Four-dimensional Localization and the Iterative Ensemble Kalman Smoother

Abstract:

The iterative ensemble Kalman smoother (IEnKS) is a data assimilation method meant for efficiently tracking the state of
nonlinear geophysical models. It combines an ensemble of model states to estimate the errors similarly to the ensemble
square root Kalman filter, with a 4D-variational analysis performed within the ensemble space. As such it belongs to
the class of ensemble variational methods. Recently introduced 4DEnVar or the 4D-LETKF can be seen as particular cases
of the scheme. The IEnKS was shown to outperform 4D-Var, the ensemble Kalman filter (EnKF) and smoother, with low-order
models in all investigated dynamical regimes. Like any ensemble method, it could require the use of localization of the
analysis when the state space dimension is high. However, localization for the IEnKS is not as straightforward as for
the EnKF. Indeed, localization needs to be defined across time, and it needs to be as much as possible consistent with
the dynamical flow within the data assimilation variational window. 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
its time integration strongly depends 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 constant and uniform
localization, (ii) the propagation through the window of a restricted set of dominant modes of the error covariance
matrix, (iii) the approximate propagation of the localization operator using model covariant local domains. These
schemes are illustrated on the one-dimensional Lorenz 40-variable model.