Water Resources Research

Identifying sources of a conservative groundwater contaminant using backward probabilities conditioned on measured concentrations

Authors


Abstract

[1] If contamination is observed in an aquifer, a backward probability model can be used to obtain information about the former position, source location, or release time of the observed contamination. A backward location probability density function (PDF) describes the possible former positions of an observed contaminant particle at a specified time in the past and can be used to identify the location of the contaminant source. A backward travel time probability density function describes the possible travel times of an observed contaminant particle from a known upgradient location to the observation location and can be used to determine the time of release of contamination from the source. Neupauer and Wilson (1999, 2001, 2002) obtained backward PDFs that depend on the observation locations and sampling times, in addition to the transport properties of the aquifer and solute. We present an approach for conditioning these backward PDFs on measured concentrations. We focus on identifying the location or release time of an instantaneous point source of contamination. We derive the conditioning equations and demonstrate some important features of the technique through the use of a hypothetical example.

1. Introduction

[2] In groundwater systems, a source of contamination may not be known to exist until contamination is observed in downgradient wells or surface water bodies. After the contamination is observed, characteristics of the source can be reconstructed using any of a variety of methods that are based on the existing distribution of contamination and on knowledge of the flow and transport processes. As a contaminant moves through the aquifer, the dispersive processes cause a loss of information about the contaminant's past; therefore, a complete reconstruction of the source characteristics is not possible. The reconstruction is further complicated by errors in measured concentrations, and uncertainty and variability in the flow and transport parameters. In this paper, we focus on identifying the location or release time of an instantaneous point source of contamination. Because the source characteristics cannot be known exactly, we represent the source location and source release time as random variables.

[3] In recent years, the source identification problem has received much attention. One type of source identification problem that has been extensively studied is the reconstruction of the release history from a known source of contamination. This problem has been addressed using linear programming and function-fitting [Gorelick et al., 1983], maximum likelihood method [Wagner, 1992], Tikhonov regularization [Skaggs and Kabala, 1994, 1999; Liu and Ball, 1999; Neupauer et al., 2000], method of quasi-reversibility [Skaggs and Kabala, 1995], geostatistical approaches [Snodgrass and Kitanidis, 1997; Butera and Tanda, 2003; Michalak and Kitanidis, 2004], classical Bayesian methods [Woodbury and Ulrych, 1996, 1998; Woodbury et al., 1998; Neupauer et al., 2000; Michalak and Kitanidis, 2003], non-linear optimization [Alapati and Kabala, 2000; Mahar and Datta, 2000], marching-jury backward beam equation method [Atmadja and Bagtzoglou, 2001], and genetic algorithms [Aral et al., 2001]. In this type of problem, the location of the contamination source is known. This differs from the problem addressed in this paper in that our approach is used to identify the location of an unknown contaminant source.

[4] Another source identification problem is the identification of the location or release time of the source, and is the problem that is addressed in this paper. Dimov et al. [1996] used concentration measurements and marginal sensitivities of concentration to source mass to identify the location of a point source of contamination in a one-dimensional domain. For an instantaneous point source in a one-dimensional domain, the ratio of the marginal sensitivity to the measured concentration will be equal to the reciprocal of the source mass at (at most) two points, one of which will be the true source location. If a second sample is taken, the ratio of its marginal sensitivity to measured concentration will also be equal to the reciprocal of the source mass at two points, including the source location. Dimov et al. [1996] identified the true source location as the point at which both ratios are equal. This approach only works if the measured concentrations, the model parameters, and the conceptual model are all accurately represented, an unlikely situation in any groundwater modeling application. Neupauer and Wilson [1999, 2001, 2002] presented an approach for developing probability density functions (PDFs) of the random location or random release time of an instantaneous point source of contamination based on one or more sampling locations or times. These PDFs are related to the marginal sensitivity of concentration to source mass. To calculate these PDFs, the observation location is treated as an instantaneous point source of probability occurring at the time of sampling, and the probability density function is propagated upgradient and backward in time to identify possible former positions (spatial distribution of location PDF at a particular backward time) or possible travel (release) times (temporal distribution of travel time PDF at a particular upgradient location) of the observed particle. These backward probabilities take into account the observation location and time, but do not account for the sampled concentration; therefore, they are based more on the spatial and temporal distribution of the sampling network than on the actual distribution of the contaminant.

[5] Other source identification problems aim to identify both the source location and release history. Mahar and Datta [2000] used non-linear optimization to estimate the release concentrations of multiple hypothetical contaminant sources over discrete time intervals using breakthrough curve data at several downstream locations. In their work, the locations of two or three sources were known, and the optimization method was used to determine the source flux at each of these sources. The source fluxes were constant over discrete time intervals, and the durations of these time intervals were assumed to be known. For some of the sources, the true source flux was zero; therefore they also tested the capability of distinguishing between locations that released contamination and those that did not. They found that the estimated source fluxes differed from the actual source fluxes by approximately 10–30%, depending on the quantity of available breakthrough curve data (number of breakthrough curves and number of samples data points in each breakthrough curve). Aral et al. [2001] and Mahinthakumar and Sayeed [2005] used genetic algorithms to identify the location and release rate of a contamination source. Aral et al. [2001] used breakthrough curve data from four observation wells in a genetic algorithm to estimate the release history of contamination from a point source with an unknown, but constrained, location. Even with measurement errors in the breakthrough curve data of up to 12%, they were able to determine the source location to within approximately 3% of the travel distance between the true source location and the most distant observation well. Mahinthakumar and Sayeed [2005] used a genetic algorithm to determine the concentration and the location of an instantaneous, non-point source with uniform initial concentration. They used discrete breakthrough curve data at several downstream wells to optimize their solution. They evaluated several different hybrid genetic algorithm-local search methods and found that some hybrid methods accurately identified the source location and concentration to within 1% of the true values for the hypothetical examples they investigated.

