Calibration‐constrained Monte Carlo analysis of highly parameterized models using subspace techniques

Abstract

[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,

On determining uncertainties of magnetic resonance sounding estimated transmissivities for groundwater modeling

“…Often this was in the order of 50% above the minimum objective function. This approach is similar to the one applied by Tonkin and Doherty (2009).…”

“…In the field case presented later, the pumped aquifer is located in a buried valley structure incised into a low permeable substratum, and there are internal boundaries inside the valley as well. Methods to analyze aquifer tests performed in such valley structures have been given by, e.g., Vandenberg (1977), or Butler and Wenzhi (1991), and in the presence of internal boundaries by van der Kamp and Maathuis (2012). However, the limitations of the analytical solutions make especially the Vandenberg solution inapplicable for the present analysis due to the long distance to some of the observation wells.…”

“…In the present case, we wish to use V 0 estimated from equation 16 to reduce the parameters space to be investigated to determine the contribution of model nonlinearity to parameter variance and covariance. According to Tonkin and Doherty (2009), this can be done by generating a set of realizations of random parameters p st . This random parameter set could be generated using the posterior parameter covariance matrix CðpÞ.…”

“…Instead, we generated random parameter values within the 99% confidence intervals estimated by the linear analysis (p st ). A new parameter set that nearly fit the data (p;) can then be calculated as (Tonkin and Doherty, 2009) …”

“…In step 1, to generate parameter sets we applied a modified version of null-space Monte Carlo (NSMC) (Tonkin and Doherty, 2009). In this nonlinear approach, the inverted model for MRS is used to define the calibration null space.…”