Global Optimal Design of Ground Water Monitoring Network Using ...

Global Optimal Design of Ground Water Monitoring Network Using Embedded Kriging by Anirban Dhar1 and Bithin Datta2

Abstract We present a methodology for global optimal design of ground water quality monitoring networks using a linear mixed-integer formulation. The proposed methodology incorporates ordinary kriging (OK) within the decision model formulation for spatial estimation of contaminant concentration values. Different monitoring network design models incorporating concentration estimation error, variance estimation error, mass estimation error, error in locating plume centroid, and spatial coverage of the designed network are developed. A bigM technique is used for reformulating the monitoring network design model to a linear decision model while incorporating different objectives and OK equations. Global optimality of the solutions obtained for the monitoring network design can be ensured due to the linear mixed-integer programming formulations proposed. Performances of the proposed models are evaluated for both field and hypothetical illustrative systems. Evaluation results indicate that the proposed methodology performs satisfactorily. These performance evaluation results demonstrate the potential applicability of the proposed methodology for optimal ground water contaminant monitoring network design.

Introduction Ground water pollution is a major concern in modern times. In reality, often possible ground water pollution is first detected in water supply wells. The amount of information available from the water supply wells is generally very limited. Process control, performance measurement, or compliance requires time consuming and costly data collection effort. Moreover, the location of these wells may not be optimal for different monitoring design objectives under consideration. Tracking the transient pollutant plume is a challenging task due to the uncertainties in predicting the complex subsurface environment and transport processes. An efficient and optimally designed monitoring

1 Corresponding author: Department of Civil Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India; 2 School of Engineering, James Cook University, Townsville, QLD 4811, Australia Received 24 November 2008, accepted 29 April 2009. Copyright © 2009 The Author(s) Journal compilation © 2009 National Ground Water Association. doi: 10.1111/j.1745-6584.2009.00591.x

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network can save millions of dollars in long-term monitoring costs. A linear mixed-integer formulation is proposed for design of ground water monitoring networks that guarantees global optimality of the design solutions. Designing ground water monitoring networks has received increasing attention in the recent past. In-depth reviews of monitoring network design studies can be found in Loaiciga et al. (1992) and ASCE Task Committee (2003). In optimization problems the formulation generally dictates the choice of algorithm. Use of genetic algorithms (Reed and Minsker 2004; Mugunthan and Shoemaker 2004; Wu et al. 2005), simulated annealing (Nunes et al. 2004; Mugunthan and Shoemaker 2004), and ant colony optimization (ACO, Li and Hilton 2005) are common due to their wide applicability. However, these algorithms cannot mathematically guarantee optimality of a solution. Hudak et al. (1995), Datta and Dhiman (1996), and Mahar and Datta (1997) used linear formulations to find an optimal solution of monitoring network design problem. Also, different decision support systems are available for monitoring network design like MAROS (Aziz et al. 2002) and GTS (Cameron and Hunter 2000). These methods use iterative solvers to find

Vol. 47, No. 6–GROUND WATER–November-December 2009 (pages 806–815)

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suitable solutions that are not truly optimal in nature, unless the search is exhaustive. In recent times different studies, for example, Reed and Minsker (2004) and Wu et al. (2005), have incorporated spatial interpolation algorithms within the framework of monitoring network design for estimation of concentration values. However, in all of these studies, spatial interpolation is incorporated as an external module to the optimization algorithm and thereby it functions as a logical operator. Different objectives along with spatial interpolation relations in a logical form result in a nonlinear formulation of the design problem. Nonlinear formulations faces major problem with discrete variables, nonconvexity, and discontinuity. They inherit the possibilities of the solutions getting stuck to some local optima. Most of the studies concentrated their attention on developing or using different optimization algorithms in order to improve solution quality. Many of the proposed formulations in literature have not adequately addressed the concern for global optimality. Linear formulations of the decision problem with associated convex search space can be one possible way to overcome this problem of global optimality. A proper optimization algorithm together with a convex search space can mathematically guarantee the global optimality of obtained solutions. Therefore, a methodology combining the spatial interpolation scheme of ordinary kriging (OK) and mixed-integer decision model based formulation is proposed for optimal monitoring network design, without sacrificing the generality of the system relations. The general problem of ground water monitoring network design can be stated as optimization of design objective(s) subject to budgetary limitation. Choice or priority of design objectives always depends on the decision maker(s). Monitoring network design may consist of single or multiple objectives. The objectives considered in this study are (1) minimization of total concentration estimation error, (2) minimization of total concentration estimation variance, (3) minimization of concentration mass estimation error, (4) minimization of error in locating plume centroid, and (5) maximization of spatial coverage. To demonstrate the implication of these objectives, single objective models are presented separately. However, objectives (3) and (4) are considered in surrogate constraint form in one of the models. In addition, the spatial interpolation technique OK is embedded within the optimization formulation for estimation of concentration values at unmonitored locations. These optimization models are solved using standard MILP (Mixed-Integer Linear Programming) software.

