Abstract
This paper introduces a new approximate solution of the optimal nonlinear filter suitable for nonlinear oceanic and atmospheric data assimilation problems. The method is based on a local linearization in a low-rank kernel representation of the state's probability density function. In the resulting low-rank kernel particle Kalman (LRKPK) filter, the standard (weight type) particle filter correction is complemented by a Kalman-type correction for each particle using the covariance matrix of the kernel mixture. The LRKPK filter's solution is then obtained as the weighted average of several low-rank square root Kalman filters operating in parallel. The Kalman-type correction reduces the risk of ensemble degeneracy, which enables the filter to efficiently operate with fewer particles than the particle filter. Combined with the low-rank approximation, it allows the implementation of the LRKPK filter with high-dimensional oceanic and atmospheric systems. The new filter is described and its relevance demonstrated through applications with the simple Lorenz model and a realistic configuration of the Princeton Ocean Model (POM) in the Mediterranean Sea.