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A geometrical approach to the synthesis of smooth anisotropic covariance operations for data assimilation
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    This article describes a fairly general approach to the synthesis of smooth spatial covariance operators, based on the application of certain geometrical properties of regular lattices. The work is motivated by the need, in data assimilation, to be able to convolve an arbitrary distribution of gridded data by a smooth covariance function, which we may assume locally to be the superposition of at most a small set of bell-shaped quasi-Guassians. This operation must be carried out repeatedly during the course of the iterative solution to the optimization problem that constitutes the major computational task of objectively assimilating new data. Since it tends to be highly non-local in all spatial dimensions, it becomes the main computational bottle-neck of the whole process. It is therefore imperative that the combination of numerical operations that collectively synthesize the covariance convolution are constructed to be as efficient as possible. With efficiency as one of the main objectives, we find that, once we have established the Gaussian form as the versatile building-block for the additive synthesis of more general covariance shapes, we are able to synthesize the Gaussian covariance profile itself via only a modest number of orientations of parallel line-smoothing filters applied sequentially. An earlier publication supplied a cursory outline of the most basic forms of these algorithms, referred to respectively as the Triad and Hexad methods in two and three dimensions. However, a great deal more about the underlying geometry and symmetry principles of these techniques has been discovered recently. This allows us to extend the methods systematically to achieve a greater degree of spatial consistency in the case of the markedly inhomogenous statistics so typical of meteorological or oceanic data analysis. The modified blended versions of these methods utilize a few additional directions of filtering and a systematic reconstruction of the collective smoothing parameters to ensure a smoother result overall. We describe here some of these formal geometrical insights and the new algorithms they have inspired and we briefly touch upon the construction of four-dimensional fully anisotropic covariances by the analogous extension of this generic approach to covariance synthesis in dimensions greater than three.

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