Estimation of Aquifer Parameters Under Transient and Steady State Conditions: 1. Maximum Likelihood Method Incorporating Prior Information

Abstract

In this series of three papers a method is presented to estimate the parameters of groundwater flow models under steady and nonsteady state conditions. The parameters include values and directions of principal hydraulic conductivities (or transmissivities) in anisotropic media, specific storage (or storativity), interior and boundary recharge or leakage rates, coefficients of head-dependent interior and boundary sources, and boundary heads. In transient situations, the initial head distribution can also be estimated if the system is originally at a steady state. Paper 1 of the series discusses some of the advantage in treating the inverse problem statistically and in regularizing its solution by means of penalty criteria based on prior estimates of the parameters. The inverse problem is posed in the framework of maximum likelihood theory cast in a manner that accounts for prior information about the parameters. Since not all the factors which contribute to the prior errors can be quantified statistically at the outset, the covariance matrices of these errors are expressed in terms of several parameters which, if unknown, can be estimated jointly with the hydraulic parameters by a stagewise optimization process. When transient head data are separated by a fixed time interval, the temporal structure of these data is approximated by a lag-one autoregressive model with a correlation coefficient that can be treated as another unknown parameter. Estimation errors are analyzed by examining the lower bound of their covariance matrix in the eigenspace. Paper 1 concludes by suggesting that certain model identification criteria developed in the time series literature, all of which are based on the maximum likelihood concept, might be useful for selecting the best groundwater model (or the best method of parameterizing a particular model) among a number of given alternatives. if • = 0, prescribed head if •--• or, and a head-dependent flux (mixed) condition otherwise. When the flow is horizontal, K can be replaced by the transmissivity tensor, 'l', and Ss by the storativity, S. We will sometimes assume that the initial head, ho, satisfies a Poisson-type steady state equation V. (K. Vho) + qo = 0 on R (4) subject to -K. Vh 0 ß n = •(H0 -h) + Q on F (5)where q0, H0, and Q0 are the initial (steady state) equivalents of q, H, and Q. The solution of (1)-(5) requires a knowledge of the "aquifer parameters" K, Ss, q, H, Q, and • over the entire flow domain, R, and its boundary, F. In practice, (1)-(5) are often solved by numerical methods such as finite differences or finite elements in which the parameters take on discrete values. We will refer to such discrete values as "model parameters." While some aquifer parameters can be measured in the field, such measurements are usually scarce and prone to error. Furthermore, the measurements are generally performed on a scale different from that required for modeling purposes, so that the measured parameters are both conceptually and numerically different from their model counterparts. Si...

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