An Optimization Approach

Transcription

1 WATER RESOURCES RESEARCH, VOL. 19, NO. 3, PAGES , JUNE 1983 Identifying Sources of Groundwater Pollution' An Optimization Approach STEVEN M. GORELICK U.S. Geological Survey, Menlo Park, California BARBARA EVANS AND IRWIN REMSON Department of Applied Earth Sciences, Stanford University, Stanford, California Least squares regression and linear programing for least absolute error estimation are each combined with groundwater solute transport simulation to identify the locations and magnitudes of aquifer pollutant sources. Pollutant sources are identified by matching simulated and measured nonreacting solute concentration data. We have assumed known hydraulic parameters but considered concentration data errors explicitly. The identification models are demonstrated and compared using two hypothetical aquifer systems, one for the steady state case and the other for the transient case. Steady state models identified unknown pipe leak locations and leak magnitudes based upon sparse and spatially distributed chloride and tritium data. The number of likely leak locations was restricted in the models by employing mixed integer programing and stepwise multiple regression. Transient models identified several annual disposal fluxes in the aquifer based upon concentration histories collected at observation wells. In this case, conservative solute concentration data were abundant and contained substantial errors. Minimizing either least absolute or least squared errors was successful in identifying pollutant sources. Furthermore, we demonstrate error analysis for the results given by either method. INTRODUCTION This paper reports two approaches to the problem of identifying groundwater pollutant source locations and magnitudes where sparse and spatially distributed data indicate that such sources exist. This research is in a formative stage. It is our intention to approach the problem of groundwater pollutant source identification by combining existing numerical simulation methods with familiar estimation techniques. The techniques and extension of linear programing and multiple regression are used for least absolute error and least squares esumauon respectively. The approach is applicable to groundwater systems that are well defined, i.e., systems for which initial and boundary conditions are known and for which parameters regarding hydraulic behavior and solute transport have been determined. Our aim is to identify system stresses under conditions of solute concentration data errors and not to address the larger and more complex problem of combined determination of groundwater parameters and system stresses. The method could find application at well-studied subsurface waste disposal facilities, landfill sites, and areas where underground pipe networks transport hazardous liquids. Linear programing has been applied to the problem of groundwater parameter identification. Kleinecke [1971] attempted to determine both the transmissivity and storage coefficient by minimizing the differences between predicted and observed groundwater heads (residuals) over time. The basin flow simulations utilized the linked Darcian flow model of Tyson and Weber [1964]. His linear programing solution was unrealistic, yielding zero values (as nonbasic variables) for parameters at some nodal locations. Neuman [1973] applied parametric linear programing to a finite element This paper is not subject to U.S. copyright. Published in 1983 by the American Geophysical Union. Paper number 3W0406. model of steady groundwater flow to determine 'a discrete set of alternative solutions to the identification problem.' The modeler then selects a particular solution from the relationship between the calibration criterion (a function of the residual errors) and the plausibility criterion (logical bounds upon the parameters). Yeh [1975] compared various identification methods, including linear programing, for a simple one-dimensional problem. The estimation of parameters for groundwater hydraulic systems has turned away from linear programing and toward statistical methods. Development and discussion of such statistical approaches may be found in the work of Cooley [1977, 1982] and Neuman and Yakowitz [1979]. The groundwater pollutant source identification problem is formulated as an optimization model. The optimization model incorporates a simulation (response) model of groundwater solute transport as a series of constraints. The concentration response matrix technique was used [Gorelick and Remson, 1982]. The objective of the optimization model is to select that set of simulated potential sources which results in simulated concentrations representing the closest match with local groundwater solute concentration data. The optimization methods of linear programing or multiple linear regression are used in conjunction with simulation to solve the identification problem. The performances of the two methods are compared. There are three significant differences between the two optimization techniques employed in this work. First, the linear programing formulation identifies pollutant source locations and magnitudes by minimizing the sum of the absolute values of the differences between simulated and measured concentrations. Linear programing is used as a tool for least absolute error estimation [Taylor, 1974; Bassett and Koenker, 1978]. The multiple regression model minimizes the sum of squares (least squares) of the above differences [Draper and Smith, 1966]. Second, the linear programing model restricts all pollutant sources to positive 779

2 780 GORELICK ET AL.' IDENTIFYING GROUNDWATER POLLUTANT SOURCES vj I I I I 1 "- CONSTANT HEAD VALUES ' i // IMPERMEABLE. S3 s],,,i,v v, v,, s sb BOUNDARY- S4 S8 VIII 48.B B 47.8 CONSTANT HEAD VALUES I I I I I 2 -' m Water Quality Observation Well Fig. ]. Hypothetical system for steady state pollutant source identification problem. Potential leak locations are indicated by I through IX. Actual leak sites are III and VIII. Hexagonal symbols indicate four observation well sites. S1 through S4 represent chloride sampling locations; S5 through S8 represent tritium sampling locations. Each grid cell is 20 m by 20 m. Head boundaries at the top and bottom are given in meters. Zero flow boundaries exist on the left and right sides. values, while the regression model has no sign restrictions. Third, control over the number of sources that may be considered as 'likely candidates' was restricted in the case of linear programing by utilizing mixed integer programing. This is in contrast to using a stepwise procedure for the multiple regression models which may in fact give a solution that is nonoptimal. Two illustrative examples apply linear optimization and numerical simulation to problems of identifying unknown pollutant sources. First, an example of steady state solute transport shows the feasibility of locating unknown pollutant sources using concentration data collected at a few locations. In this case data are sparse and spatially distributed. Second, a transient solute transport case shows how concentration histories can be used to identify the source schedules for a complex two-dimensional system in which several wells are responsible for the observed pollution. In