Wednesday, 14 January 2004
A comparison of an ensemble of positive/negative pairs and a centered spherical simplex ensemble
A new method to center the ensemble perturbations on the minimum error variance estimate of the atmospheric state are introduced and compared with the commonly used centering method of positive/negative paired perturbations. In the new method, one linearly dependent perturbation is added to a set of linearly independent perturbations to ensure that the sum of the new initial perturbations equals zero; the covariance of the new initial perturbations is equal to that of the independent initial perturbations; and all the new initial perturbations are equally likely. We illustrate the new method by applying it to the ensemble transform Kalman filter (ETKF) ensemble forecast scheme and call the resulting ensemble the spherical simplex ETKF ensemble. It is shown from multi-dimensional Taylor expansion that the symmetric positive/negative paired centering would yield a more accurate ensemble mean and covariance than the spherical simplex centering if the ensemble were large enough to span all uncertain directions. However, when the number of uncertain directions is larger than the ensemble size, the spherical simplex centering has the advantage of allowing almost twice as many uncertain directions to be spanned as the symmetric positive/negative paired centering. Spherical simplex ETKF and symmetric positive/negative paired ETKF ensembles are compared by using version 3 of the community climate model (CCM3). Each ensemble contains one control forecast and 16 perturbed forecasts. 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 accuracy of the ensemble means, the accuracy of predictions of forecast error variance and the ability of the ETKF ensembles to resolve inhomogeneities in the observation distribution were all tested. In all of these test categories, the spherical simplex ETKF ensemble was found to be superior to the symmetric positive/negative paired ETKF ensemble. The computational expense for generating spherical simplex ETKF initial perturbations is as small as that for the symmetric positive/negative paired ETKF. We also show that the seemingly straightforward centering method where centered perturbations are obtained by subtracting the average of the perturbations from individual perturbation is unsatisfactory because the covariance of the perturbations is not conserved before and after centering.