A Variational Bayesian Approach for Ensemble Filtering of Stochastically Parametrized Systems

Modern climate models use both deterministic and stochastic parameterization schemes to represent uncertainties in their physics and inputs. This work considers the problem of estimating the involved parameters of such systems simultaneously with their state through data assimilation. Standard state-parameters filtering schemes cannot be applied to such systems owing to the posterior dependency between the stochastic parameters and the “dynamical” augmented-state, defined as the state augmented by the deterministic parameters. We resort to the variational Bayesian (VB) approach to break this dependency, by approximating the joint posterior probability density function (pdf) of the augmented-state and the stochastic parameters with a separable product of two marginal pdfs that minimizes the Kullback-Leibler divergence. The resulting marginal pdf of the augmented-state is then sampled using a one-step-ahead smoothing-based ensemble Kalman filter (EnKF-OSAS), whereas a closed-form is derived for the marginal pdf of the stochastic parameters. The proposed approach combines the effectiveness of the OSAS filtering approach to mitigate inconsistency issues that often arise with the joint EnKF, with the advantage of obtaining a full posterior pdf for the stochastic parameters, which is not possible with the traditional maximum likelihood method. We demonstrate the relevance of the proposed approach through extensive numerical experiments with a one-scale Lorenz-96 model, which includes a stochastic parametrization representing subgrid-scale effects.