Most new developments in atmospheric data assimilation are based on variants of the so-called 3DVAR and 4DVAR methods. 3DVAR uses a prescribed and constant background term. 4DVAR minimizes a smoothing problem where a perfect model assumption is inconsistent with prior knowledge. This paper will discuss possibilities of including dynamically consistent and time dependent error statistics in the 3DVAR algorithm and further illustrate how the 4DVAR problem can be formulated and solved in a consistent manner.

Key words: 3DVAR, 4DVAR, model errors, predicted error statistics