WATER RESOURCES RESEARCH, VOL. 39, NO. 2, 1038, doi: /2001wr001021, 2003

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1 WATER RESOURCES RESEARCH, VOL 39, NO, 1038, doi:10109/001wr00101, 003 Marching-jury backward beam equation and quasi-reversibility methods for hydrologic inversion: Application to contaminant plume spatial distribution recovery Amvrossios C Bagtzoglou Department of Civil and Environmental Engineering, University of Connecticut, Storrs, Connecticut, USA Juliana Atmadja TAMS Consultants, an EarthTech Company, Bloomfield, New Jersey, USA Received 17 October 001; revised 30 July 00; accepted September 00; published 19 February 003 [1] In this paper, a comparison between the marching-jury backward beam equation (MJBBE) and the quasi-reversibility (QR) methods to perform hydrologic inversion and, more specifically, to reconstruct conservative contaminant plume spatial distributions is presented The MJBBE, developed by Atmadja and Bagtzoglou [000, 001a], was used to recover contaminant spatial distributions in heterogeneous porous media, while the QR method, first applied to groundwater contamination problems by Skaggs and Kabala [1995], was modified to incorporate heterogeneity and explicitly handle the advective term of the transport equation Spatially uncorrelated and correlated, stationary and nonstationary, heterogeneous dispersion coefficient fields were generated using the Bayesian nearest neighbor method (BNNM) Homogeneous and deterministically heterogeneous cases are also presented for comparison In addition, contaminant plume initial data with uncertainty were also analyzed using the MJBBE and QR methods The MJBBE is found to be robust enough to handle highly heterogeneous parameters and is able to preserve the salient features of the initial input data On the other hand, the QR method is superior in handling cases with homogeneous parameters and with initial data that are plagued by uncertainty but it performs very poorly in cases with heterogeneous media INDEX TERMS: 189 Hydrology: Groundwater hydrology; 1831 Hydrology: Groundwater quality; 183 Hydrology: Groundwater transport; 310 Mathematical Geophysics: Modeling; 330 Mathematical Geophysics: Numerical solutions; KEYWORDS: hydrologic inversion, groundwater pollution, inverse methods, Heterogeneity, marching-jury backward beam equation, quasi-reversibility Citation: Bagtzoglou, A C, and J Atmadja, Marching-jury backward beam equation and quasi-reversibility methods for hydrologic inversion: Application to contaminant plume spatial distribution recovery, Water Resour Res, 39(), 1038, doi:10109/001wr00101, Introduction [] Naturally occurring porous media are highly heterogeneous, and are characterized by spatially variable geometric and hydraulic properties With increasing demands for clean drinking water and a way to identify pollution sources accurately, it is clear that a method that can backtrack the pollution source in order to perform contaminant plume spatial distribution recovery and ultimately reconstruct the plume release time history in heterogeneous media is highly desirable For the past fifteen years, several attempts have been made to tackle one of the most difficult problems of hydrologic inversion, namely to solve the advection dispersion equation (ADE) backwards in time in order to identify the pollution source Even though this topic is of considerable interest, only recently several review papers appeared in the literature For example, Morrison [000a] presented an extensive literature review Copyright 003 by the American Geophysical Union /03/001WR00101 SBH 10-1 of commonly used environmental forensic methods for age dating and source identification In this paper, Morrison discussed environmental forensic techniques and their possible applications so that readers can decide which technique is most appropriate for their cases A more in-depth review of these techniques is given by Morrison [000b, 000c] More recently, Atmadja and Bagtzoglou [001b] presented a state-of-the-art report on mathematical methods that could be used for pollution source identification In this section we briefly summarize the literature on this topic [3] One of the early methods to backtrack the pollution source location was to run forward simulations and check the solutions with the measured/current spatial data observed Among the first to tackle pollution source identification problems using optimization approaches were Gorelick et al [1983], who formulated the groundwater pollution source problem identification as forward time simulations in conjunction with an optimization model using linear programming and multiple regressions Another optimization approach was developed by Wagner [199] in order to perform simultaneous model parameter estimation and

2 SBH 10 - BAGTZOGLOU AND ATMADJA: MJBBE AND QR METHODS FOR HYDROLOGIC INVERSION source characterization Along the same line of work, Mahar and Datta [1997, 001] developed a methodology that combines the concepts of optimal identification of a pollutant source with the optimal design of a groundwater qualitymonitoring network for an efficient identification process Mahar and Datta [000] also used a nonlinear optimization technique to estimate the magnitude, location and duration of groundwater pollution sources under transient conditions Recently, Aral et al [001] proposed a new combinatorial approach, termed the progressive genetic algorithm, to solve the nonlinear optimization model that describes the identification of contaminant source locations and release histories [4] Bagtzoglou [1990] and Bagtzoglou et al [1991, 199] were among the first to attempt solving the ADE backwards in time without relying on optimization approaches In their work, Bagtzoglou et al modeled the reversed time transport equation using the random walk particle method by reversing the advective part of the transport while leaving unchanged the dispersive part They presented a probabilistic framework to identify solute sources in heterogeneous media and using geostatistical techniques successfully assessed the relative importance of each potential source Their approach was also