An adaptive variational method for data assimilation with imperfect models

The solutions of the weak constraint data assimilation problems depend on a priori error covariance. If a priori error covariance have poor quality, a posteriori evaluation may have negative impact and solutions are not optimal. A novel variational data assimilation method is proposed, which does not assume the model is perfect, and can adaptively adjust model state without knowing explicitly the model error covariance matrix. Not by adjusting the initial condition in 4D-VAR, but by adjusting a steady gain matrix in a class of filters in this approach to yield a filter solution that minimize the norm of analysis innovation vector in a given span of time interval. The method enables very flexible ways to form some reduced order problems. A proper reduced-order problem not only reduces computational burden but leads to corrections that are more consistent with the model dynamics that trends to produce better forecast. It is shown that the optimal nudging can be reinterpreted as an example of the reduced order problems. The method is demonstrated using a simple nonlinear model (Burgers equation model) and simulated data. Full and several reduced order forms of the adaptive variational method are performed and compared with a simplified strong constraint 4D-VAR and the space variable optimal nudging scheme in assimilation-forecast experiments