Localization and the iterative ensemble Kalman smoother
Résumé
The iterative ensemble Kalman smoother (IEnKS) is a data assimilation method meant for tracking the state of nonlinear geophysical models efficiently. It combines an ensemble of model states to estimate the errors similarly to the ensemble square-root Kalman filter, with a four-dimensional variational analysis performed within the ensemble space. As such, it belongs to the class of four-dimensional ensemble variational methods. It could require the use of localization of the analysis when the state-space dimension is high. However, its localization needs to be defined across time and to be as consistent as possible with the dynamical flow within the data assimilation window where the four-dimensional variational analysis is performed. We show that a Liouville equation governs the time evolution of the localization operator, which is linked to the evolution of the error correlations. It is argued that the time evolution of the localization operator depends strongly on the forecast dynamics. Using either covariance localization or domain localization, we propose and test several localization strategies meant to address the issue: (i) a static and uniform localization, (ii) propagation through the window of a restricted set of dominant modes of the error covariance matrix and (iii) the approximate propagation of the localization operator using covariant local domains that are moved in accordance with the dynamical flow. These schemes are illustrated with the one-dimensional Lorenz 40 variable model and with a two-dimensional barotropic vorticity model. In both cases, local analysis based on the covariant local domains leads to a systematic improvement of the data assimilation performance.