extended to cases where the hydraulic conductivity field is not known with certainty with the help of conditional geostatistical simulations In the line of probabilistic approaches, Wilson and Liu [1994] solved the transport equation using stochastic differential equations backwards in time Similar to the method developed by Bagtzoglou et al, Wilson and Liu also kept the dispersion part positive and reversed the advection part An extension of their study for a two-dimensional (D) heterogeneous aquifer was reported by Liu and Wilson [1995] The results for travel time probability were in very close agreement with the simulation results from traditional forward in time methods Neupauer and Wilson [1999] later verified results from this model using the adjoint method Neupauer and Wilson [1999, 001] presented the adjoint method as a formal framework for obtaining the backward in time probabilities for multidimensional problems and more complex domain geometries Snodgrass and Kitanidis [1997] also used a probabilistic approach combining Bayesian theory and geostatistical techniques [5] Sidauruk et al [1998] presented an inverse method based on analytical solutions of contaminant transport problems The method provides a complete estimate of the dispersion coefficient, flow velocity, amount of pollutant, its initial location, and time origin However, since this method is based on an analytical solution, it works only for a very limited number of cases with homogeneous aquifers with simple geometries and flow conditions Another similar inverse analytical technique was developed by Ala and Domenico [199] with the objective to determine parameters such as the source strength and size, and the advective position of the contaminant front for the instantaneous contaminant plumes at Otis Air Force Base (AFB), Massachusetts Butcher and Gauthier [1994] used inverse modeling to estimate the residual DNAPL mass For the inverse model, Butcher and Gauthier used a tractable analytical approximation to the problem and developed additional simplifications to yield a form that is easily solved for the parameters of interest In a different field of study, namely physics, Macdonald [1995] successfully applied the nonlinear least squares (NLS) method to invert a 1D pure diffusion problem Macdonald applied the method to recover a number of Dirac-delta sources with large measurement errors in the data Alapati and Kabala [000] employed the NLS method without regularization to recover the release history of a groundwater contaminant plume from its current measured spatial distribution [6] Skaggs and Kabala [1994] proposed a different approach They attempted to directly reconstruct the history of the plume using Tikhonov regularization (TR) and studied a one-dimensional (1D) solute transport through a saturated homogeneous medium problem with a complex contaminant release history and assumed no prior knowledge of the release function In a similar effort, Samarskaia [1995] applied the TR with fast Fourier transforms to a groundwater contamination source reconstruction problem Later, Liu and Ball [1999] used Skaggs and Kabala s modified TR technique to study a contaminant release at Dover AFB, Delaware Skaggs and Kabala [1995] employed the quasi-reversibility (QR) method for the same problem solved with their TR approach In the QR method, Skaggs and Kabala solved an equation that is close to the original equation and that is stable with a negative time step A moving coordinate system was used to account for the velocity term of the ADE A problem similar to the one presented by Skaggs and Kabala [1994] was used for the QR study The QR results were less accurate than that of the TR approach, but the method is computationally less expensive Woodbury and Ulrych [1996] proposed another direct approach, namely the minimum relative entropy (MRE) inversion For the noise-free data, MRE was able to reconstruct the plume evolution history indistinguishable from the true history As for data with noise, the MRE method managed to recover the salient features of the source history Woodbury et al [1998] extended the MRE approach to reconstruct a three-dimensional (3D) plume source within a 1D constant velocity field and constant dispersivity system Recently, Neupauer et al [000] performed a study to compare the TR and the MRE methods and found that both methods perform well in reconstructing a smooth source function For an error-free step function source history, the MRE performs better than the TR On the other hand, the TR method is more robust in handling data that contain measurement errors Neupauer et al used measurement errors of 1%, 5% and 5% Skaggs and Kabala [1998] extended their study of TR using Monte Carlo simulation to answer the question of how far back one can use the Tikhonov procedure in recovering the release history of the plume, since the procedure always produces a recovered release curve that accurately reproduces the data [7] In summary, it is apparent that most of the methods currently used for recovering the contaminant plume spatial distribution and thereby reconstructing the release history of a contaminant are for homogeneous media Even though there exist methods that can handle heterogeneous parameters they are of an indirect nature; that is, they are probabilistic or inference-based Recently, Atmadja and Bagtzoglou [000, 001a] developed a method called marching-jury backward beam equation (MJBBE) to recover plume spatial distributions and the associated release time history The method was successfully applied to reconstruct the plume spatial distribution at some time between the initial

