Q. J. R. Meteorol. Soc. (2005), 131, pp. 1–999 doi: 10.1256/qj.05.66 Impact of flow-dependent analysis-error covariance norms on extratropical singular vectors By MARK BUEHNER ∗ and AYRTON ZADRA Meteorological Research Branch, Meteorological Service of Canada, Canada (Received 14 April 2005; revised 29 July 2005) SUMMARY Flow-dependent analysis-error covariances are estimated from the 128 member analysis ensembles of a pre- operational implementation of the ensemble Kalman filter. Singular vectors are then computed with the initial- time norm defined using the inverse of these error covariances and the final-time norm defined by the total energy over North America. An optimization time interval of either 24 or 48 hours is used and the horizontal resolution of the singular vectors is 3 ◦ . Provided that the analysis-error covariance estimates are accurate, these singular vectors should optimally explain the forecast error at the final time. To reduce the sampling error due to the small ensemble size, the analysis-error covariances are spatially localized. The impact of using the analysis-error covariance norm, with or without spatial localization, on the 20 leading singular vectors is measured relative to using the total-energy norm. In addition, the singular vectors are also compared with sets of 20 randomly selected members of the analysis ensembles. The results are generally consistent with those of previous studies with the maximum impact from the initial- time norm seen at the initial time. The growth is significantly reduced by using the analysis-error covariance norm instead of total energy and the smallest growth is obtained when no covariance localization is applied. The total-energy singular vectors are slightly more effective at explaining forecast error for most lead times than the analysis-error covariance singular vectors. This may point to errors in the covariance estimates, the importance of model error in affecting forecast error evolution, or shortcomings with the tangent linear model used to compute the singular vectors. The use of a random selection of members from the analysis ensembles shows an even larger impact. In contrast with the singular vectors that continually grow over the 72 h period examined, the total energy of the ensemble members initially decays during tangent linear model integrations. Also, the amount of forecast error explained by the ensemble members is significantly less than for all types of singular vectors. This suggests that, with respect to predicting forecast uncertainty over North America, the current approach of randomly selecting analysis ensemble members to initialize the ensemble prediction system may be improved. KEYWORDS: Covariance localization Ensemble Kalman filter Ensemble prediction 4D-var 1. I NTRODUCTION Ensemble forecasting and observation targeting are two areas of ongoing interest and research activity within the numerical weather prediction (NWP) community. For both applications several approaches have been used, including the singular vectors (SVs) of the tangent linear version of an NWP forecast model (e.g. Buizza and Montani 1999; Buizza et al. 1997). SVs are the most rapidly growing structures over a chosen time interval with respect to the norms specified at initial and final times. Consequently, the SVs are the structures that evolve into an optimal description of the final-time subspace (with respect to the final-time norm) that would be obtained by linearly propagating all possible initial-time perturbations of fixed amplitude (with respect to the initial-time norm). Under certain conditions, the SVs computed with an initial-time norm based on the analysis-error covariances (AEC) optimally describe the forecast error distribution at the final (optimization) time (Ehrendorfer and Tribbia 1997). Thus, such AEC SVs represent an optimal reduced-rank approximation of the full AEC matrix when propagated to the final time with the tangent linear model (TLM). This makes AEC SVs a desirable candidate for both ensemble forecasting and observation targeting, especially when the rank of the AEC matrix is much larger than the maximum number of forecast model integrations feasible. However, the quality of AEC SVs clearly depends on the accuracy of the AEC estimates. ∗ Corresponding author: Meteorological Service of Canada, 2121 Trans-Canada Hwy, Dorval, Quebec, H9P 1J3, Canada. e-mail: mark.buehner@ec.gc.ca c  Crown copyright, 2005. 1