Introduction to the Kalman filter

Abstract This chapter introduces The Kalman filter, which implements Bayesian data assimilation for linear, Gaussian systems. Its update equations can also be derived as the best linear unbiased estimator (BLUE) and its covariance. Some of the Kalman filter’s detailed properties are reviewed here: linear transformations of the state and observations, extending the state vector to include observed variables, and temporal correlation in the model or observation errors. The Kalman filter can be applied to nonlinear and non-Gaussian systems via either the extended Kalman filter or the BLUE, although both approaches are clearly sub-optimal. The ensemble Kalman filter (EnKF) employs sample covariances from an ensemble of forecasts at each update time and allows practical implementation of an approximate Kalman filter. The EnKF is consistent with a Monte- Carlo implementation of the BLUE. Many of the EnKF’s properties, including basic effects of sampling error, can be understood in the context of Kalman-filter theory.