Wang Bin, Liu Juanjuan, Liu Li, Xu Shiming, Huang Wenyu
LASG, Institute of Atmospheric Physics, Beijing, China.
Ministry of Education Key Laboratory for Earth System Modeling, Department of Earth System Science, Tsinghua University, Beijing, China.
PLoS One. 2018 Jan 19;13(1):e0191088. doi: 10.1371/journal.pone.0191088. eCollection 2018.
Localization techniques are commonly used in ensemble-based data assimilation (e.g., the Ensemble Kalman Filter (EnKF) method) because of insufficient ensemble samples. They can effectively ameliorate the spurious long-range correlations between the background and observations. However, localization is very expensive when the problem to be solved is of high dimension (say 106 or higher) for assimilating observations simultaneously. To reduce the cost of localization for high-dimension problems, an approach is proposed in this paper, which approximately expands the correlation function of the localization matrix using a limited number of principal eigenvectors so that the Schür product between the localization matrix and a high-dimension covariance matrix is reduced to the sum of a series of Schür products between two simple vectors. These eigenvectors are actually the sine functions with different periods and phases. Numerical experiments show that when the number of principal eigenvectors used reaches 20, the approximate expansion of the correlation function is very close to the exact one in the one-dimensional (1D) and two-dimensional (2D) cases. The new approach is then applied to localization in the EnKF method, and its performance is evaluated in assimilation-cycle experiments with the Lorenz-96 model and single assimilation experiments using a barotropic shallow water model. The results suggest that the approach is feasible in providing comparable assimilation analysis with far less cost.
由于集合样本不足，定位技术常用于基于集合的数据同化（例如集合卡尔曼滤波（EnKF）方法）。它们可以有效改善背景与观测值之间虚假的长程相关性。然而，当要解决的问题是高维问题（例如106或更高维）以便同时同化观测值时，定位的成本非常高。为了降低高维问题的定位成本，本文提出了一种方法，该方法使用有限数量的主特征向量近似扩展定位矩阵的相关函数，从而将定位矩阵与高维协方差矩阵之间的舒尔积简化为两个简单向量之间一系列舒尔积的和。这些特征向量实际上是具有不同周期和相位的正弦函数。数值实验表明，在一维（1D）和二维（2D）情况下，当使用的主特征向量数量达到20时，相关函数的近似扩展非常接近精确扩展。然后将新方法应用于EnKF方法中的定位，并在使用洛伦兹-96模型的同化循环实验和使用正压浅水模型的单同化实验中评估其性能。结果表明，该方法在以低得多的成本提供可比的同化分析方面是可行的。
