## 1. Introduction

[2] The increasing interest in environmental issues has led to greater attention to the quality of groundwater. The huge number of pollution events, which have occurred over the last two decades, makes the protection and restoration of groundwater of utmost importance. Scientific efforts in the subsurface flow study field have been primarily focused on flow and transport characteristics and on the corresponding parameter identification issues. Since 1990 increasing attention has been paid to the problem of recovering the release history of a pollutant. This is because the release history can be a useful tool for assessing how to share the costs of remediation of a polluted area among the responsible parties [*Skaggs and Kabala*, 1994]. Moreover, the knowledge of the release history gives information on the spread of future pollution and permits the improvement of remediation plans [*Liu and Ball*, 1999]. From a legal and regulatory point of view, it is also important to determine the release time period and the highest values of concentration released [*Snodgrass and Kitanidis*, 1997].

[3] The problem of deducing a release history from concentration data, which are measured in a limited set of locations in the aquifer, belongs to the *inverse problem* class whose solutions do not satisfy the mathematical requirements of existence, uniqueness and stability. A few methods have been proposed in literature for the recovery of a release history, using different approaches [*Skaggs and Kabala*, 1994; *Snodgrass and Kitanidis*, 1997; *Woodbury et al.*, 1998]. Among these methods, we have found that the geostatistical approach (GA), set up by *Snodgrass and Kitanidis* [1997] for a 1-D flow and transport case, is a promising tool for further practical developments.

[4] This study deals with an extension of the GA and the subsequent testing of the new developments. Some preliminary results of the first improvement of the procedure, i.e., the extension from 1-D to 2-D, are given by *Butera and Tanda* [2001]. A brief description of the GA follows to introduce the proposed extensions; for more details on the method, see *Snodgrass and Kitanidis* [1997].

[5] According to *Snodgrass and Kitanidis* [1997], the restored release history, s(t), can be considered a random process, defined through its probability density function and its statistical moments, as it cannot be determined without uncertainty, due to the dispersion phenomena; s(t) can be usefully defined as an unknown random N × 1 vector, obtained from the discretization of the unknown function. The concentration data z_{i}, observed at M locations and at time T, are related to **s**(t) through the equation **z** = **h**(**s**,**v**) + **r**, where **v** is a vector that includes the aquifer parameters and **r** is the measurement error vector (bold characters denote vector or matrix). Assuming that the aquifer parameters are known, **h**(**s**,**v**) reduces to **h**(**s**). For a conservative solute, the relation between the observed concentration **z** and the solute input **s** is linear, therefore it is possible to write **z** = **Hs** + **r** where **H** is the transfer matrix of M × N size.

[6] The methodology is applied in two steps. First, the structural analysis is performed to estimate the parameters of the random process, the mean coefficient vector (**β**) and the parameter vector of covariance function (**ϑ**). The elements of the vector **s**(t) are then determined, and the error covariance matrix is computed (for details, see *Kitanidis* [1995, 1996] and *Snodgrass and Kitanidis* [1997]). *Snodgrass and Kitanidis* [1997] suggested to apply a useful transformation, from **s** to , to constrain the solution to be nonnegative, as the nature of the concentration requires: = α(**s**^{1/α} − 1) where α is a positive number so that > −α. For the analysis of the statistical properties of s(t), given , see *Kitanidis and Shen* [1996].