Data assimilation algorithms are used to estimate the states of a dynamical system using partial and noisy observations. The ensemble Kalman filter has become a popular data assimilation scheme due to its simplicity and robustness for a wide range of application areas. Nevertheless, this filter also has limitations due to its inherent assumptions of Gaussianity and linearity, which can manifest themselves in the form of dynamically inconsistent state estimates. This issue is investigated here for balanced, slowly evolving solutions to highly oscillatory Hamiltonian systems which are prototypical for applications in numerical weather prediction. It is demonstrated that the standard ensemble Kalman filter can lead to state estimates that do not satisfy the pertinent balance relations and ultimately lead to filter divergence. Two remedies are proposed, one in terms of blended asymptotically consistent time-stepping schemes, and one in terms of minimization-based postprocessing methods. The effects of these modifications to the standard ensemble Kalman filter are discussed and demonstrated numerically for balanced motions of two prototypical Hamiltonian reference systems.

数据同化算法用于使用部分和噪声观测来估计动态系统的状态。集成卡尔曼滤波器由于其简单性和鲁棒性适用于广泛的应用领域，已成为一种流行的数据同化方案。然而，由于其固有的高斯性和线性假设，该过滤器也有局限性，这些假设可以以动态不一致的状态估计的形式表现出来。此处针对高度振荡的哈密顿系统的平衡、缓慢演化的解决方案进行了研究，该系统是数值天气预报应用的典型代表。证明标准集成卡尔曼滤波器会导致状态估计不满足相关平衡关系并最终导致滤波器发散。提出了两种补救措施，一种是混合渐近一致时间步长方案，另一种是基于最小化的后处理方法。讨论了这些修改对标准集合卡尔曼滤波器的影响，并针对两个原型哈密顿参考系统的平衡运动进行了数值演示。