Sequential State and Observation Noise Covariance Estimation Using Combined Ensemble Kalman and Particle Filters

The authors consider the joint state and parameter estimation problem for dynamical models where the system evolution is known and where the observations are linear with additive Gaussian noise whose covariance matrix depends on unknown parameters. The form of this dependence is general, both diagonal and off-diagonal elements of the covariance matrix may depend on the unknown parameters. In this situation, the ensemble Kalman filter (EnKF) cannot be applied directly to update state and parameters simultaneously. Two novel approximate Monte Carlo methods are proposed for this purpose, which are both based on the assumption that parameters and state are approximately independent after the propagation step. The first method begins with a particle filter update of the parameters followed by an EnKF update of the state. The second method first makes an EnKF update of the state based on an approximate likelihood that does not depend on the parameters, followed by a particle filter update of the parameters with weights proportional to the ratio of the correct to the approximate likelihood. To counteract sample depletion, the authors introduce algorithmic refinements like balanced sampling, kernel resampling, and increasing the number of samples for the parameters while keeping the size of the state ensemble fixed. The performance and flexibility of these methods is demonstrated in simulations with a linear Gaussian model and with the Lorenz-96 model. Provided the EnKF with all parameters known is reasonably well calibrated, then this is also true for the state estimates in the new algorithms, and the estimated posterior distributions of the parameters are consistent with the truth.