Numerical Aspects of the Application of Recursive Filters to Variational Statistical Analysis. Part I: Spatially Homogeneous and Isotropic Gaussian Covariances

Purser, R. James; Wu, Wan-Shu; Parrish, David F.; Roberts, Nigel M.
August 2003
Monthly Weather Review;Aug2003 Parts 1 and 2, Vol. 131 Issue 8, p1524
Academic Journal
The construction and application of efficient numerical recursive filters for the task of convolving a spatial distribution of "forcing" terms with a quasi-Gaussian self-adjoint smoothing kernel in two or three dimensions are described. In the context of variational analysis, this smoothing operation may be interpreted as the convolution of a covariance function of background error with the given forcing terms, which constitutes one of the most computationally intensive components of the iterative solution of a variational analysis problem. Among the technical aspects of the recursive filters, the problems of achieving acceptable approximations to horizontal isotropy and the implementation of both periodic and nonperiodic boundary conditions that avoid the appearance of spurious numerical artifacts are treated herein. A multigrid approach that helps to minimize numerical noise at filtering scales greatly in excess of the grid step is also discussed. It is emphasized that the methods are not inherently limited to the construction of purely Gaussian shapes, although the detailed elaboration of methods by which a more general set of possible covariance profiles may be synthesized is deferred to the companion paper (Part II).


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