3D‐Var Hessian singular vectors and their potential use in the ECMWF ensemble prediction system
Singular vectors are computed which are consistent with 3D-Var (three-dimensional variational) estimates of analysis error statistics. This is achieved by defining the norm at initial time in terms of the full Hessian of the 3D-Var cost function. At final time the total energy norm is used. the properties of these Hessian singular vectors (HSVs) differ considerably from total energy singular vectors (TESVs) in such aspects as energy spectrum and growth rate. Despite these differences, the leading 25 TESVs and HSVs explain nearly the same part of the 2-day forecast error. 
 
Two experimental ensemble configurations are studied. One configuration uses perturbations based on HSVs in the computation of initial perturbation, the other uses TESVs and 2-day linearly evolved singular vectors (ESVs) of two days before. the latter approach provides a way to include more stable and large-scale structures in the perturbations. Ten pairs of ensembles are compared to the operational European Centre for Medium-Range Weather Forecasts Ensemble Prediction System. the ensembles using ESVs perform slightly better. the ensembles based on HSVs show a slightly worse performance and are lacking some spread in the medium range. Possible directions to improve the computation of HSVs are discussed.