Efficient nonlinear predictive error variance for highly parameterized models

Abstract

[1] Predictive error variance analysis attempts to determine how wrong predictions made by a calibrated model may be. Predictive error variance analysis is usually undertaken following calibration using a small number of parameters defined through a priori parsimony. In contrast, we introduce a method for investigating the potential error in predictions made by highly parameterized models calibrated using regularized inversion. Vecchia and Cooley (1987) describe a method of predictive error variance analysis that is constrained by calibration data. We extend this approach to include constraints on parameters that lie within the calibration null space. These constraints are determined by dividing parameter space into combinations of parameters for which estimates can be obtained and those for which they cannot. This enables the contribution to predictive error variance from parameterization simplifications required to solve the inverse problem to be quantified, in addition to the contribution from measurement noise. We also describe a novel technique that restricts the analysis to a strategically defined predictive solution subspace, enabling an approximate predictive error variance analysis to be completed efficiently. The method is illustrated using a synthetic and a real-world groundwater flow and transport model.

Quantifying Data Worth Toward Reducing Predictive Uncertainty

“…The “error variance” of a prediction characterizes the potential for any prediction made by a calibrated model to be wrong. Error variance computation is normally based on the premise that a model has been calibrated against field data and is then used to make predictions of system behavior (Tonkin et al 2007). The “calibrated model” is endowed with one parameter field that ideally should possess a status such as that of minimum error variance which thereby guarantees its uniqueness.…”

Real‐time groundwater flow modeling with the Ensemble Kalman Filter: Joint estimation of states and parameters and the filter inbreeding problem

“…These MC type methods are very CPU intensive and, although flexible with regard to nonlinearities and system complexity, not commonly applied to very large problems (many grid cells) and multiple sources of uncertainty. A second group of methods estimates a single best solution that captures some of the details at smaller scales, namely those which are important for groundwater flow and transport predictions, and also estimates an uncertainty associated with the prediction [e.g., Hernandez et al, 2003;Vermeulen et al, 2004;Tonkin and Doherty, 2005;Tonkin et al, 2007]. Often, the calibration focuses on the uncertain hydraulic conductivities, but sometimes also other parameters like the storativities [Hendricks Franssen et al, 1999;Li et al, 2007] or recharge rate [Hendricks Franssen et al, 2004 are subject to calibration.…”

Quantifying climate and pumping contributions to aquifer depletion using a highly parameterised groundwater model: Uley South Basin (South Australia)

“…While efficient nonlinear methods have been presented and applied to highly parameterised groundwater models (e.g., Tonkin et al, 2007;Tonkin and Doherty, 2009;Herckenrath et al, 2011), linear uncertainty analysis has been shown to provide robust estimates of uncertainty, even if applied to nonlinear models (e.g., James et al, 2009;Brunner et al, 2012). Given that the objective here is to evaluate the relative magnitude of predictive uncertainty with respect to climate and pumping impacts, linear methods are considered appropriate.…”