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

Abstract

Ensemble‐based data assimilation methods often use the so‐called localization scheme to improve the representation of the ensemble background error covariance (Be). Extensive research has been undertaken to reduce the computational cost of these methods by using the localized ensemble samples to localize Be 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.

A Big Data‐Driven Nonlinear Least Squares Four‐Dimensional Variational Data Assimilation Method: Theoretical Formulation and Conceptual Evaluation

“…, , C is the spatial correlation matrix, is computed together with , and r is the selected truncation mode number (Zhang & Tian, ). The definition of “(⋅ < e > ⋅)” can also be found in Zhang and Tian ().…”

“…Unfortunately, computational resources currently allow the 4DEnVar ensemble to be about 10 2 in size, which is far less than the dimensions n x (~10 7 –10 9 ) of the model states. The small ensemble size usually results in spurious long‐range correlations of B e (Houtekamer & Mitchell, ), which could be filtered out using a localization process (Zhang & Tian, ). This process is computationally expensive but has been made more efficient by Tian et al () using a local correlation matrix decomposition approach (Zhang & Tian, ).…”

“…The small ensemble size usually results in spurious long‐range correlations of B e (Houtekamer & Mitchell, ), which could be filtered out using a localization process (Zhang & Tian, ). This process is computationally expensive but has been made more efficient by Tian et al () using a local correlation matrix decomposition approach (Zhang & Tian, ). Despite this, a finite set of samples do not represent the tangent models accurately, thus degrading the 4DEnVar performance.…”

Implementation of a Modified Adaptive Covariance Inflation Scheme for the Big Data‐Driven NLS‐4DVar Algorithm

“…Here, is the adaptive inflation parameter for i th grid ensemble; is the error variance of ; the superscripts b , o , and a denote background, estimated, and analysis, respectively (Miyoshi, ); , , and are the local observation operator, observation vector, and observation error covariance (Hunt et al, ), respectively, for the i th model grid; x b is the global background field, and are N model state perturbations (MPs); is a diagonal matrix that contains the localization weights for each observation (i.e., represented by spatial correlations between the i th model grid and its local observations ; values range from 0 to 1); “∘”indicates the Schür product of matrices P and Q , which is a matrix whose ( i , j ) entries are given by P ( i , j ) × Q ( i , j ) (Zhang & Tian, ).…”

“…In this study, we first modified the M11 inflation scheme by expanding its scope from local to global, to facilitate its application in 4DEnVar data assimilation methods. We also adopted an efficient local correlation matrix decomposition approach (Zhang & Tian, ) to compute the localization weights for each observation necessary to the scheme, thereby enhancing its computation efficiency. Then we implemented the modified M11 scheme with BD‐NLS4DVar to improve its robustness.…”