The "Chromatic Hexad" method for adaptive anisotropic covariance synthesis

We describe a systematic method to efficiently generate a spatially adaptive anisotropic covariance operator on a computational lattice for data assimilation in three dimensions. The computed self-adjoint covariance operator is formed as the product of a "square-root" factor and its adjoint. (The factor itself need not possess the self-adjointness symmetry, however). This square-root factor is generally taken to be a small sum of operators that individually resemble the quasi-Gaussian convolution obtained as the outcome of a general diffusive process acting for a finite time. A sum of this kind can produce desirable features, such as relatively fat tails to the final covariance at large separations, or negative side- lobes, which a single Gaussian term alone could not reproduce.

The key to achieving efficiency is to form each of the quasi-Gaussian terms by combining the "Hexad" geometrical decomposition principle (by which the three-dimensional operator is uniquely resolved into a sequence of possibly oblique line-operators in the lattice) with the proven method of spatial recursive filtering along the line segments that the hexad algorithm indicates. The spatial dispersion of each quasi-Gaussian contribution is quantified by the "aspect" tensor, defined as twice the product of the tensorial diffusivity and the "time" over which such diffusion must act to produce that contribution; it is also approximately the centered second moment of that quasi-Gaussian (and exactly that, in the case of perfect spatial homogeneity). The Hexad principle exploits the additive property of the general 3D diffusion operator that enables any such operator to be linearly decomposed locally into a unique hexad of (i.e., six) rank-one contributions of one-dimensional diffusion along possibly oblique grid lines whose configuration of directions are collectively bound by certain simple geometrical rules of validity. In the abstract space of the six aspect components, this resolution procedure can be interpreted geometrically as a linear projection of the aspect onto the six basis vectors associated with a valid hexad such that the resulting six projected components are all non- negative. Thus, the "cone" of all possible positive-definite aspects in this abstract linear space is effectively tiled, without gaps or overlaps, by all the "hexad cones", each one of which constitutes the set of all convex combinations of that hexad's six aspect basis vectors.

The rank-one operators that act along the oblique segments dictated by the hexad algorithm are the efficient quasi-Gaussian recursive filters used routinely for operational mesoscale data assimilation at NCEP, but now adapted to accommodate inhomogeneities of scale along the line. For spatially inhomogeneous background error-covariances constructed by this hexad method in its basic form, the several changes from one hexad to another that occur, as if at random, across interior surfaces of demarcation in the spatial domain, can unfortunately be sufficiently abrupt that they show up as loci of numerical roughness in the fields of analysis increments. The "blended hexad" method is a recently developed refinement which, at the cost of applying 13 line operations at each geographical location instead of only six, yields the quasi-Gaussian filter, with exactly the intended distribution of aspect tensors, but essentially free of discernable numerical artifacts.

A connection between formal geometrical characterizations of the hexad manipulations and some aspects of the mathematical theory of "Galois Fields" has been recently identified to provide a rigorous basis for a kind of "color coding" of all possible orientations of oblique lines in the computational lattice. In the basic hexad method, a seven-color assignment associated with one Galois field ensures that no valid hexad contains the same color of line twice; likewise, in the blended hexad, a coding of 13 colors from another Galois field will suffice to ensure that no lines of the sequence of 13 line-filters in the synthesis of a quasi-Gaussian can contain the same color twice. Thus, the coding automatically establishes a natural way to order the line operations (by their colors) so that filtering may proceed in parallel without risk of inadvertently corrupting the intermediate results of one line by the interfering operations of another that intersects it.

The steps involved in forging these geometrical ideas into a robust, practical and versatile scheme for the operational assimilation of data in three dimensions will be addressed and a brief discussion of a four- dimensional extension of the same ideas will be provided.