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The Gauss optimization technique can be used to identify the parameters of a model of a ground-water system for which the parameter-identification problem is formulated as a least-squares comparison between the response of the prototype and the response of the model. Unavoidable uncertainty in the true stress on the prototype and in the true response of the prototype to that stress will introduce errors into the parameter-identification problem. These errors are subsequently transferred to the prediction problem. The fit of the model to the data that are used to identify parameters is not a good indicator of these errors. A Monte Carlo simulation of the parameter-identification problem and of the prediction problem can be used, however, to evaluate the effects on water-level predictions of errors in the recharge (and pumpage) data used in the parameter-identification problem. INTRODUCTION Some important problems encountered in simulating the behavior of ground-water systems can be classified as either prediction problems or parameter-identification problems. The prediction problem is to estimate the response of the prototype to specific inputs by using a mathematical model of the system. The parameter-identification problem is to find an optimum set of parameters for the model from concurrent prototypical input and output observations. In the parameter-identification problem, the mathematical structure of the equation characterizing the prototype is given, but the specific values of the coefficients of the equation, the initial conditions, or the boundary conditions are unknown. These unknowns are the parameters in the parameteridentification problem. Current methods of parameter identification can be divided into direct or indirect methods (Neuman, 1973). The direct methods treat the model parameters as dependent variables in a formal boundary value problem (Stallman, 1956; Nelson, 1968; Nelson and McCollum, 1969; Kleinecke, 1971; Emsellem and de Marsily, 1971; Sagar, 1973; and Frind and Finder, 1973). With indirect methods, existing estimates of the parameters are improved iteratively (Kruger, 1961; Jacquard and Jain, 1965; Jahns, 1966; Vemuri and Karplus, 1969; Coats and others, 1970; Slater and Durrer, 1971; Yeh and Tauxe, 1971; Lovell, 1971; Bruch and others, 197A; and McLaughlin, 1974). The indirect methods are generally computationally less efficient than the direct methods. However, the indirect methods are probably more flexible with respect to the different types of models that can be used. The Gauss optimization technique (Gauss, cited in Wilde and Beightler, 1967, p. 299) is an indirect method of parameter identification. The application of this technique to the identification of the parameters of a groundwater model is demonstrated in this paper. In addition, the effects on the modeling process of errors in the input observations are analyzed. The modeling process includes both the identification of the model parameters and the use of the model to predict the response of the prototype to specific inputs. Unavoidable uncertainty in the true stress on the prototype and in the true response of the prototype to that stress will introduce errors into the parameter-identification problem. These errors are subsequently transferred to the prediction problem. The Gauss technique has been used by others to identify the parameters for models of oil reservoirs (Jahns, 1966) and ground-water systems (McLaughlin, 1974). The originality of the work in this paper is chiefly in the method of parameter discretization, the application of the Gauss technique to a field case, and the analyses of the effects of data errors. In particular applications, the mathematical equations of ground-water flow only approximately characterize the properties of the prototype. This lack of exact equivalency between the model and the prototype often leads to bias in the parameter estimates. This problem, however, is not addressed in this paper. IDENTIFICATION PROBLEM Ground-Water Model The first step in the identification of parameters is to specify the model to be used and to isolate the unknown parameters. A partial differential equation that describes the steady flow of ground water in a twodimensional isotropic aquifer is Iz <?& + ry <Ti> + o = ° <» where Q is the strength of a source or sink, Q = I Qw (xi ,yi ) 6 (x-x ) 6 (y-y^ (2) i and h is the hydraulic head, Q is the rate of recharge or withdrawl at point (x.,y.)> T is the transmissivity, and 6 is the Dirac delta function. Equation 1 is to be solved in a domain Q, which is enclosed by the boundary f = r 1UF2 . The boundary conditions are the specified head boundary condition h = £1 on T! (3) and the specified flux boundary condition 3h t TM r £2 on T2 where 3h/8n is the outward-pointing, normal derivative on f2> and £j_ and £2 are prescribed functions of location on r\ and r2 . The Galerkin-finite element method is used to solve equation 1. The solution scheme follows Finder and Frind (1972), except that linear shape functions are defined on triangular elements . Objective Function The second step in the identification of parameters is to select an objective function that compares the model response and the prototype response. Let 0 E (81, 82,..., 8.) be a vector of the parameters to be identified. A least-squares objective function is given by M •P = I [h. h.(8)] 2 where h. is the measured hydraulic head at the data point (x.,y.)r h. is the computed hydraulic head at the data point, and M is the number of data points. -Jf «• rT «• f Let the vector 6 H (6j, 62,..., 6^) be a particular point in an unbounded feasible region such that the value of the objective function is smaller than at any other point in the feasible region (McLaughlin, 1974, describes the application of the Gauss algorithm for the case of a closed feasible region). Jw The members of 8 are called the unconstrained optimum parameters. A necessary condition at 8 resulting normal equations are is that the gradient of P vanish there. The 3P aei = 0 i = 1, 2,..., K (6) or JT[h h(8)] = 0 (7)

Application of Gauss algorithm and Monte Carlo simulation to the identification of aquifer parameters