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Numerical aspects of the application of recursive filters to variational statistical analysis with spatially inhomogeneous covariance
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  • Description:
    We describe the application of efficient numerical recursive filters to the task of convolving a spatial distribution of 'forcing' terms with a quasi-Gaussian self-adjoint smoothing kernel. In the context of variational analysis, this smoothing operation is interpreted to be a either a covariance function of background error, or a contributing component to a covariance function of non-Gaussian profile formed by the superposition of a number of such quasi-Gaussian smoothing operators. A superposition of positively-weighted quasi-Gaussian smoothers enables a useful range of covariance profiles to be synthesized which, in their idealized univariate and spatially homogeneous forms, imply power spectra that exhibit tails substantially fatter than those corresponding to the single Gaussian of approximately equivalent width. As a consequence, these synthetic covariances are more suitable statistical representations of background error than single Gaussians in the typical situations where a broad dynamical range of scales contribute significantly to this error. A further expansion of the potential range of synthetic covariances is achieved by combinations that also involve the negative-Laplacians of the basic component smoothers, thus enabling the negatively-correlated side-lobes of the covariances typical of some background errors to be more faithfully modelled. The methods we describe are not restricted to the production of spatially-homogeneous covariances; by spatially modulating either the superposition weights or the digital filtering coefficients themselves, it becomes possible to synthesize operators consistent with the properties of covariances which display adaptive variations of amplitude, scale and profile shape across the geographical domain. This is clearly desirable when the background itself derives from earlier data whose spatial distribution exhibits marked inhomogeneities of density or quality, and it is probably desirable also in the case of varying synoptic regimes within the domain. Among the computational aspect of the recursive filters, we treat the problems of periodic and nonperiodic boundary conditions and an approach to achieving efficient parallelization.
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