A four-dimensional data assimilation method that provides an estimate of the model state by evolving explicitly the error covariance of the state estimate. Variants of the Kalman philtre algorithm are now being applied to atmospheric data assimilation problems. The philtre estimate is based on all data observed up to and including the current time. generalisations of the Kalman philtre exist for continuum dynamics, for nonlinear stochastic systems (e.g., extended or ensemble Kalman filters), for systems that have different types of noise, for unknown noise statistics, and for observations beyond the current time (Kalman smoothers).