稳健集合滤波及其与集合卡尔曼滤波协方差膨胀的关系
Robust ensemble filtering and its relation to covariance inflation in the ensemble Kalman filter
摘要：
基于$H_{\infty}$滤波理论，提出了一种鲁棒的集成滤波方案.最佳的$H_{\infty}$滤波器是通过最小化预定义的成本函数的上确界（或最大值），不同于卡尔曼滤波器中使用的最小方差的标准。通过设计，$H_{\infty}$滤波器比卡尔曼滤波器更鲁棒，在这个意义上，除了对应于卡尔曼滤波器的特殊情况之外，$H_{\infty}$滤波器中的估计误差通常相对于同化中的不确定性具有有限的增长率。原始形式的$H_{\infty}$过滤器包含全球的时间约束，这可能是不方便的顺序数据同化问题。因此，我们引入了一个变体，解决了一些时间局部约束，因此我们称之为时间局部$H_{\infty}$过滤器（TLHF）。通过类比集合卡尔曼滤波器（EnKF），我们还提出了集合时间-局部$H_{\infty}$滤波器（EnTLHF）的概念。我们概述了一般形式的EnTLHF，并讨论了它的一些特殊情况。特别是，我们证明了具有一定协方差膨胀的EnKF本质上是EnTLHF。从这个意义上说，EnTLHF提供了一个一般框架进行协方差膨胀的EnKF为基础的方法。我们使用一些数值例子来评估TLHF/EnTLHF的相对鲁棒性与相应的KF/EnKF方法相比。
英文摘要：
We propose a robust ensemble filtering scheme based on the $H_{\infty}$ filtering theory. The optimal $H_{\infty}$ filter is derived by minimizing the supremum (or maximum) of a predefined cost function, a criterion different from the minimum variance used in the Kalman filter. By design, the $H_{\infty}$ filter is more robust than the Kalman filter, in the sense that the estimation error in the $H_{\infty}$ filter in general has a finite growth rate with respect to the uncertainties in assimilation, except for a special case that corresponds to the Kalman filter. The original form of the $H_{\infty}$ filter contains global constraints in time, which may be inconvenient for sequential data assimilation problems. Therefore we introduce a variant that solves some time-local constraints instead, and hence we call it the time-local $H_{\infty}$ filter (TLHF). By analogy to the ensemble Kalman filter (EnKF), we also propose the concept of ensemble time-local $H_{\infty}$ filter (EnTLHF). We outline the general form of the EnTLHF, and discuss some of its special cases. In particular, we show that an EnKF with certain covariance inflation is essentially an EnTLHF. In this sense, the EnTLHF provides a general framework for conducting covariance inflation in the EnKF-based methods. We use some numerical examples to assess the relative robustness of the TLHF/EnTLHF in comparison with the corresponding KF/EnKF method.
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