An Analytical Four-Dimensional Ensemble-Variational Data Assimilation Scheme

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. Plain Language Summary In this study, a new data assimilation scheme, namely, analytical four-dimensional ensemble variational data assimilation scheme, is constructed. The analytical solution of each iteration of four-dimensional variational (4DVar) is directly given. When dealing with nonlinear problems, it completely inherits the performance of traditional 4DVar scheme, but avoids the coding and calculation of adjoint models, hence greatly improves the transferability and the calculation efficiency.