Gaussian anamorphosis in the analysis step of the EnKF: a joint state-variable/observation approach
Gaussian anamorphosis in the analysis step of the EnKF: a joint state-variable/observation approach
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DOI：
10.3402/tellusa.v66.23493
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发表时间：
2014-01-01
影响因子：
2
通讯作者：
Van Leeuwen, Peter Jan
中科院分区：
文献类型：
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作者：
Amezcua, Javier;Van Leeuwen, Peter Jan
当(1)背景分布为高斯分布，(2)状态变量与观测值通过线性算子关联，(3)观测误差具有加性且具有高斯分布时，（集成）卡尔曼滤波器的分析步骤最优。当这些条件在很大程度上被违反时，可以应用称为高斯畸变（GA）的预处理步骤。这个过程的目的是获得在某种意义上更好地满足高斯性条件的状态变量和观测值。在这项工作中，我们从联合的角度分析遗传算法，关注联合状态变量/观测空间中变换的影响。首先，我们研究了相互独立的状态变量和观测值的变换。然后，我们引入了一个有针对性的联合变换，目的是得到变换空间中的联合高斯性。我们主要关注单变量情况，并对多变量情况作简要评论。本文的一个关键点是，当(1)-(3)被违反时，使用EnKF的分析步骤将无法恢复精确的后验密度，尽管可以执行任何转换。然而，这些变换为问题的贝叶斯解提供了不同质量的近似值。通过一个贝叶斯后验可以解析计算的例子，我们评估了在应用EnKF分析步骤与不同GA选项结合后生成的分析分布的质量。当先验为高斯，观测值的边际密度接近高斯，似然为高斯混合时，目标联合变换的值特别明显。
The analysis step of the (ensemble) Kalman filter is optimal when (1) the distribution of the background is Gaussian, (2) state variables and observations are related via a linear operator, and (3) the observational error is of additive nature and has Gaussian distribution. When these conditions are largely violated, a pre-processing step known as Gaussian anamorphosis (GA) can be applied. The objective of this procedure is to obtain state variables and observations that better fulfil the Gaussianity conditions in some sense. In this work we analyse GA from a joint perspective, paying attention to the effects of transformations in the joint state-variable/observation space. First, we study transformations for state variables and observations that are independent from each other. Then, we introduce a targeted joint transformation with the objective to obtain joint Gaussianity in the transformed space. We focus primarily in the univariate case, and briefly comment on the multivariate one. A key point of this paper is that, when (1)-(3) are violated, using the analysis step of the EnKF will not recover the exact posterior density in spite of any transformations one may perform. These transformations, however, provide approximations of different quality to the Bayesian solution of the problem. Using an example in which the Bayesian posterior can be analytically computed, we assess the quality of the analysis distributions generated after applying the EnKF analysis step in conjunction with different GA options. The value of the targeted joint transformation is particularly clear for the case when the prior is Gaussian, the marginal density for the observations is close to Gaussian, and the likelihood is a Gaussian mixture.