Improving the singular evolutive extended Kalman filter for strongly nonlinear models for use in ocean data assimilation

摘要

The singular extended evolutive Kalman (SEEK) filter is one possible way of adapting the basic Kalman filtering theory to realistic situations in oceanographic (or meteorological) data assimilation. Applied to linear models or nearly linear models, the SEEK filter rank reduction of the huge error covariance matrices is fully compatible with its natural ability to retrieve the most unstable modes of the model when it is constant, and then to project the Kalman filtering process onto this unstable subspace. However, strong nonlinearities resulting in a non-constant tangent linear model can cause the SEEK filtering error to diverge. In this paper, we present new algorithms for computing the evolution over time of the low-rank image subspace of the singular error covariance matrices, so as to obtain an evolutive basis which retrieves the (now non-constant) most unstable modes of the tangent linear model. This is achieved with a computational cost comparable to that of the initial version of the SEEK filter algorithm. Applications are made which use a time-varying linear model and then the Lorenz model to show how these new versions of the SEEK filter lead to the convergence of the filtering error with a rank which is reduced as much as possible.