(1) A comparison of breeding and ETKF ensemble forecast schemes
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 about the same as 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 verifications for 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 directions spanning its 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. In contrast, the amplification of the trace of the breeding ensemble covariance matrices exceeds that of the ETKF. The ETKF ensemble mean has lower root mean square errors than the means of the breeding ensembles. The ETKF estimates of forecast error variance are considerably more accurate than those of the breeding techniques.
(2) The spherical and symmetric ETKF
One more perturbation is introduced to the K linearly independent pure ETKF initial perturbations by multiplying these K perturbations by a K × (K +1) matrix to satisfy a). the mean of the K+1 initial perturbations equal zero, b). the outer product of the K+1 perturbations is equal to the analysis error covariance matrix obtained from the K pure ETKF initial perturbations, and c). all the K+1 initial perturbations are equally likely. This ensemble is called the spherical ETKF ensemble in contrast with the symmetric ETKF ensemble where perturbations are constructed directly from the positive/negative pairs of the pure ETKF. 17-member spherical ETKF and 17-member symmetric ETKF ensembles are compared. The computational expense of the spherical ETKF is only slightly greater than that of the symmetric ETKF. It is found that while maintaining similar skills of the ensemble mean as the symmetric ETKF, the spherical ETKF produces much higher rank estimates of error covariances. In the eigenvalue spectrum of the 12-hour forecast error covariance matrix, the spherical ETKF maintains comparable amounts of variance in 15 orthogonal directions while the symmetric ETKF maintains variance in 8 orthogonal directions. The spherical ETKF better resolves the inhomogeniety of the observation distribution. The spherical ETKF estimates the forecast error variance more accurate than the symmetric ETKF.