Particle Kalman Filtering for Data Assimilation in Meteorology and Oceanography
Résumé
We describe a discrete solution of the optimal nonlinear filter that generalizes the optimality of the correction step of the ensemble Kalman filters to nonlinear systems. This approach is based on a Gaussian mixture representation of the state probability density function which results in a new particle-type filter, called particle Kalman filter (PKF), in which the standard (weight-type) particle filter correction is complemented by a Kalman-type correction for each particle using the associated covariance matrix in the Gaussian mixture. The optimal solution of the nonlinear filtering problem is then obtained as the weighted average of an ensemble of Kalman filters operating in parallel. The Kalman-type correction reduces the risk of ensemble collapse, which enables the filter to efficiently operate with fewer particles than the particle filter. Running an "ensemble of Kalman filters" is however computationally prohibitive for high dimensional systems. We first derive the popular ensemble Kalman filter as a suboptimal variant of the PKF and evaluate its performances against the PKF with the Lorenz-96 model. Then we discuss different approaches to reduce the computational burden of the PKF filter for application to atmospheric and oceanic data assimilation problems.