3 BAGTZOGLOU AND ATMADJA: MJBBE AND QR METHODS FOR HYDROLOGIC INVERSION SBH 10-3 and the final time for 1D homogeneous and deterministically heterogeneous media cases In addition, the method was also applied to recover the time release history [Atmadja and Bagtzoglou, 000, 001a] However, due to the MJBBE requirement that both initial and final conditions are known in order to solve for the time release history, the method can reconstruct the time release history only at some time between the initial and final time This is equivalent to a partial reconstruction of the time release history One of the more successful methods in recovering the contaminant release history in homogeneous media is the QR approach, first implemented by Skaggs and Kabala [1995] for groundwater pollution source identification Skaggs and Kabala [1995] claimed that the QR could be easily extended to treat heterogeneous media However, the method was never used in such cases Due to its computational efficiency and ability to be naturally extended so it can handle heterogeneous parameters, the QR method was chosen for comparison to the MJBBE Therefore, in addition to the MJBBE method, we enhanced the QR method by incorporating heterogeneity and treating the advective process explicitly In this paper we present the reconstruction of plume spatial snapshots from a present-day plume distribution using the MJBBE and QR methods in deterministically and randomly generated heterogeneous media Additionally, the MJBBE and QR methods were used to recover the plume concentration with input data that involve uncertainty A comparison between the two methods is presented Simulation of Heterogeneity in Porous Media [8] Over the past 5 years, it has become evident that the variability of natural porous media can play a tremendously important role in geohydrology [Gelhar, 1986] The spatial distribution of porous media properties and their specific type can be characterized by this variability [Tompson et al, 1987] The most important property that exemplifies the great variability in porous media is the hydraulic conductivity The hydraulic conductivity coefficient can exhibit a range of variability from 3 orders of magnitude to as high as 13 orders of magnitude in layered heterogeneous media [Freeze and Cherry, 1979] The degree and type of heterogeneity in a geological formation are usually described by statistical distributions [9] Different degrees and types of heterogeneity can be described by using different sets of variance and different structures for the stochastic fields generated, as expressed by the correlation length and the underlying covariance model Numerous methods of random field generation exist and have been applied successfully in the past For example, Smith and Schwartz [1980] assumed lognormally distributed hydraulic conductivity and used the nearest-neighbor autoregressive relation to generate the hydraulic conductivity field Ünlü et al[1990] modified and used the nearestneighbor method to generate the second-order stationary soil hydraulic properties Another very well known method is the 3D turning bands method [Mantoglou and Wilson, 198; Tompson et al, 1987, 1989] Some researchers, such as Carr and Myers [1985], Gómez-Hernández [1991], and Myers [1984] applied the joint simulation of several variables by considering a vector that consists of several random variables to obtain the cross-covariance matrix There exist also methods that use Gaussian-related algorithms For example, the sequential Gaussian simulation as explained by Johnson [1987] and Journel [1989] and the LU decomposition of the covariance matrix [Alabert, 1987; Davis, 1987] fall under this category A relatively new approach called simulated annealing has also been developed and used to simulate heterogeneity [Geman and Geman, 1984; Aarts and Korst, 1989; Deutsch, 199] Other methods also available, such as the Bayesian analysis [Kitanidis, 1986], Fast Fourier transform [Gutjahr et al, 1987], empirical Bayes regionalization method [Butcher et al, 1991], stochastic interpolation using fractional Lévy motion [Painter, 1996] and matrix decomposition technique [El-Kadi and Williams, 000] are only a nonexhaustive list of approaches The random field describing a certain property can be generated with or without conditioning Suffices to say the above listed references are but a very small fraction of the numerous worthy approaches available to treat heterogeneity and are given as examples only [10] In this paper, in addition to deterministically and randomly generated heterogeneous uncorrelated fields, we will apply the method presented by Bagtzoglou and Ababou [1997] to generate unconditional, spatially correlated Gauss-Markov random fields using the Bayesian nearest neighbor method (BNNM) The unconditional Gauss-Markov fields are generated as the numerical solutions of white noise-driven partial differential equations This method is considered here because it is very convenient and easy to code, the computational stencil implemented is identical to that of the flow equation, and it has been verified and tested before In addition, the method can be easily implemented for generating conditional simulations in multidimensional space (point, line, and plane conditioning) In this paper, application of the MJBBE and QR methods to recover the plume spatial distribution will be performed on spatially correlated heterogeneous media without conditioning The spatially correlated Gauss-Markov fields, generated by the BNNM, have the following characteristics: (1) they are stationary, and () impose a covariance structure that is dependent on dimensionality In this particular application the covariance honored is the linear exponential Details regarding the implementation of this method and verification tests are given by Bagtzoglou and Ababou [1997] [11] A 1D domain of length L = 4 discretized using a grid of x = 05 is considered for our test cases We have used both a homogeneous dispersion coefficient D = 1 as a reference case and deterministic heterogeneous dispersion coefficients with several different heterogeneity configurations of D(x) This variable dispersion coefficient accounts for deviations in the transport velocity caused by variability in dispersivity We used different values of D(x) in two different zones The two zones are: (1) outer zones for 0 x < 11 and 17 < x 4; and () inner zone for 11 x 17 The configurations are (1) D o = 1 for zone 1 and D i = 8 for zone, and () D o = 8 for zone 1 and D i = 1 for zone (Table 1) Spatially uncorrelated heterogeneous media were simulated using a randomly generated, normally distributed, dispersion coefficient with different standard deviation (s) around the mean value (m) To make the analysis comparable to the deterministic heterogeneous media, we used the same two different zones for heterogeneous D for the random nonstationary fields (Table 1)

