Several aspects of numerical weather prediction make forecasting and data assimilation particularly challenging: very high-dimensional systems, strongly non-linear (possibly chaotic) dynamics, and real-time requirements for assimilating data and physical models. One technique to update the observational data to forecasts is to use a Kalman filter (KF). As the observation and state vectors are of very high dimension --- usually of order of 10⁵ to 10⁶ --- direct implementation of KF recursions cannot be implemented.
Several variants of the KF have recently been developed relying on Monte Carlo methods. One class of approximation is the ensemble Kalman Filter (EnsKF), which updates a sample from the forecast distribution. I will briefly mention different computational characteristics of several of the EnsKF then focus on the approximation of the KF update due to sample variability in the employed matrices. The important conclusion is that ``some'' tapering,  i.e. a simple shrinkage method, must be applied to the sample forecast matrices to avoid filter divergence and to increase accuracy.