Analysis error covariance versus posterior covariance in variational data assimilation
Analysis error covariance versus posterior covariance in variational data assimilation
复制标题
变分数据同化中的分析误差协方差与后验协方差
DOI：
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发表时间：
2013
期刊：
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通讯作者：
F.‐X. Le Dimetc
中科院分区：
文献类型：
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作者：
I. Gejadze;V. Shutyaevb;F.‐X. Le Dimetc
将非线性发展模式的变分资料同化问题转化为一个最优控制问题，以寻找初始条件函数（分析）。数据包含误差（观察误差和背景误差）;因此分析中存在误差。对于轻度非线性动力学的分析误差协方差可以近似的逆Hessian的成本功能的辅助数据同化问题，和更强的非线性的“有效”逆Hessian。然而，已经注意到，从贝叶斯的角度来看，分析误差协方差不是后验协方差。虽然这两者在线性情况下是等效的，但随着非线性水平的上升，实际上差异可能变得显著。对于适当的贝叶斯后验协方差通过Hessian推导出一个新的近似和它的“有效”的对应物。在无矩阵环境下，利用Lanczos方法和预处理，给出了一种计算上述估计的方法.数值算例验证了所发展的理论由Burgers方程与非线性粘性项的模型。
The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find the initial condition function (analysis). The data contain errors (observation and background errors); hence there is an error in the analysis. For mildly nonlinear dynamics the analysis error covariance can be approximated by the inverse Hessian of the cost functional in the auxiliary data assimilation problem, and for stronger nonlinearity by the ‘effective’ inverse Hessian. However, it has been noticed that the analysis error covariance is not the posterior covariance from the Bayesian perspective. While these two are equivalent in the linear case, the difference may become significant in practical terms with the nonlinearity level rising. For the proper Bayesian posterior covariance a new approximation via the Hessian is derived and its ‘effective’ counterpart is introduced. An approach for computing the mentioned estimates in the matrix‐free environment using the Lanczos method with preconditioning is suggested. Numerical examples which validate the developed theory are presented for the model governed by Burgers equation with a nonlinear viscous term.