|Title||On the representation of initial uncertainties with multiple sets of singular vectors optimised for different criteria.|
|Publication Type||Technical memorandum|
|Secondary Title||ECMWF Technical Memorandum|
Singular vectors provide an optimal subspace for sampling initial uncertainties in the sense of maximising a linear estimate of forecast error variance. However, the optimality is only guaranteed for the particular optimisation criterion used in the singular vector computation. For instance, it could be suboptimal for forecast ranges that differ from the singular vector optimisation time. Here, two alternative approaches are discussed that account for several optimisation criteria. The first approach is a simple ortho-normalisation approach applied to multiple sets of singular vectors. Potentially, the ortho-normalisation can yield suboptimal perturbations. In response to the expected deficiency of the ortho-normalisation approach, the second approach has been developed. It yields orthogonal subspaces for different optimisation criteria without compromising optimality. For a given subspace L, consisting of a set of leading singular vectors optimised 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. The leading subspaces obtained with the two approaches are compared in two examples. First,an idealised example based on singular vectors computed for two optimisation times in the Eady model is considered. Then, both techniques are applied to initial perturbations targeted on tropical cyclones in the Ensemble Prediction System (EPS) of the European Centre for Medium-Range Weather Forecasts. The methodologies allow a consistent representation of initial uncertainties during extra-tropical transitions.