[6] A final category of source identification problems is the identification of the historical distribution of a contaminant plume. Michalak and Kitanidis [2004] used measured concentration data in a geostatistical inverse method to identify the configuration of a contaminant plume at some time in the past. With accurate concentration measurements taken at a finite number of points in a contaminant plume, they were able to reconstruct the contaminant distribution at a desired time in the past. They did not identify a source location, but rather a representation of the contaminant plume, which may represent a plume that developed from an upgradient point or distributed source, or it may represent the distributed source itself if the source were instantaneous.

[7] In this paper, we extend the work of Neupauer and Wilson [1999, 2001, 2002] by developing a method for conditioning backward location and travel time probability density functions on measured concentrations. We show that conditioning on measured concentration significantly increases the accuracy and decreases the variance of the PDFs when model and measurement errors are small, leading to an improvement in the identification of contaminant source locations and the release time of contaminants from the source. This work differs from the approaches of Mahar and Datta [2000], Aral et al. [2001], Michalak and Kitanidis [2004], and Mahinthakumar and Sayeed [2005] in that we assume an instantaneous point source and identify the unknown source location or the release time of contaminant from a known source. At this time, our approach has only been developed for an instantaneous point source and is not capable of reproducing the historical contaminant distribution.

[8] In the next section, we present equations for the unconditioned location and travel time PDFs for one or more observations. In the subsequent sections, we develop equations for conditioning these PDFs on measured concentrations, and we demonstrate the conditioning approach using a hypothetical example.

2. Unconditioned Backward Probability Density Functions

[9] If contamination is observed in an aquifer, but the source is unknown, backward location and travel time probability density functions (PDFs) can be used to describe the former (or source) location or travel (or release) time of the contamination to help in identifying the source. The backward PDFs are related to the marginal sensitivity of concentration at the sampling location to a source mass at any random location. These marginal sensitivities are adjoint states of concentration and therefore can be obtained by solving the adjoint of a forward contaminant transport model [Neupauer and Wilson, 2001, 2002].

[10] One example of a governing equation of forward contaminant transport is the advection dispersion equation (ADE) for an instantaneous point source of contamination, given by

equation image

where C is resident concentration, Dij is the i,jth entry in the dispersion tensor, vi is the component of groundwater velocity in the xi-direction (v = (v1, v2, v3)), θ is porosity (taken as spatially variable, but constant in time), qI is the source inflow rate per unit volume, CI is the source strength, qO is the sink outflow rate per unit volume, to is the source release time, xo is the source location, mo is source mass, δ(·) is the Dirac delta function, gi(t) are known boundary functions, and Γi are the boundaries.

[11] The adjoint of this equation is given by [Neupauer and Wilson, 2002]

equation image

where ψ* is an adjoint state (marginal sensitivity of concentration to source mass), τ is backward time (time prior to sampling), xw and τw are the observation location and sampling time, respectively. The form of H(x, τ) depends on the type of PDF that is desired and on the type of well that was used for sampling (pumping well or monitoring well). To obtain the adjoint state related to the location PDF, or to obtain the adjoint state related to the travel time PDF if the sample is taken at a pumping well, we use H(x, τ) = 0 [Neupauer and Wilson, 2001]. To obtain the adjoint state that is related to the travel time PDF if the sample is taken at a monitoring well, H(x, τ) is defined as [Neupauer and Wilson, 2001]

equation image

where the sample location is xw = (x1w, x2w, x3w), δ′xi(xixiw) is the derivative of the Dirac delta function with respect to xi, vertical bars denote magnitude, and aL is the longitudinal dispersivity. (See Neupauer and Wilson [2001] for more details.)

[12] The relationship between the adjoint state from (2) and the backward location probability density function is [Neupauer and Wilson, 2002]

equation image

where fX(x; τ, xw, τw) is the backward location PDF representing the random position X at backward time τ of a contaminant particle that was observed at xw at backward time τw, and ψ* is the adjoint state obtained from (2) with H(x, τ) = 0. In the PDF notation, variables to the right of the semi-colon are deterministic parameters.

[13] The relationship between the adjoint state and the backward travel time probability density function is [Neupauer and Wilson, 2002]

equation image

where fT(τ; x, xw, τw) is the backward travel time PDF representing the random backward time T at which the particle that was observed at xw at backward time τw could have been at the deterministic upgradient position x, and the adjoint state, ψ*, is the solution to (2) with H(x, τ) = 0 if the observation is made at a pumping well, and H(x, τ) is defined in (3) for a monitoring well observation.

[14] If multiple observations of contamination are made, the additional information from the multiple observations can be included in the probability density functions. Let N be the number of concentration measurements, and let {xw} and {τw} be N-length vectors of sampling locations and sampling times, respectively. The multiple-observation backward location PDF, fX(x; τ, {xw}, {τw}), describes the possible former positions of all observed contaminant particles at time τ in the past, given that all particles were at the same position at that time. It is calculated using [Neupauer and Wilson, 2005]

equation image

where fX(x; τ, xwi, τwi) is the backward location PDF for the ith sample, obtained from (4).

[15] The multiple-observation backward travel time PDF, fT(τ; x, {xw}, {τw}), describes the possible release time of the observed contaminant particles from a fixed upgradient position, x, given that all particles were released from that location at the same time. It is calculated using [Neupauer and Wilson, 2005]

equation image

where fT(τ; x, xwi, τwi) is the backward travel time PDF for the ith sample, obtained from (5).

