A comparison of variance and total‐energy singular‐vectors

Abstract

AbstractThe analysis‐error variance estimates produced by the Naval Research Laboratory Atmospheric Variational Data Assimilation System are used to constrain singular vectors (SVs) at initial time (the final‐time metric is total energy). These ‘VAR SVs’ are compared to SVs for which total energy is both the initial‐ and final‐time metrics (TE SVs). The spatial distributions and structures of the VAR SVs and TE SVs are examined and their effectiveness in explaining forecast error is compared. Consistent with previous results, the VAR SVs have relatively larger amplitude than TE SVs in regions where analysis errors are larger (such as over oceans and polar regions). In contrast to previous results, the initial‐time VAR SV energy peaks at lower altitudes than the TE SV energy, reflecting the influence of the data‐assimilation‐error variance estimates. Despite these significant differences, the TE and VAR SVs explain comparable amounts of forecast error in both linear and nonlinear contexts. By projecting the VAR SV pseudo‐inverse perturbation onto the TE SVs, it is found that the component inside the TE‐SV subspace accounts for most of the forecast‐error reduction. The component outside the TE‐SV subspace has very little impact on error reduction. It is noteworthy that for both the TE and VAR SVs, nonlinear pseudo‐inverse forecast‐corrections are better than the expected linear corrections. Both sets of SVs exhibit a relationship between singular‐vector growth and forecast‐error total energy, when averaged over many cases. This indicates that the SVs have some utility in predicting forecast‐error variance. Copyright © 2005 Royal Meteorological Society.