this case, abundant data are collected at a few locations over time. The influence of an extended period of missing data is explored. PROBLEM I: STEADY ix s2 STATE CASE Consider the following system in which groundwater pollutant sources are to be identified. A subsurface pipe lies in an unsaturated zone above a water table and carries effluent from one end to the other. Figure 1 shows the pipe as well as the groundwater head boundaries at the top and bottom of the figure. The effluent has been flowing continuously within the pipe and contains high concentrations of chloride and tritium. Several wells, also shown in Figure 1, detect both chloride and tritium. This indicates that a small leak or series of leaks in the pipe has gone undetected and pollution of the water table aquifer has resulted. The relatively small volume of effluent that has leaked into the aquifer has left the original, steady groundwater flow pattern 'unchanged.' Flow from the leak(s) through the unsaturated zone is assumed to be vertical to the water table. Both species are assumed to have no chemical interaction with the media: chloride behaves conservatively and tritium obeys first-order decay. Furthermore, it is assumed that the concentration distributions are at steady state. It is necessary to locate the leak(s) and determine the magnitude of the solute and water flux from each leak. Preliminary simulations were used to create the concentration 'data' used in this problem. First, a model was constructed to simulate a steady head distribution [Trescott et al., 1976]. The boundaries at the top and bottom were fixed at the hydraulic head values shown in Figure 1. Zero flow boundaries exist along the left and right sides. For this model a constant hydraulic conductivity of m/d was used. The aquifer is 10 m thick. Second, a steady state solute transport model using finite differences was constructed in which two sources were placed at the locations III and VIII, shown in Figure 1. The pipe contained water with 15,000 mg/1 chloride and 15,000 Ixci/1 tritium. These values were selected arbitrarily and need not have the same numerical value. Leak one (at location III) had a solute flux of 90.0 mg/s chloride (90.0/xci/s tritium), and leak two (at location VIII) had a flux of 45.0 mg/s chloride (45.0 /xci/s tritium). Given the solute concentrations in the pipe and the solute fluxes at the leaks, the water flux from the pipe may be calculated. For leak one the water flux is /d (90 mg/s divided by 15,000 mg/1 times 86,400 s/d), and for leak two it is /d. Both leaks combined represent less than 1% of the flow through the system. The longitudinal dispersivity was 40 m and the transverse dispersivity 20 m. The effective porosity was 0.3. Initially, the data were taken to be the simulated concentrations at the measurement locations and appear in Table 1. The effects of perturbing the data are considered later. Governing Equations The simulation models employed in the steady state case are based upon the steady state equations of groundwater flow and solute transport. The equation of groundwater flow in saturated media for the steady, two-dimensional, heterogeneous, anisotropic case is [Cooper, 1966; Pinder and Bredehoeft, 1968; Remson et al., 1971] where Xi, Xj O i t O. = W i, j = 1, 2 (1) transmissivity tensor, m2/d; hydraulic head, m; volume flux (source/sink) per unit area, m/d; cartesian coordinates, m. The advective-dispersive equation [Reddell and Sunada, 1970; Bear, 1972; Bredehoeft and Pinder, 1973; Konikow and Grove, 1977] is used for the general linear case of steady state, two-dimensional transport with linear decay of a single

3 GORELICK ET AL.' IDENTIFYING GROUNDWATER POLLUTANT SOURCES 781 TABLE 1. Concentration Data Indicating a Leaky Pipe Measurement Chemical Observed Number Constituent Concentrations 1 chloride 763 mg/l 2 chloride 634 mg/l 3 chloride 705 mg/l 4 chloride 663 mg/l 5 tritium 390/xci/l 6 tritium 342/xci/l 7 tritium 299/xci/l 8 tritium 284/xci/l dissolved chemical constituent in saturated porous media. It is where D 0. - (CVi) - )kc - Oxi c]xi ebb i, j = l, 2 (2) C concentration of the dissolved chemical species, mg/1 or/xci/l; au dispersion tensor, me/d; average pore water velocity in the direction of xi, m/d; b saturated aquifer thickness, m; b effective aquifer porosity, dimensionless; C' solute concentration in a fluid source or sink, mg/l or/xci/1; W volume flux (source) per unit area, m/d; first order kinetic decay rate, l/d; Xi, Xj cartesian coordinates, m. Source Identification Model A source identification model was constructed which selects pipe leak locations and leak flux magnitudes that minimize the absolute differences (or squared differences) between simulated and observed concentrations in the system of interest. The model was first formulated as a linear program and then as a multiple regression problem. In each case constraints were included that simulated solute trans- port. A concentration response matrix was developed to accomplish this. This matrix is used to describe the simulated concentrations that would result at the measurement sites as a function of the combined effects of effluent leaks along the pipe. With this information the optimization model seeks to locate those pollutant sources (leaks) and their corresponding leak magnitudes which bring the simulated concentrations and measured concentrations into agreement. Before presenting the identification model, some definitions must be presented. An ordinary residual is defined here to be the difference between a simulated concentration and a measured concentration. For a simulated concentration at a given time and location, Cs, and a corresponding observed concentration, Cobs, the ordinary residual r is Cs - Cobs = r (3) A normalized residual is defined to be the difference between a simulated concentration and an observed concentration, expressed as a frhction, where c* is a normalization divisor. The value of c* must be an estimate of the true concentration for the purpose of normalization. For example, c* could equal Cobs or some smoothed or estimated value of Cobs. This definition is appropriate for systems in which errors are proportional to the magnitude of the data. Cs -- Cob s = r,(4) C* The terms absolute residual and absolute normalized residu- al will refer to the absolute values of (3) and (4). These definitions are needed because the objective of the identification model is to minimize the sum of a function of the residuals. In all cases, if the residual equals zero for a given simulated-observed concentration pair, then perfect agreement exists between the two values. It should be noted that use of the ordinary residual introduces a bias in favor of pairs with higher numerical values. For example, suppose the difference (ordinary residual) between a simulated-observed concentration pair is 50 mg/1. If the simulated concentration is 1000 mg/1 and an observed concentration is 950 mg/1, the normalized residual using c* as 950 mg/1 is On the other hand, if the mismatch is between a simulated value of 150 mg/1 and an observed value of 100 mg/l, the normalized residual is using c* as 100 mg/l is 0.5. A model attempting to minimize the sum of absolute concentration differences (ordinary residuals) would