Optimality of 4D-Var and its relationship with the Kalman filter and Kalman smoother

Zhijin Li, I. M. Navon

The known properties of equivalence between 4D-Var and the Kalman filter as well as the fixed-interval Kalman smoother point to particular optimal properties of 4D-Var. In the linear context, the 4D-Var solution is optimal not only with respect to the model trajectory segment over the assimilation time interval, but also with respect to any model state at a single observation time level; in the batch processing method, the information in 4D-Var is fully transferred from one batch to the next by the background term; 4D-Var allows to process observations in subsets, while the final solution is optimal as all observations are processed simultaneously. These properties hold irrespective of whether the model is perfect or not. Various properties of equivalence of 4D-Var to the Kalman filter and smoother result from these optimality properties of 4D-Var. Further, we show that the fixed-lag Kalman smoother may also be constructed in an optimal way using a multiple batch-processing 4D-Var approach. Possible techniques for practical implementation of the equivalence are discussed.