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Singularity-free global telescoping grids suitable for large-scale global covariance synthesis by anisotropic recursive filtering
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  • Corporate Authors:
    National Centers for Environmental Prediction (U.S.)
  • Description:
    Background error covariances in the Gridpoint Statistical Interpolation (GSI) of NCEP's operational three-dimensional data assimilation schemes are formed by the repeated applications of recursive numerical filters along carefully selected families of parallel lines of the discretization grids used to represent the analysis state. Owing to the need for the analysis grid to be free of coordinate singularities in order that the outputs of the recursive filters remain smooth, the global version of the GSI presently employs a system of three overlapping grids. One grid patch is a cylindrical coordinate projection that covers most of the globe but excludes the poles; then a pair of stereographic projection Cartesian grid patches cover the omitted polar regions, but with a generous overlap across the edges of the cylindrical grid patch. In this way the global GSI can form portions of the analysis increments pertaining to each of the singularity-free patches with the final composition blended to remove any sharp discontinuity between them. However, for the very largest scales of covariances encountered in the highest levels of the atmosphere the width of the blending zone shared by each pair of overlapping patches is not sufficient to avoid an inevitable distortion of the shape of the synthesized covariance from the shape intended. This note describes a new grid configuration, designed to supplement the existing three-patch configuration, that allows very large scale covariances to be synthesized by the recursive filtering technique without unintended distortion. The basic idea is to map the Earth's atmosphere, regarded as a spherical shell of finite thickness, to the interior of a set of three-dimensional cubic grids in a nested configuration that allows the resolution to be increased (usually by progressive factors of two) towards the common center of the cubes. The sizes of the successive nests, and the radial transformation carrying the atmospheric shell to the cubes, are both under the user's control so that different vertical-to-horizontal aspect ratios of covariances at different altitudes can all be represented by approximately isotropic covariance shapes within the cubic grids. Although the cubic grids are free of singularities, they do possess interior boundaries which must be treated by smooth blending of only the interior contributions constructed on each of these nests, and by blending the totality with the covariance contributions synthesized on the original three-patch grid configuration. We present the theory for the construction of the nested cubic grids, together with some preliminary results obtained by adopting this approach in an experimental extension of the global GSI.

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