Generation of Global Multivariate Error Covariances by Singular-Value Decomposition of the Linear Balance Equation

Abstract

Effective data assimilation algorithms require the estimation and specification of second-moment forecast error statistics. The imposition of multivariate constraints has been found to be a particularly effective way of extracting the maximum amount of information from the observations. In the optimal interpolation (OI) algorithm, geostrophic constraints were imposed, and these have been extended to the global case in the newer three-dimensional variational (3DVAR) algorithm by the application of the linear balance equation or a Rossby?Hough expansion. This study shows that the imposition of the linear balance equation (or the Rossby?Hough expansion) in the Tropics (although mathematically attractive) is not correct and may have deleterious effects on the assimilated products. A procedure is developed, based on the singular-value decomposition (svd) of the linear balance equation, for generating global forecast error covariances in which the multivariate coupling between wind and mass is completely user controlled. Thus, two modal spaces are defined: a spectral (spherical harmonic) space for controlling the redness of the spectrum and a balance (svd) space for controlling the coupling. In this way it is possible for the rotational wind and mass error covariances to be closely coupled in line extratropics and essentially univariate at low latitudes.