Three-Dimensional Variational Analysis with Spatially Inhomogeneous Covariances

In this study, a global three-dimensional variational analysis system is formulated in model grid space. This formulation allows greater flexibility (e.g., inhomogeneity and anisotropy) for background error statistics. A simpler formulation, inhomogeneous only in the latitude direction, was chosen for these initial tests. The background error statistics are defined as functions of the latitudinal grid and are estimated with the National Meteorological Center (NMC) method. The horizontal scales of the variables are obtained through the variances of the variables and of their Laplacian. The vertical scales are estimated through the statistics of the vertical correlation of each variable and are applied locally using recursive filters. For the multivariate correlation between wind and mass fields, a statistical linear relationship between the streamfunction and the balanced part of temperature and surface pressure is assumed. A localized correlation between the velocity potential and the streamfunction is also used to account for the positive correlation between the vorticity and divergence in the planetary boundary layer. Horizontally, the global domain is divided into three pieces so that efficient spatial recursive filters can be used to spread out the information from the observation locations. This analysis system is tested against the operational Spectral Statistical-Interpolation analysis system used at the National Centers for Environmental Prediction. The results indicate that 3DVAR in physical space is as effective as 3DVAR in spectral space in the extratropics and yields superior results in the Tropics as a result of the latitude dependence of the background error statistics.