Ant Colony Optimization and Bayesian Analysis for Long-Term Groundwater Monitoring Yuanhai Li

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2006 Ant Colony Optimization and Bayesian Analysis for Long-Term Groundwater Monitoring Yuanhai Li

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COLLEGE OF ENGINEERING

ANT COLONY OPTIMIZATION AND BAYESIAN ANALYSIS FOR

LONG-TERM GROUNDWATER MONITORING

YUANHAI LI

A Dissertation submitted to the Department of Civil and Environmental Engineering in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

Degree Awarded: Summer Semester, 2006 The members of the Committee approve the Dissertation of Yuanhai Li defended on May 11, 2006.

Amy Chan Hilton Professor Directing Dissertation

Bill X. Hu Outside Committee Member

I. Michael Navon Outside Committee Member

Wenrui Huang Committee Member

Danuta Leszczynska Committee Member

Approved:

Kamal Tawﬁq, Chair Department of Civil and Environmental Engineering

The Oﬃce of Graduate Studies has veriﬁed and approved the above named committee members.

ii This thesis is dedicated to all those who have contributed to this process over the years, especially, to my parents.

iii ACKNOWLEDGEMENTS

Ph.D. program has turn out to be a long and challenging, but rewarding journey for me.

I would not have made it this far without the help of many people. My advisor Dr. Amy

B. Chan Hilton has taught me to be a hardworking and eﬀective researcher. She was always supportive of every idea I came up with for my research and helping me to improve and reﬁne them. I want to thank my doctoral committee members, Dr. Bill X. Hu, Dr. Wenrui

Huang, Dr. Danuta Leszczynska and Dr. I. Michael Navon for providing helpful suggestions and enriching my knowledge.

I am also grateful to my fellow lab-mates Omar Beckford, Satyajeet K. Iyer and Chance´e

Lundy for giving me crucial assistance and fruitful suggestions, it was enjoyable experience working with them.

Finally, I would like to show my true appreciation for the love and support of my family. I dedicate this dissertation to my parents, Xizai Li and Guishun Jin for continually encouraging me to make my best eﬀort to realize my dreams, I also appreciate my sister for so supportive and caring for me.

iv TABLE OF CONTENTS

List of Tables ...... vii

List of Figures ...... ix

Abstract ...... xii

1. INTRODUCTION ...... 1 1.1 Research Objectives ...... 1 1.2 Structure of the Thesis ...... 3

2. LITERATURE REVIEW ...... 5 2.1 Groundwater Long Term Monitoring (LTM) Problem ...... 5 2.2 LTM Spatial Optimization ...... 6 2.3 LTM Temporal Optimization ...... 8

3. INTRODUCTION TO ANT COLONY OPTIMIZATION AND BAYESIAN METHOD ...... 11 3.1 Stochastic Algorithms ...... 11 3.2 The Algorithm of ACO ...... 12 3.3 Recently Applications of ACO ...... 17 3.4 The deﬁnition of Bayesian Analysis and its applications ...... 17

4. LTM SPATIAL SAMPLING OPTIMIZATION ...... 20 4.1 Assumption of LTM Spatial Optimization ...... 21 4.2 Spatial Interpolation (IDW) ...... 22 4.3 Deﬁnitions of Data Redundancy and Overall Data loss ...... 24 4.4 Optimization Formulations for the Spatial LTM Problem ...... 26

5. LTM TEMPORAL SAMPLING OPTIMIZATION ...... 39 5.1 Bayesian Analysis Method-Prior ...... 39 5.2 Bayesian Analysis Method-Likelihood and Posterior ...... 44

6. SITE APPLICATIONS ...... 48 6.1 Spatial Test Site I - A Medium-scale Site ...... 48 6.2 Spatial Test Site II - A Large-scale Site ...... 79

v 6.3 Application of Bayesian Method for Temporal Optimization to Medium- scale Site ...... 89

7. CONCLUDING REMARKS ...... 95 7.1 Conclusions ...... 95 7.2 Future Work ...... 98

REFERENCES ...... 100

BIOGRAPHICAL SKETCH ...... 105

PUBLISHED WORKS ...... 106

vi LIST OF TABLES

5.1 Deﬁnition of SCOV values based on COV categories...... 46

5.2 Summary of SSpatial values ...... 47 6.1 Search space for medium-scale site ...... 50

6.2 Parameters used in the primal ACO-LTM algorithm for medium-scale problem 51

6.3 Summary of the optimal LTM networks identiﬁed by the primal ACO-LTM algorithm ...... 55

6.4 Parameters used in the dual ACO-LTM algorithm for medium-scale problem 63

6.5 Comparison of results from the dual ACO-LTM algorithm and complete enumeration...... 64

6.6 Parameters used in the GA algorithm for medium-scale problem ...... 72

6.7 Comparison of results from the GA algorithm and enumeration. Wells in bold font indicate those that are diﬀerent from the enumeration solution...... 72

6.8 Summary of the optimal LTM networks identiﬁed by the primal ACO-LTM, the dual ACO-LTM, the GA and Enumeration ...... 75

6.9 Summary of the CPU time (by seconds) by the primal ACO-LTM, the dual ACO-LTM, the GA and complete Enumeration (medium-scale site) ..... 76

6.10 Search space for large-scale site ...... 81

6.11 Summary of the CPU time by the dual ACO-LTM, the GA and complete Enumeration, ∗ are estimated times.(large-scale site) ...... 82

6.12 Summary of the optimal LTM networks RMSE identiﬁed by the dual ACO- LTM and the GA ...... 82

6.13 Summary of the large-scale optimal LTM networks identiﬁed by the dual ACO-LTM algorithm ...... 84

vii 6.14 Comparisons of typical results among four methods: MAROS, 3-tiered, CES, and Bayesian. Q is quarterly sampling, S is semi-annually, A is annually, B is biennially, R is Removed well, and * represents redundant well. Bolded in Bayesian are the maximum probability of posterior sampling event...... 92

viii LIST OF FIGURES

3.1 Single bridge ant experiment. After [1] ...... 14

3.2 Double bridges experiment. a) Early in the experiment. b) At a later time. After [2] ...... 15

4.1 An ant’s trail is comprised of visits to nodes that represent decision j at each well i and identiﬁes the monitoring wells retained or eliminated in the reduced network...... 28

4.2 Overview of the ACO algorithm developed for spatial optimization of LTM networks. Numbers in parenthesis correspond to the ACO-LTM algorithm steps...... 29

4.3 Cross-validation using well x, which is the closest well to the current well i, to evaluate well i’s contribution/redundancy to its local neighbors...... 30

4.4 Details of the ant activity to determine whether to retain or eliminate monitoring well i...... 31

4.5 Representation of the ACO paradigm for the LTM network optimization problem. An ant’s path of identiﬁes the redundant wells to be eliminated (solid circles) from among the existing monitoring wells (open circles). ... 35

5.1 Deﬁnition of k and k’; when k > 0, k is created from last four events, and k’ is obtained by last four events and historically the maximum concentration event...... 41

5.2 Deﬁnition of Prior Belief, P (Fi), where F are the possible sampling frequency options (Q = quarterly, S = semi-annually, A = annually)...... 43

6.1 The original 30-well LTM network and contaminant concentration plume (with contours in mg/L) based on monitoring data...... 49

ix 6.2 Comparison of REE values from solutions using a ranking method to eliminate wells...... 50

6.3 Example of the convergence of the average RMSE during (a) all iterations of the primal ACO-LTM algorithm, and (b) iterations 6-100 (zoom in of Figure a) 52

6.4 Example of the convergence of the primal objective function (number of remaining wells) ...... 53

6.5 Variation of overall data loss as quantiﬁed by the RMSE with the number of remaining wells in the optimized LTM network (primal). The average REE of the interpolated concentrations for the eliminated wells, along with error bars represented one standard deviation, is shown for comparison...... 54

6.6 Concentration contours (mg/L) based on data from the 27 wells and interpolated values at the 3 wells selected to be eliminated from the existing LTM network, resulting in an RMSE of 0.383(primal)...... 56

6.7 Concentration contours (mg/L) based on data from the 23 wells and interpolated values at the 7 wells selected to be eliminated from the existing LTM network, resulting in an RMSE of 0.637(primal)...... 57

6.8 Concentration contours (mg/L) based on data from the 21 wells and interpolated values at the 9 wells selected to be eliminated from the existing LTM network, resulting in an RMSE of 1.165(primal)...... 58

6.9 Concentration contours (mg/L) based on data from the 23 wells and interpolated values at the 7 wells selected to be eliminated from the existing LTM network using the general MAROS procedure, resulting in an RMSE of 0.721. 59

6.10 Concentration contours based on data from the original 30-well LTM network compared to the optimized 23-well network identiﬁed by (a) the developed primal ACO-LTM algorithm, and (b) a procedure based on MAROS. .... 62

6.11 Comparison of the objective function values (RMSE) for solutions identiﬁed by the dual ACO-LTM algorithm and complete enumeration...... 64

6.12 The optimal path traversed in the 27-well case from the dual ACO-LTM algorithm. The 3 redundant wells, in order of selection, are wells 25, 3 and 15. 66

6.13 The optimal path traversed in the 23-well case from the dual ACO-LTM algorithm. The 7 redundant wells, in order of selection, are wells 19, 15, 18, 3, 25, 21, and 2...... 67

x 6.14 The optimal path traversed in the 21-well case from the dual ACO-LTM algorithm. The 9 redundant wells, in order of selection, are wells 2, 11, 19, 25, 13, 3, 10, 15 and 18...... 68

6.15 Comparison of the contours (0.005, 0.01, 0.05, and 0.15 mg/L) based on the existing 30 monitoring wells and the 21-well reduced network identiﬁed by the dual ACO-LTM algorithm and complete enumeration...... 69

6.16 Comparison of the objective function values (RMSE) for solutions identiﬁed by the GA algorithm and complete enumeration...... 73

6.17 Comparison of the optimal LTM networks results (RMSE) by the primal ACO-LTM, the dual ACO-LTM, the GA and Enumeration ...... 74

6.18 21-well case of the convergence of the average RMSE during (a) all iterations of the GA and the dual ACO-LTM algorithms, and (b) iterations 2-50 (zoom in of Figure a) ...... 78

6.19 The original 53-well LTM network and contaminant concentration plume (with contours in mg/L) based on monitoring data...... 80

6.20 The objective function values (RMSE) for solutions (large-scale) identiﬁed by the dual ACO-LTM algorithm and the GA...... 83

6.21 The optimal path traversed in the 48-well case from the dual ACO-LTM algorithm. The 5 redundant wells, in order of selection, are wells 51, 28, 7, 16, and 49...... 85

6.22 The optimal path traversed in the 44-well case from the dual ACO-LTM algorithm. The 9 redundant wells, in order of selection, are wells 29, 7, 26, 28, 16, 10, 31, 49, and 23...... 86

6.23 The optimal path traversed in the 38-well case from the dual ACO-LTM algorithm. The 15 redundant wells, in order of selection, are wells 52, 10, 29, 31, 39, 17, 23, 26, 28, 7, 49, 51, 37, 15, and 16...... 87

6.24 38-well case of the convergence of the average RMSE during (a) all iterations of the GA and the dual ACO-LTM algorithms, and (b) iterations 26-50 (zoom in of Figure a) ...... 88

6.25 Historical monitoring data, (a)-(f) represent well 4, 5, 6, 8, 22, and 26 respectively...... 93

xi ABSTRACT

This dissertation presents the work of groundwater long-term monitoring optimization based on an ant colony optimization algorithm and Bayesian analysis. Groundwater long- term monitoring (LTM) is required to assess human health and environmental risk of residual contaminants after active groundwater remediation activities are completed. However, LTM can be costly because of the large number of sampling locations and frequencies that exist at a site from previous site characterization and remediation activities.

Two LTM spatial sampling optimization methods based on ant colony optimization

(ACO) algorithm were developed to identify optimal sampling networks that minimize the cost of LTM by reducing the number of monitoring locations with minimum overall data loss.

The ﬁrst method is called the primal ACO-LTM algorithm, which minimizes the number of remaining wells given the constraint on data loss quality, and it was implemented by binary decision variables. The second method is inspired by primal algorithm, and named as the dual ACO-LTM algorithm, here the role of the number of remaining wells is reversed from objective function to constraint, and this algorithm was to minimize the data loss given a

ﬁxed number of remaining wells. This dual ACO-LTM algorithm has a close analogy to the ACO paradigm for solving the traveling salesman problem (TSP). However, unlike the

xii TSP problem, in the LTM problem, the ants will not necessarily visit all the wells. The ant terminates traveling when it has visited a given number of wells equal to the described number of redundant monitoring wells. Comparisons among the primal and dual ACO-LTM, the GA, and complete enumeration show that The dual ACO-LTM algorithm showed the best performance and identiﬁed global optimal solutions.

A statistical guideline for LTM temporal redundancy problem was proposed. Instead of relying on pollutant transport simulation models, this method is a data driven analysis approach. This study uses a Bayesian statistics-based methodology to optimize the scheduling of groundwater long-term monitoring. The technique combines information from diﬀerent sets of observations over multiple sampling periods with spatial sampling optimization by ant colony optimization algorithm to provide probability distribution for future sampling schedule. Thus, the output of this method is not binary results (0/1), but fuzzy probabilistic scale (0 1) for future monitoring schedule of each individual monitoring well. The results from medium size site were compared with those from other LTM design methods, including

MAROS, CES, and 3-tiered approach. Similar but outperforming results with other methods veriﬁed that this method is a promising approach for LTM temporal problem.

xiii CHAPTER 1

INTRODUCTION

1.1 Research Objectives

Ground water is a vital resource in our environment. It replenishes our streams, rivers, habitats, and also provides fresh water for irrigation, industry and communities. How- ever, ground water is highly vulnerable to contamination from septic tanks, agricultural runoﬀ, highway de-icing, landﬁlls, pipe leaks, spills, and improper disposal practices [3].

Groundwater long-term monitoring (LTM) has been practiced for decades for groundwater remediation projects to assess the problems associated with groundwater contamination and its environmental consequences. The sampling plan may quickly become outdated and cost eﬃciency has to be considered to be optimal.

Indeed, at the beginning of long term monitoring, the redundant data are very important and necessary, because the data collection should try to discover the entire message about contaminant groundwater as much as possible. The other important issue is that the contaminant area should be very big, and there should already exit signiﬁcant number of monitoring wells. Under this condition, the LTM problem has the space to be optimized, and small contaminant area and few monitoring wells will not be considered in this research.

1 optimization of the original LTM design may be implemented by deleting the redundant data carefully and wisely. Optimization of LTM design will be considered after the plume stage may not change.

Note that remaining contaminants still exist after the contaminant remediation done.

When the contaminant remediation problem came out the ﬁrst time in United States, people believed that the contaminant sites would be completely restored as uncontaminated before, which means no further action would be needed. After several contaminant remediation projects, people realized it is impossible or infeasible to absolutely clean up all contaminants on an aquifer. Now the goal of restoration is the protection of human health and the environment.

After the closure of contaminant remediation, an LTM is still required because of remaining contaminants. So the LTM optimization should begin at later time of contaminant remediation performance, and undergo after the contaminant remediation ﬁnished.

The overall goal of LTM optimization is to reduce monitoring costs while still capturing suﬃcient information about contaminant levels and plume movement. Since the majority of monitoring costs is from sampling, the focus is to reduce number and frequency of monitoring points and to decide which contaminants and constituents require sampling. An existing monitoring network typically has more than necessary sampling locations and frequencies for the purpose of LTM. Thus LTM costs may be reduced by identifying redundant sampling locations and frequencies.

The objectives of this research are to:

Task 1: Develop an ant colony optimization (ACO) method for the LTM spatial redundancy problem.

Task 2: Develop an LTM temporal sampling optimization algorithm based on Bayesian analysis.