3. Conditioned Backward Location and Travel Time Probability Density Functions

[16] In the adjoint equation (2), information about the sampling location and time are used, but the concentration of the sample is not used. The unconditioned PDFs in (4)(7) depend only on the existence of contamination at the observation location, and not on the actual measured concentration. The concentration measurements contain additional information that can improve the characterization of the contamination source. In this section, we develop an approach to condition backward location and travel time probability density functions on measured concentrations. For the conditioned backward location PDF, we first develop a joint probability density function on source mass and source location conditioned on measurements of resident concentration, and then we find its marginal PDF of source location. For the conditioned backward travel time PDF, we develop a joint PDF on source mass and travel time conditioned on measurements of flux concentration, and then we obtain its marginal PDF of travel time. For brevity, we only derive the conditioned backward location PDF, and we present the final equation for the conditioned backward travel time PDF.

3.1. Conditioned Backward Location PDF

[17] In this section, we develop a backward location probability density function that is conditioned on measured concentration. We assume that the source of contamination is an instantaneous point source and that all flow and transport parameters are known. We also assume that the release time is known; therefore the backward location PDF represents possible source locations. We assume that N samples of contamination were observed with concentrations equation imagei = equation image(xwi, τwi), i = 1, 2, … N, where xwi is the location and τwi is the backward time at which sample i was taken.

[18] We assume that the measurements contain random measurement error, ε, such that

equation image

where equation imagei is the random measured concentration at observation i, C(xwi, τwimo, xo, τ) is the true concentration at observation i for a given release of mass mo from a source at location xo at backward time τ, and εi is the random measurement error of sample i. We assume that the error, εi, is normally-distributed with zero mean and variance σi2i ∼ N(0, σi2)), and therefore measured concentrations, equation imagei, are normally distributed with a mean equal to the true concentration and a variance of σ2, given by

equation image

Lin [2003] found that using other distributions of measurement error has little effect on the final results. We also assume that the εi are independent; therefore for a known source mass mo and a source location xo, the joint PDF on all N measured concentrations is the product of the PDFs for the individual observations given by

equation image

This joint PDF states that the measurement errors in the N sampled concentrations are independent. The measured concentrations are related to the true concentrations at the sample location, but these true concentrations are deterministic if the source location, source mass, source release time, and all flow and transport parameters are assumed to be known.

[19] Since the problem addressed in this paper is a situation in which the source mass and source location are unknown but the measured concentrations are known, (10) cannot be used directly. Instead, we use Bayes' theorem to obtain a joint PDF for the random source mass M and the random source location Xo, conditioned on the particular measured concentrations. We then find the marginal PDF over all possible values of M to obtain a probability density function on the source location.

[20] Let us define imageequation image(mo, xoequation image; τ, {xw}, {τw}) as the joint PDF of the random source mass M and the random source location Xo, for a source release occurring at time τ in the past, conditioned on the N measured concentrations, where equation image = [equation image1, equation image2, …, equation imageN], equation image = [equation image1, equation image2, …, equation imageN], {xw} = [xw1, xw2, …, xwN], and {τw} = [τw1, τw2, …, τwN] are N-length vectors of random measured concentrations, particular values of the known measured concentrations, sample locations, and sampling times, respectively. In other words, equation imagei is the concentration that is measured at location xwi at backward time τwi, and represents a particular value of the random concentration equation imagei. From Bayes' theorem, the joint PDF, imageequation image(mo, xoequation image; τ, {xw}, {τw}), can be written as

equation image

where we dropped the vectors of sampling locations and sample times from the conditional PDFs to simplify notation since they are implied by the equation image vector (see (8)). The first PDF in the numerator is the joint PDF of the N measured concentrations, given the source location and source mass, and is equivalent to (10). The second PDF in the numerator of (11) is the joint PDF of source mass and source location in the absence of any concentration information. Source mass is independent of source location in the absence of any concentration information, therefore this joint PDF can be separated into the product of the two individual PDFs, given by

equation image

The first PDF on the right hand side is a distribution on source mass alone. In the absence of any other information, we assume that this is a uniform distribution, although other distributions can be used if more information is available. The second PDF is the unconditioned backward location PDF for multiple observations, given by (6) with the random position X representing the random source location Xo.

[21] Substituting (6), (10), and (12) into (11), and integrating over the source mass domain, we obtain

equation image

where βx accounts for the uniform distribution for fM(mo; τ, {xw}, {τw}) and the denominators in (11) and (6), and is defined such that the total probability is unity.

[22] To calculate the conditioned backward location PDF using (13), it is first necessary to run one unconditioned backward location probability simulation for each observation by solving the adjoint equation (2). The resulting adjoint states are used in (4) to obtain the unconditioned PDFs, image xwiwi). Next, fequation imageimageequation imageimo, xo; τ) is calculated for each observation for a range of possible source masses and a range of possible source locations that includes the entire spatial domain of the model. This probability density function is a normal distribution defined in (9), with mean equal to the true concentration, and is conditioned on knowing the source location and source mass. In this paper, we assume that the all flow and transport parameters are known; therefore, for a given source location and source mass, the true concentration at the sampling locations can easily be calculated from, for example, a forward simulation. Instead of running one forward simulation for each possible source location, however, we use the backward model results, taking advantage of the relationship between adjoint states and concentration. By definition, the adjoint state for location probability is equal to the marginal sensitivity of resident concentration to source mass, i.e.,

equation image

where C(xww) is the concentration at location xw at backward time τw due to a source of mass mo released at backward time τ at location x. Since the system is linear, the adjoint state can also be written as ψ*(x, τ, xw, τw) = C(xw, τw)/mo(x, τ) [Neupauer and Wilson, 2002]; therefore the concentration at xw at backward time τw due to an instantaneous source of mo at location x at backward time τ is simply the product of the adjoint state and the source mass, given by C(xw, τw) = ψ*(x, τ, xw, τw) mo(x, τ). Assuming that the flow and transport parameters that were used in the model to calculate the adjoint state are correct, then this concentration represents the true concentration at location xw at backward time τ given the source mass and source location. Using the adjoint state to compute this true concentration is computationally efficient because it has already been calculated to obtain the unconditioned backward location PDF. After all of the PDFs in (13) are determined, (13) is integrated over a range of possible source masses. The resulting PDF is normalized to ensure that the total probability is unity, i.e., ∫ imageequation image(xoequation image; τ) dxo = 1.

