  Ensemble Kalman filter (EnKF) is an important data assimilation method for
high dimensional geophysical systems. Efficient implementation of EnKF in
practice often involves the localization technique, which updates each
component using only information within a local radius. This paper rigorously
analyzes the local EnKF (LEnKF) for linear systems, and shows that the filter
error can be dominated by the ensemble covariance, as long as 1) the sample
size exceeds the logarithmic of state dimension and a constant that depends
only on the local radius; 2) the forecast covariance matrix admits a stable
localized structure. In particular, this indicates that with small system and
observation noises, the filter error will be accurate in long time even if the
initialization is not. The analysis also reveals an intrinsic inconsistency
caused by the localization technique, and a stable localized structure is
necessary to control this inconsistency. While this structure is usually taken
for granted for the operation of LEnKF, it can also be rigorously proved for
linear systems with sparse local observations and weak local interactions.
These theoretical results are also validated by numerical implementation of
LEnKF on a simple stochastic turbulence in two dynamical regimes.
