On the representation of initial uncertainties with multiple sets of singular vectors optimized for different criteria

Initial uncertainties can be represented effectively in ensemble prediction systems by sampling errors in a subspace spanned by the leading singular vectors of the forecast model's tangent-linear propagator. The initial-time metric in the singular-vector computation is the inverse of the assumed analysis-error covariance matrix A. The singular vectors evolve into the leading eigenvectors of the forecast-error covariance estimate obtained by linearly propagating A to the singular-vector optimization time. In this sense, singular vectors provide an optimal subspace for sampling initial uncertainties. However, the optimality is only guaranteed for the particular optimization criterion used in the singular-vector computation. For instance, it could be suboptimal for forecast ranges that differ from the singular-vector optimization time. Here, two alternative approaches, accounting for several optimization criteria, are discussed. The first is a simple orthonormalization approach applied to multiple sets of singular vectors. Potentially, the orthonormalization can yield suboptimal perturbations. The second approach has been developed in response to the expected deficiency of the orthonormalization approach. It yields orthogonal subspaces for different optimization criteria without compromising optimality. For a given subspace L, spanned by a set of leading singular vectors optimized for the first criterion (or criteria), singular vectors are computed in the subspace orthogonal to L. The optimality properties of these subspace-singular vectors are described and proved. Subspaces obtained with the two approaches are compared in two examples. First, an idealized example based on singular vectors computed for two optimization times in the Eady model is considered. Then, both techniques are applied to initial perturbations targeted on tropical cyclones in the ECMWF Ensemble Prediction System. The methodologies allow a consistent representation of initial uncertainties during extratropical transitions. Copyright (c) 2007 Royal Meteorological Society.