Representation of correlation functions in variational assimilation using an implicit diffusion operator
Representation of correlation functions in variational assimilation using an implicit diffusion operator
复制标题
使用隐式扩散算子表示变分同化中的相关函数
DOI：
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发表时间：
2010
期刊：
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通讯作者：
A. T. Weaver
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作者：
Isabelle Mirouze;A. T. Weaver
在数据同化中，需要相关模型来表征数值网格上定义的变量的误差结构。以前的研究表明，扩散方程可以提供一个灵活和计算效率高的框架，用于表示大尺寸问题的网格点相关函数，例如在大气或海洋变分数据同化中遇到的问题。在这篇文章中，基于扩散的相关性模型的隐式公式作为传统显式公式的替代。隐式公式详细分析了一维（1D）问题，并显示出密切相关的一阶递归滤波器。在M步上积分具有常系数的1D隐式扩散方程被证明等价于将初始条件与M阶自回归（AR）函数卷积。AR函数的长度尺度和生成单位振幅（相关）函数所需的归一化因子的表达式都以M和扩散系数的形式给出。对于固定的长度尺度，高斯函数是唯一可以用常系数扩散方程的显式公式表示的函数，它是隐式扩散方程生成的AR函数的M → ∞的极限情况。讨论了扩散模型的推广，以允许相关函数的不同形状和长度尺度的空间变化。使用空间变化长度尺度的一个重要结果是归一化因子不再是常数。的归一化因子的近似表达式的有效性进行评估，以提供可行的替代方案，使用昂贵的算法，如随机化产生的估计。边界条件会扭曲边界附近的相关函数，并显着降低归一化因子的解析表达式的精度。这些问题可以通过扩散模型的直接扩展来避免，该扩展使边界有效透明，尽管该解决方案以额外应用扩散方程为代价。扩展的方法来构建二维和三维相关模型进行了讨论。皇家气象学会
Correlation models are required in data assimilation to characterize the error structures of variables defined on a numerical grid. Previous studies have shown that the diffusion equation can provide a flexible and computationally efficient framework for representing grid‐point correlation functions for problems of large dimension such as those encountered in atmospheric or ocean variational data assimilation. In this article, an implicit formulation of the diffusion‐based correlation model is presented as an alternative to the traditional explicit formulation. The implicit formulation is analyzed in detail for the one‐dimensional (1D) problem and shown to be closely related to the first‐order recursive filter. Integrating a 1D implicit diffusion equation, with constant coefficient, over M steps is shown to be equivalent to convolving the initial condition with an Mth order auto‐regressive (AR) function. Expressions for both the length‐scale of the AR function and the normalization factor required to generate unit‐amplitude (correlation) functions are given in terms of M and the diffusion coefficient. For a fixed length‐scale the Gaussian function, which is the only function that can be represented using an explicit formulation of the constant‐coefficient diffusion equation, is the limiting case as M → ∞ of the AR functions generated by the implicit diffusion equation. Generalizations of the diffusion model are discussed to allow for different shapes in the correlation function and spatial variations in the length‐scale. An important consequence of employing spatially varying length‐scales is that the normalization factors are no longer constant. Approximate expressions for the normalization factors are evaluated in terms of their effectiveness to provide viable alternatives to estimates produced using expensive algorithms such as randomization. Boundary conditions can distort the correlation functions near the boundaries and significantly degrade the accuracy of the analytical expressions for the normalization factors. These problems can be avoided through a straightforward extension of the diffusion model that makes the boundaries effectively transparent, although the solution comes at the expense of an extra application of the diffusion equation. Extensions of the method to construct two‐ and three‐dimensional correlation models are discussed. Copyright © 2010 Royal Meteorological Society