L ITERATURE R EVIEW

Groundwater transport mainly contains advection and dispersion processes, which are irreversible. Therefore, modeling the contaminant transport using reversed time is an ill-posed problem. The implications of this problem are twofold. First, the ill-posed problem is extremely sensitive to errors in the input data, so small errors in the measurement of existing plume may drastically change the recovered source release history. Second, the ill-posed problem results in unstable numerical schemes making it impossible to run transport models with reversed time and obtain an accurate contamination history (Skaggs and kabala, 1994).

Various methods were proposed in literature to solve the problem of source identification in the past two decades. Atmadja and Bagtzoglou (2001) reviewed the methods that had been developed to identify the contaminant source location and recover the time-release history. They classified the contaminant transport inversion methods into four categories. They are: direct approaches, analytical solution and regression approaches, probabilistic and geo-statistical simulation approaches, and optimization approaches.

1.2.1 Direct approaches

Various methods are also applied to solve the Fredholm integral equation in the

mathematical field. Amato and Hughes (1991) used a regularization method by minimizing the functional of the Fredholm integral equation of the first kind numerically. Conditioning on the data and the regularization parameter, this procedure was shown to be a correct regularization method. Several numerical experiments were given and comparisons with Tikhonov regularization (TR) schemes were also presented. Hansen (1992) reviewed several numerical tools that can be applied for the analysis and solution of systems of linear algebraic equations originated from Fredholm integral equations of the first kind. Those tools were developed on the basis of the singular value decomposition (SVD) and the generalized SVD which can be used to study many details of the integral equation.

Lamm (1995) generalized the idea of Beck (1985) in solving the heat flow problem and viewed that method as one in a large class of regularization methods. The solution of an ill-posed first kind Volterra equation is converted to be the limit of a sequence of well-posed second kind Volterra equations.

Skaggs and Kabala (1994) used Tikhonov regularization to solve the solute transport equations reversely and recover the spatial release history of the contaminant plumes in a one-dimensional (1-D), homogeneous system. Perhaps, TR is the most widely used technique for regularizing discrete ill-posed problems (Aster et al., 2005). Basically, TR is to transfer the ill-posed problem to a

well-posed minimization problem and find the best value of the regularization parameter via the method of Lagrange multipliers. In addition, Skaggs and Kabala (1995) also applied the quasi-reversibility (QR) method to the same problem solved by TR and employed a Monte Carlo methodology to recover the release history of an arbitrary plume in a medium with dispersive properties. Woodbury and Ulrych (1996) used minimum relative entropy (MRE) approach to recover the release history of a pollutant for 1-D transport with constant known velocity and dispersivity system. Fundamentally, MRE is an information-theoretic method in solving the problems. They showed that MRE method yields exact expressions for the expected values of the linear inverse problem and the posterior probability density function (pdf) if given prior information of an upper and lower bounds, a prior bias, and constraints in terms of measured data. Woodbury et al. (1998) extended the MRE method to recover the source release history of a three-dimensional plume. They pointed out that the relative entropy measure can indicate the reduction in uncertainty between the posterior and prior pdfs if the new information provided by the physical constrains and data.

1.2.2 Analytical solution and regression approaches

Lawson and Hanson (1995) proposed the least squares (LS) and Stark and Parker (1995) used the bounded valuables least squares (BVLS) for recovering the

release history. Aster (2005) also applied both LS, BVLS to the inverse problems and gave an example for the illustration of the recovery of the release history. The problem of solving for a least squares solution with LS and BVLS includes the minimizing or maximizing a linear function to bounds constraints and that solutions to this problem can be estimated. Sun et al. (2006) formulated a new variant of the robust least squares (RLS), called constrained robust least squares (CRLS) and allowed for imposing nonnegativity constraints, for identifying the contaminant source release histories. Originated in the field of robust identification, the RLS estimator considers the errors arising from model uncertainty and reduces the sensitivity of the optimal solution to perturbations in model and data. The authors demonstrated the use of CRLS in solving one- and two-dimensional test problems in the ill-conditioned and uncertain system and showed that CRLS gave much better performance than its classical counterpart, the nonnegative least squares.

1.2.3 Probabilistic and geostatistical simulation approaches

Butera and Tanda (2003) utilized a geo-statistical approach to identify the probability of the source location for the same problem solved by TR. Their applications focused on the case of non-point and multiple sources in a 2-D groundwater flow system of an infinite domain. Boano et al. (2005) also applied geo-statistical method to identify the contaminant sources in the river pollution

problems.

1.2.4 Optimization approaches

Sayeed and Mahinthakumar (2005) developed a parallel simulation-optimization framework including genetic algorithms and several local search approaches for solving PDE-based inverse problems. Their hybrid optimization algorithms were demonstrated to recover the groundwater contaminant source release history successfully. Newman et al. (2005) applied a hybrid method based on the simulated annealing and minimum relative entropy to estimate the magnitude and transverse spatial distribution of mass flux through a plane. When applying to a numerically generated test problem and a tracer experiment, the results demonstrated that the hybrid method is a very effective tool in inferring the contaminant mass flux probability density function, expected flux values, and confidence limits. Chen and Yeh (2006) used simulated annealing (SA) in incorporating with an exponential type of source release function and a fundamental solution of the groundwater transport equation to recover the release history of a groundwater contamination. The SA generates trial values for the parameters in the assumed release function expressed in terms of exponential functions. The simulated concentrations are then obtained from the fundamental solution with the trial source release function. While minimizing the sum of square errors between

the simulated and sampling concentrations, SA can determine the optimal

parameters of the assumed release function.