Application of anisotropic recursive filters to the generation of planetary scale background covariances

The recursive filter is an efficient numerical technique for simulating diffusion along a grid line. The NCEP’s Gridpoint Statistical Interpolation (GSI) uses it as covariance generator. A horizontally isotropic covariance can be generated by applying the filter along each grid coordinate direction, but the filter also has the ability to generate three-dimensional anisotropic covariances with the “hexad algorithm” when applied along six carefully selected, possibly oblique lines of the grid. This capability enables the synthesis of situation-dependent background covariances with shapes appropriately stretched or compressed to reflect ambient conditions.

On the other hand, the filter has the critical limitation that it must act on a grid free of polar singularities. In the global GSI this limitation is overcome by using a three-patch system of overlapping grids to avoid the polar singularities. When correlation scales are of modest extent, an adequate degree of continuity of the synthesized covariances is obtained by combining a smooth blending with the filters within the overlapping portions. But this expedient fails for covariances of very large scales such as can occur at very high altitudes. Thus it is desirable to augment the present three-patch GSI gridding with a special treatment designed to accommodate the whole globe in a horizontally uninterrupted grid system for planetary scale background error covariances above the tropopause.

For this purpose, we are developing a three-dimensional “cubic-patch”, gridded in Cartesian fashion, which contains the three-dimensional spherical shell occupied by the atmosphere’s upper layers in a configuration that avoids any coordinate singularity. The embedded sphere within the cubic grid to which a constant sigma level is projected determines its horizontal (angular) resolution according to the distance from the cube’s center. Since the scales of the atmospheric phenomena tend to be smaller at lower altitudes, we project the vertical levels in reverse order of radial distance in the cubic-patch so that the uppermost atmospheric layers with the coarsest angular resolution project to the innermost radii within the cube. But to further reduce excessive differences of the horizontal resolution at each vertical layer, we also apply a nesting of cubic grids. We use the anisotropic recursive filter to construct the covariance following the sphere to which a constant sigma level is projected.

We present results comparing the covariances generated with the nested cubic grids with the corresponding covariances generated with the standard three-patch grid configuration for some idealized horizontally-isotropic covariances of large horizontal extent. We also illustrate the effect of varying the radial projection in the cubic grid, and show how to choose a suitable radial mapping to ensure that anisotropies of these covariances within the cubic grid are minimized. Finally, we discuss how the new configuration may be integrated within a more general multigrid framework that allows covariance profiles that are not necessarily of the quasi-Gaussian type to be synthesized efficiently from superpositions of quasi-Gaussian components.