The equivalences between 4D-Var and the Kalman filter as well as the fixed-interval Kalman smoother point to particular optimal properties of 4D-Var. We theoretically show that these equivalences originate in the 4D-Var property that 4D-Var yields solutions optimal at every observation time in the entire assimilation time window. Using this property, we examine the retrospective data assimilation and quasi-continuous data assimilation, and then explore techniques for combining 4D-Var with the Kalman filter and smoother.

Key words: 4D-Var, the Kalman filter, the Kalman smoother, combination