Estimating deformations of random processes for correlation modelling in a limited area model
Estimating deformations of random processes for correlation modelling in a limited area model
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估计有限区域模型中相关建模的随机过程的变形
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发表时间：
2013
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通讯作者：
Y. Michel
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作者：
Y. Michel
协方差模型广泛用于数据同化，特别是当选择变分公式时。每个变量的自协方差通常分为方差和相关部分。尽管最近取得了进展，但这种相关性仍然经常被建模为空间静止的。本文介绍了一种新颖的方法，其中背景误差相关性通过平稳相关模型的空间变形来近似。该模型的统计校准涉及估计变形、其倒数和平稳相关性。利用计算机视觉社区之前的工作，首先以二维方式描述该问题。定向小波，也称为扭曲，能够测量不同尺度和方向的随机过程的能量。纹理梯度方程将变形过程的小波分析与局部度量变化联系起来。因此，根据一些实现，可以估计变形的雅可比行列式的局部变化。本文表明，通过求解具有非齐次系数的椭圆性质的偏微分方程，可以从变形梯度中恢复变形。讨论了边界条件的选择。然后将该方法应用于有限区域模型中背景误差相关性的建模。与大气数据同化中经常使用的其他水平相关模型（例如小波基础中的扩散方程或对角线假设）的定性相互比较表明，该方法也能够在相当程度上表示各向异性。版权所有 © 2012 英国皇家气象学会
Covariance models are widely used in data assimilation, especially when a variational formulation is chosen. Autocovariances for each variable are generally split into variance and correlation parts. Despite recent progress, the correlations are still often modelled as space‐stationary. A novel approach is introduced in this paper, where background error correlations are approximated by the spatial deformation of a stationary correlation model. The statistical calibration of this model involves estimating the deformation, its inverse and the stationary correlations. Leveraging previous work in the computer vision community, the problem is first described in two dimensions. Directional wavelets, also called warplets, are able to measure the energy of a random process at different scales and orientations. The texture gradient equation relates the wavelet analysis of a deformed process to local metric changes. From a few realizations, it is therefore possible to estimate local variations of the Jacobian of the deformation. This paper shows that a deformation can be recovered from the deformation gradient by solving partial differential equations of elliptic nature with inhomogeneous coefficients. The choice of boundary conditions is discussed. The method is then applied to the modelling of background error correlations in a limited area model. Qualitative intercomparison with other horizontal correlation models frequently used in atmospheric data assimilation, such as the diffusion equation or the diagonal assumption in a wavelet basis, suggests that the approach is also able to represent anisotropy to a fair degree. Copyright © 2012 Royal Meteorological Society