The prior covariance calculated in the Reduced-rank Sigma-Point Kalman filter (RSPKF) data assimilation method can be suboptimal as a result of finite number of sigma points, effects of sampling error, process error, and other factors. In this study, we analyse the performance of RSPKF by applying a localization scheme that combines local analysis and global generation of sigma points. The analysis at each model grid for the entire domain are updated using only the observations local to the analysis grid. The localization enables a high rank approximation of the background error covariance in a local subspace of greatly lower dimension than the global domain using a small number of sigma points. The global-analysis vector is constructed combining the local analyses at all model grid points. The global analysis-covariance matrix is generated in the ensemble subspace and the global sigma points are constructed following the RSPKF algorithm. Numerical experiments of our method utilized the Lorenz-96 model. The performance of the localization scheme is assessed in the presence of varying parameters such as the number of sigma points ($k$), inflation factor ($\varphi$), localization radius ($d$) and the number of model variables ($N_x$). When the localization is implemented, the number of sigma points required to achieve the minimum RMSE is significantly reduced compared to a case where no localization is used, for three different cases of model variables ($N_x$). We also show that the approximate number of sigma points used to obtain optimal estimate is independent of the state dimension $N_x$ of the model. This further highlights the importance of localization in RSPKF, making it a potential candidate for data assimilation in oceanic or atmospheric General Circulation Models (GCMs).

由于有限数量的 sigma 点、采样误差、过程误差和其他因素的影响，在降阶 Sigma-Point Kalman 滤波器 (RSPKF) 数据同化方法中计算的先验协方差可能是次优的。在这项研究中，我们通过应用结合局部分析和全局 sigma 点生成的定位方案来分析 RSPKF 的性能。整个域的每个模型网格上的分析仅使用分析网格本地的观测值进行更新。使用少量 sigma 点，本地化能够在比全局域低得多的维度上对背景误差协方差进行高阶近似。结合所有模型网格点的局部分析构建全局分析向量。全局分析协方差矩阵在集成子空间中生成，全局 sigma 点按照 RSPKF 算法构建。我们方法的数值实验利用了 Lorenz-96 模型。定位方案的性能是在存在不同参数的情况下进行评估的，例如西格玛点的数量（$克$), 通货膨胀因素 ($\phi$), 定位半径 ($d$) 和模型变量的数量 ($N_X$）。当实现定位时，对于三种不同的模型变量情况，与不使用定位的情况相比，实现最小 RMSE 所需的 sigma 点数显着减少（$N_X$）。我们还表明，用于获得最优估计的 sigma 点的近似数量与状态维度无关$N_X$模型的。这进一步凸显了 RSPKF 定位的重要性，使其成为海洋或大气环流模型 (GCM) 中数据同化的潜在候选者。