Session 3A.5 Efficient situation-dependent covariance generation using recursive filters and the methods of Riemannian geometry

Monday, 1 June 2009: 2:45 PM
Grand Ballroom East (DoubleTree Hotel & EMC - Downtown, Omaha)
R. James Purser, IMSG and NOAA/NCEP/EMC, College Park, MD

Presentation PDF (81.1 kB)

Recursive filters are used in NCEP’s Gridpoint Statistical Interpolation (GSI) to generate background error covariances in a numerically efficient way. The basic recursive filter operates along a grid line, in both directions sequentially, producing a response profile similar to that one would expect from a diffusive process. Therefore, by applying a sequence of such filters along lines intersecting transversally, the resulting simulated diffusion efficiently spreads the influence of the initial “input” throughout a surrounding multi-dimensional region. The ellipsoidal shape of the region significantly influenced is determined by the choice of the line orientations of the basic filters and by the strengths of the smoothing coefficients of the latter. The generated profile is approximately Gaussian at the completion of this sequence of recursive filters, but a greater variety of profiles may be constructed by repeating the sequence with different settings of correlation scales and amplitudes and summing the results. However, owing to the “implicit” nature of the basic filters, anticipating and controlling the amplitude of each quasi-Gaussian component becomes problematic when the correlation scales implied by the assumed diffusivity tensor varies across the grid.

We have recently addressed this problem by radically reformulating the filters as if they are simulations of isotropic and homogeneous diffusion on a gridded domain with a varying Riemannian metric taking the place of the formerly varying diffusivity. This new perspective allows the amplitude of diffusive components to be estimated asymptotically by the well-studied “parametrix expansion method”, which uses diagnostics of the geometry’s implied curvature, and its derivatives, to correct the amplitude formula associated with the simple Gaussian model.

This reformulation of the recursive filtering technique is being combined with a multigrid structure to allow the efficient aggregation of several quasi-Gaussian components possessing a range of scales. These developments will be described in detail at the conference.

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