## 1. Introduction

[2] In stochastic approaches to the solution of inverse problems, the unknown parameters are described through probability distributions. As such, estimation uncertainty is recognized and its importance can sometimes be determined. This is in contrast to deterministic approaches, where one seeks a single value for the unknown parameters.

[3] The geostatistical approach to inverse modeling is a stochastic, Bayesian approach. This general methodology has been used for a variety of environmental applications, including, among others, the estimation of hydraulic conductivity fields in groundwater systems based on hydraulic head measurements [*Dagan*, 1985; *Rubin and Dagan*, 1987a, 1987b, 1988, 1989; *Rubin et al.*, 1992; *Kitanidis*, 1996; *Zimmerman et al. 1998*] and the identification of the contamination history based on the current distribution of a contaminant [*Snodgrass and Kitanidis*, 1997; *Michalak and Kitanidis*, 2002, 2003].

[4] In many environmental applications, measurements are sparse and imperfect. This information sparsity can lead to high uncertainty in the estimation of the unknown function. The ability to incorporate any additional available information, such as known constraints on parameter values, would improve the precision with which an unknown function can be identified.

[5] In environmental applications, methods for dealing with nonnegativity have been almost exclusively limited to (a) the application of data transformations to the original variable, yielding a transformed variable that is defined over an infinite domain but that corresponds to the original variable defined only in the nonnegative parameter range; and (b) the use of Lagrange multipliers in constrained optimization based on probability density functions that would otherwise allow the variables to become negative.

[6] Data transformations, a standard method in statistics, have been applied in interpolation and inverse problems in, for example, *Kitanidis* [1997, p. 70] and *Snodgrass and Kitanidis* [1997]. The most common of these is the power transformation, which is defined as

where **s** is the vector of values in the original domain, is the transformed data vector, and κ is a constant selected based on the application. The commonly used logarithmic transformation is included as a special case of the power transformation, obtained at the limit of κ tending to zero. For example, it is common in hydrogeologic analysis to use the logarithm of the hydraulic conductivity field [see, e.g., *Hoeksema and Kitanidis*, 1985]. A few alternate data transformation methods have also been proposed for environmental optimization problems [see, e.g., *Kauffmann and Kinzelbach*, 1989].

[7] In general, however, data transformations render a linear inversion problem nonlinear and can lead to highly nonsymmetric probability density functions for the unknown parameter values in the untransformed space, which is unrealistic in some cases. Such difficulties have been documented for large-variance cases, for example, by *Snodgrass and Kitanidis* [1997]. Specifically, the confidence intervals tend to be unreasonably narrow for low values of the unknown function and wide for high values.

[8] The other popular method, the use of Lagrange multipliers for obtaining nonnegative best estimates of unknown functions, consists of expanding the original objective function *f*(**s**) into the Lagrange function

where **s** must satisfy the constraints *g*_{i}(**s**) = *b*_{i} or *g*_{i}(**s**) ≥ *b*_{i}, *k* is the total number of active constraints, and **λ** = (λ_{1}, λ_{2}, …, λ_{k}) are Lagrange multipliers [see, e.g., *Gill et al.*, 1986]. The solution method involves setting the derivative of the Lagrange function with respect to *s*_{j} and λ_{i} equal to zero. For inequality constraints, the Lagrange multipliers of the points corresponding to active constraints must be positive. If all that is required is a single estimate of the unknown function, then the use of Lagrange multipliers in conjunction with linear inversion methods can be justified as a means to make this estimate nonnegative. When confidence intervals or conditional realizations are needed, however, the use of this method raises some issues. Specifically, realizations obtained using the method of Lagrange multipliers will not be equiprobable samples from the assumed form of the posterior distribution of the unknown function, if this distribution did not itself implicitly assume nonnegativity.

[9] Contrary to these methods, the method presented in this paper is based on a prior probability density function that has a nonzero value only in the nonnegative parameter range. This pdf is derived based on the method of images applied to reflected Brownian motion. The probability density function of a particle undergoing Brownian motion is well known, and reflection at *s* = 0 can be enforced by using the method of images to modify the original pdf.