Toward a weak constraint 4D-Var system: application of the Burgers equation

A weak constraint observation-space atmospheric 4D-Var system is currently under development at Naval Research Laboratory (NRL) in Monterey. It is an extension of the U.S. Navy's operational three-dimensional variational (3D-Var) data assimilation system, NAVDAS (NRL Atmospheric Variational Data Assimilation System). The new system, NAVDAS-AR, where AR stands for Accelerated Representer, is similar in many respects to the incremental 4D-Var systems implemented in many other operational centers. It differs, however, in several areas. First, the minimization is sought in the observation space, where the control variable is the number of observations. Second, a linearized prediction model is used in the outer loop strategy. Finally, the perfect model assumption is only a special case in the generalized formulation. The impact of model errors is presented in a form of a four-dimensional space and time convolution of the model error covariance in NAVDAS-AR. Although the space and time convolution is generally very difficult to calculate, it can be obtained accurately and efficiently in case of some special choices of model error covariances. In this paper, we give a brief overview of the generalized observation-space 4d-Var algorithm used in NAVDAS-AR. Applying the algorithm to the Burgers equation, we then highlight the numerical procedures involved in including model error terms. We also demonstrate the benefit of including model errors on substantial improvement of the analysis.