A Geometrical Approach to Anisotropic Covariance Synthesis

R. James Purser



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.