4 SBH 10-4 BAGTZOGLOU AND ATMADJA: MJBBE AND QR METHODS FOR HYDROLOGIC INVERSION Table 1 Nonstationary, Deterministically, and Randomly Generated Heterogeneous Dispersion Coefficient Configurations Configuration m o m i s o s i l DET DET UN UN CN x CN x [1] For the stationary fields, the mean value of D is one and two different standard deviations (s = 0 and 04) are used for a Coefficient of Variation (CV) of 0 and 40%, respectively (Table ) These CV values correspond to upper bounds characteristic of dispersivity variability in unconsolidated and consolidated media, respectively For the spatially correlated fields the same mean value and standard deviations used in the case of uncorrelated fields are employed with two different spatial correlation lengths, l of x and 4 x (Table ) For the case of spatially correlated, nonstationary fields, D configurations similar to those used with spatially uncorrelated fields but with correlation length of 4 x only are used (Table 1) Within each of the two zones the dispersion coefficient field is stationary All results presented are based on single realizations and the same field is used for both forward and backward time simulations In the strict mathematical sense then the dispersion fields used in this work are not random However, the single realizations of the dispersion coefficient field are completely different from one test case to the next [13] In all the analyses, we kept the transport velocity term, u, as constant with a value of one and flow being from the left to the right of the domain In this work we are dealing with a 1D, steady state flow problem Therefore the only possibility for the transport velocity to be heterogeneous is due to the spatial variability in dispersivity, which is relatively small Even though we realize that dispersivity variability alone may not be a significant source of heterogeneity in the dispersion coefficient, we employ this heterogeneity scenario for illustration purposes and method intercomparison 3 Plume Spatial Distribution Recovery in Randomly Heterogeneous Media [14] Most attempts at quantifying contaminant transport have relied on solving some form of a well-known governing equation referred to as advection-dispersion-reaction equation or its simplest form of the ADE To reconstruct the plume spatial distribution, one needs to solve the governing equation backward in time Therefore the plume spatial distribution reconstruction is an inverse problem that is ill-posed Small variations in initial data lead to large variations in the solution of the problem A detailed discussion of the ill-posedness of the problem and the nature of the measurement errors is available in comment [Kabala and Skaggs, 1998] to the related paper by Woodbury and Ulrych [1996] and in their reply [Woodbury and Ulrych, 1998] The common strategy is to avoid solving the ill-posed problem directly and instead solve a related wellposed problem whose solution is close in some sense to the solution of the original problem Even though the methods mentioned above do not necessarily involve minimization of a norm, solutions found by these methods are typically termed regularized solutions [15] In this paper we study the advection-dispersion transport problem of a rectangular wave that is described by: @x Dx ½ ux ðþc Š; ð1þ Cx; ð 0Þ ¼ x < 13:5 and 14:5 < x L ðþ Cx; ð 0Þ ¼ C 1 ¼ 1 13:5 x 14:5 ð3þ where C is the solute concentration, D(x) is dispersion coefficient, and u(x) is the transport velocity 31 MJBBE Method for Heterogeneous Media [16] The backward beam equation (BBE) was first developed by Carasso [197] The BBE solves parabolic problems backwards in time and can be obtained by differentiating the governing equation of a parabolic equation with respect to time In addition, the BBE method was also used by Carasso [1975] in obtaining the solutions to the final value problem of Burger s equation This can be done because a solution of the parabolic equation, f t = f xx also satisfies f tt = f xxxx [17] The 1D problem is described as follows: ¼ L oc; 0 < x < L; t > 0 ð4þ Cx; ð 0Þ ¼ f 1 ðþ; x C x; T ft ¼ f ðþ; x 0 x L ð5þ Cð0; tþ ¼ 0; CL; ð tþ ¼ 0; t 0 ð6þ where L o is a differential operator in space and T ft is the terminal time of the forward problem f 1 (x) is the unknown initial concentration distribution, which is the terminal distribution for the backward problem Even though any arbitrary function could be used, in this paper we assume that f 1 (x) = 0 Since f 1 (x) is an arbitrary function, the plume spatial distribution cannot be reconstructed in the vicinity of t = 0, and errors increase as the solution is backtracked Table Homogeneous and Stationary Heterogeneous Dispersion Coefficient Configurations Configuration m s l HOM 1 0 US1 1 0 US 1 04 CS1A 1 0 x CS1B x CSA 1 04 x CSB x

5 BAGTZOGLOU AND ATMADJA: MJBBE AND QR METHODS FOR HYDROLOGIC INVERSION SBH 10-5 toward t = 0 This, as mentioned before, results in a partial reconstruction of the time release history at some time between the initial and final time In the case of a heterogeneous ADE wave transport problem, the space differential operator is @x ½ux ðþš ð7þ and f 1 (x) and f (x) are the initial and terminal data at times t = 0 and T ft, respectively [18] In the BBE method, instead of solving the ADE directly, we set where w ¼ e kt C; 0 t T ft ð8þ k ¼ 1 ln M T ft d as given by Buzbee and Carasso [1973] where d and M are bounds on the errors generated by this procedure Let such a bound be ð9þ k f T ft f k d ð10þ for the initial data, where d can be chosen to be the error between the measured and the exact values Let the bound in the resulting errors in the terminal data be k f ð0þ f 1 k M ð11þ where M is the acceptance level for the errors in the predicted terminal values Equation (9) is an exact approximation for the k parameter in the case of homogeneous media For heterogeneous cases this approximation is no longer exact but can serve as a