2 The ACO method is inspired by the fact that ants are able to ﬁnd the shortest route between their nest and a food source. This is accomplished by using pheromone trails as a form of indirect communication. Ant colony simulation techniques are adapted to minimize the number of monitoring locations in the sampling network without signiﬁcant loss of information. In the later, ant colony can decide the optimal monitoring locations and frequency and type of contaminant constituents by the historical data.

Bayesian inference provides a logical, quantitative framework for assessment of the current state of knowledge regarding the issue of interest and gathering new data to address remaining questions and then updating and reﬁning the understanding to incorporate both new and old data. Given the prior distribution, collect data to obtain the observed distribution. In practice, it is common to assume a uniform distribution over the appropriate range of values for the prior distribution. Then calculate the likelihood of the observed distribution as a function of parameter values, multiply this likelihood function by the prior distribution, and normalize to obtain a unit probability over all possible values. This is called the posterior distribution.

1.2 Structure of the Thesis

This dissertation is organized into seven chapters. The background and objectives of this research study are presented in Chapter 2. Chapter 3 introduces a stochastic optimization method, ant colony optimization (ACO). Chapter 4 provides the formulations of LTM spatial sampling optimization problem, primal and dual problem; and the development of the LTM spatial optimal algorithms based on the ACO algorithm. Chapter 5 presents an

LTM temporal sampling optimization by Bayesian analysis. Chapter 6 demonstrates the applications of LTM spatial and temporal optimization methods by two contaminant sites, a medium size and a large size. Chapter 7 summarizes the major contributions of this

3 dissertation and outlines possible future research work and enhancement.

4 CHAPTER 2

LITERATURE REVIEW

This chapter introduces general groundwater LTM problem in Section 2.1 and reviews currently existing methods for designing groundwater monitoring plans and those speciﬁcally designed for LTM problem. Methods for spatial sampling analysis (i.e., to determine the number and location of sampling points) and for temporal sampling analysis (i.e., to determine sampling frequency) are summarized in Section 2.2 and Section 2.3, respectively.

2.1 Groundwater Long Term Monitoring (LTM) Problem

Groundwater monitoring is required after the remediation stage, and in recent years, as the use of monitored natural attenuation increases, long-term monitoring (LTM) becomes more relevant and important. LTM programs are intended to assess the occurrence and migration contaminant in various media over time. The types of data of interest include chemical (such as pH, and concentrations), physical (such as temperature, and piezometric or hydraulic heads), and biological (such as type of bacteria and rate of biodegradation)

[4]. LTM can be considered as an overall quality assurance of the cleanup process, remedial action and compliance [5].

In particular, the objectives of a LTM programming include one or more of the following categories: ”(1) Validate the conclusions of a remedial investigation/feasibility study

5 (RI/FS); (2) Determine if contamination is migrating oﬀ site or oﬀ base; (3) Determine if contamination will reach a receptor (such as a drinking water supply well); (4) Track contaminants exceeding some standard; (5) Track the changes in shape, size, or position of a contaminant plume; (6) Assess the performance of a remedial system (including monitored natural attenuation); (7) Assess the practicability of achieving complete remediation; (8)

Satisfy regulatory requirements” [6]. [7] suggested that we still need ”comprehensive, eﬃcient, and robust solution methods” to solve stochastic and multi-objective subsurface optimization problems.

Groundwater remediation projects require long-term monitoring (LTM) to assess compliance of active remedial systems and post-closure sites where groundwater contamination is still present. LTM can be costly given the large number of sampling locations, frequency of monitoring, and number of constituents monitored at a given site. The U.S. Department of Energy (DOE) estimated that the total costs for monitoring at their sites where long term stewardship has been mandated may be up to $100 million per year [8]. The U.S. Navy estimated the costs of LTM and remedial active operation in Navy’s contaminated sites are from $46 million for the year 1999 to $77 million for the year 2003 [6]. This work presents the development of a methodology to optimize a groundwater-monitoring network in order to maximize cost-eﬀectiveness without compromising program and data quality.

2.2 LTM Spatial Optimization

6 LTM. Thus LTM costs may be reduced by identifying redundant sampling locations.

Two common approaches to LTM optimization are mathematical optimization and statistical analysis. In mathematical optimization, optimal LTM sampling is identified by applying search algorithms that maximize or minimize a given objective function subject to constraints. These optimization algorithms use plume predictions based on numerical simulation models of contaminant transport and groundwater flow and/or geostatistical interpolation to guide the search. Commonly used simulation models include MODFLOW [9] combined with MT3D [10] or RT3D [11]. For example, [12]optimized groundwater monitoring networks using a genetic algorithm stochastic search method combined with Monte Carlo simulation of plume movement. [13]developed an approach using simulated annealing for stochastic global optimal search and statistical methods to increase spatial accuracy of monitoring networks. [14] optimized sampling networks using geostatistical interpolation through inverse distance weighting (IDW), ordinary kriging combined with a fate-and- transport model by, and genetic algorithms [15]introduced new spatial moment constraints based on [14]to obtain robust long-term monitoring optimization designs. [16] designed an improved detection network for a landfill using a two- dimensional simulation model alone without mathematical optimization methods. [17] proposed a method combined a new myopic heuristic algorithm (MS-ER) with a contaminant transport model. The drawback of using simulation models to predict contaminant plume movement is that limited site data or complex hydrogeological conditions will lead to uncertainty in the model and input parameters, causing errors in the predications and possibly unreliable optimal monitoring networks.

Statistical methods have been applied to improve existing monitoring networks by analyzing current or historical data. [18]and [19] developed the monitoring and remediation

7 optimization system (MAROS), which is a decision support system that uses a ranking rule- based approach combined with Delaunay triangulation to reduce sampling locations and statistical methods to reduce sampling frequency. [20] and[21] developed the Geostatistical

Temporal/Spatial (GTS) optimization algorithm, which is a site speciﬁc technique based on kriging and is applicable to sites with a large number of monitoring wells. However, MAROS, and GTS are not mathematical optimization methods; they are decision support tools in which manual iterative steps rather than automated processes are used to improve existing

LTM networks in a sequential ranking procedure. As such these decision support tools do not focus on identifying global optimum solutions and do not have the beneﬁts of mathematical optimization methods, such as the ability to evaluate multiple options while considering the interactions between decision variables, objectives, and constraints simultaneously. More importantly, ranking methods that eliminate the most redundant well(s) each iteration are similar to greedy search procedures, which typically result in suboptimal and local optima solutions. 2.3 LTM Temporal Optimization

For long-term monitoring, many monitoring wells are sampled too often, so there exists some opportunities to reduce sampling frequency and, in this way save money. aforementioned simulation models are used by some researchers to predict and optimize the schedule of groundwater LTM sampling. Other than simulation models, some decision policies by statistic methods are developed to optimize LTM temporal problem.

In recent years, some approaches have been developed for and applied to the redundant sampling frequency problem, most of which rely on statistics to reduce sampling schedule.

Fro example, [22] developed cost-eﬀective sampling (CES) methodology in response to the monitoring sampling plan that many locations were being sampled too often. CES

8 evaluates the contaminant history by widely understood statistical methods - trend and variability. The trend is deﬁned as the rate of concentration change over time by regressing measured concentration, and variability is the degree of uncertainty in the concentration.

The basic sampling rationale is that the lower rate of concentration change and the degree of uncertainty, the greater the need for frequent sampling reduction. [3] recommended and compared two statistical based methodologies, the monitoring and remediation optimization system (MAROS) [18] and Three-Tiered Approach [23]. MAROS assesses the historical concentration data by a modiﬁed CES method, in which nonparametric trend analysis

- Mann-Kendall analysis - is included. The three-tiered approach is conducted in three stages: a qualitative evaluation, temporal evaluation, and spatial evaluation. In the temporal evaluation, the historical monitoring data are examined using the Mann-Kendall test. The

Geostatistical Temporal-Spatial algorithm (GTS) [21] identifies temporal redundancy by lengthening the time between sample collections. The temporal algorithm in GTS sets up time variograms using individual monitoring historical data, and the temporal redundancy is characterized by this one-dimensional axis between sampling events. [13] proposed time model by comparing the sum of the differences between time series. The redundancy depends on time series variances and temporal fluctuations. Thus maximizing the differences between time series translates to minimizing temporal redundancy in their work.

[24] developed a statistical protocol to assess to check temporal trend of groundwater contaminant. In this protocol, temporal trends are examined at three levels: overall temporal trend for the entire site, temporal trends for local regions, and individual temporal trends for single sites.

Basically, Kriging or IDW will be used to optimize the monitoring schedule. Just like spatial optimization problem, elapsed time will be considered as the distance between the

9 sample points. In the overall temporal trend analysis, average concentrations from the whole monitoring network will be considered. [25] proposed the method to estimate the average temporal autocorrelation using time of sampling as the dimension. GTS temporal optimization part sets up time variogram by each individual monitoring historical data, and the temporal redundancy is characterized by this one-dimensional axis between sampling events.

10 CHAPTER 3

INTRODUCTION TO ANT COLONY OPTIMIZATION AND BAYESIAN METHOD

The methodology used to solve the optimization problem described in Chapter 3 is a stochastic algorithm, called ant colony optimization (ACO),a new class of natural algorithms inspired by the foraging behaviour of natural ant colonies. The ﬁrst ACO system was introduced by Marco Dorigo in his Ph.D. thesis in 1992, and was called Ant System (AS).

Section 3.1 presents general knowledge about stochastic algorithm and explain the reason why stochastic algorithm is selected in this dissertation. Section 3.2 introduces the general idea of ant colony optimization by a cardinal NP-hard problem - traveling salesman problem

(TSP). Section 3.3 shows some up to date examples of ACO applications.

3.1 Stochastic Algorithms

There exist many mathematical algorithms to carry out optimal search, but generally, there are two kinds of algorithms: deterministic or stochastic. Deterministic algorithms are very traditional; all the related functions should be clearly and fully speciﬁed. Additionally, the search direction is deterministic; the terminal condition is that ﬁrst derivative of objective function should be close to zero; and depending on whether we seek the maximum or minimum problem, we should calculate the second derivative value, and verify that it is greater/less than zero. But in some optimization problems, it is impossible or infeasible to

11 specify objective function and constraints completely. On the other hand, there are some unknown quantities, which aﬀect the performance of the objective function.

By these reasons, stochastic algorithm is widely used in recent years; and it is a very new algorithms. First and second derivative functions are not required here, so the functions related do not need to be fully speciﬁed. Stochastic algorithm is especially good at combinatorial problems, which are very hard problems to solve for deterministic algorithms. Stochastic algorithms are probabilistic search method, in other words, after calculating the probability value, a random number is created. Next, the two values are compared, depending on the diﬀerence between the two values, the program can decide the search direction. Examples of stochastic algorithms are genetic algorithms (GA), simulated annealing (SA), neural nets (NN), evolutionary computation (EC).

The problem of long-term monitoring optimization is a discrete, combinatorial problem, obviously, stochastic method is expected to be a better choice. For long-term monitoring optimization problem, there may be exist some unknown quantities, for example, hydraulic conductivity distribution, the stochastic method can handle this problem very well. Even though stochastic algorithms are robust methods for uncertainty conditions, some original data in the ﬁeld are still required. We need them to approximate some local hydraulic parameters, and another big issue is that how to use these initial data to obtain optimal estimation. 3.2 The Algorithm of ACO

ACO is an evolutionary computation optimization method based on ants’ collective problem- solving ability. This global stochastic search method is inspired by the ability of a colony of ants to identify the shortest route between the nest and a food source. Individual ants contribute their own knowledge to other ants in the colony by depositing pheromones, which

12 act as a chemical ”markers,” along the paths they traverse. Through indirect communication with other ants via foraging behavior, a colony of ants can establish the shortest path between the nest and the food source over time with a positive feedback loop known as stygmergy. As individual ants traverse a path, pheromones are deposited along the trail, altering the overall pheromone density. More trips can be made along shorter paths and the resulting increase in pheromone density attracts other ants to this paths. Furthermore, shorter paths will tend to have higher pheromone densities than longer paths since pheromone density decreases over time due to evaporation [26]. This shortest path represents the global optimal solution and all the possible paths represent the feasible region of the problem. This stygmergy behavior was observed by [2], in which the movement of live ants was observed in a double bridge experiment where the longer branch is twice as long as the shorter branch.

ACO is a subset of swarm intelligence, in which ants act as an agent and analogies to behaviors of social insects are used to solve optimization problems. Ants, like bees, and wasps, are social insects. A single ant randomly chooses one path to visit from all the possible routes from the nest to food source. Individual ants can contribute their own information to the colony. Through indirect communication to other ants via pheromones and foraging behavior, a colony of ants can determine the shortest path between the nest and the food source over time. This shortest path represents the global optimal solution. The density of pheromone stimulates all these phenomena, which are deposited by individual ants as they traverse a path. In addition, pheromone decreases over time through an evaporation rate, thus a shorter path means higher pheromone density, and will attract more ants [26].

[1] tested ants’ foraging behavior by a single bridge experiment with each branch has the same length [Figure 3.1]. Initially, there is no pheromone on the either branch, and the ﬁrst few ants have the same probability to randomly select either branch. A few more ants may

13 Nest Food

Figure 3.1: Single bridge ant experiment. After [1]

select one branch, say upper branch, over the other. More ants mean a larger amount of pheromone is deposited to the upper branch, which in turn invites more ants to visit the upper branch. In this a single factor experiment, the behavior of the ants was found to be controlled by pheromone.

[2] considered the double bridge experiment, in which the longer branch is twice as long as the short branch [Figure 3.2]. The individual ants travel from the nest to the food source and return to the nest by the same pheromone driven mechanism as in the single bridge situation. Those ants that took the shortest path forward and back will return to the nest

ﬁrst, and immediately after these ants return, more pheromone is left on shorter branch than on the longer branch. By the time after ants arrive, pheromone left by the ants that selected the longer branches will have evaporated soon than in the shorter branches. As more ants deposit more pheromone to the shorter branches, and more pheromones attract more ants, the shortest path is identiﬁed. As the length ratio between the two branches increases, more ants will select shorter branch[27].

The ﬁrst ant colony simulation algorithm was developed by [28] to solve the classic traveling salesman problem (TSP)in 1992. In the TSP, the goal is to ﬁnd a closet tour of minimal length connecting n given cities, in which each city must be visited once and only once. [29] compared the results of their ACO algorithm applied to the TSP problem with

14 a) Foraging area b) Foraging area

Nest Nest

Figure 3.2: Double bridges experiment. a) Early in the experiment. b) At a later time. After [2]

a genetic algorithm (GA), which is a stochastic search method based on natural selection.

Results from several types of TSP problems show that ACO can identify solutions better than the GA. Ant colony simulation algorithms also have been developed for other classical optimization problems, including the quadratic assignment problem, vehicle routing problem, and graph-coloring problem [30].

In the TSP, the goal is to obtain a shortest path that connects all the cities. Each ant in city i is an agent who places pheromone on a visited path, and then chooses to visit the next town j with a probability that is a function of the town distance, dij, and of the pheromone density. τij represents the pheromone on edge (i, j ) at iteration t. It is updated according to the equation:

e τ(ij)(t +1)=(1 − ρ)τij(t) + ∆τij + e • ∆τij, (3.1)

where, (1 − ρ) represents the decay of pheromone between iterations t and t+1, and

15 m k ∆τ(ij) = ∆τ(ij), (3.2) k X=1 k Where, ∆τ(ij) is the change in pheromone due to ant k selecting city j ; and m is the total

k number of ants in one colony. The quantity ∆τ(ij) is given by

Q/L if ant k uses edge (i,j ) in the iteration t ∆τ k = k , (3.3) (ij) 0 else

Where Q is a constant related to the quantity of trail laid by ants, Lkis the total tour length by ant k. The third part is called elitist ant strategy, where, e is coeﬃcient

e of elitist pheromone, it is a small integer; ∆τ(ij) is the pheromone of best ant at each

e iteration,∆τ(ij) = Q/Le , where Le is the total tour length by elitist ant. This will direct ants’ colony toward a best solution with higher possibility.

