An approach to localization for ensemble-based data assimilation.

Bin Wang,Wenyu Huang,Shiming Xu,Juanjuan Liu,Li Liu,João Miguel Dias

doi:10.1371/journal.pone.0191088

Abstract

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.

Highlights

The statistical accuracy of background error is extremely important for any data assimilation scheme, and the background error covariance matrix is often estimated from ensembles[1,2,3,4]
We provide a simple comparison between the computational costs with and without localization, based on the ensemble Kalman Filter (EnKF) with simultaneous treatment for assimilating observations
We attempted to use a group of basis functions to expand the correlation function, and found that the spatial distributions of the leading eigenvectors of the correlation function are very close to the sine waves that are defined in the domain of definition

Summary

Introduction

The statistical accuracy of background error is extremely important for any data assimilation scheme, and the background error covariance matrix (the B matrix, hereinafter) is often estimated from ensembles[1,2,3,4]. To provide an intuitive understanding of the huge cost of localization in the EnKF when assimilating multi-source observations, including satellite measurements, let us consider a typical realistic NWP configuration such that mx = 107, my = 105 and n = 30. In this case, the calculation of the increment with localization takes about 3 × 1013 multiplications and about 3 × 1013 additions, much more expensive than without localization, which takes about 3 × 108 multiplications and 3 × 108 additions. We discuss our methods, the scope and limitations of this study, and some of the possible extension

Methodology

R bR l2 a

Findings

Conclusions