Least squares solution and calibration of steady state groundwater flow systems

Abstract

Groundwater models contain parameters that characterize the hydraulic response of aquifer systems. Because the parameters are never known with certainty, they are normally adjusted in a calibration step to bring the computed values of the hydraulic head into agreement with the measured values. The calibration problem can be formulated as a minimization problem with a quadratic objective function and nonlinear constraints. The quadratic objective function is formed by adding the weighted sum of the squares of measurement residuals to the weighted sum of the squares of departures from prior parameter estimates. Weighting matrices are formed by taking the inverses of the estimated covariance matrices of the head measurements and original parameter estimates. The Galerkin finite element equations are considered nonlinear constraints which are enforced by the penalty method. The proposed procedure is unique because the Galerkin finite element equations are incorporated directly into the objective function and the objective function is minimized with respect to both the parameters and the nodal values of the hydraulic heads. The solution generates an updated parameter set and the solution to the groundwater flow equation in a single step. Linearized error analysis produces the error of estimation of the heads and parameters. Results can be examined for consistency with the original statistical model and discretized governing equation.