Next, the ant can decide the next city j from i by transition rule by the following equation:

α β k [τij(t)] (ηij) Pij(t)= α β , (3.4) k∈allowed[τij(t)] (ηij)

Where, deﬁnes the ”visibility” ηijPas 1/dij, dij is the distance between two cities. α and β are parameters that control the relative important of pheromone and visibility. The translation probability is a trade oﬀ between the visibility, which is greedy heuristic strategy, and the pheromone.

[31] compared ACO with other stochastic search algorithms, including the genetic algorithm, evolutionary programming, and simulated annealing, by solving the TSP with

50, 75 and 100 cities. Results showed that ACO identiﬁed the best solution for each

TSP case. ACO algorithms also have been developed for other classical optimization problems, including the quadratic assignment problem, job-shop scheduling problem, vehicle routing problem, and graph-coloring problem [30]. Moreover, [32] proved that under certain

16 conditions, solutions from ant-based optimization converge to the global optimum with a probability close to 1.

3.3 Recently Applications of ACO

More recently, ACO algorithms have been applied to solve a wide range of engineering and science problems such as random number generators [33], autonomous decentralized shop

ﬂoor routing [34], bandwidth minimization problem in a large-scale power transmissions system [35], redundancy apportionment problem in electrical and mechanical systems (Zhao et al., 2005), and capacitated minimum spanning tree problems applied to telecommunication networks [36]. To date, a limited number of works have been published in which ACO or swarm intelligence has been used to solve water resources and hydrology problems; nevertheless, these do not focus on groundwater management and remediation design optimization problems. In water resources, [37] used ACO to optimize water distribution systems designs and [38], which used particle swarm optimization to determine pump speeds to minimize the total costs in water distribution systems. [39] used ACO to solve an inverse modeling problem of identifying unsaturated soil parameters. In this work, we focus on developing an ACO algorithm to solve this single-objective formulation. However, it may be possible to reformulate this problem into a multi-objective problem and adapt multi-objective

ACO algorithms developed for other applications [40][41][42] to this problem.

3.4 The deﬁnition of Bayesian Analysis and its applications

Bayesian logic may be applied to quantify uncertainty and inferential statistics that deals with probability inference: the theorem deﬁnes a rule for reﬁning a hypothesis by factoring in additional evidence and background information, and leads to a number representing the

17 degree of probability that the hypothesis is true. Bayesian provides a mathematical method that could be used to calculate, given occurrences in prior trials, the likelihood of a target occurrence in future trials. According to Bayesian logic, the only way to quantify a situation with an uncertain outcome is through determining its probability. The Bayesian Theorem is a means of uncertainty assessment. Based on probability theory, the theorem deﬁnes as:

posterior ∝ likelihood × prior, (3.5)

indicating that the posterior is proportional to the product of likelihood function and prior distribution. Once the normalized posterior distribution is obtained, it straightforwardly shows that the maximum probability expected event may most probably happen based on the prior belief. The limitation of Bayesian method is that it is subjective, especially for the establishment of prior belief.

Current and possible applications of Bayesian logic include in the groundwater contamination area are as follows. In groundwater contamination area, [43] proposed a variable density sampling pattern of contaminated soil and compared to uniform sampling pattern. [44] proposed risk assessment of nuclear waste disposal by Bayesian network. [45] used Bayesian approach for a flow-field modeling to determine the greatest model uncertainty at the model boundaries. [46] estimated parameters characterizing the hydrodynamic behavior of aquifers by Bayesian method. [47] characterized the the predictive uncertainty in the delineation of time-related well capture zones in heterogeneous formations by Bayesian and obtained conservative sampling densities. [48] developed probabilistic expert systems to produce a geographic distribution for the most probable sources of salinization by Bayesian belief networks methods. [49] developed Bayesian Network by means of stakeholders participation to quantify risk of over exploitation of the local aquifer. [50] assessed time-varying runoff

18 sources on the basis of chemical composition by Bayesian Markov Chain-Monte Carlo

(MCMC) analysis. [51] introduced Bayesian analysis into the decision tree to determine if additional testing would improve the available with less uncertainty. [52] measured the vulnerability of groundwater to pesticide contamination directly from monitoring observation by statistical pattern, Bayesian method.

19 CHAPTER 4

LTM SPATIAL SAMPLING OPTIMIZATION

This chapter presents two methods to determine LTM optimal sampling locations. The assumption of LTM spatial sampling optimization is introduced in Section 4.1. Section 4.2 discusses two groundwater contaminants concentration estimation techniques-one is a deterministic interpolation technique, and the other is a geostatisitcal one. Since an LTM network could be overall redundant but locally inadequate or vice versa, the deﬁnitions of local and global data loss are given in Section 4.3. Section 4.4 presents two optimization formulations, primal and dual. We formulate the LTM optimization problem in two ways: one is to minimize the number of remaining wells given constraint rules on data quality and estimation errors; the other is given the number of remaining wells, the objective is to determine the optimal combination of a reduced set of wells from among the original ground water monitoring network. Here we reverse the role of the number of remaining wells from objective function to constraint, and we call the former optimization formulation the primal problem and the latter the dual problem. Two ACO-LTM algorithms to solve the two problems are discussed in Section 4.4.1 and Section 4.4.2. Major part of these method are published in [53], [54] and [55]. The discussion below is adapted and expanded from these publications.

20 4.1 Assumption of LTM Spatial Optimization

One assumption of LTM spatial optimization in this work is that only during the later time of contaminant remediation, the groundwater monitoring design will be optimized. Also, the plume at that time should be on the stable stage or shrinking. Actually, at the beginning of long term monitoring, the redundant data are very important and necessary, because people do not have any information about the contaminant in the aquifer, the data collection should try to discover the entire message about contaminant groundwater as much as possible. The other important issue is that the contaminant area should be very big, and there should already exit signiﬁcant number of monitoring wells. Under this condition, the LTM problem has the space to be optimized, and small contaminant area and few monitoring wells will not be considered in this research.

After the performance of contaminant remediation for a certain time, the plume shape will be on a stable stage, and till that time, pretty many data will be collected. Under this condition, the original LTM design should not be followed any more, on the contrary, optimization of the original LTM design may be implemented by deleting the redundant data carefully and wisely. Optimization of LTM design will be considered after the plume stage may not change.

Note that remaining contaminants still exist after the contaminant remediation done.

21 After the closure of contaminant remediation, LTM is still required because of remaining contaminants. So the LTM optimization should begin at later time of contaminant remediation performance, and undergo after the contaminant remediation ﬁnished.

4.2 Spatial Interpolation (IDW)

Generally, there are two methods to estimate and predict an unknown concentration by current data. Also, the most challengeable issue is how to wisely and entirely use the current data to lower the residual between estimated and measured value. One of them uses numerical simulation models to calculate unknown concentrations. Absolutely, mechanism is included, but the following problems are parameters uncertainty, which will increase the residual, unless we have pretty proper parameters, but this will increase the costs. The other method is based on geostatistical analysis. One assumption of geostatistics is that statistical spatial/temporal is dependant, which is autocorrelated. Just like [56] ”The ﬁrst law of geography: everything is related to everything else, but near things are more related than distant things.” So we can estimate the unknown concentration after studying the dependence of current data. Unlike numerical simulation models, the pure geostatistical method does not require uncertain parameters from the ﬁeld, and at the same time, no mechanisms are involved, we may compromise it by optimizing current data autocorrelation.

There are many geostatistical models for estimating, inverse distance weighting (IDW) and kriging are very typical.

4.2.1 Deterministic Interpolation

The estimated value is linear combination of measured points surrounding it. The following challenging problem is how to decide the coeﬃcients (weights) of the measured points.

Depending on the way to identify the weights, generally, there are two techniques, one

22 is deterministic interpolation technique, the other is geostatisitc one. IDW is a deterministic technique, the weights are decided by mathematic functions; here they are only based on the distance between measured points and estimated location. The computing time of IDW is very fast, no assumptions are required, the method is very simple and basic. Kriging is a geostatistic estimation, which means autocorrelation is included. The weights of Kriging are identiﬁed by not only the distance between measured points and estimated location, but also the correlation among the measured points. Too many parameters are involved, which may increase the estimated residuals, and the computing time is pretty slow. Some assumptions are included, and sometimes, the data should come from a normal distribution.

To estimate a value at any unmeasured location, IDW uses the measured values surrounding the prediction location. IDW assumes that each measured point has a local inﬂuence that diminishes with distance. It weights the points closer to the prediction location greater than those farther away. The neighborhood size (n) determines how many points are included in the inverse distance weighting.

n Ci i P =1 di C0 = n 1 , (4.1) i P P =1 di

where C0 = evaluated concentration at aP location (x0, y0);

Ci =measured concentration at location (xi, yi);

di = the distance between (x0, y0) and (xi,yi);

P = power of distance;

n= number of neighbors around location (x0, y0).

4.2.2 Geo-Statistical Interpolation

There are two stationary assumptions on traditional Kriging, such as Ordinary Kriging (OK), one is called mean staionarity, and assumes that the mean is constant between samples and

23 is independent of location. The other is second-order stationarity, the assumption is that the covariance is the same between any two points that are at the same direction and same distance, and also it is independent of location, which means it dose not matter which two points you choose.

It is not appropriate to use traditional Kriging to interpolate contaminant plume by existing concentrations, the reason is that concentrations are non-stationary data, the mean and covariance vary widely across a given site, and actually, contaminant concentrations from monitoring wells are preferentially distributed near the contaminant source. Contaminant concentration fields are highly heterogeneous and anisotropic by these reasons, and estimating non-stationary phenomena such as contaminant concentrations with limited and highly sparsed data will case biased results. [57]compared 6 different interpolation methods, recommended the Quantile Kriging (QK) is the best one. QK is an extension of the OK approach, but QK overcomes the limitations of OK, such as stationary assumptions and high sensitivity to highly skewed data. The conclusion from [57] is that QK can handle non-stationary data, including contaminant concentrations. All these mean that there exists some way that we can adapt Kriging method to figure out better interpolation methods for contaminant concentration. 4.3 Definitions of Data Redundancy and Overall Data loss

Individual sampling locations are evaluated with respect to their redundancy in the LTM network in order to identify candidate monitoring wells to eliminate. The Relative Estimation

Error (REE) is developed here to quantify the spatial redundancy of measured data. The

REE is the normalized diﬀerence between the estimated and measured concentrations and is deﬁned as

24 |C − C | REE = est,i i , (4.2) min(Cest,i,Ci)

where Ci is the measured concentration of the well i and Cest,i is the estimated concentration of the well i. Note that the estimated concentration is compared with the measured value, which is assumed to be the ”true” concentration during a given sampling period. Since the diﬀerence between Cest,i and Ci is normalized by the minimum of these two values, the REE is very sensitive to the residual. REE values may range from 0 to more than

1,000. Individual monitoring wells with low REE values are potential redundant sampling locations. Acceptable REE values may vary among individual monitoring wells depending on their location. For example, since monitoring wells along the boundary of a contaminant plume help to deﬁne the extent of the plume, the acceptable REE for boundary wells may be lower than that for interior wells for these boundary wells to be considered potential redundant wells.

In addition to evaluating the importance of individual monitoring wells, the overall data loss of the reduced LTM network due to interpolating concentrations at removed redundant wells needs to be quantified. The overall data loss resulting from optimizing the number of sampling locations is quantified using the interpolation root mean square error (RMSE), which is defined as

− m Cest,i Ci 2 i=1( min(C ,C ) ) RMSE = est,i i , (4.3) sP m

where m is the number of removed wells, Ci is the measured concentration of well i, and

Cest,i is the estimated concentration of well i estimated using data from the remaining wells.

It is expected that acceptable optimized LTM networks have relatively low RMSE values.

The RMSE increases due to increased data loss from eliminating too many monitoring wells

25 and/or wells that are not redundant with respect to the overall monitoring network. 4.4 Optimization Formulations for the Spatial LTM Problem

Two formulations of the spatial LTM optimization problem are presented here. The ﬁrst

(formulation 1) is to minimize the number of monitoring locations (Equation (4.4)) while subject to a constraint on overall data loss (Equation (4.5)). In formulation 1, threshold is deﬁned for the overall data loss; decision makers may set the maximum overall data loss after optimization.

MinZ = n, where nthe number of remaining wells. (4.4)

s.t.

− m Cest,i Ci 2 i=1( min(C ,C ) ) est,i i ≤ threshold, (4.5) sP m

where m = the number of removed wells; Ci = the measured concentration of removed well i, and Cest,i = the estimated concentration of removed well i by remaining wells.

The companion ”dual” problem (formulation 2) is to minimize the data loss (Equa- tion (4.6)) given a ﬁxed number of removed wells (Equation (4.7)). The threshold in formulation 2 is deﬁned as the number of remaining wells after optimization, which decision makers may determine according to their total available budget.

− m Cest,i Ci 2 i=1( min(C ,C ) ) MinZ = est,i i , (4.6) sP m s.t.

n = threshold, (4.7)

26 4.4.1 ACO Algorithm for the Primal Problem

An ACO algorithm for solving the groundwater LTM spatial optimization problem is developed in this work. In order to adapt ACO to the LTM optimal sampling problem, heuristics for the ACO paradigm need to be made. Unlike the TSP, the path distance itself is not important in the LTM optimization problem. The developed ACO algorithm presented here is an enhancement of the initial ACO procedure for LTM optimization introduced by

[54] and [53], which is based loosely on [39]. The developed method is a mathematical optimization approach that uses IDW as the interpolation method. The overall goal of LTM optimization is to minimize the number of monitoring wells used for sampling by minimizing spatial redundancy while data loss is minimized (Equations (4.4) and (4.5)). The general representation of the LTM problem in the ACO framework is shown in Figure 4.1. Ants visit a series of locations that represent actions at the monitoring wells. When at well i (i=1, 2,

, M ) in the existing monitoring network, an ant has two choices: either eliminate the well

(j =0) or include the well (j =1) in the updated monitoring network [Figure 4.1]. As the ant travels through the monitoring network, it decides to visit location j =0 or j =1 for each well i, therefore including or excluding each well from the reduced LTM network. The number of decision variables is the total number of wells considered for optimization, with each variable having a binary option, resulting 2M combinations of monitoring well networks. A challenge in applying ACO for solving optimization problems is the development of a heuristic for the foraging and stygmergy behavior of ants. In this work, the REE at individual wells and overall RMSE of the monitoring network are used to regulate pheromone density and ant movement. The overall ACO-LTM algorithm is summarized in Figure 4.2, with the procedure is described as follows:

1. The order in which each ant visits the wells is randomized for each individual ant to

27 Food Well #

30 0 1

Legend: 9 0 1 3 Well ID number

4 0 1 0 Well is “off” (j=0)

1 1 Well is “on” (j=1) 0 1

Nest

Figure 4.1: An ant’s trail is comprised of visits to nodes that represent decision j at each well i and identiﬁes the monitoring wells retained or eliminated in the reduced network.

ensure that the colony explores different regions of search domain, thus each ant may have a different visiting order from other ants. Each ant must visit all wells before arriving to the food source from the nest (i.e. starting node), considering all wells in the original monitoring network. The initial pheromone at the beginning of the first iteration at each node is equal.

In this work, the number of ants in one colony is equal to the number of monitoring wells in the original LTM network.

2. Data redundancy is quantiﬁed using the relative estimation error (REE) (Equa- tion (4.2)). If the current well is classiﬁed as a boundary well, then the REE is calculated

ﬁrst; if and only if the REE is less than a speciﬁed boundary threshold, Tedge (in this work

28 Begin iteration

Next ant

Randomize well order

Next well

Y Boundary ?

N REE < Tedge ? N Handled by ant Y Handled by ant Retain well

N All wells visited?

Calculate RMSE and λ = (T/RMSE)2

N RMSE < T ?

Y λ≥1 λ < 1

Update pheromone trails:

N All ants finish visiting?

N Maximum iterations?

