Exploiting Laguerre Functions to Regularize Contaminant Source History Recovery Problems

The problem of recovering contaminant source histories in the subsurface represents a highly multidimensional (in reality, infinite-dimensional) inverse problem, even if the location of the source and all groundwater parameters are known precisely. Essentially, one attempts to determine a time series of concentrations at the source location based on concentrations measured down gradient. The inverse problem defined by the advection dispersion equation is known to be unstable, and cannot be solved for real data without some sort of regularizing constraint, usually enforcing temporal smoothness. A number of techniques for this task have been previously discussed in the literature, including Bayesian techniques and classical regularization techniques, such as Tikhonov regularization. Recently the authors have presented a new technique exploiting the convolution and orthogonality properties of Laguerre functions to model contaminant transport between a source and a receptor as an input-output system, using only linear algebra. This technique generalizes well to inverse modelling, an approach that will be presented for the first time. The new inverse technique works by expanding both an analytic forward model and observed monitoring well data in the same Laguerre function basis, and then computing, by linear algebra alone, the inverse solution. The inversion of the low-order terms of the Laguerre series can be shown to be numerically stable, and also to enforce smoothness on the recovered solution (implicitly containing a sort of regularizing constraint). This approach has a number of advantages over existing techniques. It allows direct estimation of the source history, obviating the need for running multiple forward models (as in MCMC Bayesian approaches), and avoids the need for selecting an arbitrary penalty functional (as in Tikhonov regularization). Simulation results comparing the performance of this approach to existing approaches will be presented.