A hybrid regularized inversion methodology for highly parameterized environmental models

Abstract

[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,

A truncated Levenberg–Marquardt algorithm for the calibration of highly parameterized nonlinear models

“…Truncating the parameter space based on these eigenvalues led to improved performance during the initial iterations, but may result in subobtimal parameter sets if the truncation limit is set too high. This difficulty bay be addressed by dynamically lowering the truncation limit, or by optimizing for superparameters (i.e., parameters that are aligned with the eigenvectors) as proposed by Tonkin and Doherty (2005).…”

“…Nevertheless, in certain applications, the objective function may contain well-justified contributions from regularization, which results in a formulation similar to the hybrid regularization methodology described by Tonkin and Doherty (2005). The problem we address is simpler in that it is only concerned with the minimization algorithm rather than the formulation of the inverse problem itself.…”

“…If truncation is merely employed to obtain a more efficient solution to an otherwise robust, overdetermined inverse problem, k can be adjusted heuristically as the minimization proceeds, so that more and more parameters may enter the calibration solution space. Tonkin and Doherty (2005) propose setting k as the number of eigenvalues that exhibit a ratio to the largest eigenvalue that is greater than 10 -6 . As an alternative to this empirical threshold value, the dimension of the calibration solution space can be chosen such that the final value of the objective function (excluding contributions from Tikhonov regularization) is commensurate with the expected measurement noise level (Finsterle and Pruess, 1995;Tonkin and Doherty, 2005;Moore and Doherty, 2005), or that a predefined maximum prediction uncertainty is not exceeded (in case the inversion is part of an estimation-prediction framework).…”

“…Tonkin and Doherty (2005) propose setting k as the number of eigenvalues that exhibit a ratio to the largest eigenvalue that is greater than 10 -6 . As an alternative to this empirical threshold value, the dimension of the calibration solution space can be chosen such that the final value of the objective function (excluding contributions from Tikhonov regularization) is commensurate with the expected measurement noise level (Finsterle and Pruess, 1995;Tonkin and Doherty, 2005;Moore and Doherty, 2005), or that a predefined maximum prediction uncertainty is not exceeded (in case the inversion is part of an estimation-prediction framework). Evidently, choosing the appropriate truncation level is not a straightforward task, may be problem dependent, and thus needs some further experimentation.…”

An update of the Death Valley regional groundwater flow system transient model, Nevada and California

“…PEST provides two basic types of regularization for underdetermined inverse problems: Tikhonov regularization and a method based on Truncated Singular Value Decomposition (TSVD) known as SVD-assist (Tonkin and Doherty, 2005;Doherty, 2010;Doherty and others, 2010). For this study, Tikhonov regularization was the method of choice because correlation and instability within DVRFS v. 2.0 are the result of the presence of physical processes (Hill and Østerby, 2003), as well as a large number of hydraulic properties.…”