A statistical approach to the inverse problem of aquifer hydrology: 3. Improved solution method and added perspective

Abstract

Paper 1 of this sequence presented a new statistically based approach to the problem of estimating spatially varying aquifer transmissivities on the basis of steady water level and flux data. Paper 2 described a case study in which the new method had been applied to actual field data from the Cortaro Basin in Southern Arizona. The purpose of paper 3 is to introduce a new efficient method of solution which works under a much wider range of conditions than the method employed in papers 1 and 2. The new method is based on a variational theory developed by Chavent (1971), which is extended here to the case of generalized nonlinear least squares. The method is implemented numerically be a finite element scheme. The inverse problem is posed in terms of log transmissivities instead of transmissivities and is solved by a Fletcher‐Reeves conjugate gradient algorithm in conjunction with Newton's method for determining the step size to be taken at each iteration. The method does not require computing sensitivity coefficients, and one may therefore expect it to result in considerable savings of both computer storage and computer time. Posing the problem in terms of log transmissivities is shown to have important advantages over the traditional approach, not the least of which is guaranteeing that the computed transmissivities will always be positive. The paper includes a theoretical analysis of the effect that various errors corrupting the data and the model may have on the final log transmissivity estimates. This analysis shows that small errors in the model and in the flow rate and sink/source data have only a minor influence on the log transmissivity estimates and therefore can often be disregarded. On the other hand, low‐amplitude noise in the water level data may cause these estimates to become unstable and therefore must always be filtered out during the solution of the inverse problem. Two theoretical examples are included to demonstrate the ability of the new method to deal with artificial noise of a relatively large amplitude, derived from a given stochastic model. The results demonstrate that the inverse method may be capable of computing log transmissivity estimates with an error variance which is significantly smaller than that of the original (or prior) log transmissivity data. The variance reduction achieved in this manner is shown to depend on the quantity and quality of data describing the flow regime. Finally, it is shown that undercalibration (when the variance of the computed residuals exceeds the error variance of the ‘Observed’ water levels) and/or overcalibration (when the variance of the residuals is less than the error variance of the ‘Observed’ water levels) may lead to relatively poor results, whose error variance can be so large as to render the inverse solution useless.