Monitoring Network Design Models The optimization models are formulated utilizing objectives mentioned earlier and OK constraints. Let us consider to be the set of all possible monitoring locations. In reality, contains infinite member of locations. NGWA.org

Generally, a finite set P of potential monitoring locations is chosen from , depending on technical feasibility. However, monitoring is performed for a selected set M ( M ⊆ P ⊂ ). Mathematical formulations of different models are discussed in the following sections. Optimal Network Design Model-I (ONDM-I) The concentration estimation error for the obtained design as compared to the potential monitoring network design consisting of all potential locations provides a measure of the designed network efficiency. The objective can be stated as minimization of summation of absolute normalized deviation of actual concentration from estimated concentration based on present monitoring plan for all potential monitoring locations; that is, Nw cj act − cj (1) Minimize c + η j act j =1

where cj is the concentration at potential monitoring location j , estimated using potential monitoring wells according to monitoring plan, act denotes actual value of any attribute, η is a small constant real number (η ∈ [0, 1]), and Nw is the number of potential monitoring locations (number of candidates in P ). The formulation is difficult to solve using classical optimization algorithms as the objective function (Equation 1) is nondifferentiable at cj = cj act . The objective function (Equation 1) can be converted to linear form using the following decomposition: Minimize

Nw

− + j + j

(2)

j =1

subject to: + j

≥

+ j

≥

− j ≥ − j ≥

⎫ cj act − cj ⎬ cj act + η ,∀j ∈ {1, . . . , Nw } ⎭ 0 ⎫ cj − cj act ⎬ cj act + η ,∀j ∈ {1, . . . , Nw } ⎭ 0

(3)

(4)

− where + j and j are positive real variables. Monitoring networks are long-term investments. Therefore, the cost associated with a particular network design is important. The budgetary limitation can be imposed as a constraint: Nw

χj = P

(5)

j =1

where χ j is a binary decision variable indicating whether a monitoring well is to be monitored at potential monitoring location j , a value of 1 indicates monitoring, and P is the total number of wells to be monitored (number of candidates in M ). The prime challenge is to incorporate OK equations within the formulation for spatial interpolation of concentration values. OK (Deutsch and Journel 1992) is the most

A. Dhar and B. Datta GROUND WATER 47, no. 6: 806–815

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widely used among the kriging methods. The method estimates a value at a point in a region based on the data available in the neighborhood of the estimation location. Let us consider a problem of estimating the value of a continuous attribute z at any unmonitored location u using n(u) number of z-data available in the neighborhood. The OK estimator Z ∗ (u) can be written as a linear combination of the n(u) random variables (RVs) Z(uα ) (Goovaerts 1997): Z ∗ (u) =

n(u)

λα (u)Z(uα )

(6)

