  Contemporary data assimilation often involves more than a million prediction
variables. Ensemble Kalman filters (EnKF) have been developed by geoscientists.
They are successful indispensable tools in science and engineering, because
they allow for computationally cheap low ensemble state approximation for
extremely large dimensional turbulent dynamical systems. The practical finite
ensemble filter like EnKF necessarily involve modifications such as covariance
inflation and localization, and it is a genuine mystery why they perform so
well with small ensemble sizes in large dimensions. This paper provides the
first rigorous stochastic analysis of the accuracy and covariance fidelity of
EnKF in the practical regime where the ensemble size is much smaller than the
large ambient dimension for EnKFs with random coefficients. A challenging issue
overcome here is that EnKF in huge dimensions introduces unavoidable bias and
model errors which need to be controlled and estimated.
