Unbiased ensemble square root filters
Unbiased ensemble square root filters
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DOI：
10.1016/j.physd.2008.01.005
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发表时间：
2008-06-15
影响因子：
4
通讯作者：
Nichols, Nancy K.
中科院分区：
文献类型：
--
作者：
Livings, David M.;Dance, Sarah L.;Nichols, Nancy K.
有一个已建立的框架，将一类集合卡尔曼滤波器算法描述为平方根滤波器（SRFS）。这些算法通过更新一体协方差矩阵的状态估计和平方根来产生分析。预测协方差的矩阵平方根被另一个矩阵进行了多个矩阵，从而给出了分析协方差的基质平方根。后型后层次的选择不是唯一的，并且可以用任何正交矩阵乘以另一个也是SRF的方案。并非所有这种过滤器都具有与预测合奏的期望关系：分析集合平均值可能不等于分析状态估计值，因此在分析集合的传播中可能存在伴随的分析集合的短缺，如集合协方差矩阵所示。这表明需要一个限制版的合奏父亲概念，我们称之为公正的合奏SRF。本文提供了一组通用的必要条件，使该方案以分析状态估计（无偏见）为中心。其中一些结果已经在文献中的其他地方发表。本文将它们与新的结果一起融合在一起，并提供了简单的证明和示例，以及对无偏见SRF的数学描述。许多（但不是全部）发表的集合SRF算法满足我们建立的标准。尽管这些条件并非完全治愈，并且无法处理诸如模型和观察错误之类的独立偏见来源，但它们将来对集合SRF的设计师应该有用。 （c）2008 Elsevier B.V.保留所有权利。
There is an established framework that describes a class of ensemble Kalman filter algorithms as square root filters (SRFs). These algorithms produce analyses by updating a state estimate and a square root of the ensemble covariance matrix. The matrix square root of the forecast covariance is post-multiplied by another matrix to give a matrix square root of the analysis covariance. The choice of post-multiplier is not unique and can be multiplied by any orthogonal matrix to give another scheme that is also a SRF. Not all filters of this type bear the desired relationship to the forecast ensemble: the analysis ensemble mean may not be equal to the analysis state estimate and consequently there may be an accompanying shortfall in the spread of the analysis ensemble as expressed by the ensemble covariance matrix. This points to the need for a restricted version of the notion of an ensemble SIRE, which we call an unbiased ensemble SRF. This paper provides a generic set of necessary and sufficient conditions for the scheme to be centred on the analysis state estimate (unbiased). A few of these results have already been published elsewhere in the literature; this paper brings these together with new results and provides simple proofs and examples, as well as a mathematical description of the set of unbiased ensemble SRFs. Many (but not all) published ensemble SRF algorithms satisfy the criteria that we establish. While these conditions are not a Cure-all and cannot deal with independent sources of bias such as model and observation errors, they should be useful to designers of ensemble SRFs in the future. (C) 2008 Elsevier B.V. All rights reserved.