Dynamical evolution of the error statistics with the SEEK filter to assimilate altimetric data in eddy-resolving ocean models

The Singular Evolutive Extended Kalman (SEEK) filter introduced by Pham et al. is applied to a primitive equation model in order to reconstruct the mesoscale circulation typical of the mid-latitude ocean from altimetric data, The SEEK filter is a variant of the Kalman-filter algorithm based on two concepts: the order reduction of the initial error covariance matrix, and the dynamical evolution of the reduced-order basis. This makes the method potentially suitable for problems with a high number of degrees of freedom. Previous work has shown the ability of a steady version of the filter to improve the vertical structure of the ocean thermocline in the case of the quasi-linear dynamics associated with the equatorial tropical Pacific Ocean, and the need to combine the dynamical evolution of the basis with an adaptive scheme in a mid-latitude ocean model of the Gulf Stream region. This work examines the potential advantages of the dynamical evolution of the basis functions with simple assimilation experiments. It demonstrates the ability of the method to propagate in time the statistical properties of the system when the filter is initialized properly. However, the lack of robustness of the filter is investigated theoretically and experimentally, showing the need to consider variants of the method when the filter is not properly initialized.