The inverse problem of estimating the conductivity function from head observations is generally ill posed: Many conductivity functions are consistent with the data. It is widely accepted now that a well-defined estimate can be obtained only if additional information about the function structure is introduced into the problem formulation. This work presents a method to obtain a stable and reasonable estimate that utilizes only the data and the flow or transport model with the minimum possible suppositions about the unknown function or its structure. The motivation is to develop a solution that has only characteristics that are traced directly to the data and the flow or transport model, without taking advantage of spatial continuity or other prior information. The solution is obtained by minimizing the upper bound to the error, or, in a stochastic conceptual framework, as the most likely solution given the data. This solution, although generally not the most accurate since it neglects to utilize structural information that may be available, is of fundamental importance and may be useful as a benchmark. For example, by comparing this solution with other solutions, one can become aware of how prior information or the model of spatial structure affects the solution to the inverse problem.¿ 1997 American Geophysical Union