End Program

Figure 4.2: Overview of the ACO algorithm developed for spatial optimization of LTM networks. Numbers in parenthesis correspond to the ACO-LTM algorithm steps. 29 x i

Figure 4.3: Cross-validation using well x, which is the closest well to the current well i, to evaluate well i’s contribution/redundancy to its local neighbors.

Tedge = 0.1), then the decision of whether to keep or eliminate this boundary well will be determined by ants by continuing onto step 3 [Figure 4.2]. Otherwise, the boundary well is retained in this ant’s solution. Since wells on the edge of the LTM network are important for plume delineation, the probability of eliminating these wells from the monitoring network is lower than that of interior wells. Evaluation of the REE is part of the heuristic developed for the LTM problem to allow ants to make decisions about individual wells. An individual well’s contribution to or impact on its neighboring monitoring locations is evaluated via a cross- validation procedure. Ant k identiﬁes well x, which is the closest well to its current location well i [Figure 4.3]. The REE at well x is calculated twice: REE(x,0) is determined using the concentration at well x interpolated without the data from the current well i, and REE(x,1) is determined by including the concentration data from current well i. A local assessment of current well i’s contribution to the local area is made by comparing the REE(x,0) and

REE(x,1) values. A low REE(x,0) value indicates that well i provides little additional

30 Calculate REE(x,0) and REE(x,1)

τ i α ()0( REEx ))0,( β iP )0,( = τ i α ()0( REEx ))0,( β +τ i α ()1( REEx ))1,( β

Create random number, R [0,1]

R ≤ P(i,0) ?

Y N Delete the well

Keep the well

Figure 4.4: Details of the ant activity to determine whether to retain or eliminate monitoring well i.

information (low contribution) when used to estimate the concentration at well x, whereas a high REE(x,0) value indicates that well i may be important to well x. Moreover, a large diﬀerence between REE(x,0) and REE(x,1) indicates that well i contributes to interpolating the concentration at well x.

3. An ant decides its next step based on a weighted combination of the current pheromone density along a segment and the redundancy of a well. It is this balance between these two heuristics that diﬀerentiates ACO from other search methods or a sequential ranking method, which takes a greedy approach to identifying redundant wells. Ant k decides to visit option

31 j =0 or j =1 for well i, indicating that well i is excluded or included the overall monitoring network, respectively, based on a probability function deﬁned by:

[τ(i0)]α[REE(x, 0)]β P (i, 0) = , (4.8) [τ(i0)]α[REE(x, 0)]β + [τ(i1)]α[REE(x, 1)]β

where and are the pheromone levels during the current iteration for the cases that represent eliminating well i (j =0) or retaining well i (j =1), respectively, and α and β are parameters that weight the pheromone and REE components. The values P(i,0) and P(i,1) represent the probability well i will be eliminated or retained in k’s monitoring network

[Figure 4.4]. Since P(i,0) + P(i,1) = 1, only P(i,0) is directly calculated. Note that if well i is eliminated, then well i will no longer considered as a neighbor or existing well by ant k for additional well assessments in the current iteration.

4. Steps 2-4 are repeated for each well in the existing LTM network for ant k, in the order randomly assigned in step 1.

5. After ant k has visited all wells in the existing LTM network, the overall data loss, which quantiﬁes the overall quality of the reduced set of monitoring wells, is evaluated by calculating the interpolation RMSE. The concentrations at the removed wells are estimated using data from the remaining wells. The RMSE is aﬀected by the number and spatial distribution of removed wells.

6. Steps 2-5 are repeated for all other ants in the colony, with the path of all ants aﬀected by the pheromone deposited along trails during the previous iteration.

7. After all ants have completed their tours in the iteration, the pheromone trails are updated. The pheromone for decision j at well i during iteration t+1 is updated according to:

32 e τ(ij)(t +1)=(1 − ρ)τij(t) + ∆τij + e • ∆τij, (4.9)

where ρ is the pheromone decay coeﬃcient (0 6 ρ 6 1); is pheromone during iteration t for well i and case j (j =0 indicates well i is eliminated; j =1 indicates well i is included).

The RMSE is calculated using newly interpolated concentrations using the data from the remaining wells. The pheromone density deposited on the path taken by ant k depends on how well the RMSE constraint (Equation (4.5)) is met. The term λ is a penalty for violating the RMSE constraint (Equation (4.5)). For a solution that violates the RMSE constraint

(RMSE >T ), the value of λ is less than one, decreasing the pheromone density of this ant’s trail. The term λ is the penalty for RMSE constraint violation (Equation (4.5)). Similarly, an ant’s solution is rewarded if it is a feasible. If RMSE is less than or equal to T, then the value of λ is greater than or equal to one. In the second term of Equation (4.9), is the total change in pheromone associated with decision j at well i, which is deﬁned as

m k ∆τ(ij) = ∆τ(ij), (4.10) k X=1 k where ∆τ(ij) is the change in pheromone due to ant k selecting decision j at well i; and

k m is the total number of ants in one colony. The quantity ∆τ(ij) is given by

k Q ∆τ(ij) = 2 , (4.11) nk in the case when ant k uses well i with decision j during the current iteration, Q is

2 a constant related ant pheromone density, and nk is the number of remaining wells in ant k’s monitoring network. The third term in Equation (4.9) allows for elitism, in which the pheromone of the ant with the best solution found so far is included in the pheromone update. In this term, e is an integer parameter and is the pheromone of the best path (from

33 this elitist ant) found so far, where is the number of remaining wells in the elitist ant’s solution. This elitism term helps direct future iteration ants toward a good solution with higher probability.

8. Continue to the next iteration t+1 using the updated pheromone trails and return to step 1. This entire process continues for a preset number of iterations [Figure 4.2].

This iterative procedure of individual ants traversing possible paths and pheromone updating guides the stochastic search to the optimal solution.

4.4.2 ACO Algorithm for the Dual Problem

An ACO algorithm for solving the groundwater LTM spatial optimization problem is presented in this work. The developed ACO-LTM algorithm is analogous to the ACO paradigm for the classic traveling salesman problem (TSP) developed in [27]. In the ACO-

TSP paradigm, each ant at city i is an agent who places pheromone on a visited path, and then chooses to visit the next city j with a probability that is a function of the distance between cities i and j (dij) and the pheromone density on this path. In the TSP, the distance between cities and the order in which they are visited are significant and affect the solution quality (i.e., total path length). However in the ACO optimization problem, the distance between wells and the order in which they are visited are not explicitly relevant to the objective function and constraints. Nevertheless, distance between wells is significant in concentration estimation (Equation (4.1)). In the ACO-LTM paradigm, ants select wells to include in the reduced LTM network based on the relative importance of a monitoring well to its neighboring wells. An ant at well i may choose from multiple wells not already selected to visit at each step along its path [Figure 4.5] based on a local error resulting from removing next well j (ηj) and the pheromone density along path ij. A summary of the developed ACO-LTM algorithm is described below.

34 Nest

Figure 4.5: Representation of the ACO paradigm for the LTM network optimization problem. An ant’s path of identiﬁes the redundant wells to be eliminated (solid circles) from among the existing monitoring wells (open circles).

1. An ant’s starting point is from a fake point, an imagined nest[Figure 4.5]. The order in which an ant visits the wells is stochastically determined, depending on the pheromone density and relative error value. Each individual ant will visit only the speciﬁed number of wells (n), which become the redundant wells of the reduced LTM network.

2. In this step, the ant decides which well will be selected among the candidates. A set of relative errors ηj is calculated for current well i and its neighbors. Calculate the error one by one as follows. Suppose well j is one well in the candidate list and well i is the center well where the ant is currently located. The concentration at well j is estimated through

IDW interpolation and Cest,j is the estimated concentration of center well j. The relative error from redundant candidate well j (ηj) is characterized by

35 |Cest,i − Ci| ηi = , (4.12) min(Cest,i,Ci) 3. The next monitoring well j in the reduced LTM network is selected from among the candidate list stochastically based on the above relative error (ηj) and pheromone deposited in the individual path ij ( τij). The probability well j is chosen when an ant currently is at well i (pij) is deﬁned by

α β (τij) (ηj) Pij = α β , (4.13) l∈L(τil) (ηl) where and b are parameters ( α = 1P and β = -1)

4. After a well among the candidates list is selected, the well is marked as a redundant well. Then this well becomes the current well for this ant.

5. Repeat steps 2 through 4, until the number of visited wells is equal to the predetermined number of redundant wells.

6. After an ant has visited the prespeciﬁed number of wells (m), the overall data loss of the reduced LTM network is determined. The data loss is due to estimating the concentrations at the eliminated wells based on information from the remaining wells. The overall goal of the LTM optimization is to reduce this overall loss, which is quantiﬁed by the root mean square error (RMSE) of the estimated concentrations of removed wells.

7. Repeat steps 1 through 6, until all individual ants in this colony ﬁnish the tour, and one iteration is implemented then.

8. The pheromone for each segment τij of an ant’s path is updated for iteration t+1 by the following rule:

e τ(ij)(t +1)=(1 − ρ)τij(t) + ∆τij + e • ∆τij, (4.14)

36 where ρ = pheromone evaporation rate; τij(t) = the pheromone density for path ij during current iteration t; and ∆τ = Q/RMSE, in which Q is a parameter (Q = 100), and the goal is to minimize the RMSE value. The idea here is that after an ant ﬁnishes a tour, if the total data loss RMSE value is low, then the pheromone density of the path of the ant will be high. Lower total data loss means higher density of pheromone, which will attract more ants to follow the same path. The term e includes elitism to the ACO search, where e is a parameter that is the number of elitist ants in an iteration; and e is the pheromone density of the recent elitist ant. The elitist ant is the ant which has obtained the best solution so far. An old elitist ant’s pheromone will be replaced by an ant with a lower RMSE value.

9. Return to step 1 to implement the next iteration. The procedure terminates after a speciﬁed number of iterations.

4.4.3 Comparison of the two Algorithms

The primal and dual problems are formulated for the LTM optimization with both of them solved by ACO. In the primal problem, the objective is to minimize the monitoring cost

(i.e., number of monitoring locations) constrained by a predetermined acceptable data loss; while in the dual problem, the objective is to lower the data loss, while constrained by

ﬁxed monitoring costs (i.e., number of remaining wells). In both formulations, the goal is to identify the optimal combinatorial set of remaining wells from among the original monitoring network.

The ACO algorithm for the primal problem has similarities to the genetic algorithm, which also is heuristic stochastic search method and is based on natural selection. Basically, each individual ant visits all the wells in the monitoring network, and each well has two stages. Each ant has only two choices at any well - keep the well, or delete the well from the monitoring network. The algorithm here is a combinatorial problem with binary variables.

37 The advantage of this algorithm is it is easy to implement. Fuzzy decision variables may be included in the code to account for uncertainty.

In the algorithm developed for the dual problem, each individual ant has more choices in comparison to the primal problem algorithm. Each ant can choose from among a set of candidate wells. This ACO algorithm has a close analogy to the traveling salesman problem (TSP). ACO algorithms have been developed to eﬃciently and successfully large

TSP problem. However, unlike the TSP problem, in the LTM problem, the ants will not visit all the wells. The ant terminates traveling when it has visited a given number of monitoring wells. This ACO algorithm for the dual problem requires more computational memory and time.

38 CHAPTER 5

LTM TEMPORAL SAMPLING OPTIMIZATION

This chapter presents a Bayesian statistics-based methodology to optimize the scheduling of groundwater long-term monitoring. The technique combines information from diﬀerent sets of observations over multiple sampling periods with spatial sampling optimization by ant colony optimization algorithm to provide probability distribution for future sampling schedule. Prior part of Bayesian method for LTM is introduced in Section 5.1, and likelihood and posterior parts is given in Section 5.2.

5.1 Bayesian Analysis Method-Prior

One assumption of this work is that the contaminated site is at or near the final stages of clean-up or undergoing natural attenuation. Sites with active remediation are not considered here for optimization since the concentrations may be significantly fluctuating and the contaminant plume shape may be not stable. During the site assessment stage in which initial data collection is the goal, reducing sampling frequency should not be considered. Moreover, this Bayesian procedure is based on the condition that there exists enough historical data at a given monitoring location and consequently, temporal redundant data may come out.

This Bayesian method deals with conditional probability, to each variable, F, with monitoring data xj, a conditional probability is obtained. Monitoring data xj represent observed data of well j, and xj will be systematically analyzed on the assumption that

39 well j was sampled at diﬀerent sampling frequency, Fi. In this work, the sampling frequency options considered include quarterly, semi-annually, and annually. According to the Bayesian

Theorem, posterior predictive distributions are obtained by integrating the product of the likelihood and prior:

π(xi/Fi)P (Fi) P (Fi/xi)= , (5.1) i[π(xi/Fi)P (Fi)]

where Fi = variable of sampling frequency,P the value is quarterly, semi-annually or annually; xj = historical concentration data of well j, xj = (xj,1, xj,2, xj,3, xj,4, ..., xj,m) and m is the number of historical ground water monitoring sampling events; P (Fi) = prior belief as a probability of each variable F ; P (Fi/xj) = Bayesian posterior predictive distribution and expresses the probability of each sampling frequency variable Fi given the observed data of well j ; π(xj/Fi) = refers to as likelihood model assuming that well j was sampled with the condition of a diﬀerent sampling frequency Fi.

In adapting Bayesian analysis for the LTM temporal optimization problem, the challenge is identifying how to include monitoring data into the prior belief, P (Fi) and likelihood model, π(xj/Fi). Note that since the number of monitoring data are limited because of costs of monitoring, it would not be appropriate to derive a distribution density function of P (Fi) by using such a sparse data set. Thus instead of assuming a distribution the prior belief,

P (Fi) is obtained through expert knowledge, such as from trends of historical concentration data which describes the rate of change in concentration over time. Since the value of

P (Fi) ranges between 0 and 1, the trend is normalized by a factor equal to several times of the maximum contaminant level (MCL) of the contaminant. Site speciﬁc goals would help determine this speciﬁc factor of MCL. Note that i P (Fi) = 1, or in this work P(quarterly) + P(semi-annual) + P(annual) = 1. P

40 Linear (k) Linear (k') 0.1600 0.1400 0.1200 0.1000

l k' 0.0800 mg/ 0.0600 0.0400 0.0200 k 0.0000 2 3 4 5 6 7 8 9 10 11 12 13 14 15 quarter year

Figure 5.1: Deﬁnition of k and k’; when k > 0, k is created from last four events, and k’ is obtained by last four events and historically the maximum concentration event.

To determine the prior belief, P (Fi), ﬁrst the preliminary normalized slope k is calculated using concentration data from the latest four sampling events (Figure 5.1). The result is a unitless value of k which is either greater than, equal to, or less than zero (Figure 5.1 and

Figure 5.2). When k is equal to zero, it implies that there was no change in the concentration over time in the recent four sampling events, and thus it is reasonable to reduce sampling frequency to annual sampling in this case.

Next, a amplified slope, k’, is determined by considering the historical extreme concentration event, such as the maximum or minimum concentration. The variable k’ is defined as follows. When k < 0, the amplified slope k’ is calculated using the historical maximum concentration event with the latest four sampling events. When k > 0, k’ is determined using the historical minimum concentration event and four latest sampling events. Note that the absolute value of k’ is greater than absolute value of k, and k’ can be considered the

41 overall trend of concentration samples that is biased towards recent concentrations. Since k’ is considered the ampliﬁed slope, in order to assure that the absolute value of k’ is greater than k, k′ = min(k, k′) when k < 0, and k′ = max(k, k′) when k > 0 (Figure 5.2). In the extreme case, when absolute value of k’ is greater than one, the change of concentration

(increasing or decreasing) is categorized as signiﬁcantly high; in this case, the sampling frequency should be increased to the highest frequency, which is quarterly in this work.

Ideally, most absolute value of individual monitoring well’s k’ should in the range of 0 and

1, but the factor of MCL used to normalize k is not a very high value. When −1 < k′ < 0, this represents an overall decreasing concentration trend, and 0 < k′ < 1 implies an overall increasing concentration trend over time (Figure 5.2).

