Computationally Efficient Algorithms for Parameter Estimation and Uncertainty Propagation in Numerical Models of Groundwater Flow

Abstract

Finite difference and finite element methods are frequently used to study aquifer flow' however, additional analysis is required when model parameters, and hence predicted heads are uncertain. Computational algorithms are presented for steady and transient models in which aquifer storage coefficients, transmissivities, distributed inputs, and boundary values may all be simultaneously uncertain. Innovative aspects of these algorithms include a new form of generalized boundary condition' a concise discrete derivation of the adjoint problem for transient models with variable time steps' an efficient technique for calculating the approximate second derivative during line searches in weighted least squares estimation; and a new efficient first-order second-moment algorithm for calculating the covariance of predicted heads due to a large number of uncertain parameter values. The techniques are presented in matrix form, and their efficiency depends on the structure of sparse matrices which occur repeatedly throughout the calculations. Details of matrix structures are provided .for a two-dimensional linear triangular finite element model. INTRODUCTION Modeling of regional groundwater flow involves more than the straightforward calculation of piezometric heads. An investigation may also require (1) the estimation of model parameters using both historical head measurements and prior information (i.e., the inverse problem) and (2) an assessment of prediction uncertainty. Many papers have addressed the largescale inverse problem [e.g., Neuman, 1980a; Cooley, 1982], and others have addressed the estimation of the covariance of predicted heads due to uncertain parameters [e.g., Delhomme, 1979; Smith and Freeze, 1979; Dettinger and Wilson, 1981; Clifton and Neuman, 1982]. The purpose of this paper is to describe a systematic approach to the design of computational procedures for carrying out both of these steps. The proposed procedures have evolved from the observation that sensitivities of heads to parameters are fundamental to all the necessary calculations. Sensitivities are simply the first derivatives of heads with respect to parameters. They are required for traditional sensitivity analysis, for calculation of the gradient of the objective function in the inverse problem, for second derivative calculations during line searches in the inverse problem, for calculation of the posterior covariance of parameters estimated in the inverse problem, and finally for calculating the covariance of predicted heads due to uncertainty in the valu es of parameters. The sensitivities appear time and time again, but in many cases it is possible to use alternative computational procedures which avoid thei r explicit evaluation. In these cases, the sensitivities are implicitly contained in other matrices, and the use of alternative procedures can greatly improve the speed of calculations. The matrices which implicitly contain sensitivities are defined by (18) and (20) below and are denoted by [K•h(0)] and [A•,•,h(k)] for steady state and tra...

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