have no preference between the two pairs. It is clear, however, that the large relative difference in the second pair (a normalized residual of 0.5) should not be disregarded in many cases, particularly when errors are proportional to concentration values. A normalized residual emphasizes the relative mismatch in each simulated-observed concentration pair. Normalized residuals were employed in this work. The influence of employing ordinary residuals upon the model results will be discussed later in this paper. The identification model seeks to minimize the sum of the absolute normalized residuals (or their squares) in order to determine the pipe leak locations and leak magnitudes. For the linear programing model the objective is Min z = {et}{ r,} (5) where { r,) is a vector containing the absolute values of the normalized residuals r, corresponding to the ith simulatedobserved concentration pair and {et) is a row vector of ones (1,'-., 1). In linear programing absolute values can be expressed as the difference between two components, each of which is a nonnegative variable. The normalized residual becomes rni-- Ll i -- U i (6) Minimizing the sum of absolute residual components insures that at most one value (either ui or vi) will be in the solution for each pair because this will always result in an improved objective value. The identification model is formulated as the following linear programing problem: subject to Min Z = {et}{u} + (7) [Rl{q}- [/]{u} + [/]{v} = {e} (8) {q} -> {0} (9)

4 782 GORELICK ET AL.: IDENTIFYING GROUNDWATER POLLUTANT SOURCES where {u} -> {0} {v} -> {0} [R] rectangular matrix of m constraints, n unknowns, and containing the concentration response information for both chloride and tritium; (q) vector of unknown source flux magnitudes, q forj = (1, ß ß ß, n); n is the number of potential sources; [/] m-dimensional identity matrix; (u) vector of positive normalized residuals ui for i = (1, ß ß ß, m); m is the number of measurements; (v) vector of negative normalized residuals vi for i = (1,...,m); {e t} row vector of ones; {e} right-hand side vector which is the vector (1, ß ß ß, 1) and reflects normalized concentration data. Similarly, the identification problem may be formulated as the following problem in multiple regression: subject to Min Z' = {et}{rn 2} (12) [R]{q}- [/]{r,} = {e} (13) where {r,} is a vector of normalized residuals. It is important to note that in this case data are so sparse that the multiple regression model does not have enough degrees of freedom to obtain a value for every potential leak site. There is one more unknown than there are constraini ng equations. Therefore, at least one regression coefficient q; must be set to zero in order to obtain a solution. The concentration response matrix [R] was developed in the following manner. Each of the nine finite difference cells in which the pipe is located was considered a potential leak location. For each cell, unit leaks were simulated for both chloride and tritium (18 simulation runs). The concentrations resulting at the measurement sites from each simulation show the response if a leak were to have occurred at only one location. The concentration at any location is a linear function of the unit leaks. Each row of the concentration response matrix is composed of concentrations corresponding to a particular measurement site and which result from a unit leak at each location. Finally, each row is divided by the measured concentration corresponding to that row. This is done to normalize the concentrations in accordance with the definition of the normalized residual. Each optimization model superposes the unit sources in such a way as to minimize the sum of the normalized residuals. The result is identification of those leak locations and leak magnitudes which result in the best match between observed and simulated concentrations at the measurement sites. Initial Results The source identification model was constructed for the sample problem. The MPS/III [Ketron Inc., 1979] was used to solve the linear program. The multiple regression problem was solved using the general linear modeling procedure available on the SAS package [Statistical Analysis Systems (SAS) Institute, Inc., 1979]. Results are presented in Table 2. The source fluxes are expressed as water volume lost from the pipe per day. The leaks were located by the models. The ' TABLE 2. Leak Locations and Magnitudes Predicted Using Linear Programing and Multiple Regression Identification Models Predicted Leak Magnitudes Potential True Leak Regression Source Magnitudes, LP Results, Results, Locations l/d l/d l/d I II III IV V VI VII VIII IX Total flux total residual was brought to in the linear programing model. It may be noted that the linear programing solution did establish small sources at other locations. These spurious small sources appearing in the solution are due to roundoff error. Because there are too few degrees of freedom to solve the multiple regression problem, the coefficient for a leak at site 9 was fixed at zero. Therefore the multiple regression results in Table 2 are predicated upon setting the leak at site 9 to zero. Other solutions to the normal equations are possible, depending upon which site is assigned a zero leak magnitude. In the particular solution shown in Table 2, sites which were not in fact sources were assigned very small flux magnitudes. In this case, alternate solutions in which one of these sites had been set to zero would not differ substantially. The two major sources were identified with values that were within 1% of the correct values. However, the lack of sign restrictions on the variables enabled 'negative sources' to be established at other sites. For the linear programing model the objective function of minimizing the total of the absolute normalized residuals is satisfactory only if the measured data are quite accurate. This is because the only means that the model has of introducing one pollutant leak into the solution is to set one residual equal to zero. If the data contain measurement errors, it may be impossible for the proper number of residuals, corresponding to the true number of sources, to go to zero. This is supported by the following analysis. We have a system of m constraints and n unknowns, where n is greater than m. There can be at most m basic variables in any linear programing solution. If p sources are responsible for the pollution, then the model can allow these p sources, as well as m minus p nonzero residuals. The remaining p residuals will be nonbasic and therefore set to zero. There must be a perfect match between simulated and observed concentrations for p measurements. Similarly, there is a lack of degrees of freedom in the multiple regression model. There can be no more than m = 8 sources identified in the solution. Yet there are nine potential sources. Although only two true sources exist and therefore only two regression coefficients must be nonzero, errors in the data will tend to force all variables into the solution. Then the solution can become extremely sensitive to inaccuracies in the data. This problem is compounded by the fact that the regression model does not restrict the signs on the variables and large spurious negative values are possible.

