Model Reduction and Parameter Estimation in Groundwater Modeling

Water resources systems management often requires complex mathematical models whose use may be computationally infeasible for many advanced analyses. The computational demand of these analyses can be reduced by approximating the model with a simpler reduced model. Proper Orthogonal Decomposition (POD) is an efficient model reduction technique based on the projection of the original model onto a subspace generated by full-model snapshots. In order to implement this method, an appropriate number of snapshots of the full model must be taken at the appropriate times such that the resulting reduced model is as accurate as possible. Since confined aquifers reach steady state in an exponential manner, a simple exponential function can be used to select snapshots for these types of models. This selection method is then employed to determine the optimal snapshot set for aunit, dimensionless model. The optimal snapshot set is found by maximizing the minimum eigenvalue of the snapshot covariance matrix, a criterion similar to those used in experimental design. The resulting snapshot set can then be translated to any complex, real world model based on a simple, approximate relationship between dimensionless and real-world times. This translation is illustrated using a basin scale model of Central Veneto, Italy, where the reduced model runs approximately 1,000 times faster than the full model. Accurate reduced modeling can be significantly beneficial for advanced analyses such as parameter estimation. A new parameter estimation algorithm is proposed that is an extension of the quasilinearization approach where the governing system of differential equations is linearized with respect to the parameters. The resulting inverse problem therefore becomes a quadratic programming problem (QP) for minimizing the sum of squared residuals; the solution becomes an update on the parameter set. This process of linearization and regression is repeated until convergence takes place. POD is applied to reduce the size of the linearized model, thereby reducing the computational burden of solving each QP. In fact, this study shows that the snapshots need only be calculated once at the very beginning of the algorithm, after which no further calculations of the reduced-model subspace are required. The proposed algorithm therefore only requires one linearized full-model run per parameter at the first iteration followed by a series of reduced-order QPs. The method is applied to a groundwater model with about 30,000 computation nodes where as many as 15 zones of hydraulic conductivity are estimated