, SanDiego, (1992).

[14] Christakos, G., and V .R. Raghu, Modeling and estimation of space/time waterquality parameters, CMR News, U. of North Carolina at Chapel Hill 2 (1994),14{22.

[15] Cleveland, T. G., and W. W-G. Yeh, Sampling network design for transportparameter identi�cation, J. Water Resour. Plng. and Mgmt., 116 (1990), 764{783.

[16] Cleveland, T. G., and W. W-G. Yeh, Optimal con�guration and scheduling ofground-water tracer test, J. Water Resour. Plng. and Mgmt., 117 (1991), 37{51.

[17] Cressie, N. A., Statistics for Spatial Data, Wiley and Sons, New York, (1993).

[18] Dagan, G., Stochastic modeling of groundwater ow by unconditional and con-ditional probabilities, 2. The solute transport., Water Resour. Res., 18 (1982),835{848.

[19] Dagan, G., Solute transport in heterogeneous porous formations, J. Fluid Mech.,

145 (1984), 151{177.

[20] Dagan, G., Flow and Transport in Porous Formations, Springer Verlag, NewYork, (1989).

[21] Dagan, G., I. Butera, and E. Grella, Impact of concentration measurements uponestimation of ow and transport parameters: The Lagrangian approach, Water

Resour. Res., 32 (1996), 297{306.

[22] Dagan, G., and S. P. Neuman, Nonasymptotic behavior of a common Eulerianapproximation for transport in random velocity �elds, Water Resour. Res., 27

(1991), 3249{3256.

[23] de Marsily, G., and J.-P. Delhomme, Geostatistics: from mining to hydrology,in: Proc. 1989 Nat. Con. on Hydraulics Engineering, ed. by Ports, M. A., ASCE,New York (1989), pp. 659{666.

[24] Deutsch, C. V., and A. G. Journel, GSLIB, Geostatistical Software Library and

User's Guide, Oxford University Press, New York, (1992).

[25] Domenico, P. A., and F. W. Schwartz, Physical and Chemical Hydrology, JohnWiley and Sons, New York, N.Y., (1990).

[26] Egbert, G. D., and Lettenmaier D. P., Stochastic modeling of the space-timestructure of atmospheric chemical deposition, Water Resour. Res., 22 (1986),165{179.

151

[27] Ezzedine, S., and Y. Rubin, A geostatistical approach to the conditional estima-tion of spatially distributed solute concentration and notes on the use of tracerdata in the inverse problem, Water Resour. Res., 32 (1996), 853{861.

[28] Freeze, R. A., J. Massman, Smith L., T. Sperling, and B. James, Hydrogeologicaldecision analysis: 1. A framework, Ground Water, 28 (1990), 738{766.

[29] Freeze, R. A., T. Sperling, B. James, J. Massman, and L. Smith, Hydrogeologicaldecision analysis: 4. The concept of data worth and its use in the developmentof site investigation strategies, Ground Water, 30 (1992), 574{588.


[31] Gelhar, L. W., A. L. Gutjahr, and R. L. Na�, Stochastic analysis of macrodis-persion in a strati�ed aquifer, Water Resour. Res., 15 (1979), 1387{1740.

[32] Gnedenko, B. V., The Theory of Probability, Chelsea, New York, (1963).

[33] Graham, W. D., and D. McLaughlin, Stochastic analysis of nonstationary sub-surface solute transport: 1. Unconditional moments, Water Resour. Res., 25

(1989), 215{232.

[34] Graham, W. D., and D. McLaughlin, Stochastic analysis of nonstationary subsur-face solute transport: 2. Conditional moments, Water Resour. Res., 25 (1989),2331{2355.


Res., 27 (1991), 1345{1359.

[36] Hoel, P. G., S. Port, and C. Stone, Introduction to Stochastic Processes, WavelandPress, Inc., Illinois, (1972).

[37] Isaaks, E. H., and R. Mohan Srivastava, Applied Geostatistics, Oxford UniversityPress, New York, (1989).

[38] James, B. R., and S. M. Gorelick, When enough is enough: The worth of moni-toring data in aquifer remediation design, Water Resour. Res., 30 (1994), 3499{3513.


[40] Jinno, K., A. Kawamura, T. Ueda, and H. Yoshinaga, Prediction of the concen-tration distribution of groundwater pollutants, in: Groundwater Management:

Quantity and Quality (Proceedings of the Benidorm Symposium, October 1989).IAHS publ. no. 188, (1989), pp. 131{142.

