Estimating Model Error Covariances using Particle Filters

COCUCCI T.; VAN LEEUWEN, PETER JAN; LUCINI M.; PULIDO M.

Viena

Conferencia; European Geophysical Conference; 2019

European Geophysical Union

State-space models are the framework in data assimilation to mathematically describe the hidden state of a sys-tem by combining observations with constraints from a physical model. The formulation of these models usuallyinvolves statistical parameters that do not rely on physical constants and therefore must be estimated, since theyplay a central role in the performance of the data assimilation method. In particular, model error and observationerror covariance matrices describe the second-order statistical properties of the system and observation stochasticequations, respectively. The model error covariance matrixQis the least constrained statistical parameter since itdepends on the model physics imperfections. Moreover, a misspecification ofQhas a strong impact on the com-putation of the probability density functions involved in a particle filter algorithm, leading to an unreliable andinaccurate inference.In this work, we propose the combination of the Expectation-Maximization algorithm (EM) with an efficient par-ticle filter to estimate the model error covariance matrixQ, using a batch of observations over a time window. Theproposed method encompasses two stages: the expectation step, in which a particle filter is used with the presentestimate of the model error covariance to find the probability density function that maximises the likelihood, fol-lowed by a maximization step in which this expectation is maximised as function of the model error covariance.The model evidence is written in terms of the sequential marginal likelihoods and therefore the likelihood maxi-mization requires a particle filter and a particle smoother is not needed. Since the problem is highly nonlinear ananalytical solution for this maximum is not available so that we use a fixed point iteration for the maximizationstep. We show that this methodology converges to the true model error covariance in stochastic twin experimentsusing a linear model and the Lorenz-96 system, but at different rates and with different accuracies depending onthe system parameters. The extension to online estimation using the Expectation-Maximization algorithm is alsodiscussed and evaluated