42 Evaluate k value

k<0 k=0 k>0

P(Q)=P(S)=0, P(A)=1

k’ calculated by maximum k’ calculated by minimum concentration and latest concentration and latest four sampling events four sampling events

k’=min(k, k’) k’=max(k, k’)

Evaluate k’ value

-11 0

P(Q)=1, P(S)=P(A)=0 P(Q)=-k’ P(Q)=k’ P(S)= (1+k’)/3 P(S)=2*(1-k’)/3 P(A)=2*(1+k’)/3 P(A)=(1-k’)/3

Figure 5.2: Deﬁnition of Prior Belief, P (Fi), where F are the possible sampling frequency options (Q = quarterly, S = semi-annually, A = annually).

43 5.2 Bayesian Analysis Method-Likelihood and Posterior

The likelihood π(xj/Fi), combines both temporal and spatial components of well j on the condition that it is sampled with frequency of Fi. This likelihood is determined by the product of the coeﬃcient of variance (COV) score of historical concentration data, SCOV , and spatial redundancy score for each individual well, SSpatial:

π(xi/Fi)= SCOV • SSpatial, (5.2)

The deﬁnitions of SCOV and SSpatial systems will be given as follows.

The coeﬃcient of variance, COV, is an indicator of data scatter. It is often called the relative standard deviation since it takes into account the mean of the data set. A higher

COV implies a larger variation among the historical monitoring data. The COV is deﬁned as

σ COV = , (5.3) X where σ is the sample set standard deviation, and X is the sample set average. When estimating uncertainty of historical monitoring data, a low COV percentage implies a high probability of over sampled for the individual well. The threshold between high variability and low variability is 1.0, which is a commonly used empirical value cited in the literature, such as from the CES method [22]. For each individual monitoring well, the COV is calculated three times using each of the three assumed sampling frequency plans, quarterly, semi-annually, and annually. For the quarterly sampling plan, the COV is calculated using all historical monitoring data xj =(xj,1, xj,2, xj,3, xj,4, ..., xj,m). For the semi-annual sampling

′ plan, the COV is calculated twice - once using xxj,1 =(xj,1, xj,3, ..., xj,2n−1) and again using

44 ′ xj,2 = (xj,2, xj,4, ..., xj,2n). For the annual sampling plan, the COV is calculated four times

′′ ′′ using four data subsets, which are xj,1 = (xj,1, xj,5, ..., xj,2n−1), xj,2 = (xj,2, xj,6, ..., xj,2n),

′′ ′′ ′ ′ xj,3 = (xj,3, xj,7, ..., xj,2n+1), and xj,4 = (xj,4, xj,8, ..., xj,2n+2). Note that xj,1 xj,2 = xj and ′′ ′′ ′′ ′′ S xj,1 xj,2 xj,3 xj,4 = xj . Seasonal ﬂuctuation is considered in the annual sampling COV sinceS eachS data subsetS represented the contaminant concentration sampled at the same time

′′ annually; for example, the data subset xj,1 is the set of concentrations of well j sampled during the ﬁrst quarter each year. In the cases when multiple COV values are calculated

(e.g., semi-annual and annual), the highest COV value is considered the sampling COV value for that sampling frequency. The higher/highest value of COV is selected to represent variability since this represents a conservative sampling frequency reduction. If the COV value is greater than one, then the data set is categorized as high variability. Depending on the COV category (e.g., low or high variability), a score, SCOV , is assigned. The values of SCOV used in this work for LTM temporal optimization are deﬁned and summarized in Table 5.1. The score values were determined based on a series of tests and sensitivity analysis. For example, in the extreme cases, when all the three COV values are high, the quarterly SCOV is set to a value of 4 and the SCOV for the other two sampling options are set to 1, which indicates that frequent monitoring is encouraged. On the other hand, if all the three COV values are categorized as low variability, then the annual SCOV is set to 4 and other two scores are set to 1, and thus low monitoring frequency is preferred. Since low

COV values for all three sampling frequency options mean low variability overall, reducing the monitoring frequency is reasonable.

The spatial redundancy of each individual well is determined a priori. Examples of methods used for spatial optimization of LTM monitoring networks include the monitoring and remediation optimization system (MAROS) [18], the three-tiered approach [23], geo-

45 Table 5.1: Deﬁnition of SCOV values based on COV categories.

Quarterly Semi-annual Annual high (4) high (1) high (1) high (4) high (1) low (1) high (4) low (1) high (1) high (4) low (1) low (1) low (4) high (1) high (1) low (4) high (1) low (1) low (1) low (4) high (1) low (1) low (1) low (4)

statistical temporal-spatial algorithm (GTS) [21], space optimization portion for network optimization [13], genetic algorithms (GAs) [14], and ant colony optimization (ACO) [53],

[58] and [59]. Through these methods, a spatially redundant monitoring well is no longer sampled and in essence removed from the monitoring network. The number of redundant wells among the whole monitoring site can be preset according to the budget and site speciﬁc goals for LTM. The values of the spatial score, SSpatial, depends on whether each well is categorized as redundant and are summarized in Table 5.2. The idea here is that if a particular well is considered redundant, then it will be given a higher score in lower monitoring frequent sampling plan; in other words, a redundant well may have a high probability of low monitoring frequency. Since the likelihood value of each individual well is the product of score of COV and score of spatial redundancy (Equation (5.2), in order to assure the same weighting of spatial component as temporal component in likelihood part, the same order of magnitude for the SCOV and SSpatial values are suggested. Thus, higher

SCOV and SSpatial values on low frequent sampling plan expect higher likelihood value of low frequent monitoring.

Once values for SCOV and SSpatial are deﬁned based on Table 5.1 and Table 5.2, the

46 Table 5.2: Summary of SSpatial values

Redundant well? Quarterly Semi-annual Annual Yes 1 2 4 No 1 1 1

likelihood of each possible sampling schedule is calculated (Equation (5.2). Substitute likelihood π(xj/Fi) and prior belief Fi at the condition of diﬀerent sampling frequency Fi into Equation (5.1), each SSpatial posterior predictive distribution π(xj/Fi) is obtained. Note that the posteriors of each possible sampling schedule are conditional probabilities based on observed well j ’s historical data and are determined by the normalized product of the prior and likelihood.

47 CHAPTER 6

SITE APPLICATIONS

Two site applications are presented in this chapter to demonstrate the performance of the spatial optimization methods developed in this thesis for optimizing LTM plans. The ﬁrst site is a military facility which represents a medium-scale site with 30 active monitoring wells, while the second site is a large-scale site in a sub area contaminated in Savannah

River Site (SRS) with 53 active monitoring wells. In medium-scale site, the results from the primal ACO-LTM algorithm, the dual ACO-LTM algorithm and a GA were compared with enumeration. The best performance algorithm, the ACO-LTM dual algorithm was tested with the large-scale site for further veriﬁcation and observation of scalability, since the numeration is not practical for large site, the results were compared with those from the

GA. The Bayesian Analysis were mainly tested through the application to the medium-scale site as temporal optimization test.

6.1 Spatial Test Site I - A Medium-scale Site

6.1.1 Site Description

A ﬁeld site in the Upper Aquifer at the Fort Lewis Logistics Center in Pierce County,

Washington [3] is used as an example application and case study in this work. This same site was used in a study comparing other LTM network improvement approaches [3]. The existing LTM network is optimized using the developed ACO approach. The contaminant

48 Figure 6.1: The original 30-well LTM network and contaminant concentration plume (with contours in mg/L) based on monitoring data.

of concern is trichloroethylene (TCE), which was used as a degreasing agent at the site until

1970’s and has a maximum contaminant level (MCL) of 0.005 mg/L. Regular monitoring was conducted during the period between November 1995 and October 2001 quarterly. Data from the September 2000 monitoring period is used in this work to optimize the existing LTM network of 30 monitoring wells [Figure 6.1]. The direction of groundwater is upper left, and the monitoring wells are distributed along groundwater ﬂow direction and concentrated on the location of contaminant source. Geologic units under the Logistics Center consists primarily of outwash sand and outwash gravel ([3]).

By inspection of the existing data, some wells may be categorized as redundant or important by calculating the REE values based on estimating the concentration at each well

49 29 wells 27 wells 25 wells 23 wells 21 wells

1.2

1.0

0.8

0.6 REE 0.4

0.2

0.0 Well 3 Well 15 Well 19

Figure 6.2: Comparison of REE values from solutions using a ranking method to eliminate wells.

Table 6.1: Search space for medium-scale site

Remaining wells Search space 3 27 C30=4,060 4 26 C30=27,405 5 25 C30=142,506 6 24 C30=593,775 7 23 C30=2,035,800 8 22 C30=5,852,925 9 21 C30=14,307,150 using the other 29 data points. For example, wells 3, 15, and 19 each have low REE values that are less than 0.3. However, well 3 is deﬁned as boundary well [Figure 6.1] and may not be presumed to be a redundant well. On the other hand, some wells have REE values over 1,000, such as well 12 and 16, and may be considered to be very important to the whole monitoring network since their values cannot be estimated reliably based on neighboring data. More

50 importantly, using a ranking of each well’s REE values with concentrations interpolated with data from the other 29 wells can signiﬁcantly underestimate the actual REE for individual wells [Figure 6.2]. This is because the impacts of eliminating wells on individual wells are not taken into account. This demonstrates that mathematical optimization is necessary to simultaneously analyze the eﬀects of eliminating multiple wells and results in identifying solutions with better performance than those found through a ranking approach.

The search space of medium-scale site are listed in Table 6.1. Three stochastic algorithms, the primal ACO-LTM, the dual ACO-LTM, and the GA were developed and tested by this site. The results were compared with enumeration and further detailed discussion are followed.

6.1.2 The Primal ACO-LTM Algorithm Results

The developed ACO algorithm for solving the optimization formulation deﬁned by Equa- tion (4.4) and Equation (4.5) was applied to the Fort Lewis case study [3]. In this work, the set of ACO parameters were identiﬁed by prior evaluations and based on the guidelines presented in the literature [27]. The parameter values used in this work are summarized in

Table 6.2.

Table 6.2: Parameters used in the primal ACO-LTM algorithm for medium-scale problem

Parameter Value Number of neighbors for IDW, n 8 Exponent parameter in IDW, p 2 α parameter 0.5 β parameter -1 Initial pheromone 0.1 Total pheromone, Q 500 Elitism parameter, e 5 Pheromone evaporation rate, ρ 0.5

51 (a) 350

300

250

200

150

Average RMSE 100

0 0 20406080100 Iteration

(b) 1.5

1.4

1.3

1.2 Average RMSE

1.1

1 020406080100 Iteration

Figure 6.3: Example of the convergence of the average RMSE during (a) all iterations of the primal ACO-LTM algorithm, and (b) iterations 6-100 (zoom in of Figure a)

52 Avg Best

22 Number of Remaining Wells Remaining of Number 20 0 20406080100 Iteration

Figure 6.4: Example of the convergence of the primal objective function (number of remaining wells)

Results indicate that the developed algorithm successfully identiﬁed feasible solutions that satisfy the speciﬁed overall data loss (RMSE) constraint Equation (4.5)). Figure 6.3 and Figure 6.4 show a typical convergence of the average RMSE and best feasible solution of a colony during each iteration of the ACO search for the case when the RMSE threshold

(T) is 1.5. For this example, an ant colony consists of 30 individual ants each iteration.

In other words, the activity of each ant colony is equivalent to a search with a colony containing a single ant over 30 iterations, with each ant following a diﬀerent random order of visiting all wells and updating the pheromone density every 30 iterations or ant tours. Thus while Figure 6.3 indicate that the solution converged after 5 iterations for this case, this is approximately equivalent to a search using 150 iterations of an ACO using a single-ant colony. In the early iterations, the average RMSE of the colony was high [Figure 6.3], with values over 100, and no feasible solutions identiﬁed. After a few additional iterations the

53 RMSE Avg REE

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4

AverageREE or RMSE 0.2 0.0 20 22 24 26 28 Number of Remaining Wells

Figure 6.5: Variation of overall data loss as quantiﬁed by the RMSE with the number of remaining wells in the optimized LTM network (primal). The average REE of the interpolated concentrations for the eliminated wells, along with error bars represented one standard deviation, is shown for comparison.

average RMSE of the colony rapidly decreased to a range between 1.26 and 1.37 [Figure 6.3], while the best feasible ant solution (i.e., minimum number of remaining wells) in the colony decreased to a range between 22 and 23 wells [Figure 6.4]. During these later iterations, variations of 8% of the average RMSE and 4% of the minimum number of wells occurred for this example case. The average number of wells remaining in the solutions found during the early iterations were lower [Figure 6.4] since these included infeasible solutions in which a high RMSE resulted from eliminating too many wells [Figure 6.3].

Optimal solutions for several cases of number of remaining wells were identiﬁed by solving the LTM optimization formulation with diﬀerent values in the overall data loss threshold (T) constraint (Equation (4.5)). As expected, results indicate that as the number of remaining

54 Table 6.3: Summary of the optimal LTM networks identiﬁed by the primal ACO-LTM algorithm

Remaining Reduction Wells Eliminated RMSE Average Wells in Wells (Std Dev) REE 27 10 % 2, 15, 19 0.383 0.361 (0.157) 26 13 % 2, 10, 11, 25 0.595 0.578 (0.166) 25 17 % 10, 11, 15, 19, 25 0.559 0.523 (0.219) 24 20 % 2, 10, 11, 15, 19, 25 0.545 0.515 (0.197) 23 23 % 2, 3, 10, 11, 15, 18, 25 0.637 0.562 (0.323) 22 27 % 2, 10, 11, 13, 15, 18, 19, 25 0.725 0.649 (0.345) 21 30 % 2, 10, 11, 13, 15, 18, 19, 23, 25 1.165 0.862 (0.832)

wells decreases, the average REE for the eliminated wells and overall RMSE increase nonlinearly [Figure 6.5 and Table 6.3]. Since concentrations at wells eliminated from the monitoring network are estimated using the remaining wells, fewer remaining wells may increase the REE of each deleted well, and consequently, the RMSE increases. The RMSE values here are the lowest for a given number of remaining wells based on a minimum of

50 replicate runs of the ACO for each case. Other reduced networks with the same number of wells were identified by the ACO; however, the resulting RMSE values varied depending on the specific set of wells that remain. For the cases with 25 and 26 remaining wells, the reduced LTM networks identified by the ACO are near-optimal in terms of RMSE compared to the solution for the 24-well case. However, the solutions for the 24-, 25-, and 26-well cases are all similar; the average REE for the 25- and 26-well solutions fall within one standard deviation of the average REE for the 24-well solution (Table 6.3). Figure 6.5 indicates that after reducing the number of wells in the LTM network to a certain point, the trade-off between eliminating additional wells and maintaining a desirable or acceptable level of error changes such that the resulting increase in overall RMSE becomes too large. For example in this problem, eliminating more than 8 wells out of the original 30 results in a sharp increase

55 Figure 6.6: Concentration contours (mg/L) based on data from the 27 wells and interpolated values at the 3 wells selected to be eliminated from the existing LTM network, resulting in an RMSE of 0.383(primal).

in the RMSE as well as the average and standard deviation of the REE of the interpolated concentrations [Figure 6.5 and Table 6.3].

The reduced LTM networks identiﬁed by the ACO also were evaluated using concentration contours resulting from the cases with 27, 23, and 21 remaining wells (Figure 6.6,

Figure 6.7, and Figure 6.8, respectively). The contours were developed using interpolated concentrations calculated using the same IDW method employed in the ACO algorithm at the eliminated wells and the measured data at the remaining monitoring wells. Results show that the eliminated wells are distributed in the areas inside the plume boundary and among clustered wells. Diﬀerences between the contours resulting from the reduced networks and the original contours were minor for the cases with 23 and 27 remaining wells (compare

Figure 6.7 and Figure 6.6 with Figure 6.1). In the extreme case with 21 remaining wells,

56 Figure 6.7: Concentration contours (mg/L) based on data from the 23 wells and interpolated values at the 7 wells selected to be eliminated from the existing LTM network, resulting in an RMSE of 0.637(primal).

which represents a 30% reduction in monitoring wells, there is some contour area loss at the

0.05 mg/L concentration isopleth (compare Figure 6.1 and Figure 6.8). This result verifies that the trade-off between well reduction and data loss through interpolation error may not be beneficial or desirable when too many wells are eliminated, as also indicated in Figure 6.5 and Table 6.3. Decision makers need to assess what their error tolerance is and whether this higher data loss, as indicated by the contour map [Figure 6.8], is a reasonable trade-off for cost savings through eliminating wells. Thus the RMSE value alone may not be sufficient to evaluate a solution; additional analysis such as through contour mapping, which shows the

RMSE and individual REE qualitatively, may be necessary.