5 GORELICK ET AL.: IDENTIFYING GROUNDWATER POLLUTANT SOURCES 783 To illustrate these problems, very small errors were introduced into the data and the models were rerun. The perturbed data are in Table 3. The errors were selected for illustrative purposes and were bracketed within 5% of the original data. Furthermore, all data were assumed to be in error. The results of the identification model are in Table 4. The two sources are not identified accurately. Using the linear programing formulation, the principal source is identified but the leak magnitude is underestimated by 41%. The second source is displaced to an adjacent location and its magnitude is overestimated by 15%. The total flux from the pipe was overestimated by 13%. Finally, other significant sources appear in the solution. This particular multiple regression model gave wildly incorrect results. It identified sources at incorrect locations with extremely large magnitudes and gave three large negative sources. It is clear that the solution is sensitive to slight errors in the data. Additional constraints must be placed upon the solution using either of the two optimization techniques. This must be done to restrict the number of unknowns that can be considered in any particular model solution. Reformulation One possible approach is to reformulate the problem. Under certain conditions a leak may be an unlikely event. Several leaks would be even more improbable. Therefore one might wish to identify the smallest number of sources which explain the measured concentrations within a specified tolerance. Minimizing the sum of the absolute normalized residuals would be a secondary (subordinate) objective. This may be formulated as the following linear program: subject to Min Z = rt{et}{p} + e{et}{,} + e{et}{v} (14) [Rl{q}- [/]{u} + [/]{v} = {e} (15) [/]{q}- ILl(p} -< 0 (16) {u) < {u*) {v} -< {v*} (18) {q) -> 0 (19) {u} > 0 (20) {v} 0 (21) TABLE 3. Perturbed Concentration Data for Steady State Identification Problem Mea- Chemi- Concensure- cal tration ment Constit- Data With Number uent Errors Percent Error 1 chloride chloride chloride chloride tritium tritium tritium tritium [L] {u*} {v*} diagonal matrix whose entries are very large values, larger than any value in {q}; the value 1000 was used in this problem; error tolerance associated with the positive residual; error tolerance associated with the negative residual. This is a mixed integer problem which is similar in form to a fixed charge problem. The primary objective is to minimize the number of sources and the secondary objective is to minimize the sum of the absolute normalized residuals. The constraints specify that for each leak entered into the solution a large penalty is to be assessed in the objective function. Furthermore, the residuals are bracketed to be within specified tolerances. Stepwise multiple regression may be used in an analogous formulation. The regression model determines the best single candidate leak, the best two candidate leaks, and so on. The results of each of these models may then be compared. Use of the stepwise procedure overcomes the problem of too few degrees of freedom by considering only a few likely leak locations. Error brackets, like (17) and (18), are not included for any simulation-measurement pair when using stepwise regression. Results Using Revised Models The sample problem with perturbed data (Table 3) was used to test the revised models. For the linear programing model, two runs were attempted using the mixed-integer where {e} E [R] {q} [/] Pi-- 0 or 1 i = (1,- ß ß, n) (22) vector of leak location indices containing elements Pi for i = (1, ß ß ß, n), where the values of Pi are restricted to 0 or 1 for n potential sources; large scalar multiplier associated with the primary objective; the value of 100 was used in this problem; vector of positive residuals; vector of negative residuals; fight-hand side vector which is the vector (1, ß ß ß, 1); small scalar multiplier associated with the secondary objective; a value of 1.0 was used in the sample problem; concentration response matrix; vector of unknown leak flux magnitudes; identity matrix; TABLE 4. Predicted Leak Locations and Magnitudes Using Perturbed Data Predicted Leak Magnitudes Potential True Leak Regression Source Magnitudes, LP Results, Results, Locations l/d l/d l/d I II III IV V VI VII VIII IX Total Flux

6 , 784 GORELICK ET AL.: IDENTIFYING GROUNDWATER POLLUTANT SOURCES TABLE 5. Predicted Leak Locations and Magnitudes Using Revised Models Potential Source Locations True Leak Magnitudes, l/d Mixed Integer Model Results Predicted Leak Magnitudes Contin- One Two Three Run 1 Run 2 uous Leak Leaks Leaks Stepwise Regression Model Results Four Leaks Five Leaks I II III IV V VI VII VIII IX Total Flux programing code MISTIC [Ketron Inc., 1979]. In the first, the tolerances which bracket the residuals were established at Therefore the model must find the smallest number of leaks that minimize the sum of absolute residuals, and no simulated-measured concentration pair may differ by more than 5%. The second run was identical to the first however one error bracket was narrowed. The simulated concentra- tion corresponding to measurement number '2' (see Table 3) was constrained to be within 3% of the measured value. All other tolerances were left at 5%. For the stepwise multiple regression model, one set of runs was executed for solutions corresponding to a range of 1 to 9 leaks. The R squared maximizing procedure available on SAS [SAS Institute, Inc., 1979] was used. The results using the revised models appear in Table 5. Also in Table 5 are the results of the 'continuous solution' to the mixed integer formulation. The continuous solution is the optimal solution to the mixed integer problem without enforcing the integer (0 or l) constraints. The results indicate that the revised formulation can be successful if the error brackets on the data are suitably restrictive. In the first run two leaks (the correct number) were identified and the leak magnitudes were close to the true values. Unfortunately, the smaller of the two leaks is displaced one grid cell. The minimal sum of residuals associated with the solution (0.192) was in fact less than the residual that would have resulted had the 'true' leak locations and leak magnitudes been identified (0.239). The second run, in which the error brackets for measurement 2 were reduced from 5% to 3%, lead to both the correct two locations