A consistent interpretation of the stochastic version of the Ensemble Kalman Filter,Quarterly Journal of the Royal Meteorological Society

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A consistent interpretation of the stochastic version of the Ensemble Kalman Filter
Quarterly Journal of the Royal Meteorological Society ( IF 3.0 ) Pub Date : 2020-06-17 , DOI: 10.1002/qj.3819
Peter Jan Leeuwen _{1,

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Affiliation

Ensemble Kalman Filters are used extensively in all geoscience areas. Often a stochastic variant is used, in which each ensemble member is updated via the Kalman Filter equation with an extra perturbation in the innovation. These perturbations are essential for the correct ensemble spread in a stochastic Ensemble Kalman Filter, and are applied either to the observations or to the modelled observations. This paper investigates if there is a preference for either of these two perturbation methods. Both versions lead to the same posterior mean and covariance when the prior and the likelihood are Gaussian in the state. However, ensemble verification methods, Bayes' Theorem and the Best Linear Unbiased Estimate (BLUE) suggest that one should perturb the modelled observations. Furthermore, it is known that in non‐Gaussian settings the perturbed modelled observation scheme is preferred, illustrated here for a skewed likelihood. Existing reasons for the perturbed observation scheme are shown to be incorrect, and no new arguments in favour of that scheme have been found. Finally, a new and consistent derivation and interpretation of the stochastic version of the EnKF equations is derived based on perturbing modelled observations. It is argued that these results have direct consequences for (iterative) Ensemble Kalman Filters and Smoothers, including “perturbed observation” 3D‐ and 4D‐Vars, both in terms of internal consistency and implementation.

中文翻译：

对Ensemble Kalman滤波器的随机形式的一致解释

Ensemble Kalman过滤器广泛用于所有地球科学领域。通常使用随机变量，其中每个集合成员通过卡尔曼滤波器方程式进行更新，并且在创新中会产生额外的干扰。这些扰动对于在随机的集成卡尔曼滤波器中进行正确的整体散布是必不可少的，并且可以应用于观测值或建模观测值。本文研究这两种扰动方法中的任何一种是否存在偏爱。当先验和似然在该状态下时，这两种形式都导致相同的后验均值和协方差。但是，集成验证方法，贝叶斯定理和最佳线性无偏估计（BLUE）建议人们应该干扰建模的观测值。此外，众所周知，在非高斯环境中，最好采用扰动的建模观测方案，此处以偏斜可能性进行说明。扰动观测方案的现有原因被证明是不正确的，没有发现支持该方案的新论据。最后，基于扰动的建模观测结果，得出了对EnKF方程的随机形式的新的一致的推导和解释。有人认为，这些结果会对（迭代）集成卡尔曼滤波器和平滑器产生直接影响，包括内部一致性和实现方面的“扰动观测” 3D和4D变量。尚未发现支持该方案的新论据。最后，基于扰动的建模观测结果，得出了对EnKF方程的随机形式的新的一致的推导和解释。有人认为，这些结果会对（迭代）集成卡尔曼滤波器和平滑器产生直接影响，包括内部一致性和实现方面的“扰动观测” 3D和4D变量。尚未发现支持该方案的新论据。最后，基于扰动的建模观测结果，得出了对EnKF方程的随机形式的新的一致的推导和解释。有人认为，这些结果会对（迭代）集成卡尔曼滤波器和平滑器产生直接影响，包括内部一致性和实现方面的“扰动观测” 3D和4D变量。

更新日期：2020-06-17

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