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.

中文翻译：

---

解析四维集合变分数据同化方案

二维变分（4DVar）方案的使用受到静态背景误差协方差和伴随模型的限制。在四维整体变分数据同化方案（4DEnVar）的混合框架中，能够避免4DVar中的正切线性模型和伴随模型，如今已成为下一代数据同化方法的前沿研究课题，设计了一种解析4DEnVar（A-4DEnVar）方案。首先，导出了背景误差协方差的显式演化的解析表达式。该表达式同时在同化窗口上收集观测的创新，并通过时间互协方差将信息传播到初始背景场。其次，估算伴随模型 时间协方差的构建是通过将集合成员相对于从初始条件开始集成的模型状态集中化来实现的。第三，引入了迭代线性搜索过程以最小化成本函数，以更新分析字段直至收敛。进行了基于带有三个变量的Lorenz混沌模型的双实验，以验证A-4DEnVar方案。与常规4DVar的比较表明，即使具有很长的同化窗口和稀疏的观测结果，A-4DEnVar的准确性也相当。同化结果还表明，A‐4DEnVar方案可以以非常小的整体大小实现，这意味着在没有切线和伴随模型的情况下，可以很容易地将其合并到使用的数据同化系统中。