152

[41] Journel, A. G., and C. J. Huijbregts, Mining Geostatistics, Academic Press,London, (1978).

[42] Kavanaugh, M. C., et al., Alternatives for ground water cleanup, NationalAcademy Press, Washington, D.C., (1994).

[43] Knopman, D. S., and C. I. Voss, Behavior of sensitivities in the one-dimensionaladvection-dispersion equation: implications for parameter estimation and sam-pling design, Water Resour. Res., 23 (1987), 253{ 272.

[44] Knopman, D. S., and C. I. Voss, Further comments on sensitivities, parameterestimation, and sampling design in one-dimensional analysis of solute transportin porous media, Water Resour. Res., 24 (1988), 225{238.

[45] Knopman, D. S., and C. I. Voss, Discrimination among one-dimensional modelsof solute transport in porous media: implications for sampling design, Water

Resour. Res., 24 (1988), 1859{1876.

[46] Knopman, D. S., C. I. Voss, and S.P. Garabedian, Sampling design for ground-water solute transport: tests methods and analysis of Cape Cod tracer tastedata, Water Resour. Res., 27 (1991), 925{949.

[47] Li, S.-G., and D. McLaughlin, A nonstationary spectral method for solvingstochastic groundwater problems: unconditional analysis, Water Resour. Res.,

27 (1991), 1589{1605.


[49] Loaiciga, H. A., R. J. Charbeneau, L. G. Everett, G. E. Fogg, and B. F. Hobbs,Review of ground-water quality monitoring network design, J. Hydr. Engrg., 118(1992), 11{37.

[50] McGrath, W. A., Sampling Network Design to Delineate Groundwater Contam-

inant Plumes. Ph. D. Thesis, University of Vermont, Burlington, (1997).

[51] McGrath, W. A., and G. F. Pinder, Delineating a contaminant plume boundary,in: Computational Methods in Water Resources X, ed. by Peters, A. et al., KluwerAcademic, (1994), pp. 317{324.


[53] Neuman, S. P., Eulerian-Lagrangian theory of transport in space-time nonsta-tionary velocity �elds: Exact nonlocal formalism by conditional moments andweak approximation, Water Resour. Res., 29 (1993), 633{645.

153

[54] Neuman, S.P., Levin, O., Orr, S., Paleologos, E., Zhang, D., and Zhang, Y.-K., Nonlocal representations of subsurface ow and transport by conditionalmoments. In Computational Stochastic Mechanics (chapter 20), edited by Cheng,A.H.-D., Yang, and C.Y., (1993), pp. 451{473.

[55] Papoulis, A., Probability, Random Variables, and Stochastic Processes, McGraw-Hill Kogakusha, Tokyo, (1965).

[56] Pettyjohn, W. A., Cause and e�ect of cyclic changes in ground-water quality,Ground-Water Monitoring Review, 2 (1982), 43{49.

[57] Ripley, B. D., Stochastic Simulation, John Wiley, New York, (1987).


[59] Rouhani, S., Variance reduction analysis, Water Resour. Res., 21 (1985), 837{846.

[60] Rouhani, S., and T. J. Hall, Geostatistical schemes for groundwater sampling,J. Hydrology, 103 (1988), 85{102.

[61] Rouhani, S., and T. J. Hall, Space-time kriging of groundwater data, in: Geo-statistics, Vol. 2, ed. by Amstrong, M., Kluwer Academic, Dordrecht (1989),pp. 639{650.

[62] Rouhani, S., and H. Wackernagel, Multivariate geostatistical approach to space-time data analysis, Water Resour. Res., 26 (1990), 585{591.

[63] Rubin, Y., Prediction of tracer plume migration in disordered porous media bythe method of conditional probabilities, Water Resour. Res., 27 (1991), 1291{1308.

[64] Shafer, J. M., M. D. Varljen, and H. Allen Werhmann, Identifying temporalchange in the spatial correlation of regional groundwater contamination, in: Proc.1989 Nat. Conf. on Hydraulics Engineering, ed. by Ports, M.A., ASCE, New York(1989), pp. 398{403.

[65] Smith, L., and R. Allan Freeze, Stochastic analysis of steady state groundwater ow in a bounded domain 1. One-dimensional simulations, Water Resour. Res.,

15 (1979), 521{528.

[66] Smith, L., and F. W. Schwartz, Mass transport, 1. A stochastic analysis ofmacroscopic dispersion, Water Resour. Res., 16 (1980), 303{313.

