## 1. Introduction

[2] When determining the effect of historical groundwater contamination, the distribution of a plume at a given point back in time is often required to establish exposure of wells or individuals to the contaminant. For example, in the case of *Woodrow Sterling et al. versus Velsicol Chemical Corporation* [1986], a class of people who owned property in the vicinity of a chemical waste burial site sought damages for personal injury and damages to their property suffered when water in their home wells became contaminated by hazardous chemicals escaping from Velsicol's site. Velsicol admitted that some of the wells were contaminated with chemicals from its waste burial site, but disputed that all members of the class had been exposed and did not agree with the plaintiffs as to the intensity and duration of exposure. Therefore the case centered on estimating the past distribution and concentration of the chemical plume, in order to determine concentrations in the plaintiffs' wells at given times [*Michalak*, 2001]. Emerging inverse modeling methods can be applied to solve such problems.

[3] One set of inverse methods is based on geostatistical principles and allows for the estimation of unknown functions based on the dual criterion of reproducing available observations while maintaining an assumed correlation structure. Methods falling under this category have been used for some time for estimating subsurface hydraulic conductivity or transmissivity distributions based on hydraulic head and other data [e.g., *Kitanidis and Vomvoris*, 1983; *Kitanidis*, 1995; *Zimmerman et al.*, 1998]. More recently, these types of methods have also been applied to contaminant release history identification in groundwater systems [*Snodgrass and Kitanidis*, 1997; *Michalak and Kitanidis*, 2002, 2003, 2004]. In this paper the geostatistical method is extended to the estimation of the antecedent distribution of a contaminant at a given point back in time, making it applicable to cases such as the one described.

[4] In the geostatistical approach to inverse modeling, the solution involves the calculation of a sensitivity matrix relating each point in the discretized unknown function to each observation, which typically requires one forward run for each point in the discretized unknown function. Because inverse problems associated with groundwater systems are typically strongly underdetermined, in the sense that the number of points in the discretized unknown function *m* is greater than the number of available measurements *n*, the computational cost of calculating the sensitivity matrix can be prohibitive. This is especially true when the function to be estimated is itself multidimensional. In this work the adjoint state method is used to efficiently populate the full sensitivity matrix by solving a series of adjoint problems instead of the traditional approach of solving a series of forward problems. The combination of the adjoint and geostatistical methodologies makes the identification of a multidimensional contaminant distribution in a heterogeneous domain feasible.

[5] Note that throughout this paper, we will use the term “historical contaminant distribution” to describe the plume at a single, given point in the past. We avoid using the term prior distribution so as to prevent confusion with the terms “prior” and “posterior,” which have a different, very specific definition in the context of Bayesian inverse modeling. Also, although in the presented applications the historical distribution will be recovered for a single point in time, the presented algorithm could directly be applied for a series of times, yielding a time-dependent description of the history of a plume.