In most operational implementations of 4D-Var, it is assumed that the model used in the data assimilation process is perfect or, at least, that errors in the model can be neglected when compared to other errors in the system. We consider here the question of how model error could be accounted for in four dimensional variational data assimilation.
We present three approaches for the formulation of weak constraint 4D-Var: estimating explicitly a model error forcing term, estimating a representation of model bias or, estimating a four dimensional model state as the control variable. The consequences of these approaches with respect to the implementation and the properties of 4D-Var are discussed.
We show that 4 D-Var with an additional model error representation as part of the control variable is essentially an initial value problem and that its characteristics are very similar to that of strong constraint 4D-Var. Taking the four dimensional state as the control variable however leads to very different properties. In that case, weak constraint 4D-Var can be interpreted as a coupling between successive strong constraint assimilation cycles. A possible extension towards long window 4D-Var and possibilities for evolutions of the data assimilation system are presented.
Finally, we show some results obtained in the IFS when estimating model error as a forcing term. In particular, we show its impact in the stratosphere at the winter pole where a temperature bias is present in the model. We also show examples of interaction between the model error term and observation bias.