α=1

monitoring wells (denoted by i) can be written as: ⎧ Nw ⎪ ⎪ ⎪ ⎪ λβ (uj ) C(ui − uβ ) + μ(uj ) − C(ui − uj ) ⎪ ⎪ ⎪ ⎪β=1 ⎪ ⎪ ⎪ ⎪ ≤ M(1 − χ i + χ j ) ⎪ ⎪ ⎪ ⎪ Nw ⎪ ⎪ ⎨ λβ (uj ) C(ui − uβ ) + μ(uj ) − C(ui − uj ) β=1 ⎪ ⎪ ⎪ ⎪ ≥ −M(1 − χ i + χ j ) ⎪ ⎪ ⎪ Nw ⎪ ⎪ ⎪ ⎪ ⎪ λβ (uj ) + χ j = 1; ∀i ∈ {1, . . . , Nw }, ⎪ ⎪ ⎪ ⎪ β=1 ⎪ ⎪ ⎩ j ∈ {1, . . . , Nw }

with

(12) n(u)

and λα (u) = 1

(7)

α=1

where λα (u) is the weight assigned to a realization of the RV Z(uα ) at spatial location α. Error variance, σ 2E (u), can be expressed as:

⎧ ⎪ ⎪ ⎪−Mχ i ≤ λi (uj ) ≤ Mχ i ⎪ ⎪ ⎪ −M(1 − χ j ) ≤ λi (uj ) ≤ M(1 − χ j ) ⎪ ⎪ ⎪ ⎨−M ≤ μ(uj ) ≤ M Nw ⎪ ⎪ ⎪ c = c χ + λi (uj ) ci act ⎪ j j act j ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎩∀i ∈ {1, . . . , N }, j ∈ {1, . . . , N } w

σ 2E (u)

∗

= Var{Z (u) − Z(u)}

(8)

The estimation objective can be stated as minimization of the estimation or error variance (Equation 8) under the nonbias condition (Equation 7). This results in the OK system with n(u) + 1 linear equations with equal number of unknowns (Goovaerts 1997): ⎧ n(u) ⎪ ⎪ ⎪ λβ (u)C(uα − uβ ) + μ(u) ⎪ ⎪ ⎪ ⎪ ⎨β=1 = C(uα − u) , ∀α ∈ {1, . . . , n(u)} ⎪ ⎪ n(u) ⎪ ⎪ ⎪ ⎪ λβ (u) = 1 ⎪ ⎩

(9)

β=1

where C(uα − uβ ) covariance of attribute z is between location uα and uβ (α and β denotes spatial location); μ(u) is a variable. The resulting minimum error variance can be written as (Goovaerts 1997):

σ 2E (u) = C(0) −

n(u)

λα (u)C(uα − u) − μ(u)

(10)

α=1

Covariance C(h) and variogram γ (h) with lag distance of h are related by (Isaak and Srivastava 1989): C(h) = γ (∞) − γ (h)

(11)

Thus equations for estimation of concentration value at j th potential monitoring well based on neighboring 808

(13)

w

where M is a very large positive number. Equation 12 represents the kriging equations in linear mixed-integer form (see Supporting Information). In Equation 13 the first three constraint sets are used to bound the values of λi (uj ) and μ(uj ). The fourth and final constraint is for calculating the estimated value. It can be noted that if a potential monitoring location j is chosen as an actual monitoring location; then the estimated value turns out to be the actual value. The third constraint in Equation 12 along with the first two constraints of Equation 13 forces the λi (uj ) values to zero if monitoring is performed at location j . It is worth mentioning that use of big-M creates convex decision space. A more general description of the big-M method is given in Hooker (2000). The single objective optimization model Optimal Network Design Model-I (ONDM-I) consists of the objective function (Equation 2) together with the constraint set (Equation 3), (Equation 4), (Equation 5), (Equation 12), (Equation 13). It contains the decision vari− ables + j , j , cj , μ(uj ), λi (uj ), and χ j . Optimal Network Design Model-II (ONDM-II) The variance term is often considered synonymous with uncertainty. It signifies the amount of uncertainty with which concentration values are estimated. An objective function can be formulated as minimization of total estimation variance of spatial concentration all around the area under study, including the potentials monitoring locations where a monitoring well is not to be installed as per the design. The total estimation variance minimization objective can be written as: Minimize


Nw j =1

σ 2E (uj )(1 − χ j ) +

NP

σ 2E (uj )

(14)

j =Nw +1

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subject to: σ 2E (uj ) = C(0) −