An interesting result of the 23-well case is that boundary well 3 was eliminated from monitoring network. The REE of well 3 when calculated using all remaining wells identiﬁed

57 Figure 6.8: Concentration contours (mg/L) based on data from the 21 wells and interpolated values at the 9 wells selected to be eliminated from the existing LTM network, resulting in an RMSE of 1.165(primal).

by the ACO-LTM process is 0.0939, which satisﬁes the boundary well REE constraint (Tedge

= 0.1). However, if the concentration at well 3 is interpolated using the original data set from the other 29 wells, its REE would be 0.206. The error was reduced by removing wells through spatial optimization. The reason is that well 18, which is one of the nearest neighbors to well 3, has a measured concentration of 0.027 mg/L, which is an order of magnitude smaller than the measured concentration at well 3, which is 0.1 mg/L. This violates the assumption of IDW that the closer two points are spatially, the more similar their data are. When well 3 is estimated with data from well 18, the concentration at well 18 has a very high weight in the IDW interpolation due to its proximity to well 3, which increased the REE of well 3. By removing well 18, the REE of well 3 was reduced. This result highlights

58 Figure 6.9: Concentration contours (mg/L) based on data from the 23 wells and interpolated values at the 7 wells selected to be eliminated from the existing LTM network using the general MAROS procedure, resulting in an RMSE of 0.721.

that the interpolation method can play an important role in quantifying the redundancy of sampling locations. Log-transformation of the concentration data may resolve this issue.

Evaluation runs using log-transformed concentration values indicate that this is eﬀective in cases with more remaining wells. Nevertheless, the IDW interpolation method used in this work may be replaced by other methods. One advantage of the ACO algorithm is that it is modular, allowing for interpolation, prediction, and simulation methods to be incorporated.

In addition, this result highlights the potential sensitivity of the solutions to the constraints in this example.

The same LTM optimization example problem also was solved following a procedure based on the MAROS procedure for sampling optimization outlined in [18]. In particular,

59 the MAROS procedure was used here to sequentially eliminate wells using rankings of slope factor (SF) values. The SF is defined as the normalized difference between the measured and estimated log-concentrations and is used to quantify an individual well’s contribution to the entire monitoring network [18]. Note that for comparison to the ACO algorithm, the estimated concentrations were interpolated using IDW here. In MAROS a well may be considered redundant if the value of SF is smaller than a specified value. However, the SF is not very sensitive to differences between Cest,i and Ci. One reason is that both concentration values are log-transformed, which reduces the range of the Cest,i and Ci values and introduces data loss before the network reduction procedure is implemented. The second reason is that the concentration difference is normalized by the maximum of Cest,i and Ci, reducing the range of SF to between 0 and 1. The case with 23 wells remaining in the monitoring network was analyzed [Figure 6.9]. The SF of the seventh removed well, which is well 18, was 0.211 for the case with 23 remaining wells. A limitation of MAROS is that the impact of removing additional wells on previously removed wells is not explicitly considered. Thus there is a possibility that the SF of previously removed wells may increase and even might violate SF thresholds, such that a well should not be considered a redundant well. For example, well 15 was eliminated during the fourth iteration with a SF value of 0.120, but its SF value increased to 0.169 after 3 more wells were deleted. Therefore the estimated concentration and error

(SF) of previously removed wells needs be recalculated each iteration. Eliminating the well with the lowest SF value each iteration is analogous to a greedy search procedure. There is potential that if a well other than the well with the lowest SF is selected for elimination then additional wells may be removed without violating overall constraints and thresholds.

In MAROS, after a well is eliminated based on SF the overall information loss metrics

Average Concentration Ratio (CR) and Area Ratio (AR) are calculated. However, these are

60 averaged metrics, and individual estimated errors of wells removed in previous iterations are not explicitly assessed.

When comparing the results from the ACO and the aforementioned procedure based on MAROS for the case with 23 remaining wells (Figure 6.7 and Figure 6.9, respectively), it is seen that the ACO results are very similar to and slightly better than the MAROS- based results. The resulting overall RMSE (Equation ??eq:RMSE)) is 0.637 for the ACO solution and 0.721 for the MAROS-based solution. The difference is due to the location of two removed wells. In both solutions, the same five wells were identified as redundant: wells

2, 10, 15, 18, and 25; the other two wells chosen by MAROS-based procedure are wells 13 and 19, and wells 3 and 11 by the ACO. Concentration contours based on the data from sampled wells and interpolated values at the deleted wells for the 23-well networks identiﬁed by the ACO and MAROS-based procedure are shown in Figure 6.10, respectively. These contours are overlain on the contours created using data from the original 30-well network.

The contours from the ACO solution slightly overestimate contaminant contour regions, in particular at the 0.01 mg/L isopleth, in comparison to the contours developed using data from original 30-well LTM network. The ACO results are slightly more conservative compared to the minor underestimation of the 0.005 mg/L isopleth of the MAROS-based solution.

Overall, the two sets of resultant contours match the original contours very well.

6.1.3 The Dual ACO-LTM Algorithm Results

In this work, The parameter values used in the dual ACO-LTM algorithm summarized in

Table 6.4. Each ant colony is comprised of 30 ants, and the ACO search is continued over

50 iterations. The algorithm is implemented using MATLAB 7 (MathWorks 2004). By solving the LTM optimization problem with varying desired number of remaining wells (n, m + n = 30) using this ACO-LTM algorithm, optimal reduced LTM networks with diﬀerent

61 (a)

(b)

Figure 6.10: Concentration contours based on data from the original 30-well LTM network compared to the optimized 23-well network identiﬁed by (a) the developed primal ACO-LTM algorithm, and (b) a procedure based on MAROS.

62 numbers of remaining wells are identiﬁed.

Table 6.4: Parameters used in the dual ACO-LTM algorithm for medium-scale problem

The optimal solutions identiﬁed by the dual ACO-LTM algorithm for the cases with 27 to

21 remaining wells were the global optima, as indicated by comparison with solutions from complete enumeration (Table 6.5 and Figure 6.11). The optimal RMSE for the solutions found by the dual ACO-LTM algorithm for diﬀerent cases of number of remaining wells illustrate that the RMSE nonlinearly increases as the number of remaining wells decreases

[Figure 6.11]. All results are the best ones from among 50 multiple runs for each case solved using diﬀerent random number seeds.

Results indicate that the developed dual ACO-LTM algorithm is efficient and effective in solving the LTM optimization problem, requiring less than 30 × 50 = 1, 500 function evaluations. This is significantly less than the 14,307,150 evaluations used in enumeration to solve the 21-well case.

Additionally, the results indicate that solutions that use a ranking of individual errors are not guaranteed to be global optima. For example, the optimal solution for the 26-well case is not comprised of the entire set of wells from the optimal solution for the 27-well case plus one additional well (Table 6.5). In fact, only two common wells exist (wells 15 and 25)

63 ACO-LTM Enumeration

0.7 0.65 0.6 0.55

RMSE 0.5 0.45 0.4 0.35 21 22 23 24 25 26 27 # of Remaining Wells

Figure 6.11: Comparison of the objective function values (RMSE) for solutions identiﬁed by the dual ACO-LTM algorithm and complete enumeration.

Table 6.5: Comparison of results from the dual ACO-LTM algorithm and complete enumeration.

# Wells Redundant (Removed) Wells RMSE, RMSE, Remaining In the order of selection ACO-LTM Complete Enumeration 27 25, 3, 15 0.3806 0.3806 26 25, 15, 19, 2 0.4067 0.4067 25 25, 3, 15, 21, 2 0.4709 0.4709 24 2, 3, 19, 15, 18, 25 0.5001 0.5001 23 19, 15, 18, 3, 25, 21, 2 0.5301 0.5301 22 25, 2, 3, 15, 18, 10, 19, 11 0.5768 0.5768 21 2, 11, 19, 25, 13, 3, 10, 15, 18 0.6515 0.6515

between the optimal solutions for the 26- and 27-well cases.

Figure 6.12, Figure 6.13 and Figure 6.14 illustrate the path of each individual ant traverses in their optimal path (solution) for the 27-well, 23-well and 21-well cases. Notice that the ants do not simply move from one well to a nearby well. Occasionally the movement is quite signiﬁcant, in which other regions of the network are explored. This is expected since

64 removing too many wells from the same region could lead to considerable estimation errors.

Additionally, the solutions identified by the ACO are evaluated by comparing the resulting concentration contours with the original contours from the existing 30-well LTM network [Figure 6.15]. Two sets of contours are compared in Figure 6.15, which include the contours that result from the 30 original wells and the optimal monitoring wells identified by the ACO-LTM for the 21-well case. The contours for the 21-well solution use the concentration data from the 21 remaining monitoring wells and the interpolated values for the 9 redundant wells. Both sets of contours are very similar, with only minor differences in a few locations [Figure 6.15], indicating that the 30% reduction in number of monitoring wells is reasonable for this case.

65 Figure 6.12: The optimal path traversed in the 27-well case from the dual ACO- LTM algorithm. The 3 redundant wells, in order of selection, are wells 25, 3 and 15.

66 Figure 6.13: The optimal path traversed in the 23-well case from the dual ACO- LTM algorithm. The 7 redundant wells, in order of selection, are wells 19, 15, 18, 3, 25, 21, and 2.

67 Figure 6.14: The optimal path traversed in the 21-well case from the dual ACO- LTM algorithm. The 9 redundant wells, in order of selection, are wells 2, 11, 19, 25, 13, 3, 10, 15 and 18.

68 Figure 6.15: Comparison of the contours (0.005, 0.01, 0.05, and 0.15 mg/L) based on the existing 30 monitoring wells and the 21-well reduced network identiﬁed by the dual ACO-LTM algorithm and complete enumeration.

69 6.1.4 The GA Search for Spatial Optimization

Genetic Algorithms (GAs) are adaptive heuristic search algorithm premised on the evolutionary ideas of natural selection and genetic. The basic concept of GAs is designed to simulate processes in natural system necessary for evolution, specifically those that follow the principles first laid down by Charles Darwin of survival of the fittest. As such they represent an intelligent exploitation of a random search within a defined search space to solve a problem. The genetic algorithm is a computer simulation of such evolution where the user provides the environment (function) in which the population must evolve.

John Holland, from the University of Michigan, began his work on genetic algorithms in the early 1960s. Holland had a double aim : to improve the understanding of natural adaptation process, and to design artiﬁcial systems having properties similar to natural systems.

The basic idea is as follow : the genetic pool of a given population potentially contains the solution, or a better solution, to a given adaptive problem. This solution is not ”active” because the genetic combination on which it relies is split between several subjects. Only the association of diﬀerent genomes can lead to the solution.

Holland method is especially eﬀective because he not only considered the role of mutation

(mutations improve very seldom the algorithms), but he also utilized genetic recombination,

(crossover): these recombination, the crossover of partial solutions greatly improve the capability of the algorithm to approach, and eventually ﬁnd, the optimum.

To use a genetic algorithm, decision variable set must be represented to a problem as strings (or chromosomes). Each sting is made up of a series of characters, which represent a coding of the decision variable set. The genetic algorithm then creates a population of strings randomly and applies genetic operators such as mutation and crossover to evolve

70 strings in order to ﬁnd the best one(s).

The three most important aspects of using genetic algorithms are: (1) definition of the objective function, (2) definition and implementation of the genetic representation, and (3) definition and implementation of the genetic operators. Once these three have been defined, the generic genetic algorithm should work fairly well.

The objective function and constraint were deﬁned as:

− m Cest,i Ci 2 i=1( min(C ,C ) ) MinZ = est,i i , (6.1) sP m s.t.

m = Sgoal, (6.2)

where, where m is the number of removed wells, and the goal is to minimize the IDW estimatd errors at individual wells, given a ﬁxed number of monitoring wells. The Z function is deﬁned as RMSE pattern previously in Equation (4.3).

The next step in formulating a GA is to determine how to code the GA decision set. For coding convenience, the length of the string is equal to the number of total monitoring wells, and the coding is based on binary. The value of each bit (0/1) represents the situation of each well (off/on). For example, 11100010 means that number 1, 2, 3, and 7 wells are on, and number 4, 5, 6, and 8 are off. The fitness value was defined as

10 • RMSE•| m − S | if m =6 S fitness = goal goal , (6.3) RMSE else where,RMSE is the goal function in Equation (6.1); m = number of zeros in each GA string; Sgoal = number of expected removed wells in monitoring network. Indeed, the ﬁtness function is a penalty formulation based on the Multiplicative Penalty Method presented in

71 [60]. If the number of removed wells from the GA is not equal to expected number, the

fitness value is increased by 10 times, in other words, the penalty coefficient is 10. In this work, the lower fitness value, the better string it is. After selection, crossover, and mutation, the fittest strings from the population are obtained. Note that elitism strategy is processed at each generation, the best string so far, which has the lowest fitness value, is copied to the next generation, which means the elitism string is always remained in each generation.

The parameter values used in this GA are summarized in Table 6.6, the population size is

30 strings, just like the number of an ant colony; the crossover pattern here is one point crossover, the possibility of crossover is 0.85.

Table 6.6: Parameters used in the GA algorithm for medium-scale problem

Parameter Value Population 30 Maximum generations 50 Mutation possibility 0.05 Crossover possibility 0.85

Table 6.7: Comparison of results from the GA algorithm and enumeration. Wells in bold font indicate those that are diﬀerent from the enumeration solution.

# Wells Redundant (Removed) RMSE, RMSE, Remaining Wells GA Complete Enumeration 27 3,15,25 0.3806 0.3806 26 2,15,19,25 0.4067 0.4067 25 2,10,15,19,25 0.5027 0.4709 24 2,3,10,11,21,25 0.5632 0.5001 23 2,3,10,15,18,19,25 0.5470 0.5301 22 2,10,13,15,18,19,21,25 0.7334 0.5768 21 23,9,10,15,18,19,21,25,27 0.8680 0.6515

72 GA Enumeration

0.9 0.85 0.8 0.75 0.7 0.65

RMSE 0.6 0.55 0.5 0.45 0.4 0.35 21 22 23 24 25 26 27 # of Remaining w ells

Figure 6.16: Comparison of the objective function values (RMSE) for solutions identiﬁed by the GA algorithm and complete enumeration.

The solutions generated by the GA, as compared to the enumeration, are summarized in Table 6.7 and Figure 6.16 respectively. The GA was coded by MATLAB 7 (MathWorks

2004), and the results are the best set through an entire GA run from 50 multiple runs of diﬀerent random number seeds. Figure 6.16 shows the GA results are reasonable, overall, the more wells removed, the higher the RMSE value, but the results of 24 and 23 well cases slightly violate the trend - the RMSE value of 23 well case is a little higher than 24 well case; and the results of 22 and 21 well cases are a little high. In the following section, all the results from the four algorithms are compared and analyzed.

