Which Is Better, an Ensemble of Positive–Negative Pairs or a Centered Spherical Simplex Ensemble?

Abstract

New methods to center the initial ensemble perturbations on the analysis 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 initial perturbations to ensure that the sum of the new initial perturbations equals zero; the covariance calculated from the new initial perturbations is equal to the analysis error covariance estimated by the independent initial perturbations, and all of the new initial perturbations are equally likely. The new method is illustrated by applying it to the ensemble transform Kalman filter (ETKF) ensemble forecast scheme, and the resulting ensemble is called the spherical simplex ETKF ensemble. It is shown from a multidimensional Taylor expansion that the symmetric positive-negative paired centering would yield a more accurate forecast ensemble mean and covariance than the spherical simplex centering if the ensemble were large enough to span all initial uncertain directions and thus the analysis error covariance was modeled precisely. 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. The performances of the spherical simplex ETKF and symmetric positive-negative paired ETKF ensembles are compared by using the Community Climate Model Version 3 (CCM3). Each ensemble contains 1 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 about as small as that for the symmetric positive-negative paired ETKF. Also shown is that the seemingly straightforward centering method, in which centered perturbations are obtained by subtracting the average of the perturbations from each individual perturbation, is unsatisfactory because the covariance estimated by the uncentered perturbations is not necessarily conserved after centering.

Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter

“…The use of the symmetric square root to determine W a fromP a (as compared to, for example, a Cholesky factorization, or the choice described in [4]), is important for two main reasons. First, as we will see below, it ensures that the sum of the columns of X a is zero, so that the analysis ensemble has the correct sample mean (this is also shown for the symmetric square root in [44]). Second, it ensures that W a depends continuously oñ P a ; while this may be a desirable property in general, it is crucial in a local analysis scheme, so that neighboring grid points with slightly different matricesP a do not yield very different analysis ensembles.…”

Assessment of a Nonlinear Ensemble Transform Filter for High-Dimensional Data Assimilation

“…Following Wang et al (2004), the unique symmetric square root should be chosen to compute the transform matrix in Eq. (4).…”

Controlling model error of underdamped forecast models in sparse observational networks using a variance‐limiting Kalman filter

“…In particular we use the ETKF (Bishop et al, 2001;Tippett et al, 2003;Wang et al, 2004), which seeks a transformation S ∈ R k×k such that the analysis deviation ensemble Z a is given as a deterministic perturbation of the forecast ensemble Z f via Z a = Z f S. Details of the implementation can be found in Bishop et al (2001), Tippett et al (2003) and Wang et al (2004). Alternatively one could choose the ensemble adjustment filter (Anderson, 2001) in which the ensemble deviation matrix Z f is pre-multiplied with an appropriately determined matrix A ∈ R N×N .…”

“…In a perfect model data assimilation scheme, unrealistic overestimation of the error covariances occurs in sparse observational networks as a finite size effect; it was shown that, for short observational intervals of up to 10 h, the VLKF produces superior skill when compared to the standard ensemble transform Kalman filter (ETKF) (Bishop et al, 2001;Tippett et al, 2003;Wang et al, 2004). Here we will apply the VLKF in the case of an imperfect, underdamped forecast model, and for large observation intervals.…”