and flux magnitudes within 2% of the true values. Finally, one might take note of the results of the continuous solution to the revised formulation. While this solution was not successful in identifying the flux magnitudes, the two leak locations correspond to the two largest values for the leaks found by the model. The continuous solution does not involve integer programing and is far simpler to obtain. In this case it roughly pointed to the true source locations. Experimentation was inadequate to determine whether this will generally be the case. The stepwise regression model gave rise to nine solutions which must be compared. Table 5 shows five of the solutions. The true leak pattern was determined by the two-leak model. In this case comparison of statistics is of very limited value because of the small number of concentration data. However, the R squared criterion for the one-variable model was and increased immediately to for the twovariable model, indicating a substantial improvement in model performance. The three-variable model identified the two correct sources but showed one small negative source. Successive models showed large erroneous values, misidentitled the true sources, and contained very large negative values. PROBLEM II: TRANSIENT CASE The combined optimization-simulation identification model may be applied to cases involving transient pollutant migration in which pollutant sources are identified from concentration data collected over time. The transient problems differ from steady state cases in two important ways. First, pollutant sources may occur over both space and time. This means that the time period during which groundwater pollution occurred as well as the source location must be identified. Second, because concentration measurements are collected over time at various locations, there is not a serious problem of data sparsity. Even if some data records are missing for certain periods at certain water quality observation sites, solute concentration data are not sparse when compared to the steady state case. For the transient case the matrix system is one in which the number of constraints (548 measurements) is much greater than the number of unknowns (20 disposal events). We again compare the two identification methods which utilize linear programing and multiple regression respectively. We apply the methods to a hypothetical physical system in which five waste disposal wells may have operated during any of 4 years. Figure 2 shows the hypothetical groundwater system to which the methods were applied. This is the same system used by Gorelick [ 1982]. It is an irregular aquifer with impermeable boundaries on the left and right and nonuniform steady flow from top to bottom of the figure. The system parameters are summarized in the figure caption. The pond recharges the aquifer with fresh water and accounts for about 19% of the flow through the system. The initial concentration of the conservative solute in the pond and in the aquifer is zero. In addition to the five disposal sites, there are three water quality observation sites at which samples have been collected and analyzed over time. The concentration response matrix method employing the

7 GORELICK ET AL.' IDENTIFYING GROUNDWATER POLLUTANT SOURCES 785 Y of disposal events and the magnitude of disposal fluxes were selected so as to enhance pollutant plume interference. This was done to provide a stringent test of the source identification models. It is important to note that upon visual inspection of the concentration histories in Figure 3 one might conclude that two or three disposal events were responsible for the observed pollution. In fact there were 12 disposal events. As an initial test of the identification models, cases were run using exact data, i.e., data with no measurement errors. Both the linear programing and the multiple regression models properly identified the 12 disposal events. These results demonstrated that neither method suffers from severe numerical problems in the transient case. The exact data were then perturbed to simulate the effects of measurement errors. Each perturbed datum was assumed to be sampled from a normal distribution having the exact datum as its mean, and a standard deviation equal to one tenth the magnitude of the datum. This is well within the range of precision of chloride analysis techniques [Thurnau, 1978; or Greenberg eta!., 1981]. This amounts to assuming that all data were analyzed with the same relative precision; larger concentration values would have greater concentration measurement errors. Figure 3 shows the exact and I I ß Potential sources 91.4 m ß Observation sites Fig. 2. Hypothetical groundwater system for the transient pollutant source identification problem. Disposal sites are numbered 1 through 5. Observation well sites are labeled A, B, C. Total flow through the system is 24.2 l/s, including 4.3 l/s of recharge from the pond. initial solute concentration is zero. Hydraulic conductivity is 0.01 cm/s. Aquifer thickness is 30.5 m. Effective porosity is 0.2. Longitudinal and transverse dispersivities are 7.6 and 2.3 m, respectively n,- I-- Z '" 500 z o -- EXACT DATA n SITE A n PERTURBED solute transport simulation model of Konikow and Bredehoeft [1978] was utilized. That method was explained by Gorelick [1982] and will not be repeated here. The equation governing steady groundwater flow has already been presented (see (1)). In this case, instead of (2), the equation of transient solute transport was employed. This equation is OC at O( O- x ) 0 C'W = OX -- i Dij -- OX (CVi) hc i 6ooo n [] SITE B i,j= 1,2 (23) where t is time. It is assumed that waste disposal in the subsurface does not materially affect the groundwater hydraulic head distribution. Therefore changes in the groundwater velocity field induced by such disposal can be ignored. Under this assumption, the product of the liquid volume disposal rate and the solute concentration of the waste may be treated as a single value called the 'disposal flux.' The disposal fluxes, having units of grams per seconds, are identified over four annual periods at the five disposal facilities. The concentration histories (data) at observation sites A, B, and C were developed using simulation, linear superposition, and data perturbation. Twelve 1-yr disposal events gave rise to the observed concentration histories. The timing Fig [] SITE C I YEARS Pollutant concentration data for the transient problem.

