This paper investigates an approximation scheme of the optimal nonlinear
Bayesian filter based on the Gaussian mixture representation of the state
probability distribution function. The resulting filter is similar to the
particle filter, but is different from it in that, the standard weight-type
correction in the particle filter is complemented by the Kalman-type correction
with the associated covariance matrices in the Gaussian mixture. We show that
this filter is an algorithm in between the Kalman filter and the particle
filter, and therefore is referred to as the particle Kalman filter (PKF). In
the PKF, the solution of a nonlinear filtering problem is expressed as the
weighted average of an "ensemble of Kalman filters" operating in parallel.
Running an ensemble of Kalman filters is, however, computationally prohibitive
for realistic atmospheric and oceanic data assimilation problems. For this
reason, we consider the construction of the PKF through an "ensemble" of
ensemble Kalman filters (EnKFs) instead, and call the implementation the
particle EnKF (PEnKF). We show that different types of the EnKFs can be
considered as special cases of the PEnKF. Similar to the situation in the
particle filter, we also introduce a re-sampling step to the PEnKF in order to
reduce the risk of weights collapse and improve the performance of the filter.
Numerical experiments with the strongly nonlinear Lorenz-96 model are presented
and discussed.Comment: Accepted manuscript, to appear in Monthly Weather Revie