A Reassessment of the Groundwater Inverse Problem

Abstract

This paper presents a functional formulation of the groundwater flow inverse problem that is sufficiently general to accommodate most commonly used inverse algorithms. Unknown hydrogeological properties are assumed to be spatial functions that can be represented in terms of a (possibly infinite) basis function expansion with random coefficients. The unknown parameter function is related to the measurements used for estimation by a “forward operator” which describes the measurement process. In the particular case considered here, the parameter of interest is the large‐scale log hydraulic conductivity, the measurements are point values of log conductivity and piezometric head, and the forward operator is derived from an upscaled groundwater flow equation. The inverse algorithm seeks the “most probable” or maximum a posteriori estimate of the unknown parameter function. When the measurement errors and parameter function are Gaussian and independent, the maximum a posteriori estimate may be obtained by minimizing a least squares performance index which can be partitioned into goodness‐of‐fit and prior terms. When the parameter is a stationary random function the prior portion of the performance index is equivalent to a regularization term which imposes a smoothness constraint on the estimate. This constraint tends to make the problem well‐posed by limiting the range of admissible solutions. The Gaussian maximum a posteriori problem may be solved with variational methods, using functional generalizations of Gauss‐Newton or gradient‐based search techniques. Several popular groundwater inverse algorithms are either special cases of, or variants on, the functional maximum a posteriori algorithm. These algorithms differ primarily with respect to the way they describe spatial variability and the type of search technique they use (linear versus nonlinear). The accuracy of estimates produced by both linear and nonlinear inverse algorithms may be measured in terms of a Bayesian extension of the Cramer‐Rao lower bound on the estimation error covariance. This bound suggests how parameter identifiability can be improved by modifying the problem structure and adding new measurements.

Advances in the use of observed spatial patterns of catchment hydrological response

“…The last ten years have seen the development of several new methods for the calibration and testing of models using time-series data. Sorooshian and Gupta [92] and McLaughlin and Townley [75] present reviews of approaches to calibration and automated optimisation in surface and groundwater models respectively. They highlight a range of problems such as the presence of multiple optima, strong correlations between parameters, and the subjectivity associated with choice of objective functions.…”

A multiscale approach for estimating solute travel time distributions

“…The linearization process is well documented in the stochastic groundwater literature [14,17,18]. A simple approach is to assume that the deviations df x Y dh x , and ds are small.…”

“…It is helpful for estimation purposes to distinguish dependent variables (such as hydraulic head and travel time) from independent variables (such as hydraulic conductivity) which are more di cult to observe or model [14]. Independent variables are the primary sources of uncertainty in hydrologic estimation problems and are often characterized as random ®elds with speci®ed statistical properties.…”

“…A reasonable and widely used choice for the optimal estimate is the linear leastsquares estimate. This estimate z is the conditional mean of z given y, written as E zjy , when the G Á operators are linear and z and v are normally distributed [4,14]. Moreover, in this case the estimation error covariance is equal to the conditional covariance of z given y.…”

Data fusion modeling for groundwater systems

“…Ž . Dougherty 1996, McLaughlin and Townley 1996and Daniel and Willsky 1997 , the limitations still remain. Part of our contribution is to recognize that the difficulty is the Ž .…”