SENSITIVITY ANALYSIS OF FORECAST ERRORS AND THE CONSTRUCTION OF OPTIMAL PERTURBATIONS USING SINGULAR VECTORS

The sensitivity uf forecast errors to initial conditions is used to ex amine the optimality of perturbations constructed from the singular ve ctors of the tan ent propagator of the European Centre for Medium-Rang e Weather Forecasts model. Sensitivity and pseudo-inverse perturbation s based on the 48-h forecast error are computed as explicit linear com binations of singular vectors optimizing total energy over the Norther n Hemisphere. It is assumed that these perturbations are close to the optimal perturbation that can be constructed from a linear combination of these singular vectors. Optimality is measured primarily in terms of the medium range forecast improvement obtained by adding the pertur bations a posteriori to the initial conditions. Several issues are add ressed in the context of these experiments, including the ability oi s ingular vectors to describe forecast error growth beyond the optimizat ion interval, the number of singular vectors required, and the implica tions of nonmodal error growth. Supporting evidence for the use of sin gular vectors based on a total energy metric for studying atmospheric predictability is also presented. In general, less than 30 singular ve ctors capture a large fraction uf the variance of the Northern Hemisph ere sensitivity pattern obtained from a T63 adjoint model integration, especially in cases oi low forecast skill. The sensitivity patterns f or these eases tend to he highly localized with structures determined by the dominant singular vectors. Forecast experiments with these pert urbations show significant improvements in skill in tile medium range, indicating that singular vectors optimized for a short-range forecast continue to provide a useful description of error growth sell beyond this time. The results suggest that ensemble perturbations based on 10 -30 singular vectors should provide a reasonable description of the me dium-range forecast uncertainty, although the inclusion of additional singular vectors is likely to be beneficial. Nonmodality is a keg cons ideration in the construction of optimal perturbations. There is virtu ally no projection between the contemporaneous unstable subspaces at t he end or one forecast! trajectory portion and the beginning of a seco nd, consecutive portion. Sensitivity and ensemble perturbations constr ucted using the evolved singular vectors from a previous (day-2) forec ast are suboptimal for the current (day+0) forecast initial conditions . It is argued that these results have implications for a range of iss ues in atmospheric predictability including ensemble weather predictio n, data assimilation, and the development of adaptive observing techni ques.