8 , 786 GORELICK ET AL.' IDENTIFYING GROUNDWATER POLLUTANT SOURCES perturbed concentration data at the three observation sites. The errors in some cases are as large as 23% of the true data values. Formulation The formulation of transient pollutant source identification models using simulation combined with linear programing or multiple regression are structured identically to those presented for the steady state case (see (7) through (13)). However, the definitions of the decision variables (disposal fluxes), the concentration response matrix, and the righthand side vector are somewhat different. These changes are needed because the problem is transient and uses smoothed concentration values for the values of c* in the normalized residuals (4). The new definitions are {q} [R] {c'} annual disposal fluxes at the five disposal sites during each of 4 years; concentration response matrix representing the normalized concentration histories at the water quality monitoring sites; normalized concentration histories at the water quality monitoring sites; this equals {Cobs/C* } and replaces the right-hand side vector {e} in (13) and (15). Figure 4 shows the perturbed and smoothed concentration data at the observation wells. The data were smoothed using a repeated sequence of running means, running medians, Hanning weighted running means, and residual smoothing and recombination. The smoothing methods are based upon those described by Tukey [1977] and the particular smoothing routine was developed by Chaffee [1980]. For the purpose of normalization the smoothed values provide an adequate estimate of the true concentrations shown in Figure 3. The ratios of perturbed values to smoothed values have a mean of and a standard deviation of for the 548 measurements. These compare well with a mean of 1.0 and a standard deviation of 0.1 for the ratios of perturbed to true concentrations. Given our definition of normalized residual (4), bias in the mean for c* must be small. This bias is less than 1% using the smoothed values. Results values appeared in the solutions, but these are conspicuous artifacts and may be disregarded. All true source values were contained within the 95% confidence intervals. The small erroneous values had relatively large intervals which contained zero. Figure 5 shows a plot of the R squared criterion for the 20 successive solutions returned by the stepwise procedure. The R squared value reaches a sill after seven sources (independent variables) have been added. This indicates that early choices of sources have considerable impact upon the model's ability to account for observations and these are the most reliable choices. This is consistent with the results using linear programing, in which eight annual pollutant events were described and the remainder, while properly located, contained large errors in magnitude. The presence of a sill means that the R squared statistic gives no information about the true number of sources but does point to a lower bound on the number of sources as the sill is reached. Confidence intervals can be constructed about the estimated flux values returned by the linear programing model as well as the multiple regression model. For the least squares case, standard errors are given as square roots of the SMOOTHED DATA o PERTURBED DATA [] SITE A -? [31 - SITE B Table 6 shows the results of the source identification models using either linear programing or stepwise multiple regression. The linear programing problem was solved using MINOS [Murtagh and Saunders, 1980], and the stepwise regression was solved using the R squared maximizing routine of SAS [SAS Institute, Inc., 1979]. It was not necessary to include integer constraints in the linear program, as demonstrated by the model performance. The linear programing model located all the true sources and gave disposal flux values of zero where there were no sources. It is apparent that the incorporation of abundant data (548 measurements) in this transient problem is effectively a replacement for the integer constraints required under the sparse data conditions of the steady state case using linear programing. The determination of disposal flux values was accurate to within 10% for eight of the sources; the remaining four values, while properly located, had magnitudes that were determined with errors ranging from 11 to 32%. The stepwise regression was also successful in identifying all of the correct sources. Some small positive'and negative o I 0 SITE C YEARS Fig. 4. Pollutant concentration data for transient problem using smoothed data.