3.2. Conditioned Backward Travel Time PDF

[23] The derivation of the conditioned backward travel time probability density function follows a similar approach as in the previous section. We assume that the location of the instantaneous point source of contamination is known, but the release time is unknown. We assume that N samples of flux concentration were made, and we assume that the measurements contain normally-distributed random error, ε, such that

equation image

where equation imageif = equation imagef(xwi, τwi) for i = 1, 2, … N is the measured flux concentration of the ith observation (at location xwi at backward time τwi), Cf(xwi, τwimo, xo, τ) is the true flux concentration at observation i given a release of mass mo at source location xo and backward time τ, and εi is the random measurement error (εi ∼ N(0, σi2)). The measured flux concentration is normally distributed with mean equal to the true concentration, given by

equation image

Assuming that the measurement error of the N measurements are independent, a joint PDF of all measured concentrations is the product of the individual PDFs (16) for each measurement, given by

equation image

[24] To obtain the conditioned backward travel time PDF, we first define fM,Tequation image(mo,τ∣equation imagef; xo) as the joint PDF of source mass and travel time conditioned on the N measured concentrations, where equation imagef = [equation image1f, equation image2f, … equation imageNf] and equation imagef = [equation image1f, equation image2f, …, equation imageNf] are N-length vectors of random measured flux concentrations and the particular values of the known measured concentrations. Using Bayes' theorem, and assuming that all the N measurement errors are independent, that source mass and travel time (or release time) are independent, and that the distribution of source mass is uniform, the joint PDF can be written as

equation image

Integrating this expression over the source mass domain produces the conditioned travel time PDF, given by

equation image

where βτ is defined such that the total probability is unity.

[25] To calculate the conditioned backward travel time PDF using (19), it is first necessary to obtain unconditioned backward travel time PDFs for each observation by solving the adjoint equation (2). The resulting adjoint states are used in (5) to obtain the unconditioned backward travel time PDFs. Next, fequation imageM,T(cifmo, τ; xo) is calculated from (16) for each observation for a range of possible source masses. The true concentration, which is used in (16), is related to the adjoint state. By definition, the adjoint state for travel time probability is equal to the marginal sensitivity of flux concentration to source mass, i.e.,

equation image

where Cf(xw, τw) is the concentration at location xw at backward time τw due to a source of mass mo released at backward time τ at location x, which can also be written as ψ*(x, τ, xw, τw) = Cf(xw, τw)/mo(x, τ) [Neupauer and Wilson, 2002]. Assuming that the flow and transport parameters used in the model are correct, the true flux concentration at location xw at backward time τ given the source mass and source location is Cf(xw, τw) = ψ*(x, τ, xw, τw) mo(x, τ). These PDFs are substituted into (19), which is integrated over a range of possible source masses to evaluate the integral in (19). Finally, βτ is obtained by ensuring that the total probability is unity, i.e., ∫fTequation image(τ∣equation imagef; xo) dτ = 1.

4. Example

[26] In this section, we present a hypothetical example to illustrate the implementation of the conditioned backward probability model and some important features of the model results. The hypothetical aquifer is a two-dimensional, rectangular, confined aquifer, with specified head conditions at the east and west boundaries, and no-flow boundaries at the north and south (see Figure 1). The heterogeneous transmissivity field, which was generated using sequential Gaussian simulation [Deutsch and Journel, 1992] with parameters from Table 1, is shown in Figure 2. Flow is essentially from east to west, with a weak pumping well in the western part of the aquifer. The head distribution, obtained using MODFLOW-2000 [Harbaugh et al., 2000], is shown in Figure 3. Parameter values used in the simulation are shown in Table 1.

Figure 1.

Two-dimensional aquifer and boundary conditions used in the example problem.

Figure 2.

Transmissivity distribution for example problem. Units are m2/d.

Figure 3.

Hydraulic head (in m) distribution (dashed lines) and concentration distribution (solid lines) at 800 days after release from the source. The square denotes the source location; the solid circle denotes the pumping well location; and open circles denote monitoring well locations. The contour interval for concentration is 0.02 g/m3.

Table 1. Flow and Transport Parameters for the Example Problem
ParameterValue
Mean of log-transmissivity8.64 m2/d
Variance of log-transmissivity0.2
Correlation length60 m
Aquifer thickness1 m
Pumping rate8.64 m3/d
Spatial discretization5 m
Longitudinal dispersivity, αL10 m
Transverse dispersivity, αT2 m
Porosity, θ0.3
Source mass, mo1 kg

4.1. Forward Model Results

[27] Suppose 1 kg of a conservative contaminant was released instantaneously at t = 0 from a source at (x1o, x2o) = (397.5 m, 177.5 m). The contaminant distribution at 800 days after release from the source, obtained using MT3DMS [Zheng and Wang, 1999], is shown in Figure 3. Parameter values used in the simulation are shown in Table 1. After 800 days, the plume has traveled downgradient, and some of the contaminant has been extracted at the pumping well. The highest concentration is slightly northeast of the pumping well; the leading edge of the plume has passed the pumping well, and the trailing edge of the plume is slightly downgradient from the source. We sampled the contaminant concentration at four locations: the pumping well (Location 1 in Figure 3) and three monitoring wells (Locations 2–4 in Figure 3). The locations, sample times, and measured concentrations are shown in Table 2. The sampling times are presented in both forward time (tw, relative to the source release time) and backward time (τw, relative to the latest sampling event; τw = 800 days-t). Both resident and flux concentration are presented since location PDF is related to resident concentration and travel time PDF is related to flux concentration. For the pumping well, resident and flux concentrations are equal because we assume that the concentration gradient approaches zero at the well [see Chen and Woodside, 1988].

