Study and tests of weak constraint 4D-Var

Current operational implementations of 4D-Var rely on the assumption that the numerical model representing the evolution of the atmospheric flow is perfect, or at least that model errors are small enough to be neglected compared to other errors in the system. Weak constraint 4D-Var with a model error forcing control variable has been implemented and tested within the YH4DVAR system in this paper. The preliminary test results show that weak constraint 4D-Var is affordable in an operational environment for a global forecasting system. Imposed weak constrain dose not affect the convergence of the cost function over much, they are descended to a reasonable low level after certain iterations in the minimization processes. And the gradient of cost function seems dropping as fast in the weak constrain as in the strong constrain conditions.