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A Geometrical Approach to Anisotropic Covariance Synthesis

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R. James Purser
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SAIC at NOAA/NCEP/EMC
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Abstract:
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In order to overcome the limitations of the isotropic background error
covariances currently being used in operational meteorological data
assimilation, an effort is under way at NCEP to develop new grid-based
assimilation schemes for global and regional domains accommodating fully
anisotropic and spatially inhomogeneous covariances. In the context of an
assimilation it is customary to treat covariances not as explicit matrices
but rather as parameterized operators. In the grid domain such operators
take the form of smoothing filters and it is natural to synthesize them
by the sequential application of filters of a particularly simple form.

The "Triad" and "Hexad" algorithms are recently developed methods for two,
and three dimensions respectively, by which anisotropic covariance operators
of approximately Gaussian shape may be efficiently constructed as sequences
of one-dimensional pseudo-diffusive smoothing operators applied along
a precisely determined set of generalized (possibly oblique) lines of the
assimilation grid. When the covariance shape changes from location to location
it is found that the simplest forms of the triad and hexad algorithms can
exhibit unsightly numerical artifacts wherever the set of active generalized
grid directions implied by these algorithms is forced to change. However,
by exploiting symmetries and some geometric methods that relate to the
abstract algebraic theory of "Galois Fields", it is possible to systematically
construct "blended" generalizations of the basic algorithms free of the
aforementioned defects and only a little less efficient.

This talk will focus on these geometrical aspects of the construction of the
blended triad and hexad algorithms and will touch briefly on the prospects
for extending these same techniques to four dimensions.