Table 2. Sample Locations, Times, and Concentrations
Sample NumberLocationSampling TimeConcentration
tw, dτw, dResident, g/m3Flux, g/m3
1(197.5 m, 177.5 m)750501.33 × 10−11.33 × 10−1
2(247.5 m, 197.5 m)80001.66 × 10−11.58 × 10−1
3(147.5 m, 222.5 m)80002.30 × 10−23.01 × 10−2
4(202.5 m, 242.5 m)750501.44 × 10−21.65 × 10−2

4.2. Backward Location Probability Model Results

[28] Suppose that the source location is unknown, but we suspect that it was an instantaneous release 800 days prior to the most recent sampling event. We can use the observations in the unconditioned and conditioned backward probability models to identify the source location.

4.2.1. Unconditioned Backward Location PDFs

[29] We ran four unconditioned backward location probability simulations, one for each observation, by using MT3DMS to solve (2) with H(x, τ) = 0. We used the results in (4) to obtain the unconditioned backward location probability density functions shown in Figure 4. For these simulations, we define τ = 0 to correspond with the time of the most recent sampling event, i.e., at t = 800 days after the source release, which corresponds to the sampling time of Samples 2 and 3. The numerical implementation procedure is described in Neupauer and Wilson [2004].

Figure 4.

Unconditioned backward location probability density functions at τ = 800 days for (a) Sample 1; (b) Sample 2; (c) Sample 3; (d) Sample 4; (e) all four samples; and (f) Samples 1 and 3. The square denotes the source location. Sample locations are numbered and shown with a circle (open circle for monitoring well; solid circle for pumping well).

[30] The PDFs in Figure 4 show the possible former positions of the observed contaminant particles at backward time τ = 800 days. For Samples 1 and 2, the unconditioned location probabilities indicate that the true source location is a likely source location; however, the results show that the most likely source locations of Samples 3 and 4 are downgradient and slightly to the north of the true source location. The positions of the PDFs are controlled by the locations of the observations and by the flow field. Since Samples 3 and 4 are taken from the northern edge of the plume, their resulting backward location PDFs fall north of the true source location.

[31] Using (6), we combined the four individual unconditioned backward location probability density functions into one multiple-observation PDF, shown in Figure 4e. The results show that the most likely source location is downgradient and north of the true source location; and the magnitude of the PDF at the true source location is negligible. The position of this PDF is controlled by the locations of the observations. Notice that three of the four observations (Samples 1, 3, and 4) are taken near the leading edge of the plume, while Sample 2 is taken near the center of the plume. No information is available from the trailing edge of the plume; therefore, the unconditioned backward probability model results behave as if no contamination existed in that region. In addition, each observation is weighted equally, therefore the position of the unconditioned multiple-observation backward location PDF is approximately in the middle of the four individual PDFs shown in Figures 4a–4d. For comparison, the multiple observation unconditioned backward location PDF for Samples 1 and 3 is shown in Figure 4f. Again, the most likely source location is downgradient of the true source location because both samples were taken from the leading edge of the plume. Notice that the variances of the PDFs decrease as the number of observations increases. The variance of the two-observation PDF (Figure 4f) is smaller than the variance of any of the single-observation PDFs (Figures 4a–4d); and the variance of the four-observation PDF (Figure 4e) is smaller than the variance of any of the other PDFs.

4.2.2. Conditioned Backward Location PDFs

[32] We used (13) to calculate the conditioned backward PDFs for four different combinations of two to four observations. Solution of (13) requires integration over the source mass domain. In theory, the source mass domain is mo > 0; however, in practice, numerical integration must be performed over a smaller range. For this example, our source mass domain was 100 g < mo < 4000 g. A discussion of the selection of the source mass range is presented in the next section.

[33] The conditioned backward location PDFs are plotted in Figure 5 for standard deviation of measurements errors of σ = 0.005 g/m3, and the statistics of the PDFs are presented in Table 3. For this small σ, the most likely source location (location of maximum value of PDF, mode in Table 3) is at or within 5 m of the true source location in both the x and y directions for all cases, except when only Samples 3 and 4 are used. This is in contrast to the unconditioned PDFs, in which even the four-observation backward location PDF did not identify the true source location as a highly likely source location. Also, the variances of the conditioned PDFs are smaller than the variances of the unconditioned PDFs. These results demonstrate the conditioning on measured concentrations improves the accuracy and reduces the variance of the backward location PDFs.

Figure 5.

Conditioned backward location probability density functions at τ = 800 days using σ = 0.005 g/m3 for (a) Samples 1 and 2; (b) Samples 1 and 3; (c) Samples 3 and 4; (d) Samples 1, 2, and 3; (e) Samples 1, 3, and 4; and (f) all four samples. The square denotes the source location. Sample locations are numbered and shown with a circle (open circle for monitoring well; solid circle for pumping well).

