An Efficient Local Correlation Matrix Decomposition Approach for the Localization Implementation of Ensemble‐Based Assimilation Methods

Ensemble‐based data assimilation methods often use the so‐called localization scheme to improve the representation of the ensemble background error covariance ( B e ). Extensive research has been undertaken to reduce the computational cost of these methods by using the localized ensemble samples to localize B e by means of a direct decomposition of the local correlation matrix C . However, the computational costs of the direct decomposition of the local correlation matrix C are still extremely high due to its high dimension. In this paper, we propose an efficient local correlation matrix decomposition approach based on the concept of alternating directions. This approach is intended to avoid direct decomposition of the correlation matrix. Instead, we first decompose the correlation matrix into 1‐D correlation matrices in the three coordinate directions, then construct their empirical orthogonal function decomposition at low resolution. This procedure is followed by the 1‐D spline interpolation process to transform the above decompositions to the high‐resolution grid. Finally, an efficient correlation matrix decomposition is achieved by computing the very similar Kronecker product. We conducted a series of comparison experiments to illustrate the validity and accuracy of the proposed local correlation matrix decomposition approach. The effectiveness of the proposed correlation matrix decomposition approach and its efficient localization implementation of the nonlinear least‐squares four‐dimensional variational assimilation are further demonstrated by several groups of numerical experiments based on the Advanced Research Weather Research and Forecasting model.