1 Department of Earth Sciences, Faculty of Science, Aarhus University, Aarhus University2 Department of Geoscience, Science and Technology, Aarhus University3 Watermark Numerical Computing4 Department of Geoscience, Science and Technology, Aarhus University
A significant practical problem with the pilot point method is to choose the location of the pilot points. We present a method that is intended to relieve the modeler from much of this responsibility. The basic idea is that a very large number of pilot points are distributed more or less uniformly over the model area. Singular value decomposition (SVD) of the (possibly weighted) sensitivity matrix of the pilot point based model produces eigenvectors of which we pick a small number corresponding to significant eigenvalues. Super parameters are defined as factors through which parameter combinations corresponding to the chosen eigenvectors are multiplied to obtain the pilot point values. The model can thus be transformed from having many-pilot-point parameters to having a few super parameters that can be estimated by nonlinear regression on the basis of the available observations. (This technique can be used for any highly parameterized groundwater model, not only for models parameterized by the pilot point method.) A synthetic model is used to test and demonstrate the application of the method for a case with a highly heterogeneous log-transmissivity field to be estimated from a limited number of hydraulic head observations. It is shown that the method produces a smoothly varying spatial parameter field, and that the fit of the estimated log-transmissivity field to the real field varies with the parameterization specification (i.e. the density of pilot points and the number of estimated super parameters), and that the structural errors caused by using pilot points and super parameters to parameterize the highly heterogeneous log-transmissivity field can be significant. For the test case much effort is put into studying how the calibrated model's ability to make accurate predictions depends on parameterization specifications. It is shown that there exists no unique parameterization specification that produces the smallest possible prediction error variance for all eight studied predictions simultaneously. However, a reasonable compromise of parameterization can be made. It is further shown that it is possible to choose parameterization specifications that result in error variances for some predictions that are greater than those that would be encountered if the model had not been calibrated at all. Test case predictions that have this "problem" are all dependent on the field conditions near an inflow boundary where data is lacking and which exhibit apparent significant nonlinear behavior. It is shown that inclusion of Tikhonov regularization can stabilize and speed up the parameter estimation process. A method of linearized model analysis of predictive uncertainty and of prediction error variance is described. The test case demonstrates that linearized model analysis can be used prior to groundwater model calibration to determine a parameterization specification that produces (close to) minimum possible error variance for predictions that do not behave like seriously nonlinear functions. Recommendations concerning the use of pilot points and singular value decomposition in real-world groundwater model calibration are finally given. (c) 2008 Elsevier Ltd. All rights reserved.
Advances in Water Resources, 2008, Vol 31, Issue 4, p. 674-700
pilot point; singular value decomposition; parameter estimation; calibration; prediction error; nonlinearity; Tikhonov regularization