Table 3. Spatial Statistics of the Conditioned Backward Location PDFs
Samplesx Directiony Direction
Mode, mMean, mVariance, m2Mode, mMean, mVariance, m2
1,2397.5403.83510177.5180.5135
1,3397.5395.31050177.5175.5555
3,4317.5304.42120207.5215.8746
1,2,3397.5398.240.8177.5177.61.64
1,3,4402.5395.6313182.5175.5165
1,2,3,4397.5397.327.9177.5177.61.39

[34] The conditioned backward PDFs for Samples 1 and 2 (Figure 5a) and for Samples 1 and 3 (Figure 5b) have long and narrow shapes, owing to the unknown source mass. Samples 1 and 2 are both taken near the center of the plume and their concentrations are similar (equation image1 = 0.133 g/m3 for Sample 1, and equation image2 = 0.166 g/m3 for Sample 2). If the source mass were much larger than the true source mass, the observed concentrations at locations 1 and 2 could be achieved in two different cases – if the source location were near the observation locations and the samples were taken from the trailing edge of the plume; or if the source location were far from the observation locations and the samples were taken from the leading edge of the plume. The outer regions of the long, narrow conditioned PDF result from the larger source masses. For samples taken from the trailing limb of the plume, if the source mass were decreased, the observed concentrations would only be possible if they were taken closer to the center of the plume. This would require that the plume position is shifted upgradient, which could only occur if the source location were shifted upgradient. Therefore, as the random source mass decreases, the possible source locations shift closer to the true source location. This accounts for the long shape of the conditioned PDFs in Figures 5a and 5b.

[35] A similar pattern is seen for the conditioned PDF for Samples 3 and 4 (Figure 5c); however, the true source location is not shown to be a likely source. This occurs because of the combination of the low measured concentrations for these samples and the locations of the samples within the plume. Both samples are taken from the northern edge of the plume; therefore the unconditioned PDFs for each of these samples indicate likely source locations to the north of the true source location. The true source location and source mass combination is obviously one of the possible combinations that would produce the measured concentrations at these two locations. The magnitude of the PDF at the true source location is imageequation image(xoequation image; τ) ≈ 2.7 × 10−5 m−2, which is ≈9% of the maximum value of the PDF. Because the value of the unconditioned PDF is so low at the true source location for these two samples, there are other more likely combinations of source mass and source location that would produce the measured concentrations, as shown in Figure 5c.

[36] This example illustrates that the results are more accurate if the samples are spread throughout the plume or are taken where concentrations are higher. In practice, we may not know the full extent of the plume, therefore we do not know if the samples are taken from throughout the plume or from just one region of the plume. The variances of the PDFs provide a measure of the uncertainty in the identified source location. Samples 3 and 4 are taken from the same region of the plume, and the variances in both the x and y directions of the resulting conditioned backward location PDF are relatively large indicating less certainty in the identified source location. The variance of the conditioned PDF is strongly affected by the measurement error. In Figure 5, the standard deviation of measurement error was assumed to be σ = 0.005 g/m3, which is equivalent to approximately 3–4% error in the measured concentrations for Samples 1 and 2, and 20–35% error for Samples 3 and 4. The effects are observable in the conditioned PDFs in Figure 5 which show a narrow spread (small variance) for the conditioned PDF for Samples 1 and 2 (Figure 5a), a slightly larger spread for Samples 1 and 3 (Figure 5b), an even larger spread for Samples 3 and 4 (Figure 5c).

[37] As more samples are used in the conditioning process, the conditioned PDF becomes more accurate and less uncertain (smaller variance). Figure 5d shows the conditioned PDF for Samples 1, 2, and 3. Although the conditioned PDFs for Samples 1 and 2 (Figure 5a) and for Samples 1 and 3 (Figure 5b) both select the true source location as a possible source location, the variances of the PDFs are large, and therefore the PDFs include many other possible source locations. Adding the third sample greatly reduces the variance of the conditioned PDF (Figure 5d) and still maintains the accuracy. Figure 5e shows the conditioned PDF for Samples 1, 3, and 4. This PDF has a larger variance than the conditioned PDF for Samples 1, 2, and 3, because Samples 1, 2, and 3 are spread throughout the plume while Samples 1, 3, and 4 capture mostly the leading edge. When all four samples are used (Figure 5f), the conditioned PDF identifies the vicinity of the true source location as the only likely source area.

[38] Figure 6 shows two- and four-observation conditioned backward location PDFs (solid lines), for three different standard deviations of measurement error (σ = 0.05 g/m3 in subplots a and b, σ = 0.5 g/m3 in subplots c and d, and σ = 5 g/m3 in subplots e and f). As measurement error increases, the accuracy of the PDFs declines and the variance increases. In the limit, as σ → ∞ (which is approached for σ = 5 g/m3), the conditioned PDF approaches the unconditioned PDF.

Figure 6.

Conditioned backward location probability density functions at τ = 800 days using various groups of samples and measurement errors. The left-hand column shows results for Samples 1 and 3. The right-hand column shows results for all samples. The top row has moderate error (σ = 0.05 g/m3); the middle row has large error (σ = 0.5 g/m3); and the bottom row has error approaching infinity (σ → ∞). The square denotes the source location. Sample locations are numbered and shown with a circle (open circle for monitoring well; solid circle for pumping well).

4.2.3. Verification of the Conditioned Backward Probability Model

[39] To demonstrate that the conditioned backward probability model accurately identifies the source location, we performed Monte Carlo simulation to identify the most likely source locations for 10,000 sets of random samples. For each simulation, we use a sample taken at the pumping well at τ = 50 days, and samples at three random locations taken at τ = 0 days. This is based on a scenario that first a single observation of contamination was made at the pumping well. Then, to delineate the plume, additional samples were taken at a later time. Since it is likely that additional samples would be taken in the vicinity of the pumping well, we randomly selected the new sampling locations from a joint normal distribution centered at the pumping well location. We used a larger standard deviation (50 m) in the x direction than the y direction (standard deviation of 20 m), assuming that the sampling locations would be chosen based on knowledge of the flow direction. In other words, if the flow direction is known to be in the −x direction on average, the extent of the plume is likely to be larger in the x direction than in the y direction. The sampled concentrations were taken as the true concentration with added random measurement error that was assumed to be normally-distributed with zero mean and a standard deviation of σ. For each simulation, we identified the most likely source location as the location of the maximum value of the conditioned backward location PDF. The results are shown in Figure 7 for n = 10000 simulations. Figure 7a shows the locations of the samples (circles). Figures 7b–7d show contours representing the regions within which the identified source location falls for 95% (outer contour), 90%, 80%, 70%, 60%, and 50% (inner contour) of the simulations using σ = 0.005 g/m3 (Figure 7b), σ = 0.01 g/m3 (Figure 7c), and σ = 0.05 g/m3 (Figure 7d). With small measurement error (Figure 7b, with σ = 0.005 g/m3), the most likely source location obtained from the conditioned backward location PDF falls within a few meters of the true source location for almost all sets of sample locations, indicating that the conditioned backward probability model can accurately identify the source location. As the measurement error increases, the confidence regions become larger, indicating a loss of accuracy in the identification of the source location. Note however that even for moderate measurement error (e.g., Figure 7d), the 50% confidence region still contains the true source location.