6.1.5 Summary and Conclusions of Spatial Sampling Algorithms

The same LTM spatial optimization problem was solved by four methods: the primal

ACO-LTM, the dual ACO-LTM, the GA, and complete enumeration. All three stochastic

73 algorithms were run using 50 trial, and they were set up using the same set of 50 diﬀerent random number seeds, in order to fairly compare the three algorithms. Note that our implementation of the GA does not solve exactly the same formulation as the optimization problem solved by the ACO; the problem sizes are diﬀerent, with the GA problem size being bigger. Because the GA solutions can have more or fewer number of redundant wells; in other words, more combinations are in the search space, not just those for the given number of redundant wells. The best results of each 50 runs and the results from complete enumeration are summarized and graphed in Table 6.8 and Figure 6.17 respectively, which show that dual

ACO-LTM algorithm can identify all global optimal results. Especially it outperformed the other two stochastic algorithms in reﬁning the local search near global optimal, both of these two algorithms can identify low RMSE results, but their results were a little worse than the results from the dual ACO-LTM algorithm.

GA Enumeration Primal ACO-LTM Dual ACO-LTM 1.2 1.15 1.1 1.05 1 0.95 0.9 0.85 0.8 0.75 RMSE 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 21 22 23 24 25 26 27 # of Remaining w ells

Figure 6.17: Comparison of the optimal LTM networks results (RMSE) by the primal ACO-LTM, the dual ACO-LTM, the GA and Enumeration

74 Table 6.8: Summary of the optimal LTM networks identiﬁed by the primal ACO-LTM, the dual ACO-LTM, the GA and Enumeration

Remaining RMSE of Primal RMSE of Dual RMSE of RMSE of Wells ACO-LTM ACO-LTM GA Enumeration 27 0.3834 0.3806 0.3806 0.3806 26 0.5954 0.4067 0.4067 0.4067 25 0.5585 0.4709 0.5027 0.4709 24 0.5453 0.5001 0.5632 0.5001 23 0.6367 0.5301 0.5470 0.5301 22 0.7245 0.5768 0.7334 0.5768 21 1.1650 0.6515 0.8680 0.6515

The reason is that the primal ACO-LTM algorithm and the GA are very similar, they are based on binary algorithm (0/1), their results converge to minimum by one penalty function, the algorithm is controlled by one element. But in the dual ACO-LTM algorithm, it is based on TSP algorithm, there are two functions by which each individual ant is controlled, one is the local area searching function for ants, each well was evaluated and compared with other candidate wells, in the GA no such step and in the primal ACO-LTM algorithm, since only two choices (on/oﬀ) for each well, no such kind of local wells comparison was involved; the other is the overall evaluation, all the three algorithms have this step. From algorithm mechanism, the dual ACO-LTM was expected to have superior performance to other two algorithms.

In addition to solution quality of the primal ACO-LTM, the dual ACO-LTM and the GA algorithm, the CPU times from the three algorithms and complete enumeration are listed in

Table 6.9. Practically, there is no diﬀerence among the the three stochastic algorithms by

CPU time; the CPU time of the primal ACO-LTM algorithm was the longest, the reason is that each ant has to visit all the wells in this algorithm, which means more iterations and more calculations. But in the dual ACO-LTM algorithm, the number of wells to be visited

75 by an ant is preset, in other words, each ant will only visit a certain number of wells, so less iterations are involved here. For complete enumeration, as the search space is increased, the CPU time is huge, especially in the case of 21 remaining wells, it took a little more than two entire days (48.01hr) for a computer to identify the result, in this case, complete enumeration is not practical and economical. In the large-scale (53 wells) in the following section, it may be impossible for complete enumeration by CPU time, it may take up to more than two thousand years in CPU time.

Table 6.9: Summary of the CPU time (by seconds) by the primal ACO-LTM, the dual ACO-LTM, the GA and complete Enumeration (medium-scale site)

Remaining CPU time (s) of CPU time (s) of CPU time (s) of CPU time (s) of Wells primal ACO-LTM dual ACO-LTM GA complete enumeration 27 646.61 20.03 25.58 26.61 26 646.51 26.55 25.49 208.03 ≈ 3.47 min 25 646.42 32.98 25.29 1231 ≈ 20.52 min 24 645.61 38.88 25.22 5636.9 ≈ 1.567hr 23 646.27 44.66 24.84 21390 ≈ 5.942hr 22 646.44 50.6 25.51 66206 ≈ 18.4hr 21 646.88 55.9 25.98 172840 ≈ 48.01hr

As mentioned before, the cases with 27 through 21 results from the dual ACO-LTM algorithm, in RMSE pattern, indicate that the dual ACO-LTM can obtain the global optimal results for this medium-scale problem among the three stochastic algorithms (Table 6.8).

Computationally the stochastic algorithms were extremely cheap (Table 6.9) compared with complete enumeration, especially, for the dual ACO-LTM algorithm, in the cases with 21 and

22, and the CPU times were 55.9 seconds by the dual ACO-LTM vs. 48.01hr by complete enumeration with around 3086 times diﬀerence and 50.6 seconds vs. 18.4hr with around

1308 times diﬀerence respectively but with exact the same results. In order to understand

76 the way the ACO-LTM functions, convergence of RMSE value was studied by 21-well case.

Figure 6.18(a,b) show the comparison of the GA and the ACO-LTM convergences of best solution at each generation/iteration. Figure 6.18(a) presents that the ACO-LTM can identify pretty good solution even from ﬁrst iteration. Unlike the GA, the solution from the

ACO-LTM does not converge to a single common solution; on the contrary, the ACO-LTM can create new, maybe improving solutions, which means a high diversity among solutions

[Figure 6.18(b)]. This non-convergence property is the advantage of the ACO-LTM, since it tends to prevent the algorithm from being trapped in local optima.

All above suggest that the dual ACO-LTM is a promising method and can be applied and work very well on a ﬁeld site. Since the results of the dual ACO-LTM algorithm were the best, this algorithm was tested by a large-scale site for further veriﬁcation and observation.

77 GA Dual ACO-LTM (a) 350

300

250

200 RMSE 150

Best Best 100

0 0 1020304050 Iteration

10 (b) 9 8 7 6 5 RMSE 4 3 Best Best 2 1 0 0 1020304050 Iteration 2 through 50

Figure 6.18: 21-well case of the convergence of the average RMSE during (a) all iterations of the GA and the dual ACO-LTM algorithms, and (b) iterations 2-50 (zoom in of Figure a)

78 6.2 Spatial Test Site II - A Large-scale Site

6.2.1 Site Description

A large-scale site is used here to test the dual ACO-LTM algorithms. Since the results of dual algorithm are better than the GA and ACO primal algorithm, a large-scale site is tested to further observe the performance of the dual ACO-LTM algorithms. The contaminated site is one sub area of Savannah River Site (SRS), named as M-Area HWMF, located in the northwest portion of the Savannah River Site (SRS). SRS’s primary mission since its inception until the early 1990s was production and separation of plutonium and tritium for use in national defense programs. M-Area HWMF consists of the M-Area Settling Basin; a seepage area, overﬂow ditch, and inlet process sewer line; and Lost Lake, a shallow upland depression, and the general groundwater direction is northeast. From 1958 until 1985, this site received waste water that contained volatile organic solvents used for metal degreasing as well as chemical constituents and depleted uranium from fuel fabrication processes in M

Area ([61]).

Among several contaminant institutes, TCE is selected as COC in this work, and in this contaminant site, four aquifers exist underground, and Lower Lost Lake Aquifer Zone is selected. The total number of monitoring wells installed in Lower Lost Lake Aquifer Zone is 53 [Figure 6.19]. Since the remediation activities in this area are still in process, only

LTM spatial optimization is considered in large-scale site, and temporal analysis will not be conducted. Indeed, currently, the contaminant site is still on the active remedial eﬀorts, and the monitoring of COC is not long enough to do temporal redundant analysis, so only spatial optimization is tested in this site as large-scale problem for the performance of the dual ACO-LTM algorithm on the assumption of snap-shot, also spatial optimization results may help for the adjustment of future monitoring strategy.

79 Figure 6.19: The original 53-well LTM network and contaminant concentration plume (with contours in mg/L) based on monitoring data.

80 6.2.2 The Dual ACO-LTM Algorithm & the GA Results

The search space of large-scale site are listed in Table 6.10, now the search space was huge, more than six trillion. In large-scale, the set of the dual ACO-LTM algorithm parameters were still based on the guidelines presented in the literature [27] and the same set parameter values with medium-scale problem were used (Table 6.4) except the number of ant in a colony, it was increased to 50 from 30, the reason is that large-scale site requires more ants to do optimal search.

Table 6.10: Search space for large-scale site

Remaining wells Search space 5 48 C53=2,369,685 7 46 C53=154,143,080 9 44 C53=4,431,613,550 11 42 C53 =76,223,753,060 13 40 C53 =841,392,966,470 15 38 C53 =6,250,347,750,920

In order to verify the results from the dual ACO-LTM algorithm, this site was also spatial optimized by and compared with only the GA algorithm, and not by complete enumeration.

The reason is that the search space was increased tremendously, complete enumeration was not practical. Table 6.11 shows the CPU time of the ACO-LTM, the GA and complete enumeration. The CPU time of complete enumeration was approximated on the assumption that the CPU time is proportional to search space. For the large-scale site problem, the complete enumeration was not practical from the case of 46 remaining wells, it might take

22 days to ﬁnish the enumeration, and in the extreme case of 38 remaining wells, the CPU time might be even more than two thousand years with the search space of more than six trillion.

81 Table 6.11: Summary of the CPU time by the dual ACO-LTM, the GA and complete Enumeration, ∗ are estimated times.(large-scale site)

Remaining CPU time (s) of CPU (s) time of CPU time of Wells GA dual ACO-LTM complete enumeration 48 67.28 86.54 47114s ≈ 13.09hr 46 79.66 120.79 528.11hr ≈ 22d∗ 44 89.76 153.34 15192hr ≈ 633d∗ 42 88.32 184.11 10890d ≈ 29.84yr∗ 40 94.99 212.21 120198d ≈ 329yr∗ 38 123.18 240.52 862906d ≈ 2446yr∗

The results from the dual ACO-LTM algorithm and the GA were listed and graphed by

Table 6.12 and Figure 6.20 respectively, they were the best results from 50 multiple runs by diﬀerent random number seeds. The same GA code, which was tested by medium-scale, was used here and the parameters were remained no change except the population number, it was 50 for this large-scale problem, same as the number of ants in a colony.

Table 6.12: Summary of the optimal LTM networks RMSE identiﬁed by the dual ACO-LTM and the GA

Remaining RMSE Dual RMSE GA Wells ACO-LTM 48 0.1462 0.1669 46 0.2089 0.4346 44 0.412 0.8423 42 0.5165 1.0821 40 0.7634 1.8401 38 1.1179 1.8458

Only the 48 wells case of complete enumeration was processed and the CPU time was around 13 hours. The result of enumeration was 0.1486, which was also identiﬁed by the dual ACO-LTM algorithm, it indicated that global optimum can be obtained by the dual

82 GA Dual ACO-LTM 2 1.8 1.6 1.4 1.2 1 0.8 RMSE 0.6 0.4 0.2 0 38 40 42 44 46 48 # of remaining w ells

Figure 6.20: The objective function values (RMSE) for solutions (large-scale) identiﬁed by the dual ACO-LTM algorithm and the GA.

ACO-LTM algorithm in the 48 wells case. The dual ACO-LTM algorithm outperformed the

GA by quality of solutions of all cases with 48 wells through 38, especially in case of 40, it improved up to 122% in RMSE value, 0.8276 vs. 1.8401. It implied that the dual ACO-LTM algorithm can reﬁne the solution quality more than the GA in local search near the global optimum, and the GA could only obtain some poor solutions close to global optimum.

Table 6.13 shows the identiﬁed redundant wells by the order of selection in the case of 48 remaining wells through case of 38. Note that results from the dual ACO-LTM algorithm still showed there do not exist complete common wells unnecessarily, for example, the case with 48 remaining wells, well 51, 7, 16, 29 and 49 were selected, but in case with 46 remaining wells instead of selecting all the ﬁve wells of 48-case with two more well, well 16, 7, 23, 49,

83 28, 29, and 26 were selected, in other words, between 46-case and 48-case, there were only four common wells, 7, 16, 29 and 49. It still implied that manually deleting the lowest individual estimated error cannot guarantee best solution quality. Figure 6.21, Figure 6.22 and Figure 6.23 presents the actual estimated plume maps and the paths of ants by the cases of 48, 44 and 38 remaining wells respectively. Compared with original 53 wells plume map

[Figure 6.19], there were some minor diﬀerence in the three plume maps with Figure 6.19, and the distribution of redundant wells were even as expected. Only in the extreme data loss case, by 38 remaining wells, the highest concentration isopleth was split, but in general, the contour of 38 case can still describe the contaminant concentration distribution. So the solutions from the dual ACO-LTM algorithm can be considered reasonable. Figure 6.24(a,b) shows the convergences of the GA and the dual ACO-LTM algorithms, and the dual ACO-

LTM can identify satisﬁed results earlier than the GA. When search space is huge, dual

ACO-LTM outperforms the GA in terms of quality of results signiﬁcantly, and still hold diversity among the results [Figure 6.24(a)]. All these indicate that the dual ACO-LTM algorithm can still obtain high quality solutions in large-scale problem as expected.

Table 6.13: Summary of the large-scale optimal LTM networks identiﬁed by the dual ACO- LTM algorithm

Remaining wells Redundant (Removed) Wells (in order of selection) 48 51, 28, 7, 16, 49 46 52, 7, 16, 23, 28, 38, 49 44 29, 7, 26, 28, 16, 10, 31, 49, 23 42 29, 7, 49, 23, 26, 52, 10, 51, 28, 16, 31 40 52, 7, 10, 16, 23, 26, 28, 29, 31, 37, 39, 49, 51 38 52, 10, 29, 31, 39, 17, 23, 26, 28, 7, 49, 51, 37, 15, 16

84 Figure 6.21: The optimal path traversed in the 48-well case from the dual ACO- LTM algorithm. The 5 redundant wells, in order of selection, are wells 51, 28, 7, 16, and 49.

85 Figure 6.22: The optimal path traversed in the 44-well case from the dual ACO- LTM algorithm. The 9 redundant wells, in order of selection, are wells 29, 7, 26, 28, 16, 10, 31, 49, and 23.

86 Figure 6.23: The optimal path traversed in the 38-well case from the dual ACO- LTM algorithm. The 15 redundant wells, in order of selection, are wells 52, 10, 29, 31, 39, 17, 23, 26, 28, 7, 49, 51, 37, 15, and 16.

87 GA Dual ACO-LTM (a) 4000 3500 3000 2500

RMSE 2000 1500

Best Best 1000 500 0 0 1020304050 Iteration

3.5 (b) 3

2.5 RMSE 2 Best

1.5

1 26 36 46 Iteration 26 through 50

Figure 6.24: 38-well case of the convergence of the average RMSE during (a) all iterations of the GA and the dual ACO-LTM algorithms, and (b) iterations 26-50 (zoom in of Figure a)

88 6.2.3 Summary and Conclusions

The dual ACO-LTM algorithm have been applied to a large-scale problem with 53 monitoring wells in Section 6.2 by the case of 38 through 48 wells. Since the search space was very huge in this large-scale problem with more that six trillion, the results were compared with the GA only because of impractical of complete enumeration in CPU time. From the comparison with the GA, the higher quality solutions by RMSE pattern were identiﬁed by the dual

ACO-LTM algorithm, and the estimated contours compared with original contour and the distribution of redundant wells indicated that all the solutions were reasonable and the overall data loss after optimization by each case were still acceptable. According to above, the following conclusion can be drawn for large-scale sites: the dual ACO-LTM algorithm can eﬀectively identify redundant monitoring wells and it is very eﬃcient computationally when complete enumeration is not feasible.

6.3 Application of Bayesian Method for Temporal Optimization to Medium-scale Site

This work was applied to the Upper Aquifer at the Fort Lewis Logistics Center in Piece

County, Washington [3], and the contaminant of concern selected in this work was a degreasing agent, trichloroethylene (TCE), which has a MCL of 5µg/L. At this site 30 monitoring wells were sampled quarterly. Using the available data from [3], data from

14 sampling events, from the second quarter in 1998 to the third quarter in 2001, for each monitoring well were used. During a few monitoring events, there was incomplete monitoring data, such as the fourth quarter in 1998 for wells 1, 5, 7, 14, 16, and 24, and wells 29 and 30 for the third quarter in 1999. Through prior spatial optimization, 30% wells (9 wells) were identiﬁed as redundant using ant colony optimization (ACO) approach [59]. In calculating the prior belief, P (Fi), k was normalized by twelve times the MCL of TCE, or

89 60µg/L/yr. This factor was determined through sensitivity analysis and prior assessment.

