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Xuguang Wang
and
Craig H. Bishop

Abstract

The ensemble transform Kalman filter (ETKF) ensemble forecast scheme is introduced and compared with both a simple and a masked breeding scheme. Instead of directly multiplying each forecast perturbation with a constant or regional rescaling factor as in the simple form of breeding and the masked breeding schemes, the ETKF transforms forecast perturbations into analysis perturbations by multiplying by a transformation matrix. This matrix is chosen to ensure that the ensemble-based analysis error covariance matrix would be equal to the true analysis error covariance if the covariance matrix of the raw forecast perturbations were equal to the true forecast error covariance matrix and the data assimilation scheme were optimal. For small ensembles (∼100), the computational expense of the ETKF ensemble generation is only slightly greater than that of the masked breeding scheme.

Version 3 of the Community Climate Model (CCM3) developed at National Center for Atmospheric Research (NCAR) is used to test and compare these ensemble generation schemes. The NCEP–NCAR reanalysis data for the boreal summer in 2000 are used for the initialization of the control forecast and the verifications of the ensemble forecasts. The ETKF and masked breeding ensemble variances at the analysis time show reasonable correspondences between variance and observational density. Examination of eigenvalue spectra of ensemble covariance matrices demonstrates that while the ETKF maintains comparable amounts of variance in all orthogonal and uncorrelated directions spanning its ensemble perturbation subspace, both breeding techniques maintain variance in few directions. The growth of the linear combination of ensemble perturbations that maximizes energy growth is computed for each of the ensemble subspaces. The ETKF maximal amplification is found to significantly exceed that of the breeding techniques. The ETKF ensemble mean has lower root-mean-square errors than the mean of the breeding ensemble. New methods to measure the precision of the ensemble-estimated forecast error variance are presented. All of the methods indicate that the ETKF estimates of forecast error variance are considerably more accurate than those of the breeding techniques.

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