On the reliability of ensemble variance in subspaces defined by singular vectors

In a statistically consistent ensemble forecast, ensemble variance agrees with the mean squared error of the ensemble mean when looking at a large sample of independent realizations. Here, the first goal is to introduce a new diagnostic that quantifies the reliability of ensemble variance in subspaces spanned by singular vectors of the model propagator. The second goal is to apply this technique to ensemble forecast experiments with the European Centre for Medium-Range Weather Forecasts (ECMWF) Integrated Forecasting System (IFS) in order to gain new insight into essential approximations made when the sources of forecast uncertainties are represented with initial perturbations based on the leading initial singular vectors. Diagnostics are based on extratropical singular vectors optimized for growth of total energy over 48 hours. For the initial perturbations as well as the singular vector diagnostics, two versions of the tangent-linear model are compared, which differ in terms of the accuracy of the tangent-linear approximation. When model uncertainties are not represented, initial perturbations based solely on the leading singular vectors can achieve reliable ensemble variances in the subspace spanned by the singular vectors evolved to their optimization time. However, it is impossible to achieve at the same time reliable total variances with up to about 100 singular vectors in the extratropics. Far more singular vectors would be required to do so. Ensembles using initial perturbations based on up to 100 leading singular vectors with amplitudes tuned such that total variance matches total error suffer from overdispersion in the subspace spanned by the leading singular vectors. In the operational configuration of the ECMWF ensemble, the singular vector initial perturbations introduce overdispersion in the subspace of the leading 50 extratropical singular vectors.