Nw

λi (uj )C(ui − uj ) − μ(uj ) (15)

i=1

where NP is the total cells in the study area including the potential monitoring points or cells. This nonlinear formulation can be converted into equivalent linear form as: Minimize

NP

σj

(16)

j =1

subject to: σj ≥ σ 2E (uj ) − Mχ j , σj ≥ σ 2E (uj ), σj ≥ 0,

⎫ ∀j ∈ {1, . . . , Nw } ⎬ ∀j ∈ {Nw + 1, . . . , NP } ⎭ ∀j ∈ {1, . . . , NP } (17)

where σj is a positive real variable. This formulation requires estimation of μ(uj ), λi (uj ) values at all discrete NP locations in the space, including potential monitoring well locations. The equations for estimation of concentration value at j th location (other than potential monitoring well locations) based on neighboring monitoring wells (denoted by i) can be written as: ⎧ Nw ⎪ ⎪ ⎪ ⎪ λβ (uj )C(ui − uβ ) + μ(uj ) − C(ui − uj ) ⎪ ⎪ ⎪ ⎪ β=1 ⎪ ⎪ ⎪ ⎪ ≤ M(1 − χ i ) ⎪ ⎪ ⎪ ⎪ N w ⎪ ⎪ ⎨ λβ (uj )C(ui − uβ ) + μ(uj ) − C(ui − uj ) (18) β=1 ⎪ ⎪ ⎪ ⎪ ≥ −M(1 − χ i ) ⎪ ⎪ ⎪ N ⎪ w ⎪ ⎪ ⎪ ⎪ λβ (uj ) = 1; ∀i ∈ {1, . . . , Nw }, ⎪ ⎪ ⎪ ⎪β=1 ⎪ ⎪ ⎩ j ∈ {Nw + 1, . . . , NP } and ⎧ −Mχ i ≤ λi (uj ) ≤ Mχ i ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨−M ≤ μ(uj ) ≤ M Nw ⎪ cj = λi (uj )ci act ⎪ ⎪ ⎪ ⎪ i=1 ⎪ ⎩ ∀i ∈ {1, . . . , Nw }, j ∈ {Nw + 1, . . . , NP }

where t is the mass estimated using all potential monitoring locations at any time t, n is the porosity of aquifer material, and represents space. The objective can be stated as the minimization of absolute normalized deviation of mass based on all monitoring wells from estimated mass based on present monitoring plan. t all − t est t (21) Minimize e0,0 = t all where all is the value estimated using all potential monitoring wells and est is the value estimated using potential monitoring wells according to present monitoring plan. The equivalent linear formulation can be written as: − Minimize + m + m

(22)

subject to: ⎫ t all − t est ⎬ t all ⎭ + m ≥0

(23)

⎫ t est − t all ⎬ ≥ t all ⎭ − m ≥ 0

(24)

+ m≥

and

− m

− where + m and m are positive real variables. The single objective optimization model ONDMIII consists of Equation 22 together with the constraint set (Equations 23, 24, 5, 12, 13, 18, and 19). It contains − the decision variables + m , m , cj , μ(uj ), λi (uj ), and χ j .

(19)

The single objective optimization model ONDM-II consists of Equation 16 together with the constraint set (Equations 17, 18, 19, 5, 12, and 13). Concentration calculation cj in Equations 13 and 19 is not required for this formulation. It contains the decision variables σj , σ 2E (uj ), μ(uj ), λi (uj ), and χ j . Optimal Network Design Model-III (ONDM-III) Spatiotemporal plume characterization is a major objective from a remediation point of view. Without a NGWA.org

scientifically designed network it is not possible to assess the spatiotemporal extent of a contaminant plume. This study deals with spatial characterization of a contaminant plume. Spatial moments signify the overall distribution of concentration in the aquifer. Zeroth moment provides information about the contaminant mass in the aquifer. It can be expressed as:

n c(x, y, t) d (20) t =

Optimal Network Design Model-IV (ONDM-IV) Spatial coverage of the network is also a major concern. Information should come from all over the study area. A distance criterion is the most common for quantifying spatial coverage. If interwell distances are summed and maximized, then it will ensure the desired goal. This is, in other words, minimization of total interwell inverse distance for all possible combinations. This objective can be written as: Minimize :

