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

Abstract

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.

Stochastic representations of model uncertainties at ECMWF: state of the art and future vision

“…In section 4, results were based on a fixed representation of initial uncertainties corresponding to the operational configuration of the ECMWF medium‐range/extended‐range ensemble. The initial perturbations are based on perturbations constructed from the EDA and singular vectors (Buizza et al , 2008; Leutbecher and Lang, 2014). The perceived optimal model uncertainty representation will under‐ (over)estimate the effect of actual model errors if the initial perturbations sample a wider (tighter) distribution than the distribution of actual initial errors.…”

Spectral analysis of forecast error investigated with an observing system simulation experiment

“…Initial perturbations can be constructed such that such decay is not observed, such as by using singular vectors (Farrell, 1988;Buizza et al, 1993;Palmer et al, 1998), but in this example this would just mask the real cause of error growth, namely model error, and therefore likely misrepresent some characteristics of the latter. In this pair of experiments, a large portion of the initial perturbations apparently project onto non-growing or even decaying short-term singular vectors, which is unsurprising given what is known of the spectra of such singular vectors (Errico et al, 2001) and the projections of analysis error on them (Leutbecher and Lang, 2014).…”

Introducing independent patterns into the Stochastically Perturbed Parametrization Tendencies (SPPT) scheme

“…The perturbed members arise due to the representation of initial state uncertainties via a combination of the ensemble of data assimilations (EDA: Isaksen et al , 2010) and singular vectors (see e.g. Leutbecher and Lang, 2014). Operationally, forecast uncertainties due to the model integrations are represented by the (standard) SPPT scheme and the Stochastic Kinetic Energy Backscatter (SKEB) scheme (Berner et al , 2009).…”

“…This uncertainty is represented by perturbing the initial conditions of the different ensemble forecast members, for example by using a singular vector approach (see e.g. Leutbecher and Lang, 2014) or by producing an ensemble of data assimilations (Isaksen et al , 2010).…”