A statistical parameter estimation method using singular value decomposition with application to Avra Valley aquifer in southern Arizona

Abstract

Inverse modeling of aquifers usually involves identification of effective parameters such as transmissivities over a finite number of subregions, or zones. Theoretical restrictions on the maximum size of a zone for which such effective transmissivities can be properly defined and the desire to obtain a good resolution of the spatial variability of transmissivity may suggest that the aquifer be divided into numerous small zones. Considerations of parameter identifiability, on the other hand, may require that the number of unknown transmissivities be limited. To satisfy both requirements, an inverse approach has been developed in which the number of zones can be as large as deemed necessary on the basis of hydrogeological considerations. However, instead of trying to estimate a similar number of transmissivities, a smaller number of surrogate parameters, which are defined as linear combinations of the original log transmissivities, is estimated. The optimum number and definition of the surrogate parameters are determined through a singular value decomposition of a matrix arising from the linearization of the inverse problem. A "resolution matrix" and an "information density matrix" can also be obtained from the singular value decomposition. The resolution matrix is indicative of parameter identifiability and is valuable in deciding whether specific log-transmissivity zones should be lumped with their neighbors or left intact. The information density matrix shows how well the model can reproduce each measured hydraulic head value and may be used to determine the relative worth of each datum point for parameter estimation. This, in turn, may suggest discontinuing the collection of certain data and/or starting to collect data at other points in the aquifer. The methodology is illustrated by using data from the Avra Valley aquifer of southern Arizona.