Estimation of model error covariances for nonlinear dynamical systems using Particle Filters and the Expectation-Maximization algorithm

LUCINI, MARÍA MAGDALENA; VAN LEEWUEN, PETER JAN; PULIDO, MANUEL ARTURO

Workshop; 13TH INTERNATIONAL ENKF WORKSHOP; 2018

Model error covariances play a central role in the perfomance of particle filters applied to nonlinear state-space models. However, these covariances are largely unknown in most of the applications, including models used in geophysics, engineering and epidemiology. In this work, we propose the combination of the Expectation-Maximization algorithm (EM) with an efficient particle filter to estimate the model error covariance.A batch of observations is used to find the elements of this model error covariance. However, because the relation between the observations and the model error covariance is via the evolving state of the system under this covariance matrix, this is a complicated problem. Based on the EM algorithm principles, the proposed solution method encompasses two stages: the expectation stage, in which a particle filter is used with the present estimate of the model error covariance as given to find the probability density function that maximises the likelihood, followed by a maximization stage in which this expectation is maximised as function of the elements of the model error covariance. Since the problem is highly nonlinear an analytical solution for this maximum cannot be found. We explore two different approximations to find a solution, an online method in which the parameters are included in a Bayesian framework and estimated sequentially using a short batch of observations, and an offline method for a long batch of observations which could be useful for tuning the assimilation system and so finding optimal static parameters. Both methodologies are shown to be converging towards true model error covariances in twin experiments using the Lorenz-96 system,but at different rates and with different accuracies depending on system parameters. We explore the causes for these differencesand discuss application to high-dimensional systems