N w −1

Nw 1 χiχj di,j

(25)

i=1 j =i+1


809

where di,j is the distance between potential monitoring locations i and j . The distance objective is nonlinear. This objective can be converted to linear equivalent form as: Minimize :

N w −1

Nw

dist i,j

(26)

i=1 j =i+1

subject to: ⎧ 1 ⎨dist ≥ − M(1 − χ i ) − M(1 − χ j ) i,j di,j ⎩ dist i,j ≥ 0

(27)

where dist i,j is positive real variable. If the mass estimation error minimization objective is implicitly incorporated as constraints, then the following set of constraints can be defined: t all − t est ≤ ε0,0 t all (28) t est − t all ≤ ε0,0 t all where ε0,0 is the upper limit of the mass estimation error. Physically speaking, the first moment represents the center of mass (centroid) of the contaminant plume in the aquifer. There are two moments in the 2-D Cartesian coordinate system. Mathematically these can be expressed as:

1 t n c(x, y, t) x d (29) μ1,0 = t and μt0,1 =

1 t

n c(x, y, t) y d

(30)

Error in locating the plume centroid can be incorporated as a constraint with the formulation: μt all − μt est 1,0 1,0 t e1,0 = (31) ≤ ε1,0 μt1,0 all and t e0,1

μt all − μt est 0,1 0,1 = ≤ ε0,1 μt0,1 all

(32)

where ε1,0 and ε0,1 are the upper limits on the centroid estimation error. Assuming all spatial points are located in a positive quadrant (in 2D) of the Cartesian coordinated system, for a given ε 1,0 , Equation 31 can be decomposed into a linear set of equations: ⎫ μt1,0 all t est − n c(x, y, t) x d est ⎪ ⎪ ⎬ ≤ ε1,0 μt1,0 all t est (33) t n c(x, y, t) x d est − μ1,0 all t est ⎪ ⎪ ⎭ t ≤ ε1,0 μ1,0 all t est Similarly for the first moment in y, we can write: ⎫ μt0,1 all t est − n c(x, y, t) y d est ⎪ ⎪ ⎬ ≤ ε0,1 μt0,1 all t est (34) n c(x, y, t) y d est − μt0,1 all t est ⎪ ⎪ ⎭ ≤ ε0,1 μt0,1 all t est 810

The single objective optimization model ONDM-IV consists of Equation 26 together with the constraint set (Equations 27, 28, 33, 34, 12, 13, 18, and 19). It contains the decision variables dist i,j , cj , μ(uj ), λi (uj ), and χ j . The optimal design models are solved using standard optimization software CPLEX (ILOG 2000). These are implemented in C++, using ILOG CPLEX 7.0 Concert Technology (ILOG 2000) application programming interface. Due to the linear nature of the formulations, the optimization algorithm produces a global optimal solution.

Application of Developed Models On the basis of design goals, monitoring network design can be classified into (a) design and installation of a new monitoring network; (b) monitoring network design by choosing a subset of existing monitoring locations; and (c) augmentation of an existing network for additional information gathering. In order to evaluate the performance of the developed methodology, two applications are demonstrated. The first one is installation of new monitoring network for a hypothetical illustrative system. The second one is monitoring network design by choosing a subset of existing monitoring locations in field problem. Hypothetical Illustrative Application The proposed methodology is applied for the design of monitoring networks in an illustrative study area (300 × 100 × 5 m). The aquifer is confined, with 2D flow and transport processes (Figure 1). The aquifer parameters are given in Table 1. Some of the parameter values are taken from Mugunthan and Shoemaker (2004). A single pollutant is assumed to be active throughout the simulation horizon. Flux from the pollutant source during a planning horizon is assumed to be constant. The pollutant of concentration 500 mg/L is assumed to leak from the source location. The injection rate of polluted water into the aquifer is assumed to be 40 L/h. In this study it is assumed that the flow and transport models are based on the finite difference flow code MODFLOW (Harbaugh et al. 2000) and its solute transport companion MT3DMS (Zheng and Wang 1999). The illustrative area is discretized as 20 × 10 × 5 m. The entire study area is covered by 150 cells. Initially the aquifer is assumed to be contaminant free. Initial head values are generated using steady-state simulation of the flow model for imposed boundary conditions. A hydraulic conductivity field is generated from uniform distribution within the given bounds (Figure S1). Concentration contours at the end of 1095 days along with potential monitoring well locations are shown in Figure 1. Ground water flows from left to right with constant head boundaries on left and right ends of the aquifer, and no flow boundaries are specified at the top and bottom. Contaminant transport is modeled for advection, dispersion, and diffusion processes under general hydrologic conditions. Representative contaminant plumes obtained as solution of the numerical simulation models represent


