中文摘要：

集合数值天气预报的关键问题就是如何生成有效的初始扰动。奇异向量反映了初始扰动在大气系统相空间中演变发展的最不稳定方向，基于奇异向量产生的集合样本是模拟概率密度函数的最合理方法。以非静力、半隐式半拉格朗日GRAPES-Meso中尺度数值预报模式为基础，采用Lanczos迭代算法，利用GRAPES-Meso的切线伴随模式计算GRAPES奇异向量。为了检验求得的奇异向量的正确性，提出了两种检验奇异向量正确性的方法：一是比较计算的奇异值的一致性；二是依据特征向量在矩阵变换中的方向不变性意义，验证GRAPES奇异向量空间结构的正确性。最后研究了不同的时间间隔对GRAPES奇异向量的影响，结果表明GRAPES奇异向量在36小时的最优时间间隔误差增长速度最快，这表明在非静力、半隐式半拉格朗日格点模式中利用切线伴随技术计算奇异向量是可行的。

英文摘要：

One of the most difficult problems in ensemble forecasting is how to generate effective initial perturbations. The SVs （Singular Vectors） reflect the evolution of the initial perturbations along the most unstable directions in the atmospheric phase space. Therefore, to generate the ensemble members based on the SV technology is the most reasonable approach for estimating the evolution of the atmospheric probability density function. In this paper, based on the GRAPES＇ non-hydrostatic, semi-implicit and semi-Lagrangian mesoscale model, the SVs with the norm of total energy are calculated with the Lanczos algorithm and the tangent and adjoint version. Two approaches are designed to verify the correctness of GRAPES SVs; one is to compare the consistence of the singular values and the other is to validate the rationality of the spatial structure of the computed SVs according to its invariability of the matrix transformation. Finally, the impact of different time intervals on the SVs has been studied, and the results show that GRAPES SVs reach the maximum growth of error in perturbations at the time interval of 36 hours. These indicate that it is feasible to generate the initial perturbations by computing SVs using the GRAPES non-hydrostatic, semi-implicit and semi-Lagrangian model and this will establish the basis for the development of a GRAPES ensemble prediction system in the future.