9 GORELICK ET AL.' IDENTIFYING GROUNDWATER POLLUTANT SOURCES 787 TABLE 6. Disposal Fluxes Predicted Under Perturbed Concentration Data by Linear Programming and Stepwise Regression Models Year at Site 1 Year at Site 2 Year at Site 3 Year at Site 4 Year at Site I I I I Actual rates ll LP results 51 Stepwise results 1 variable 2 variables 52 4 variables 53 6 variables 52 8 variables variables variables variables variables variables % confidence intervals, +/- All values are in grams per second ll II II ll l0 ll l ll.l ll 12 ll 13 II 12 elements along the diagonal of the covariance matrix o2([rt][r]) - (24) where 02 is the mean squared error and [R t] is the transpose of the concentration response matrix [R]. For the model employing least absolute errors, the asymptotic covariance matrix is [Bassett and Koenker, 1978], 1 4[f(0)]2 ([Rt][R]) -I (25) where f(0) is the value of the density at the median. Therefore standard errors and confidence intervals for the linear programing case can be constructed using (25) if nonnegativity constraints are ignored. Having obtained confidence intervals for the least squares case, the corresponding values for the least absolute error case can be calculated simply. Confidence intervals using least absolute error estimation are the same as those for least squares estimation times the factor 1/(2o-f(0)). For the solution displayed in Table 6 this factor is In our case, standard errors for least absolute error estimation are greater than those for least squares estimation because least squares estimation is optimal when errors follow a Gaussian distribution. Given the perturbed data in Figure 3, one might turn toward data smoothing to reduce the scatter due to measurement error. This is justified as long as the physical phenomena underlying the data ought to produce smooth responses. The influence of data smoothing upon the performance of the transient pollutant source identification models was inspected. It should be noted that smoothing violates the statistical assumptions of the regression model by potentially inducing unknown bias, artificially removing variance, and inducing serial correlation among the data. While the regression model statistics will therefore be meaningless, the use of the model to identify the best linear least squares regression coefficients remains valid. Table 7 summarizes results of the stepwise regression and the linear programing identification models when smoothed data were used. Both models behaved similarly under both original and smoothed data. The linear programing solution indicated some small erroneous sources, but they were almost two orders of magnitude less than the true source values and could be ignored. There was a notable improvement in prediction of the largest source values occurring at site 3 during years 1 and 2 under smoothed data. There was a poorer prediction at site 3 during year 3. The stepwise regression results differed only slightly from the case with unsmoothed data. In general, smoothing did not change the results. The fact that the solutions were insensitive to alterations in the data due to smoothing provides a promising inference that missing data could be handled by interpolation. While not tested here, it is likely that the pollutant source identification models would be insensitive to errors in such interpolated values. In field problems it is possible that monitoring of water quality may be intermittent. Monitoring may not begin until after a substantial water quality problem has arisen. We investigated this matter by eliminating the first 5 years of data from the concentration histories at the three water quality observation sites. The results of the multiple regression model and the linear programing model using perturbed data appear in Table 8. The linear programing solution correctly located 11 of the true sources but omitted one true source and introduced two spurious sources. The interval of missing data includes those months when the effects of injection at site 1 during year 1 R I i I I I I i I i I i i I NUMBER OF VARIABLES Fig. 5. R squared values as a function of the number of sources (variables) included in the stepwise multiple regression solution. The model used normalized residuals.

10 788 GORELICK ET AL.' IDENTIFYING GROUNDWATER POLLUTANT SOURCES TABLE 7. Disposal Fluxes Predicted Under Smoothed, Perturbed Concentration Data by Linear Programing and Stepwise Regression Models Year at Site 1 Year at Site 2 Year at Site 3 Year at Site 4 Year at Site Actual rates LP results Stepwise results 1 variable 2 variables 3 variables 4 variables 6 variables 8 variables 10 variables 12 variables 14 variables 17 variables 20 variables O All values in grams per second. are most pronounced. Without these data the model cannot identify it. The most serious problem in the missing data case is the large spurious source entered at site 5 during year 1, coupled with an underestimate of injection at site 5 during year 3. Apparently some low concentration values were misinterpreted as the trailing edge of a large source. When ordinary residuals were used instead of normalized residuals for this case of missing data, the source at site 5 during year 1 disappeared. This indicated that low concentration values were the cause of the problem. The stepwise regression solution identified many of the correct sources but gave some large negative sources. During the early addition of variables, all sources consistently appearing in the solutions were correctly located. In later models, variables were added in excess of the true number of active sources, and negative sources were a serious problem. Large spurious sources were added in pairs, with a large negative source apparently offsetting the effects of a large, spurious positive source. The very large confidence intervals for these paired spurious sources helps to identify them as such. The paired appearance of these spurious sources suggested that constraining the regression to eliminate negative sources would also suppress large, spurious positive sources. The regression was rerun to eliminate negative sources. After eliminating negative values during two successive runs, all remaining sources were positive and all sources whose 95% confidence intervals did not contain zero were true sources. These revised results appear at the bottom of Table 8. Effect of Normalization In this section we investigate the effects of using ordinary residuals rather than normalized residuals in the source identification models. Use of ordinary residuals would be TABLE 8. Disposal Fluxes Predicted Under Perturbed Concentration Data by Linear Programing and Stepwise Regression Models When Data for the First Five Years are Missing Year at Site 1 Year at Site 2 Year at Site 3 Year at Site 4 Year at Site I I Actual rates LP results Stepwise results 1 variable 2 variables 4 variables 6 variables 8 variables 10 variables 12 variables 14 variables 17 variables 20 variables 95% confidence intervals +/- Revised multiple regression 95% confidence intervals +/ * * * * * * * ll All values are in grams per second. *Indicates variables with negative values eliminated from regression model during successive model runs.