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Figure 1. Location of source and monitoring wells including concentration (in mg/L) contour for illustrative study area.

Table 1 Different Parameters for Hypothetical Illustrative Study Area Parameter

Unit

Value

Length Width Depth Hydraulic conductivity Porosity Longitudinal dispersivity Transverse dispersivity

m m m m/d — m m

300.00 100.00 5.00 2.00–7.00 0.30 0.30 0.20

future contamination scenarios to be monitored. Concentrations (expected) at potential monitoring locations are considered to be known. The present work does not incorporate the parameter uncertainties involved. Although, it can be extended by incorporating multiple plume realizations corresponding to statistically generated parameter sets. Variogram or spatial structure identification is the first step for kriging interpolation. In deriving the spatial structure, the nugget effect is not considered. In general any kind of variogram structure can be utilized in the

present framework even with the nugget effect. Also, standardized variogram is used. Spatial variability structure (variogram) is identified using Surfer® (Golden Software Inc. 2004). A spherical variogram model is used to fit the concentration data. Mathematically the model can be expressed as: ⎧ 3 ⎪ h h ⎨S 1.5 − 0.5 if h ≤ a a a (35) γ (h) = ⎪ ⎩ S if h > a where γ (h) is the semi-variogram, S is the scale for the structured component of the variogram, h is the anisotropic separation distance, and a is the range. Values for different variogram parameters are identified as S = 1.0 and a = 90 m. The values of NP and Nw are 150 and 30, respectively. As the potential locations are well spread over the illustrative study area, reference mass and centroid values are calculated using all potential monitoring locations, from which maximum level of information can be collected. ONDM-I considers the objective of minimizing concentration estimation error. The concentration estimation error objective function is considered for potential monitoring well locations only. Table 2 presents the optimal well locations. It can be noted from Table 2 that the concentration estimation error increases with the decreases in number of monitoring wells.

Table 2 Optimal Solutions Corresponding to Different Models for Hypothetical Illustrative Study Area ONMD-I

ONMD-II

ONMD-III

P

OFV

WNSD

OFV

WNSD

OFV

WNSD

29 28 27

3.6080 × 10−03 3.5818 × 10−02 8.8506 × 10−02

21 11–21 11–17–21

39.746 40.593 41.449

19 17–21 6–17–21

1.5173 × 10−04 3.8324 × 10−06 3.2990 × 10−05

21 24–26 10–19–20

OFV, objective function value; WNSD, wells not specified as per monitoring design.

