Study of weak constraint 4dvar with model error forcing control variable

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In the traditional implementations of four-dimensional variational data assimilation (4dvar for short), it is assumed that the model used is perfect. However the model error in the model can directly affect the accuracy of data assimilation. The weak constraint 4dvar is an effective way of correcting and estimating the model error in 4dvar. In this paper, an approach to weak constraint 4dvar with model error forcing control variable is studied and implemented in the one-dimensional shallow water equations. The results show that when the model error cannot be ignored, the prediction error with the weak constraint 4dvar is smaller than with the traditional 4dvar in both the assimilation window and the prediction period, and the improvement with weak constraint 4dvar is more obvious in the condition with large model error. Also, the weak constraint 4dvar approach to estimating model error captures some basic features of model error including the magnitude and the characteristic of distribution.