11 GORELICK ET AL.' IDENTIFYING GROUNDWATER POLLUTANT SOURCES 789 TABLE 9. Disposal Fluxes Predicted Under Perturbed Concentration Data Using Ordinary Residuals Year at Site I Year at Site 2 Year at Site 3 Year at Site 4 Year at Site 5 I I I I I Actual rates LP results 51 Stepwise results I variable 2 variables 51 4 variables 51 6 variables 51 8 variables variables variables variables variables variables 51 95% confidence intervals +/- I All values are in grams per second I I l appropriate for cases in which one is not willing to assume that errors are proportional to the magnitude of the values. Here we evaluate the effect of using ordinary residuals when in fact errors are proportional to the magnitude of the pollutant concentrations. The transient case with perturbed data was solved using ordinary residuals. The results are presented in Table 9. The linear programing solution located all of the true sources, adding two small obviously spurious sources. The determination of disposal flux values was accurate for about five of the sources. The magnitudes of the remaining seven values were poorly determined. The stepwise regression added successive sources until negative sources began to appear. The presence of these negative sources diminishes the credibility of the results. The magnitudes of the true sources are overestimated to compensate for negative sources, and the negative sources force the exclusion of a true source. Figure 6 shows the progression of R squared for the stepwise regression using ordinary residuals. The sill is reached after five or six sources have been added. This is again consistent with the linear programing results where only five sources were properly identified. The 95% confidence intervals are much greater for this model, and one true source (site 4 during year 3) is not strictly positive over its entire range. In this case, confident determination of the true sources cannot be made from inspection of the regression coefficients and their standard errors. These results indicate that the linear programing model identifies pollutant sources better than the multiple regression model when ordinary residuals are used instead of normalized residuals. There are two central reasons for this. First, incorporating nonnegativity constraints in the linear programing model gives prior information that is not provided to the regression model. In this sense, comparison with a constrained regression model would have been better. However, constrained regression problems are nonlinear, and consequently stepwise procedures do not exist. Second, least absolute error estimation is more robust than least squares estimation [Bassett and Koenker, 1978]. Large individual residuals can dominate the solution when their values are squared. It should be noted that the use of ordinary residuals tends to weight the peak concentration values and effectively to disregard the limb and tail values. Minimizing absolute values places more emphasis on the off-peak values than does least squares estimation. SUMMARY AND DISCUSSION This study has inspected two techniques, linear programing and multiple regression, which when combined with numerical simulation of linear solute transport were used as tools for identifying unknown sources of groundwater pollution. Two different cases were addressed. The first was steady state transport in which tracer pollutants emanated from a leaking pipe system just above the water table. The problem here was to isolate the likely leak locations and the magnitudes of each leak. Concentration data were considered to be extremely sparse, and all measured values were considered to be somewhat in error. The second was a transient case in which waste disposal at several facilities was responsible for pollution observed over time at a few locations. The problem was to identify those sites, those time periods, and those disposal flux magnitudes which caused the observed pollution. In this case, concentration histories provided a substantial quantity of data. However, they contained large random measurement errors. Pollutant source identification models were developed that minimized various functions of differences between 1.0 R ø' o g 1 o 1 g 2o N MBE OF VARIABLES Fig. 6. R squared values as a function of the number of sources (variables) included in the stepwise multiple regression solution. The model used ordinary residuals.

12 ß 790 GORELICK ET AL.: IDENTIFYING GROUNDWATER POLLUTANT SOURCES measured and simulated concentrations. All identification models contained simulation models of groundwater solute transport as constraints. For the steady state case, in which data were sparse and there were more unknowns than constraining equations, an additional restriction was required to determine the fewest possible leaks that best explained the data. To accomplish this, a mixed integer programing model was developed, as was a stepwise multiple regression model. While both models were capable of identifying the pollutant sources, the mixed integer program required stringent error brackets about the data and the regression model displayed spurious negative values which detracted from the true solution. Stepwise models with more than three variables had large negative values and were quite sensitive to errors in the data. Incorporating nonnegativity constraints in the linear programing formulation provides significant prior information to the model. For the transient case the linear programing and regression models properly identified the pollutant sources and the disposal episodes but contained some errors in the determination of disposal flux magnitudes. The relative abundance of data over the steady state case permitted solutions without integer (fewest source) restrictions. The transient case models performed fairly well in light of erroneous data, data smoothing, and an extended episode of missing data. Estimation was comparable using either multiple regression or linear programing. Error analysis can be conducted using either technique. If errors follow a Gaussian distribution, then least squares is the preferred method. If errors follow other distributions, least absolute error estimation using linear programing may be preferable because it is more robust and restricts variables to positive values. One drawback of the linear programing model is that it required about 30 times as much CPU time as the stepwise multiple regression on an IBM 370/3081. This study assumed no uncertainty in the physical parameters of the aquifer. Field systems are never so well defined. Representing uncertainty in the physical parameters would involve perturbing the concentration response matrix, which appears as constraints in the identification models. It may also involve nonlinear optimization techniques. This study was a necessary first step in the larger task of developing pollutant source identification techniques that maintain their integrity under physical, as well as data measurement, uncertainty. Acknowledgments. We gratefully acknowledge Richard L. Cooley and Brent M. Troutman of the U.S. Geological Survey for their helpful suggestions and insights during the development of this work. We thank Paul Switzer of the Stanford Statistics Department for his most valuable advice. We thank David L. Freyberg of the Stanford Department of Civil Engineering and James R. Slack of the U.S. Geological Survey for their keen review comments. We also thank Ketron, Inc., Wes Winkler, and the Stanford Department of Operations Research for permitting use of the MPS/III for this research. This study was supported by the U.S. Geological Survey and National Science Foundation grant CME The use of computer and software brand names in this report is for identification purposes only and does not imply endorsement by the U.S. Geological Survey. REFERENCES Bassett, G., and R. Koenker, Asymptotic theory of least asolute error regression, J. Am. Stat. Assoc., 73(363), , Bear, J., Dynamics of Fluids in Porous Media, Elsevier, New York, Bredehoeft, J. D. and G. F. 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