Figure 7.

Results of Monte Carlo simulations for conditioned backward location PDFs. (a) Sample locations used in Monte Carlo simulations; (b)–(d) regions within which the most likely source location is contained for 95% (outer contour), 90%, 80%, 70%, 60%, and 50% (inner contour) of the simulations using (b) σ = 0.005 g/m3, (c) σ = 0.01 g/m3, and (d) σ = 0.05 g/m3. The square denotes the source location, and the white triangle denotes the pumping well location. Note that the scale of the axes for Figure 7a is different than the scale of the axes for Figures 7b–7d.

[40] For small measurement error (Figures 7b and 7c), the shape of the confidence regions are approximately elliptical with the major axis aligned in the general direction of groundwater flow. The locations of the samples affect the shape and position of the conditioned backward PDFs (see Figure 5); however, with small measurement error, the most likely random source locations are constrained to be near the true source location. For moderate measurement error (Figure 7d), the shape of the confidence regions are more complex. The contours are tightly-spaced to the north and east of the true source location, and more spread out to the west and south of the true source location. This shape can be explained by the positions of the samples (Figure 7a) relative to the center of mass of the contaminant plume (Figure 3). Four sample locations were used for each simulation: the pumping well, and three random sample locations taken from a two-dimensional normal distribution centered at the pumping well. For some simulations, the three random sample locations were south and west of the pumping well, where concentrations are relatively low (see Figure 3). Because of the low concentrations, the error level of σ = 0.05 g/m3 is on the same order as the concentrations; therefore conditioning on these measured concentrations add very little information. This leads to a PDF with a large variance and whose position is controlled more by the location of the samples than by their concentrations. The unconditioned backward location PDFs for samples taken south and west of the pumping well fall west of the true source location due to their proximity to the pumping well. For these reasons, if the three random samples are taken southwest of the pumping well, the position of the most likely source location (mode of the conditioned backward location PDF) falls southwest of the true source location and is quite variable. This is observed in Figure 7d for x < 400 m where the boundaries between successive confidence regions are quite spread out. This effect is not observed for smaller levels of measurement error (σ = 0.005 g/m3 and σ = 0.01 g/m3 for Figures 7b and 7c, respectively) because the measured concentrations are large relative to the measurement error, and therefore they contribute useful information about the source location.

4.3. Backward Travel Time Probability Model Results

[41] Suppose that an instantaneous point source of contamination is known to be located at (x1o, x2o) = (397.5 m, 177.5 m), but the release time of contamination from the source is unknown. We can use observations of flux concentration in the unconditioned and conditioned backward probability models to identify the release time, which is related to the travel time of the observed contaminant particles from the source to their observation locations.

4.3.1. Unconditioned Backward Travel Time PDFs

[42] We ran four unconditioned backward travel time probability simulations, one for each of the observations shown in Table 2, by using MT3DMS to solve (2) with H(x, t) = 0 for the pumping well observation (Sample 1) and H(x, t) defined in (3) for the monitoring well observations (Samples 2–4). The resulting adjoint states were used in (5) to obtain the unconditioned backward travel time probability density functions shown in Figure 8. For these simulations, we again define τ = 0 as the time of the most recent sampling event (sampling time for Samples 2 and 3).

Figure 8.

Unconditioned backward travel time probability density functions at (x1o, x2o) = (397.5 m, 177.5 m). (a) Single-observation PDFs. (b) Multiple-observation PDFs. Note that the scales on the vertical axes of the two plots are different.

[43] The PDFs in Figure 8 show the possible release times of the observed contamination at the true source location, (x1o, x2o) = (397.5 m, 177.5 m). The single-observation PDFs are shown in Figure 8a. The true release time of τ = 800 days is a possible release time for all four samples, although the variances of the PDFs are large. The most probable release times for Samples 1–4 are 670 days, 510 days, 1150 days, and 1045 days, respectively. These release times differ from the true release time by as much as one year. The area under each backward travel time PDF describes the probability that the observed particle was ever at the true source location. Of the four sample locations, Sample 2 is closest to the true source location, and therefore is the most likely of the four samples to have been at the true source location; this is consistent with the large area under the PDF for Sample 2. On the other hand, Sample 4 is more distant and is on an advective flow path that does not pass near the true source location, therefore, it is the least likely of the four samples to have been at the true source location, as can be seen by the small area under this PDF.

[44] Using (7), we combined the individual unconditioned backward travel time probability density functions into multiple-observation PDFs shown in Figure 8b. These PDFs are conditioned on all of the particles' having been at the true source location at some time in the past; therefore, the total probability (area under the PDF) is unity. The multiple-observation PDF for Samples 1 and 3 (solid line in Figure 8b) has a smaller variance than all of the single-observation PDFs (Figure 8a). The most likely release time of the two observed particles, assuming they were released from the true source location at the same time, is 860 days, which is closer to the true release time than any of the single-observation PDFs. When all four observations are considered (dashed line in Figure 8b), the most likely release time is 790 days, which is very close to the true release time.