very good estimate or starting point for searching for k [19] Taking first the derivative of equation (8) with respect to time and substituting equation (1), @x @x ½ux ðþ Š k w ð1þ Taking the derivative of equation (1) with respect to time once more, we get the auxiliary problem of @x @x ½ux ðþš k w ð13þ [0] For the purpose of solving a backward problem, we set t = T ft T bi as our present-day or initial time and t =0= T fi T bt as our terminal time (ie, the terminal condition of a forward solution, T ft, will be equal to the initial condition of a backward solution, T bi, and vice versa) The corresponding initial and terminal conditions for the auxiliary problem are: wx; ð T bi Þ ¼ e ktbi f ; wx; ð 0Þ ¼ e ktbt f 1 ¼ 0 ð14þ Boundary conditions associated with (13) are: w t ð0; t wð0; tþ ¼ wl; ð tþ ¼ 0; t 0 ð15þ Þ ¼ w t ðl; @x ½ux ðþš k t 0 w ¼ 0; ð16þ [1] Let ½ux ðþ Š kþ Using the central difference approximation for @x terms and denoting by w n a vector of w values at all discrete spatial locations at time step n, (13) becomes: ðw nþ1 w n þ w n 1 Þ t A w n ¼ 0 n ¼ 1; ; N t 1 ð17þ w Nt ¼ e ktbi f ; w 0 ¼ 0; ð18þ where N t is the number of discrete temporal points A = P,and the approximation for P is: P ¼ 1 x D iþ 1 þ D i 1 D 3 iþ 1 D i 1 D iþ 1 þ D i 1 D iþ D i 1 D iþ 1 þ D i 1 D iþ 1 5 D i 1 D iþ 1 þ D i u iþ1 u i 1 0 u iþ1 1 x k½iš u i 1 0 u iþ1 5 u i 1 0 ð19þ D i 1 is the interfacial value of D, obtained by using harmonic averaging P is a N g N g matrix, where N g is the number of grid points and [I ] is the identity matrix N g N g In matrix form, equation (17) becomes: where ¼ 6 4 w ¼ 0 ð þ A t Þ I I ð þ A t Þ I ð0þ I ð þ A t Þ I I ð þ A t Þ ð1þ is a N t N g N t N g matrix [] Once we solved the auxiliary problem and obtained w(x, t), to get the solutions sought, we apply Cx; ð tþ ¼ e kt wx; ð tþ ðþ [3] The BBE method was modified and enhanced to solve the ADE within a contaminant source identification

6 SBH 10-6 BAGTZOGLOU AND ATMADJA: MJBBE AND QR METHODS FOR HYDROLOGIC INVERSION context and with an improved computing effort requirement Detailed development and testing of the MJBBE is given by Atmadja and Bagtzoglou [000, 001a] 3 Quasi-Reversibility Method for Heterogeneous Media [4] The implementation of the QR method to recover the plume release history was originally proposed by Skaggs and Kabala [1995] who solved the ADE with homogeneous parameters only Due to its superiority, compared to the MJBBE, in solving the spatial distribution in the case of homogeneous media and its computational efficiency and natural extendability, we apply this method to heterogeneous media for comparison purposes In order to accomplish this, we modified and extended the QR method for heterogeneous media Unlike Skaggs and Kabala s formulation, which employed a moving coordinate system to account for the advective term of the ADE, we applied the QR method with the advective term being explicitly handled in the ADE [5] The QR method was first developed by Lattes and Lions [1969] They proposed this method to solve irreversible partial differential equations with reversed time They solved the diffusion equation with reversed time by replacing the diffusion operator: with the @t r er 4 ð3þ ð4þ where e is a small positive stabilization constant and r stands for the Laplacian operator Incorporating the QR method into a 1D heterogeneous ADE, @x @x ½ux 4 ð5þ Similar to the MJBBE method, where k is an important parameter, in the QR method, e is the most important stabilization parameter [6] We used a central finite difference approximation in space and Crank-Nicolson scheme in time for equation (5) Discretization of equation (5) leads to: ½CŠ jþ1 ð½šþ I Þ ¼ ½CŠ j 𽚠I Þ ð6þ where [C ] j+1 and [C ] j are the concentration vectors at time j + 1 and j, respectively, [I ] is the identity matrix N g N g, and is t x D þ t et 4x U x E Matrices D, U, and E are all N 4 g N g and are defined as: 3 D iþ 1 D i 1 D iþ 1 D i 1 D iþ 1 D i 1 D iþ 1 D ¼ D i 1 D iþ 1 D i 1 D iþ 1 D i 1 D iþ 1 D i 1 ð7þ 0 u iþ1 u i 1 0 u iþ1 U ¼ 6 4 u i 1 0 u iþ1 u i E ¼ ð8þ ð9þ 4 Results and Discussion [7] For all simulations, we ran the forward ADE simulation up to time t = T ft = and assumed this as our presentday configuration We also saved the forward results at different snapshots in time and treated these as our exact plume distribution for comparison purposes Using the MJBBE and QR methods, the concentration configuration at t = T ft = T bi is taken as f (x) Since in the MJBBE method one needs to assume the spatial distribution of the attribute of interest at t = 0 to be equal to some arbitrary function, the true spatial distribution at t = 0 will be impossible to be recovered In our analysis, we use the spatial distribution of the attribute of interest at t = 0 to be equal to zero Therefore in our synthetic examples, we assumed our terminal time to be t = T bt = 1 (ie, the furthest back we can hope to recover is t = 1) and there is no recoverable concentration distribution for t < 1 In this case, our total time of simulation becomes T sim = T bi T bt = 1 Terminal time t = 1 was chosen, because as mentioned earlier, the errors in the solution grow as the plume is marched backward toward t = 0 In using the MJBBE method there exists a trade off between accuracy of the solution and the total recovery time We want to recover the spatial distribution from t = to t = 1, which is approximately half of the duration of the forward simulation The MJBBE can be readily applied to plume recovery for different target times Details regarding the implementation of the method are given by Atmadja and Bagtzoglou [001a] [8] We solve equation (1) subject to () and (3) using the MJBBE and QR methods with x = 05 and t = 00 The following two error measurements are used as indicators of the MJBBE and QR performance: (1) mass error, normalized by the exact mass, as given by the corresponding forward simulation result, e M ¼ Mass f Mass b Mass f 100%; ð30þ () concentration peak error, normalized by the exact peak concentration, as given by the corresponding forward simulation result, e p ¼ maxðc f Þ maxðc b Þ maxðc f 100%; Þ ð31þ where superscripts f and b stand for forward and backward numerical values, respectively Errors of the results were compared at different snapshots backward in time, namely