[67] Smith, L., and F. W. Schwartz, Mass transport, 2. Analysis of uncertainty inprediction, Water Resour. Res., 17 (1981), 351{369.

154

[68] Smith, L., and F. W. Schwartz, Mass transport, 3. Role of hydraulic conductivitydata in prediction, Water Resour. Res., 17 (1981), 1463{1479.

[69] Solow, A. R., and S. M. Gorelick, Estimating monthly stream ow values bycokriging, Mathematical Geology, 18 (1986), 785{809.

[70] Steel, R. G. D., and J. H. Torrie, Principles and Procedures of Statistics,McGraw-Hill, New York, (1960).

[71] Stein, M., A simple model for spatio-temporal processes, Water Resour. Res.,

22 (1986), 2107{2110.

[72] Sun, N., Inverse Problems in Groundwater Modeling, Kluwer Academic, Dor-drecht, The Netherlands, (1994).

[73] Tang, D. H., and G. F. Pinder, Simulation of groundwater ow and mass trans-port under uncertainty, Advances in Water Resources, 1 (1977), 25{30.

[74] Tang, D. H., and G. F. Pinder, Analysis of mass transport with uncertain pa-rameters, Water Resources Research, 15 (1979), 1147{1155.

[75] Tucciarelli, T., and G. Pinder, Optimal data acquisition strategy for the devel-opment of a transport model for groundwater remediation, Water Resour. Res.,

27, (1991), 577{588.

[76] Van Geer, F., C. B. Te Stroet, and Z. Yangxiao, Using Kalman �ltering toimprove and quantify the uncertainty of numerical groundwater simulations. 1.The role of system noise and its calibration., Water Resour. Res., 27 (1991),1987{1994.

[77] Van Genuchten, M. Th., and W. J. Alves, Analytical Solutions of the One-

Dimensional Convective-Dispersive Solute Transport Equation, Tech. Bull. 1661,U.S. Dept. of Agric., Washington, D.C., (1982).

[78] Wagner, B. J., Sampling design methods for groundwater modeling under un-certainty, Water Resour. Res., 31 (1995), 2581{2591.

[79] Woldt, W., and I. Bogardi, Ground water monitoring network design using mul-tiple criteria decision making and geostatistics, Water Resour. Bull., 28 (1992),45{62.

[80] Wolfram Research, Mathematica 3.0 Standard Add-on Packages (1996), WolframMedia/Cambridge University Press.

[81] Yangxiao, Z., C. B. Te Stroet, and F. Van Geer, Using Kalman �ltering toimprove and quantify the uncertainty of numerical groundwater simulations. 2.Application to monitoring network design.,Water Resour. Res., 27 (1991), 1995{2006.

155

[82] Yeh, W. W.-G., Review of parameter identi�cation procedures in groundwaterhydrology: the inverse problem, Water Resour. Res., 22 (1986), 95{108.

[83] Yu, Y.-S., M. Heidari, and W. Guang-Te, Optimal estimation of contaminanttransport in ground water, Water Resources Bulletin, 25 (1989), 295{300.

[84] Zeger, S. L., Exploring an ozone spatial time series in the frequency domain, J.of the Am. Stat. Assoc., 80 (1985), 323{331.

[85] Zhang, D., and Neuman Shlomo P., Eulerian-Lagrangian analysis of transportconditioned on hydraulic data 1. Analytical-numerical approach, Water Resour.

Res., 31 (1995), 39{51.

[86] Zhang, D., and S. P. Neuman, Eulerian-Lagrangian analysis of transport con-ditioned on hydraulic data 2. E�ects of log transmissivity and hydraulic headmeasurements, Water Resour. Res., 31 (1995), 53{63.

[87] Zhang, D., and S. P. Neuman, Eulerian-Lagrangian analysis of transport condi-tioned on hydraulic data 3. Spatial moments, Water Resour. Res., 31 (1995),65{75.

[88] Zhang, D., and S. P. Neuman, Eulerian-Lagrangian analysis of transport con-ditioned on hydraulic data 4. Uncertain initial plume state and non-Gaussianvelocities, Water Resour. Res., 31 (1995), 77{88.

[89] Zou, S., and A. Parr, Optimal estimation of two-dimensional contaminant trans-port, Ground Water, 33 (1995), 319{325.

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.

Documents

mmc2.geofisica.unam.mxmmc2.geofisica.unam.mx/cursos/geoest/Articulos/Geostatistics/Cos… · Ac kno wledgemen ts Thank y ou Jos e Luis for our patience and supp ort during all these