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ONDM-II considers the objective of minimization of the concentration estimation variance. Total variance is calculated using all the points including potential monitoring well locations. Solution results prescribed (Table 2) for this model are different compared to ONDMI solutions. Variance reduction is in effect equivalent to reduction in uncertainty. The total uncertainty increases steadily with the removal of potential monitoring locations from network. However, as is evident from the objective function (Equation 14), a base value of the variance would always remain, which is greater than zero. Therefore, the objective function showing total estimation variance does not decrease below a base value of 38.923. ONDM-III considers the objective of minimizing mass estimation error. Mathematically, mass is calculated as zeroth moment using the concentration at all NP cells of the aquifer. The result (Table 2) shows that when 29 locations out of 30 potential locations are selected as optimal locations, the mass estimation error is larger compared to the case when 28 wells are selected. It is due to the constrained nature of the problem and the spatial interpolation scheme. This signifies that mass estimation error is dependent not only on the configurations but also on the number of wells. Although solution results are different as compared to ONDM-I and ONDM-II, it is interesting to note that the only monitoring location not chosen is the same location not chosen by ONDM-I solution when 29 wells are selected. ONDM-IV considers multiple objectives of monitoring. However, some of the implicit objectives are included as surrogate constraints. This model ensures maximum spatial coverage while maintaining contaminant plume characterization errors within bounds. Both mass and first moments are included for plume characterization purposes. When spatial interpolation constraints and upper bound on the spatial moment estimation errors are incorporated, the configuration of wells changes. For upper limits on the plume characterization errors, ε0,0 = 0.001, ε1,0 = 0.005, and ε0,1 = 0.005, the obtained optimal design solution excluded locations 7–11–17–18–19–21. The resulting objective function value is 3.58965. Using the spatial coverage objective, without plume characterization error constraint, the optimal prescribe locations exclude 6–9–14–16–19–24 and the corresponding objective function value is 3.53451. Therefore, sacrifice in spatial coverage objective function (3.58965–3.53451) ensures plume characterizations are within permissible bounds (Table 3).

utilized for application. Relevant information is available in U.S. EPA (2004). This particular study area is identical to the area utilized by Li and Hilton (2005). With limited information, the stationary plume condition is assumed to be as reported in Li and Hilton (2005). The performance is evaluated only for ONDM-I. To compare the performance of the proposed methodology with the study of Li and Hilton (2005), the relative estimation error (REE avg ) is defined as: REEavg =

N w −P |cj est − cj act | 1 (Nw − P ) min(cj est , cj act )

(36)

j =1

Overall data loss for monitoring well removal is quantified using root mean square error (RMSE ): N w −P |cj est − cj act | 2 1 RMSE = (Nw − P ) min(cj est , cj act ) j =1

(37) Concentration contours based on a single monitoring event along with existing monitoring wells are shown in Figure 2. A spherical variogram model is used to fit concentration data. Values for different parameters are identified as S = 1.0 and a = 762 m. Based on concentration data it is observed that the nugget effect is negligible. Use of standardized variogram would permit the specification of a lower value of M = 1.0, while satisfying convexity condition. The performance monitoring network design problem depends to a large extent on the geometric configuration of available observation wells. Table 4 shows the configuration of removed wells and corresponding errors for ONDM-I with different P values. It is evident from comparison of errors with Li and Hilton (2005) that RMSE and REE avg performance of the proposed kriging based method appears to be consistent and better as errors increase with the decrease in number of monitoring wells. In all the evaluations RMSE is higher than REE avg because of the fact that RMSE uses squared deviation terms.

Table 3 Attribute Comparison with and without Spatial Interpolation Constraints Number of Variables

Field Application In order to illustrate field applicability of the developed methodology, an upper aquifer long-term monitoring well network at the Fort Lewis Logistics Center in Pierce County, Washington (U.S. EPA 2004), which was polluted by the degreasing agent trichloroethylene (TCE), is considered. Remedial measures are intended to restore water quality standards at down-gradient compliance points, to less than 5 ppb within 30∼40 years (U.S. EPA 2004). Existing 30 wells with a monitoring event on 9/1/2000 are 812

Attribute

ONDM-IV

WSIC

OFV t e0,0 t e1,0 t e0,1 WNSD

3.58965 0.000991 0.003772 0.004891 7–11–17–18–19–21

3.53451 0.094201 0.008937 0.00212 6–9–14–16–19–24

WSIC, ONDM-IV without spatial interpolation constraints; OFV, objective function value; WNSD, wells not specified as per design.


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Figure 2. The 30-well monitoring network and contaminant concentration contours (in mg/L) using actual data of 9/1/2000.

Table 4 Comparison with the Study of Li and Hilton (2005) for Field System OK-Scenario P

% Reduction in wells

27 26 25 24 23

10 13 17 20 23

% % % % %

Li and Hilton (2005): ACO

REE avg

RMSE

WTE

REE avg

RMSE

WTE

0.11425 0.12856 0.21662 0.28371 0.32319

0.12389 0.13736 0.27779 0.33667 0.36643

9–15–20 9–13–15–20 3–9–13–15–20 2–3–13–15–20–23 3–9–13–15–18–20–23

0.361 0.578 0.523 0.515 0.562

0.383 0.595 0.559 0.545 0.637

2–15–19 2–10–11–25 10–11–15–19–25 2–10–11–15–19–25 2–3–10–11–15–18–25

WTE, wells to be eliminated.