The Bayesian method will be processed annually and model result will be updated according to new sampling schedule data to reﬂect the dynamic change of the time series data.

6.3.1 Bayesian analysis results

The results from the application of the Bayesian method to this ﬁeld site data were compared with the results from three other methods: MAROS, Three-Tiered approach, and CES. The results from MAROS were reevaluated using the MAROS software [18] with the data set described above, and the CES method described in [22] was applied The results from three- tiered approach presented here were taken from [3], and may include a slightly diﬀerent data set in which there are more than 14 sampling events for each monitoring well.

Overall, the four sets of results from the diﬀerent methods were very similar. However, the Three-Tiered approach had a tendency to suggest lower frequency monitoring schedules compared to the other three methods. With the Bayesian method, the results are a probabilistic scale for each possible sampling plan, quarterly, semi-annually, and annually, and thus the Bayesian method provides more information than other three methods. On the other hand, the other three methods provide deterministic recommendations for sampling frequency at each monitoring location. Thus an advantage of the Bayesian method is that it presents a probabilistic evaluation of multiple sampling plans, allowing decision makers to assess alternative sampling options.

All 30 wells’ results were summarized and compared in Table 6.14 for more details.

Results from select monitoring wells, which represent a variety of data trend scenarios, are presented and compared in Figure 6.25. For well 4, the Bayesian method resulted in a probability of 0.5 for the semi-annual sampling plan. With the other three methods a reduced sampling frequency also was recommended, with the Three-Tiered approach suggesting that

90 well 4 be removed. The concentrations in well 4 were low but still fluctuating (Figure 6.25(a)), and thus well 4 should be sampled with a low frequency; the four results could be considered similar. For well 5, since there was no change in concentration over time for quite a while (Figure 6.25(b)), the Bayesian method resulted in the annual sampling plan with a probability of 1.0; this is a reasonable result and is similar to the results of the other three methods. The result from the Bayesian method for well 6 was significantly different from the results of the other three methods. The Bayesian method resulted in quarterly sampling with a probability of 0.44, while the other three results recommended an annual sampling plan. The concentrations at well 6 increased recently (Figure 6.25(c)), instead of reducing sampling frequency, the sampling plan of well 6 should be kept as quarterly, so the result from

Bayesian was reasonable for well 6. Well 8 is considered a redundant well spatially and has a decreasing reducing concentration trend (Figure 6.25(d)). The Bayesian method resulted in a 0.87 probability of annual sampling for well 6, which is consistent with the other three results. The concentration trend for well 22 sharply increased during the recent sampling events (Figure 6.25(e)), and thus all four methods resulted in a quarterly sampling plan, which were very reasonable. The historical data for well 26 shows a decreasing and stable trend during the more recent sampling periods, which reasonably allows for reducing the monitoring frequency (Figure 6.25(f)). The Bayesian method resulted in annual sampling with a probability of 0.88, and similar results also were obtained from other three methods.

91 Table 6.14: Comparisons of typical results among four methods: MAROS, 3-tiered, CES, and Bayesian. Q is quarterly sampling, S is semi-annually, A is annually, B is biennially, R is Removed well, and * represents redundant well. Bolded in Bayesian are the maximum probability of posterior sampling event.

well ID MAROS 3-tiered* CES Bayesian 1 A B A Q(0.05) S(0.63) A (0.32) 2 * S R A Q (0.22) S(0.62) A (0.16) 3 * A A A Q (0.05) S (0.19) A (0.76) 4 A R S Q (0.25) S (0.5) A (0.25) 5 B B A Q (0) S (0) A (1.00) 6 A A A Q(0.43) S (0.38) A (0.19) 7 A R A Q (0) S (0) A (1.00) 8 * A A A Q (0.08) S (0.05) A(0.87) 9 S A A Q (0.27) S (0.24) A(0.48) 10 A A A Q (0.46) S (0.18) A (0.36) 11 * Q R Q Q (1.00) S (0) A (0) 12 A B A Q (0) S (0) A (1.00) 13 A A A Q (0.17) S (0.10) A(0.73) 14 A R A Q (0) S (0) A (1.00) 15 A R A Q(0.42) S (0.19) A (0.38) 16 A R A Q(0.97) S (0.01) A (0.02) 17 Q A A Q (1.00) S (0) A (0) 18 * A R A Q (0.01) S (0.06) A(0.93) 19 * S A A Q (1.00) S (0) A (0) 20 A R A Q(0.86) S (0.01) A (0.13) 21 * Q A A Q (1.00) S (0) A (0) 22 Q Q Q Q (1.00) S (0) A (0) 23 Q R S Q (0) S (0) A (1.00) 24 A R A Q (0.01) S (0.33) A (0.66) 25 * A B A Q (0.03) S (0.06) A(0.91) 26 A B A Q (0.01) S (0.11) A(0.88) 27 * A A A Q (0.01) S (0.05) A(0.94) 28 A S A Q (0.01) S (0.06) A(0.88) 29 A B A Q (0) S (0) A (1.00) 30 A S A Q (0.01) S (0.11) A(0.88)

92 0.2000 0.0008

0.1500 0.0006

0.1000 0.0004

0.0500 0.0002 Concentration Concentration 0.0000 0.0000 23456789101112131415 23456789101112131415 (a) Time (b) Time

0.0150 0.0800

0.0600 0.0100 0.0400 0.0050 0.0200 Concentration Concentration 0.0000 0.0000 23456789101112131415 23456789101112131415 (c) Time (d) Time

2.0000 0.0025

1.5000 0.0020 0.0015 1.0000 0.0010 0.5000 0.0005 Concentration Concentration 0.0000 0.0000 23456789101112131415 23456789101112131415 (e) Time (f) Time

Figure 6.25: Historical monitoring data, (a)-(f) represent well 4, 5, 6, 8, 22, and 26 respectively.

93 6.3.2 Summary and Conclusions

A method for LTM temporal problem optimization under certain assumptions was implemented and compared with three approaches, MAROS, 3-tiered, and CES. The prior belief was established by focusing more on the recent data, and the likelihood portion combined thoroughly analysis over historical sampling data trend and variance as well as spatial optimization results from ACO algorithm. A guide to the systematic analysis of site information based on prior monitoring frequency belief was provided and expert judgment was allowed to be incorporated in a transparent way. The posterior sampling plan was obtained by prior and likelihood with fuzzy scale (0-1) probability. Similar but outperforming results with other methods veriﬁed that this Bayesian method is a promising approach for an

LTM temporal problem. However, it should be recognized that more factors such as onsite conditions need to be take into account and the prior belief and the likelihood need to be more objective in the future.

94 CHAPTER 7

CONCLUDING REMARKS

7.1 Conclusions

Many ground water remediation sites, especially large ones, require long-term monitoring to assess the performance of remediation and immigration of contaminants. In recent years, people realize that it is not practical to restore some sites completely after many remediation projects, so people focus more on ground water monitoring to assure not posing an unacceptable risk to human health or the environment. However, long-term monitoring

(LTM) can be costly given the large number of sampling locations, frequency of monitoring and number of constituents monitored at given sites. The main issue of current LTM problem is that many locations show little or no change in concentration over time, which implies that many of the monitoring wells were over sampled unnecessarily. Thus selecting the appropriate and cost eﬃcient sampling techniques to measure the data without compromising program quality is important.

Previous approaches motivated towards cost-eﬀective design of LTM sampling have contributed to reducing LTM costs, but some potential problems remain. First, the drawback of simulation model-based methods is that limited site data or complex hydrogeological conditions will lead to uncertainty in the model and input parameters, causing errors in the predications and possibly unreliable optimal monitoring networks. Second, a limited amount of statistical LTM cost eﬀective sampling optimal policies are available and very few have

95 been combined with powerful optimization algorithms, such as ACO and GA.

Two LTM spatial sampling optimization methods were developed in this dissertation.

The goal of spatial sampling optimization is to determine the optimal combination of a reduced set of wells from among the original ground water monitoring network with minimum

RMSE value. The IDW interpolation is used to estimate the concentration of removed wells by remaining wells, and the overall data loss is quantiﬁed by RMSE. The ﬁrst method is called primal ACO-LTM algorithm, which minimizes the number of remaining wells given the constraint on data loss quality. In this algorithm, each individual ant has two stages

(0/1) or choices of decisions, and it is very similar with the GA, which is a combinatorial optimization method with binary decision variables. The second method is the dual ACO-

LTM algorithm; here the role of the number of remaining wells is reversed from objective function to constraint, and this algorithm is to minimize the data loss given a ﬁxed number of remaining wells. This dual ACO-LTM algorithm has a close analogy to the ACO paradigm for solving the traveling salesman problem (TSP). However, unlike the TSP problem, in the

LTM problem, the ants will not necessarily visit all the wells. The ant terminates traveling when it has visited a given number of wells equal to the described number of redundant monitoring wells.

The two ACO-LTM algorithms were tested on data from a medium-scale ﬁeld site and compared with the GA and complete enumeration, and the performance of the dual ACO-

LTM was additionally evaluated and compared with the GA for a large-scale site. The dual

ACO-LTM algorithm showed the best performance and identiﬁed global optimal solutions.

The reason is that the dual ACO-LTM algorithm is based on TSP, and each monitoring well is selected among the candidate list, so candidate well selection is involved in ant activity.

However, the primal ACO-LTM algorithm and the GA are binary methods, the selection

96 of each monitoring well is decided randomly, thus there is no such well selection criteria.

Computationally, the dual ACO-LTM algorithm is signiﬁcantly eﬀective, especially, when complete enumeration was not practical for the large-scale site.

In addition, an algorithm based on Bayesian analysis was developed for the temporal optimization of LTM sampling schedules. The Bayesian analysis method provides recommendations for reasonable reduced sampling frequency for each sampling location. An initial estimate of the recommended schedule of each individual well is obtained by slope of trends, and they are considered as prior monitoring frequency of each individual well. Likelihood is proposed by combining historical monitoring concentration trend and variance, as well as spatial optimization results. According to the Bayesian Theorem, posterior predictive distributions are obtained by integrating the product of the likelihood and prior. The more ﬂat of concentration trend, the lower variability and the more likely it belongs to the redundant well, and the lower monitoring frequency. Since the dual ACO-LTM algorithm was superior to other two stochastic algorithms, the redundant wells identiﬁed by the dual

ACO-LTM algorithm were combined with Bayesian analysis. This Bayesian method was applied to a site application with 30 monitoring wells, and the results were compared with other approaches including MAROS, CES, and the three-tiered approach. With the Bayesian method, the results are a probabilistic scale for each possible sampling plan (e.g., quarterly, semi-annually, and annually), and thus the Bayesian method provides more information than other three methods. More intuitive and outperforming results with Bayesian method veriﬁed that this method is a promising approach for the LTM temporal optimization problem.

The methods developed in this dissertation are statistical strategies combined with a powerful optimization algorithm, ACO-LTM. These methods can identify global optimum solutions and systematically analyze the historical monitoring data by a Bayesian logic. They

97 are suitable for all sites, especially large-scale sites. These methods are promising approaches and will eﬀectively and eﬃciently reduce the LTM costs.

7.2 Future Work

This research has established the foundation for methods for improving and optimizing LTM sampling programs based on ant colony optimization and Bayesian methods. Future work for enhancement to this work to improve the performance and applicability include the following.

More complex behaviors and types of ants may be included in the ACO-LTM to improve the performance. For example, each individual ant may be able to pick up or drop off food under different circumstance, meanwhile, in each ant colony, some ants act as soldiers, and others are nurses, of course, different behaviors and types of ant means different pheromone, for example, selecting redundant wells and inserting some additional wells in some place where monitoring wells are too scarce. In this work, a simple and basic ACO is adapted for an LTM problem and only one type of pheromone, which is selecting redundant wells.

In this work, the interpolation method is IDW, but IDW has some potential problem in an LTM problem, especially, in contaminant plume, the basic assumption, ’the closer, the more related’, is not always true. Alternative interpolation methods, which can describe the plume shape properly, should be developed and substituted into an ACO algorithm.

The protocols part for the temporal LTM problem in the Bayesian method need to be reﬁned. Instead of a linear function between concentration and time, an alternative description of concentration trends should be characterized, and the seasonal ﬂuctuation and on site conditions should be involved and combined into trends.

Additionally, spatial and temporal optimization should reﬂect dynamic change of contaminant plume, and be considered more realistically; for the spatial problem, instead of the assumption that the plume shape is static, physical factors (such as ﬂow direction,

98 sources of pollutants, and landscape, geological structure...) may be involved. For the temporal problem, the sampling schedule will be updated annually to reflect the dynamic change of the time series data. The spatial optimization and temporal analysis of an LTM should be modified to optimize remediation performance monitoring. The performance of the remediation can be evaluated only by monitoring, as data are collected and site knowledge increases, the monitoring scheme should be changed properly. So the cost-effective sampling optimization can be processed and involved before site-closure; in other words, the spatial and temporal optimization of monitoring can be extended before an LTM stage.

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104 BIOGRAPHICAL SKETCH

Yuanhai Li

Yuanhai Li was born on June 25th,1975, in Antu County, Jilin Province, P.R.China. In the fall of 1999, he completed his Bachelors degree in Environmental Engineering Department at Tsinghua University. He obtained his Masters degree in summer of 2002, from the

Department of Civil Engineering at Shantou Unversity, P.R.China. He enrolled in the doctoral program in Environmental Engineering at FSU in the fall of 2002.

Yuanhai’s research interests include groundwater long-term monitoring optimization, stochastic optimizations, GIS application in environmental engineering, and waster water treatment design.

105 PUBLISHED WORKS

The following are refereed papers based on the work in this dissertation.

Li, Y. and A. B. Chan Hilton. Reducing Spatial Sampling in Long-Term Groundwater

par Monitoring Networks using Ant Colony Optimization. International Journal of

Computational Intelligence Research, ISSN 0973-1873 Vol.1, No. 1-2 (2006), pp. 19-

28.

Li, Y. and A.B. Chan Hilton. Optimal Groundwater Monitoring Design Using an Ant

Colony Optimization Paradigm. Environmental Modeling and Software, in press.

The following are conference papers based on the work in this dissertation.

Li, Y. and A. B. Chan Hilton. Bayesian Statistics-Based Procedure for the Groundwater

Long-Term Monitoring Temporal Problem. Proceedings of the ASCE EWRI 2006

World Water & Environmental Resources Congress, May 2006, Omaha, NE: ASCE,

5 pp.

Li, Y. and A. B. Chan Hilton. An Algorithm for Groundwater Long-Term Monitoring

Spatial Optimization by Analogy to Ant Colony Optimization for TSP. Proceedings

of the ASCE EWRI 2006 World Water & Environmental Resources Congress, May

2006, Omaha, NE: ASCE, 6 pp.

106 Chan Hilton, A.B. and Y. Li. Optimal groundwater sampling network design through

ant colony optimization. Proceedings of the Genetic and Evolutionary Computation

Conference (GECCO 2005) (June 25-29, 2005,Washington, DC). ACM, 6 pp.

Li, Y. and A. B. Chan Hilton. Analysis of the Primal and Dual Problem for Long Term

Groundwater Monitoring Spatial Optimization. Proceedings of the ASCE EWRI 2005

World Water & Environmental Resources Congress, Anchorage, AK: ASCE, 11pp.

Li, Y., A. B. Chan Hilton, and L. Tong. Development of ant colony optimization for long-

term groundwater monitoring. Proceedings of the ASCE EWRI 2004 World Water

& Environmental Resources Congress, Salt Lake City, UT: ASCE, 10 pp.

107