Optimal Prediction of Forecast Error Covariances through Singular Vectors

Optimal perturbations, also referred to as singular vectors (SVs), currently constitute an important guideline for the generation of initial ensembles to be used for ensemble prediction. The optimality of these perturbations refers to their property of maximizing prespecified quadratic measures of error growth, given that tangent-linear error evolution is assumed. The goal of ensemble prediction is the accurate prediction of the uncertainty of forecasts made with dynamical numerical weather prediction models. In the present paper the theoretical justification for the use of SVs in ensemble prediction systems is investigated. It is shown that, in a tangent-linear framework, SVs—constructed using covariance information valid at the initial time—evolve into the eigenvectors of the forecast error covariance matrix valid for the end of the optimization interval. As such, SVs represent the most efficient means for predicting the forecast error covariance matrix, given a prespecified number of allowable (tangent-linear) model integrations. Such optimal prediction is of particular importance in light of the fact that the forecast error covariance matrix is summarizing important information about the probability density function of the model state at a given future time. Based on the above result, optimal covariance prediction through appropriately determined SVs is demonstrated here for a three-dimensional Lorenz model, as well as for a barotropic model of intermediate dimensionality, both within a perfect-model framework. In the case of the barotropic model it is found that less than 15% of the SVs suffice to account for more than 95% of the total final error variance. Viewed differently, at least 80% of the final error variance is accounted for by retaining those SVs that are amplifying in terms of an enstrophy norm. In addition, variances and covariances predicted through SVs agree closely with independently obtained Monte Carlo estimates, as long as the tangent-linear approximation is sufficiently accurate. Further, the problem of approximating the forecast error covariance matrix in the presence of a state-independent model-error representation is briefly considered. The paper is concluded with a summary of the results and a discussion of their possible implications on data assimilation procedures and on the further development of ensemble prediction systems.