The computational requirement associated with any formulation of the design problem is an important consideration. It is basically dependent on number of variables involved. Table 5 shows the exact number of variables involved in optimization for different formulations. To show the complexity pattern, the logarithmic value of iteration number (ItCnt) required for convergence is plotted against remaining number of wells for ONDM-I (Figure 3). Running time is not quoted anywhere, because different systems configuration will lead to different CPU time. For example, some of the runs took 72 h for solving the ONDM-I in SunOS 5.8.

Conclusions A set of models are developed for optimal design of a ground water quality monitoring network. The purpose NGWA.org

Table 5 Number of Variables Comparison for Different Models Number of Variables Model

Real

ONDM-I ONDM-II ONDM-III ONDM-IV

Nw (Nw + 4) NP (Nw + 3) NP (Nw + 2) + 2 NP (Nw + 2) + 12 Nw (Nw − 1)

Integer Nw Nw Nw Nw

of the developed models is to find an appropriate set of monitoring locations that can estimate the pollutant plume approximately, or reduce uncertainty in terms


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Figure 3. Variation in computational ONDM-I for field system.

complexity

for

of concentration estimation, under budgetary limitations. Without budgetary limitation there is no need of optimization. Presented formulations consider plume construction error reduction in terms of reduction in mass estimation error, and error in locating centroid of the plume. Concern about the global optimality of resulting solutions is addressed by converting the nonlinear optimization model to a linear optimization model. A number of modifications and transformations are utilized to ensure a linear formulation of the monitoring network design model. Linear models ensure global optimality of solutions obtained. It is worth mentioning that the linearization neither affects the generality of the problem nor compromises in terms of system relations. The objective of monitoring may generally differ based on site-specific requirements. The solutions for the optimal design models are dependent on the objectives specified, although each solution is globally optimum for the models used. Due to the independent nature of objectives, it is difficult to specify a single criterion for meaningful comparison of different model performances. Also, the constraints define different feasible spaces for solution. However, these limited solution results and evaluations should help decision makers choose a proper model for design of the monitoring network. The proposed logic-based mixed-integer linear formulation methodology is applied to a field and a hypothetical illustrative system. Performance of the proposed methodology is also evaluated for a field system utilizing available concentration data. Two different performance error terms RMSE and REE avg are defined to quantify the efficiency of the designed network. The limited performance evaluation results only show the potential applicability of the developed methodologies. The proposed methodology yields a single objective solution at a time. However, it can be used to generate the Pareto front (i.e., multiobjective or nondominated solutions) for multiple objectives of design. Pareto front is a guideline to the decision makers but the ultimate outcome of decision-making process is a single solution based on decision makers’ preference ordering. 814

Choice of spatial interpolation scheme is an important issue as no interpolation scheme works well for every situation. Reed et al. (2004) compared several spatial interpolation schemes using perchloroethylene (PCE) data and concluded that quantile kriging (QK) is the most robust one. Mugunthan and Shoemaker (2004) found that inverse distance weighting (IDW) and ordinary kriging (OK) performs much better as compared to QK for their problem. In the present work, OK is used as the spatial interpolation technique. Use of different spatial interpolation algorithm will definitely need a change in approach for a linear formulation. Due to dynamic nature of the pollutant plumes, typically monitoring network design is a time varying problem. The present work focuses on development of monitoring network design models for a single time period. However, the proposed methodology can be extended to a transient case by adopting a dynamic design (Dhar and Datta 2007).

Supporting Information Supporting Information may be found in the online version of this article: It includes logical explanation for Equation 12 and random hydraulic conductivity field utilized for hypothetical study. Table S1. Implications of the expression (12) Figure S1. Random hydraulic conductivity field utilized (m/day) Please note: Wiley-Blackwell are not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the article.

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