7 BAGTZOGLOU AND ATMADJA: MJBBE AND QR METHODS FOR HYDROLOGIC INVERSION SBH 10-7 Figure 1 Comparison between MJBBE and QR methods for a homogeneous case at t b =09T sim 0% and 90% back in time (t b ) for consistency purposes However, only the 90% back in time results are presented for the sake of brevity All results are presented for x [0, 30] because no concentration changes are detected past x = 30 [9] For the homogeneous case, we were able to use a value of parameter k estimated by equation (9) with, M = k C(x, 0)k =1, d =10 9, and T bi = The estimated k value was 10 for this configuration For heterogeneous cases we were no longer able to use the same value of k However, this value served as a good estimate and starting point for the subsequent search The k value for heterogeneous cases was obtained by minimizing the mass error involved Since for nonreactive contaminant plumes mass is conserved, one can run the problem backward one time step, and check for the mass error The k value that minimizes the mass error was used to run the problem further backward Similarly, in the case of the QR method, the value of e was obtained by performing the simulation backwards for several time steps and the smallest value of e that did not introduce concentration fluctuations and gave a reasonable mass error was used 41 Homogeneous and Deterministically Heterogeneous Dispersion Field [30] For the homogeneous medium case, D(x) is set to be constant equal to one for the whole domain The plume spatial distribution recovered by both the MJBBE and QR methods is almost indistinguishable (Figure 1) However, our study indicated that the QR method is superior to the MJBBE in conserving the mass of the plume This can be seen in Figure 1 where, even though both methods give good plume shape preservation 90% back in time (ie, t b = 09T sim ), the QR method gives a better accuracy in terms of the mass error (0003%) and peak error (858E-9%) On the other hand, at t b =09T sim, the MJBBE method generated a mass error of 00% and peak error of 05% [31] The simplest case of heterogeneity that was tested in this paper was generated by assigning different zones of dispersion coefficients, as explained in Section Despite the fact that the QR method is superior in solving the homogeneous cases, the MJBBE method is more accurate and more robust in handling heterogeneity A value of k = 87 was used to recover the spatial distribution from the beginning to the end of the backward simulation The MJBBE method is able to preserve the initial data with a very good accuracy Although the value of the errors does not vary greatly from one case to another, the MJBBE gives better results in heterogeneous media where the inner zone has a higher value of D (configuration DET1) The mass errors for the DET1 configuration are the same as the homogeneous case, with values of 0005% at t b =0T sim and 00% at t b =09T sim On the other hand, the peak errors are slightly better for the DET1 case The peak error for DET1 at t b =0T sim is 003% compared to 008% for the homogeneous case Further back in time, at t b =09T sim, the peak error for DET1 was still better than that of the homogeneous case with a value of 0% for the heterogeneous case DET1 compared to 05% for the homogeneous case (Figure versus Figure 1) When the value of D(x) in the inner zone is lower (DET), the mass error is increased to 001% at t b =0T sim and to 007% at t b =09T sim Similarly, the peak errors are worse compared to the DET1 case, with values of 009% at t b =0T sim and 035% at t b =09T sim [3] In contrast to the homogeneous case, for the QR method we can no longer use one value of e to reconstruct the plume spatial distribution from initial to terminal time Moreover, we were unable to find a single value of e to match the peak concentration For consistency of comparison between the MJBBE and QR methods, we used one value of e for the backward simulation In addition to difficulty in finding an e value that gives acceptable mass and peak errors, one can see that the QR results start exhibiting fluctuations and give negative concentrations around x = 7 (Figure ) We were able to keep the mass error to be small ( 0018%) at t b =0T sim, but the peak error grows to 317% Although the mass error is still small at t b =09T sim (08%), the peak error grows very rapidly to 5818% In addition, the location of the maximum peak concentration generated by the QR method was shifted to the left compared to the forward result One downside of the mass-conservativeness of the QR method is that it creates negative concentrations due to undershoot errors The heterogeneity also affects the QR method by smoothing out the results, thereby missing the salient features of the reconstructed plume Similar to the MJBBE in configuration DET, the QR method also generated worse mass and peak errors and concentration fluctuations Figure Comparison between MJBBE and QR methods for a heterogeneous D configuration (DET1) at t b =09T sim

8 SBH 10-8 BAGTZOGLOU AND ATMADJA: MJBBE AND QR METHODS FOR HYDROLOGIC INVERSION Table 3 Summary of Errors, Stabilization k and e Values for Various Heterogeneous Dispersion Coefficient Fields for the MJBBE and QR Methods MJBBE e M,% e P,% QR e M,% e P,% Configuration k 0T sim 09T sim 0T sim 09T sim e 0T sim 09T sim 0T sim 09T sim HOM E <1E-8 <1E-8 DET DET US E US E-5 494E CS1A E-6 338E CS1B E-6 138E CSA E CSB E UN UN CN CN These results are summarized in tabular form only for the sake of brevity Another controlling parameter in the QR method is the time step In our analysis, using a smaller time step improved the mass error Decreasing the time step t 0 times, from t = 00 to 0001 improved the mass error from 509% to 051% at t b = 09T sim for the DET configuration However, the peak error remained to be 339% and the concentration fluctuations persisted For the analysis we performed, reducing the t by a factor of 0, increased the CPU time by 3 times Finally, unlike the MJBBE, the QR method was not able to preserve the shape