高维集合卡尔曼滤波
High Dimensional Ensemble Kalman Filter
AI总结：
本文提出了高维集合卡尔曼滤波（HD-EnKF）方法，通过配备一致的预测误差协方差和Kalman Gain矩阵估计量，解决了高维状态变量下的理论问题，并证明了其在模型误设下的优越性。
摘要：
集合卡尔曼滤波（EnKF）作为一种基础的数据同化方法，已广泛应用于科学与工程的许多领域。当状态变量具有高维特征并伴随物理模型的高分辨率观测时，EnKF的一些关键理论问题仍有待研究。本文提出了几种高维EnKF（HD-EnKF）方法，配备了重要预测误差协方差和Kalman Gain矩阵的一致估计量。随后研究了在固定维和高维状态变量下EnKF的理论性质，给出了分析状态相对于Kalman滤波提供的潜在oracle状态的一步和多步均方误差，并深入揭示了预测误差协方差对EnKF准确性的作用。此外还考虑了物理模型误设下的数据同化准确性。在Lorenz-96和Shallow Water Equation模型上的数值研究表明，所提出的HD-EnKF算法优于标准EnKF和广泛使用的膨胀方法。
英文摘要：
The Ensemble Kalman Filter (EnKF), as a fundamental data assimilation approach, has been widely used in many fields of the sciences and engineering. When the state variable is of high dimensional accompanied with high resolution observations of physical models, some key theoretical aspects of the EnKF are open for investigation. This paper proposes several high dimensional EnKF (HD-EnKF) methods equipped with consistent estimators for the important forecast error covariance and Kalman Gain matrices. It then studies the theoretical properties of the EnKF under both fixed and high dimensional state variables, which provides one-step and multiple-step mean square errors of the analysis states to the underlying oracle states offered by the Kalman Filter and gives the much needed insight to the roles played by the forecast error covariance on the accuracy of the EnKF. The accuracy of the data assimilation under the misspecified physical model is also considered. Numerical studies on the Lorenz-96 and the Shallow Water Equation models illustrate that the proposed HD-EnKF algorithms outperform the standard EnKF and widely used inflation methods.
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