The usage of four‐dimensional variational (4DVar) scheme is limited by the static background error covariance and the adjoint model. In a hybrid frame of the four‐dimensional ensemble‐variational data assimilation scheme (4DEnVar), being able to avoid the tangent linear and adjoint models in the 4DVar and nowadays developed into a cutting‐edge research topic of the next‐generation data assimilation methods, an analytical 4DEnVar (A‐4DEnVar) scheme is designed. First, an analytical expression for explicit evolution of the background error covariances is derived. The expression collects the innovation of observations over an assimilation window simultaneously and propagates information to the initial background field by temporal cross covariances. Second, to estimate the adjoint model, the temporal covariances are constructed with ensemble members being centralized with respect to the model states integrated from the initial condition. Third, an iterative linear search process is introduced to minimize the cost function to update the analysis field until convergence. Twin experiments based on the Lorenz chaos model with three variables are conducted for the validation of the A‐4DEnVar scheme. Comparisons to the conventional 4DVar show that the A‐4DEnVar is comparable in accuracy even with a long assimilation window and sparse observations. The assimilation results also show that the A‐4DEnVar scheme can be implemented with a very small ensemble size which means that under circumstances without the tangent linear and adjoint models it can be easily incorporated into data assimilation systems in use.

Institute of Mathematics of the Czech Academy of Sciences

In this paper, we provide the necessary and sufficient conditions under which a sequence of cosine operators is topologically multiply recurrent or recurrent on Orlicz spaces. Furthermore, we study several interesting special cases.