of the input data It appears that the QR method tends to smooth out the results and cannot reproduce the sharp interfaces in the true solution at dispersion zone boundaries While matching the peak value, larger negative concentrations are created by the method Summary of the errors for the deterministically heterogeneous dispersion field cases can be found in Table 3 4 Random Stationary Heterogeneous Dispersion Field [33] The next level of complexity in creating the heterogeneity is to randomly perturb the dispersion coefficient around the mean values We generated both uncorrelated and correlated stationary heterogeneous fields (Table ) Configuration US1 is the uncorrelated field and it has a mean dispersion coefficient value of 1 with a CV of 0% The errors created by MJBBE are comparable to the homogeneous dispersion case with a D value of 1 (Figure 3b) The small variability in D (Figure 3a) did not affect the recovered spatial distribution drastically The results for t b = 09T sim, indicated that the mass error remains the same (00%) However, the peak error for the US1 configuration is slightly better than the homogeneous case with a value of 045% compared to 05% for the homogeneous case (Figure 3b versus Figure 1) Increasing the CV of the D field to 40% (configuration US) for the uncorrelated field does not affect the performance of the MJBBE The mass errors remained the same as those of configuration US1 with a value of 0005% at t b =0T sim and 00% at t b = 09T sim Similarly, the peak errors remained the same as those of configuration US1 [34] For the QR method, a value of e = 1 was used in configuration US1 The method still shows the mass conservativeness with a mass error of 101E-4% at t b =0T sim and 001% at t b =09T sim (Figure 3b) Whereas the mass is Figure 3 Comparison between MJBBE and QR methods for a heterogeneous D configuration (US1) (a) dispersion coefficient field, (b) result at t b =09T sim

9 BAGTZOGLOU AND ATMADJA: MJBBE AND QR METHODS FOR HYDROLOGIC INVERSION SBH 10-9 we were not able to use the same value of e for the QR method Due to the change in e, from 1 for the uncorrelated fields to 05 for the correlated field with l =x, the peak error (099%) is smaller at t b =0T sim compared to the uncorrelated field However, the peak position lags behind the forward solution The peak position phase lag is more pronounced further back in time at t b =09T sim (Figure 4b) and again the peak error (1589%) is better compared to that of the uncorrelated field (188%) Increasing the correlation length to l = 4x (configuration CS1B) affected the behavior of the QR method Using the same value of e = 05, the peak position phase lag improved and it is less than that of configuration CS1A In addition to the peak position, the negative concentrations are also less pronounced at t b =09T sim Eliminating the negative concentrations brings the peak concentration upward and, Therefore the peak error is also improved from 1589% (Figure 4b) to 1336% (Figure 5b) As a consequence of the better-constrained spatially correlated field, the performance of the QR method is better for larger correlation lengths This is consistent with the poor QR behavior for deterministic heterogeneous cases where sharp dispersion boundaries exist Summary of the mass and peak errors for stationary fields can be found in Table 3 [37] The effect of increasing the CV to 40% for the correlated field with l = x (configuration CSA) on Figure 4 Comparison between MJBBE and QR methods for a heterogeneous D configuration (CS1A) (a) dispersion coefficient field, (b) result at t b =09T sim conserved, the peak error is significantly large Even at t b = 0T sim the value of the error is already 550% and it grows further back in time t b =09T sim to 188% (Figure 3b) Resembling the deterministically heterogeneous cases, the QR method generates negative concentrations especially further back in time Figure 3b shows the concentration fluctuations around x = 8 to 11 and x = 0 to Unlike the MJBBE method, increasing the CV to 40% makes the QR peak errors worse The peak error at t b =0T sim was 945% and at the later time (t b = 09T sim ), the peak error was increased to 44% Once again we noticed the negative concentrations created by the method at x =0to3andx = 8 to 11 These fluctuations are larger for the US configuration compared to those of configuration US1 [35] In the MJBBE analysis, for the correlated heterogeneous field with l =x and CV of 0% (configuration CS1A), we were able to use the same value of k as we used for the uncorrelated case The mass errors are the same as for the uncorrelated D field (Figure 4b versus 3b) However, the recovered peak errors are now slightly larger than those of the uncorrelated field, with peak errors of 009% and 047% at t b = 0T sim and t b = 09T sim, respectively Increasing the correlation length to l = 4x, did not affect significantly the mass or the peak errors compared to those of the l =x field (Figure 5b) [36] While we were able to use the same value of k for uncorrelated and correlated fields in the MJBBE method, Figure 5 Comparison between MJBBE and QR methods for a heterogeneous D configuration (CS1B) (a) dispersion coefficient field, (b) result at t b =09T sim

10 SBH BAGTZOGLOU AND ATMADJA: MJBBE AND QR METHODS FOR HYDROLOGIC INVERSION UN1 Further back in time, the mass error is consistently good, with a value of 00% (Figure 6b) Even though the peak error grows about almost ninefold from 003% at t b = 0T sim to 06% at t b =09T sim, the error value is still significantly less than 1% [39] The QR method does not perform well in the case of nonstationary heterogeneous fields In heterogeneous media, the mass conservative nature of the QR still holds However, the value of the errors is no longer in the neighborhood of 10 5 such as in the stationary cases The mass error for configuration UN1 increased from 00% at t b =0T sim to 035% at t b =09T sim (Figure 6b) The mass error increase does not seem to be too significant However, we observed a large concentration fluctuation especially further back in time (Figure 6b) Whereas mass errors are increased slightly, the peak errors were