4.3.2. Conditioned Backward Travel Time PDFs

[45] We used (19) to calculate the conditioned backward travel time PDFs for four different combinations of two to four observations, using a source mass domain of 100 g < mo < 4000 g. The conditioned backward travel time PDFs are plotted in Figure 9 for standard deviation of measurements errors of σ = 0.005 g/m3. In all cases, except for when only Samples 3 and 4 are used, the most likely release time (mode) and the mean of the PDFs are very close to the true release time of τ = 800 days prior to sampling (see Table 4). In addition, the standard deviations are small (<32 d for five of the six cases), indicating that conditioning on measured concentrations, even with just two samples, greatly improves the accuracy of the backward travel time PDFs. Samples 3 and 4 are both taken from the leading edge of the plume, and have low concentrations, which do not provide sufficient information to identify the release time. As the number of samples increases, the standard deviation of the conditioned travel time PDF decreases, as a result of the additional information available.

Figure 9.

Conditioned backward travel time probability density functions at (x1o, x2o) = (397.5 m, 177.5 m). (a) Pairs of samples. (b) Groups of three or four samples.

Table 4. Statistics of the Conditioned Backward Travel Time PDFs
SamplesMode, dMean, dStandard Deviation, d
1,280080032
1,380380026
3,49521080270
1,2,380079918
1,3,480580324
1,2,3,480280017

[46] Figure 10 shows two- and four-observation conditioned backward travel time PDFs, for four different standard deviations of measurement error (σ = 0.005 g/m3, σ = 0.05 g/m3, σ = 0.5 g/m3, and σ = 5 g/m3). As measurement error increases, the accuracy of the PDFs declines and the variance increases. In the limit, as σ → ∞ (which is approached for σ = 5 g/m3), the conditioned PDF approaches the unconditioned PDF in Figure 8b. For σ = 0.5 g/m3 and σ = 5 g/m3, the PDFs are visually indistinguishable.

Figure 10.

Conditioned backward travel time probability density functions at (x1o, x2o) = (397.5 m, 177.5 m) for four different values of standard deviation of measurement error. (a) Samples 1 and 3. (b) All samples. For σ = 0.5 g/m3 and σ = 5 g/m3, the PDFs are visually indistinguishable.

5. Implementation of the Conditioned Probability Model

[47] To calculate the conditioned probability density functions, it is necessary to identify a range of possible source masses. Although the final conditioned backward location or travel time PDF does not contain a source mass, a necessary intermediate step is the calculation of a joint PDF on source mass and either source location or source release time. If the range of source masses is too large, the procedure will be computationally inefficient. On the other hand, if the range of source masses is too small, it may exclude values that could produce the range of measured concentrations.

[48] One method for selecting the bounds of the source mass range is to determine the masses that would be required to obtain each of the measured concentrations if the true source location were at the most likely source location identified in each of the unconditioned backward location PDFs. The relationship between the backward location PDF and source mass can be obtained by combining (4) and (14) to obtain

equation image

Using the peak values of each of the unconditioned backward location PDFs (from Figure 5) and the resident concentrations in Table 2, we obtained estimates of source mass from (21) of 890 g/m, 660 g/m, 72 g/m, and 43 g/m. The true source mass can be no smaller than the minimum of these values; therefore the minimum value of (21) should be the lower bound of the source mass range. The upper bound can be taken as a multiple of the maximum value of (21). For our example, we used an upper bound of approximately five times the maximum value obtained from (21).

6. Conclusion

[49] Backward modeling is an efficient approach for characterizing sources of groundwater contamination based on observations of contamination made at one or more locations in a contaminant plume. A backward location probability density function describes possible source locations or former positions of observed contamination; while a backward travel time probability density function describes possible travel times for the observed contaminant particle to travel from an upgradient position, possibly the source, to the observation location. In the paper, we present an approach for conditioning these backward PDFs on measured concentrations. The derivation of the conditioned PDFs is based on Bayes' theorem. The resulting expression for the conditioned PDF depends on the unconditioned PDFs and a probability density function for the measured concentration. We have assumed that this latter PDF is a normal distribution with mean equal to the true concentration at the observation location and time and standard deviation (σ) that accounts for measurement error.

[50] We demonstrated the conditioning method using a hypothetical example of contaminant transport in a two-dimensional, heterogeneous aquifer. We first ran a forward transport simulation with an instantaneous point source of contamination, generating a plume from which we obtained four samples of resident and flux concentrations. We obtained unconditioned backward location and travel time PDFs for each of the four samples, and for combinations of the four samples. Finally, we conditioned these PDFs on the measured concentrations. The method requires one backward simulation for each observation to produce the unconditioned PDFs. These simulations can be carried out using standard groundwater flow and transport codes; we used MODFLOW-2000 and MT3DMS. Additional post-processing is needed for the conditioning step.

[51] The results show that when measurement error (σ) is small, the conditioned backward location PDFs identify a narrow band or small patch of possible source locations. As long as the samples are obtained from throughout the plume, e.g., they are not taken from just the leading edge of the plume, the conditioned PDFs include the true source location. Likewise, for small σ, the conditioned backward travel time PDFs identify a small range of possible release times that generally includes the true release time if the samples on which we condition are obtained from throughout the plume. As the number of observations increases, the variance of the conditioned PDF decreases. As σ increases, the variance of the conditioned PDFs increases and the accuracy decreases. In the limit as σ → ∞, the measured concentrations provide no information, and the conditioned PDF approaches the unconditioned PDF.

[52] At this time, the approach presented in this paper is only applicable to an single instantaneous point source. Preliminary work on the extension of this approach to multiple instantaneous point sources of contamination is presented by Lin [2003]. Additional work is needed to generalize the approach to distributed and continuous sources.

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