4DEnVar with an Iterative Nonlinear Forecast Model

Ensemble-4DVar (En4DVar) and 4D-Ensemble-Var (4DEnVar) are the popular ensemble-based variational data assimilation methods for the numerical weather prediction. En4DVar calculates the gradient of the cost function with the adjoint of tangent linear forecast model. On the other hand, 4DEnVar calculates the gradient of the cost function using the ensemble perturbations instead of the adjoint of tangent linear forecast model. When the gradient is the nonlinear function of the control variables, neither En4DVar nor 4DEnVar can necessarily minimize the original cost function because they generally approximate the nonlinear forecast model with the tangent linear forecast model and the ensemble perturbations, respectively. In this study, we developed "4DEnVar with iterative nonlinear forecast model" (hereafter, 4DEnVar-IF), which calculates the nonlinear forecast model in each iteration to obtain the original cost function and its nonlinear gradient.

The 4DEnVar-IF applied to the Lorenz63 model succeeded to make the cost function smaller than those of En4DVar and general 4DEnVar . The 4DEnVar-IF applied to Japan Meteorological Agency nonhydrostatic model-based 4DVar data assimilation (JNoVA) system was also consistent with the results obtained with the Lorenz63 model in the single-observation assimilation experiments. In the assimilation of multiple real observations with 4DEnVar-IF of the small horizontal localization scale, however, the cost function was not sufficiently minimized probably because the time evolution of the horizontal localization ignored in 4DEnVar-IF.