increased significantly to 47% at t b =0T sim and 5600% at t b =09T sim [40] For configuration CN1, where we have a correlation length l = 4x, the MJBBE results showed that the correlation length does not have any effect on the MJBBE performance Both mass and peak errors remained the same, with an exception of the peak error at t b =0T sim where the value increased from 003% to 0034% for UN1 and CN1, respectively Furthermore, the MJBBE performance is consistent with the previous results, namely it preserves the shape of the input data Figure 6 Comparison between MJBBE and QR methods for a heterogeneous D configuration (UN1) (a) dispersion coefficient field, (b) result at t b =09T sim the MJBBE performance is negligible The same mass errors were observed as for configuration CS1A at t b = 0T sim (Table 3) However, at t b =09T sim, the mass error is better for the higher CV case with a value of 003% compared to 009% for configuration CS1A On the other hand, the peak error at t b = 09T sim becomes worse for configuration CSA where the CV is higher The value of the peak error was increased from 047% for configuration CS1A to 055% for configuration CSA Increasing the correlation length to l = 4x (CSB) improved the peak error slightly at t b =09T sim but the mass error is worse (Table 3) The QR method performs better for the higher CV correlated field (CSA) in terms of the peak error, especially at t b =09T sim The peak error was improved from 747% to 066% Similarly for configuration CSB where the correlation length is l =4x, the peak error at t b = 09T sim was improved from 1336% to 671% (Table 3) 43 Random Nonstationary Heterogeneous Dispersion Field [38] Four different configurations were analyzed for the nonstationary heterogeneous fields (Table 1) When we deal with a configuration with zone 1 having a lower value of m than zone, as is the case for configurations UN1 and CN1 (Figures 6a and 7a), the MJBBE created smaller errors The mass error is only 0005% at t b =0T sim for configuration Figure 7 Comparison between MJBBE and QR methods for a heterogeneous D configuration (CN1) (a) dispersion coefficient field, (b) result at t b =09T sim

11 BAGTZOGLOU AND ATMADJA: MJBBE AND QR METHODS FOR HYDROLOGIC INVERSION SBH CN1 We observed an improvement of about to 6% compared to the uncorrelated spatially distributed media At t b =0T sim, the peak error was 333% and at t b = 09T sim, the value was 503% (Figure 7b) [4] In our heterogeneous configuration UN where m o > m i, the MJBBE exhibits the worst mass and peak errors This is consistent with our observation for the deterministically heterogeneous media The value of the mass error is 001% for t b =0T sim and it grows to 006% for t b =09T sim Adding spatial correlation to the heterogeneous field, which translates into increasing the correlation length from 0x to 4x (configuration CN), improved the performance of the MJBBE in terms of the mass error However, the MJBBE generated a worse peak error at t b =09T sim compared to the uncorrelated field [43] Unlike the MJBBE method, the QR method performed better when m o > m i Att b =0T sim the peak errors for the uncorrelated field (UN) are 003% and 44% at t b =0T sim and t b =09T sim, respectively We were not able to obtain a value for e that can reconstruct the spatial distribution with a good accuracy No matter what value of e we used, we always observed concentration fluctuations, which are more pronounced at later times For the correlated field, both mass and peak errors are better than that of the uncorrelated one (Table 3) We also observed that the peak concentration recovered by the QR method lags compared to the forward simulation, especially at t b =09T sim [44] Consistent with the previous observation for configuration CN1, the correlation length helps the performance of the QR method in terms of the values of the mass and peak errors Even though the values of the errors for the QR method are improved by adding spatial correlation, the concentration fluctuations are worse We were able to obtain a peak error of 179% at t b =09T sim, but the price to be paid is the increase in the observed negative concentrations Summary of the errors, optimal k, and e values for both methods can be found in Table 3 44 Initial Data Uncertainty Effect [45] In the analyses of Sections 41, 4, and 43 we have been evaluating examples with exact initial data In reality, however, the contaminant plume input data are never exact since there exist some measurement errors To account for measurement errors, we added random deviates to the exact data The initial data with uncertainty are: Figure 8 Comparison between MJBBE and QR results for 1% uncertainty in the initial data at (a) t b =0T sim,(b)t b =06T sim, and (c) t b =09T sim [41] The QR method gives the worst peak errors for cases where m i > m o (configuration UN1 and CN1) Similar to the MJBBE results, in the case of the QR method, the correlation length does not affect the mass errors at t b = 0T sim However, the correlation length helps the QR method performance further back in time At t b =09T sim, the mass error was improved from 035% for UN1 to 09% for CN1 (Figure 6b versus 7b) The effect of correlation length in the domain is more pronounced for the concentration peak of the recovered plume The peak errors are better for configuration C uncert ðt bi Þ ¼ C exact ðt bi ÞþwzC exact ðt bi Þ ð3þ where C exact (T bi ) is a vector of exact concentration values at t = T bi, w is a scaling factor, and z is a vector of random deviates with mean of zero and standard deviation of one A truly additive error was not used in this work since on the average it would result in more negative concentrations, which is not realistic We applied three different levels of uncertainty to the homogeneous dispersion field case These correspond to scaling factors of 001, 005, and 01 The results showed that the QR method is more robust than the MJBBE method when the initial data are plagued by measurement errors [46] The MJBBE method was able to recover the spatial distribution up to t b =06T sim when the initial data involve a 1% uncertainty (w = 001) Even though we were able to