[1] A hybrid approach to the regularized inversion of highly parameterized environmental models is described. The method is based on constructing a highly parameterized base model, calculating base parameter sensitivities, and decomposing the base parameter normal matrix into eigenvectors representing principal orthogonal directions in parameter space. The decomposition is used to construct super parameters. Super parameters are factors by which principal eigenvectors of the base parameter normal matrix are multiplied in order to minimize a composite least squares objective function. These eigenvectors define orthogonal axes of a parameter subspace for which information is available from the calibration data. The coordinates of the solution are sought within this subspace. Super parameters are estimated using a regularized nonlinear Gauss-Marquardt-Levenberg scheme. Though super parameters are estimated, Tikhonov regularization constraints are imposed on base parameters. Tikhonov regularization mitigates over fitting and promotes the estimation of reasonable base parameters. Use of a large number of base parameters enables the inversion process to be receptive to the information content of the calibration data, including aspects pertaining to small-scale parameter variations. Because the number of super parameters sustainable by the calibration data may be far less than the number of base parameters used to define the original problem, the computational burden for solution of the inverse problem is reduced. The hybrid methodology is described and applied to a simple synthetic groundwater flow model. It is then applied to a real-world groundwater flow and contaminant transport model. The approach and programs described are applicable to a range of modeling disciplines.Citation: Tonkin, M. J., and J. Doherty (2005), A hybrid regularized inversion methodology for highly parameterized environmental models, Water Resour. Res., 41, W10412,
[1] We describe a subspace Monte Carlo (SSMC) technique that reduces the burden of calibration-constrained Monte Carlo when undertaken with highly parameterized models. When Monte Carlo methods are used to evaluate the uncertainty in model outputs, ensuring that parameter realizations reproduce the calibration data requires many model runs to condition each realization. In the new SSMC approach, the model is first calibrated using a subspace regularization method, ideally the hybrid Tikhonov-TSVD ''superparameter'' approach described by Tonkin and Doherty (2005). Sensitivities calculated with the calibrated model are used to define the calibration null-space, which is spanned by parameter combinations that have no effect on simulated equivalents to available observations. Next, a stochastic parameter generator is used to produce parameter realizations, and for each a difference is formed between the stochastic parameters and the calibrated parameters. This difference is projected onto the calibration null-space and added to the calibrated parameters. If the model is no longer calibrated, parameter combinations that span the calibration solution space are reestimated while retaining the null-space projected parameter differences as additive values. The recalibration can often be undertaken using existing sensitivities, so that conditioning requires only a small number of model runs. Using synthetic and real-world model applications we demonstrate that the SSMC approach is general (it is not limited to any particular model or any particular parameterization scheme) and that it can rapidly produce a large number of conditioned parameter sets.Citation: Tonkin, M., and J. Doherty (2009), Calibration-constrained Monte Carlo analysis of highly parameterized models using subspace techniques, Water Resour. Res., 45, W00B10,
mined problems, both linear and nonlinear. PEST tools for calculating contributions to model predictive uncertainty, as well as optimization of data acquisition for reducing parameter and predictive uncertainty, are presented. The appendixes list the relevant PEST variables, files, and utilities required for the analyses described in the document.
The idea that models should be as simple as possible is often accepted without question. However, too much simplification and parsimony may degrade a model's utility. Models are often constructed to make predictions; yet, they are commonly parameterized with a focus on calibration, regardless of whether (1) the calibration data can constrain simulated predictions or (2) the number and type of calibration parameters are commensurate with the hydraulic property details on which key predictions may depend. Parameterization estimated through the calibration process is commonly limited by the necessity that the number of calibration parameters be smaller than the number of observations. This limitation largely stems from historical restrictions in calibration and computing capability; we argue here that better methods and computing capabilities are now available and should become more widely used. To make this case, two approaches to model calibration are contrasted: (1) a traditional approach based on a small number of homogeneous parameter zones defined by the modeler a priori and (2) regularized inversion, which includes many more parameters than the traditional approach. We discuss some advantages of regularized inversion, focusing on the increased insight that can be gained from calibration data. We present these issues using reasoning that we believe has a common sense appeal to modelers; knowledge of mathematics is not required to follow our arguments. We present equations in an Appendix, however, to illustrate the fundamental differences between traditional model calibration and a regularized inversion approach.
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