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Chaotic dynamics and the role of covariance inflation for reduced rank Kalman filters with model error

机译：具有模型误差的降阶卡尔曼滤波器的混沌动力学和协方差膨胀的作用

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摘要

The ensemble Kalman filter and its variants have shown to be robust for data assimilation in high dimensional geophysical models, with localization, using ensembles of extremely small size relative to the model dimension. However, a reduced rank representation of the estimated covariance leaves a large dimensional complementary subspace unfiltered. Utilizing the dynamical properties of the filtration for the backward Lyapunov vectors, this paper explores a previously unexplained mechanism, providing a novel theoretical interpretation for the role of covariance inflation in ensemble-based Kalman filters. Our derivation of the forecast error evolution describes the dynamic upwelling of the unfiltered error from outside of the span of the anomalies into the filtered subspace. Analytical results for linear systems explicitly describe the mechanism for the upwelling, and the associated recursive Riccati equation for the forecast error, while nonlinear approximations are explored numerically.
机译：集合卡尔曼滤波器及其变体已显示出对高维地球物理模型中的数据同化具有鲁棒性的能力，并且具有相对于模型维的极小尺寸的局部化。但是，估计协方差的秩降低表示会留下未过滤的大尺寸互补子空间。利用反向Lyapunov向量的过滤动力学特性，本文探索了一个以前无法解释的机制，为协方差膨胀在基于集合的卡尔曼滤波器中的作用提供了新颖的理论解释。我们对预测误差演化的推导描述了未过滤误差从异常范围之外到过滤子空间的动态上升。线性系统的分析结果明确描述了上升过程的机理，以及与预测误差相关的递归Riccati方程，同时对数值逼近进行了非线性探索。

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