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

TLDR

Groundwater flow models require estimation of parameters such as anisotropic hydraulic conductivities, storativity, recharge/leakage rates, source coefficients, and boundary heads, and in transient cases the initial head distribution. The study presents a method to estimate groundwater flow model parameters under both steady‑state and transient conditions. The authors formulate the inverse problem as a maximum‑likelihood estimation that incorporates prior information through penalty terms, jointly estimate unknown covariance parameters with hydraulic parameters via stagewise optimization, model temporal correlation with a lag‑one autoregressive process, and assess estimation errors using the lower bound of the covariance matrix in eigenspace. They conclude that maximum‑likelihood‑based model identification criteria from time‑series analysis can aid in selecting the most appropriate groundwater model or parameterization among alternatives.

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.