The “Chromatic Hexad” methodThe “Chromatic Hexad” methodfor adaptive covariance synthesisfor adaptive covariance synthesis

R. James PurserR. James PurserSAIC at NOAA/NWS/NCEP/EMCSAIC at NOAA/NWS/NCEP/EMC

Camp Springs, MDCamp Springs, MD

21st Conference on Weather Analysis and Forecasting/17 th Conference on Numerical Weather Prediction. 1st-5th August 2005, Washington D.C.

The Focus of this Talk:The Focus of this Talk:To describe and discuss some fairly specific geometricalfeatures of the algorithms we are developing at NCEP inorder to be able to implement, in an efficient way, geographically adaptive statistical assimilation techniquesfor our operational atmospheric (and oceanic?) models.

Acknowledgment: Much of the work described herebenefitted from the collaboration with, and feedbackfrom, Dave Parrish and Wan-Shu Wu. Also, the continuing support and encouragement of Geoff DiMego,Steve Lord and John Derber is greatly appreciated.

The topic we discuss is the creation of fast algorithms for the synthesis of covariance convolution operators on a regular computational lattice. In 3DVAR we wish to be able to use covariances that are not necessarily horizontally isotropic or spatially homogeneous. Rather, by exploiting a less restrictive set of covariance models, we wish to be able to gain at least some of the advantages of a 4DVAR scheme, but within the less costly 3DVAR framework.

It is possible to code a fairly efficient line-smoother by employing standard recursive (line-implicit) techniques. Many filtering shapes may be simulated by this method. However, the only shape that it makes sense to simulate, if we want to build up a useful three-dimensional filter, is the Gaussian because it is only this special shape that, by sequential transverse applications, fills outa three-dimensional form free of the imprint of the underlying grid orientation – contours are ellipsoidal.

ANY Gaussian in D dimensions can be synthesized by D line smoothers at (in general) carefully selected oblique orientations. However, we work on a grid where the set of practical smoothing orientations is limited.

Can we still synthesize any Gaussian by operations confined to the lines of the given grid?

Triads and HexadsTriads and HexadsThe affirmative answer to this question led to: • the “Triad” filtering algorithm in two dimensions and • the “Hexad” algorithm in three dimensions.

In general we claim that, in D dimensions, any homogeneousGaussian may be synthesized on a uniform grid by appropriate Gaussian line smoothers applied sequentially along (D*(D+1))/2generalized (possibly oblique) grid directions.

Moreover, it is possible to show that in D=2 or D=3 (but not forD > 3) dimensions, the combination of orientations and line smoothing “spreads” is UNIQUE in these minimal algorithms.

The Basic Hexad AlgorithmThe Basic Hexad AlgorithmSix lines are configured as the diameters of a minimalskewed grid-imbedded cuboctahedron (a figure with 8triangles alternating with 6 parallelograms).

Alternatively, by Legendre-Fenchel duality, the sixorientations may be characterized by the normals ofa minimal skewed (dual-)grid-embedded rhombic dodecahedron (12-hedron).

The centered second moment “aspect tensor”, with 6 independent components, is uniquely projected to 6 “smoothing weights” on each of the 6 directions of the hexad. But weights of an arbitrary hexad might not be all positive for a given aspect tensor.

What then?

Hexad transition rulesHexad transition rules Identify the most negative weight.Identify the most negative weight. Remove the offending vertex-pair of the Remove the offending vertex-pair of the

cuboctahedron.cuboctahedron. Create the replacement pair by extending Create the replacement pair by extending

outwards by *2 the twin centers of the old outwards by *2 the twin centers of the old quadrilaterals that did quadrilaterals that did notnot contain the old contain the old disqualified vertices.disqualified vertices.

A new cuboctahedron is formed!A new cuboctahedron is formed! Proceed until Proceed until all all weights are >0.weights are >0.

+1

+2+6

-1

-2-6

+3-5

+4

+3

+4

-5

+7

-6

+2

+6

-3

-2

-7

+1+1

+3+3

-6-6

+5+5-3-3

-4-4

-1-1

+6+6-5-5

+2+2

+1 +4+2 +5+3 +6

+3 +5+4 +7-6 -2

+1 +4+1 +4+2 +5+2 +5+3 +6+3 +6

+g(7) = +g(2) + g(3) – g(1)

The rule for the replacement of g(1), -g(1):

Chromatization via Galois FieldsChromatization via Galois FieldsEvery generalized grid line may be systematically assignedone of a finite palette of “colors” according to the non-nullelements of an appropriate Finite, or “Galois” Field.

Each hexad never contains the same color twice.(I.e., lines of the same color never cross in the analysis grid). Perform the filter sequence by color and spread the work-load without fear of data conflict.

The simplest chromatic hexad scheme has 7 colors [GF(8)].A “blended hexad” scheme, 13 colors, uses GF(27).Also, the algorithm design exploits the symmetries of GF(8) to ensure invariance of the labels of lines of a given hexadregardless of the path taken by the algorithm to reach it.

Repeated exponentiationof lambda is done,modulo-2 and modulo-the primitive polynomial.

By repeating the pattern of “colors” (the non-zero elements of GF(8)) in 2*2*2 blocksthat cover the entire grid of lattice line generators, we find that each generatortakes one of seven colors and the six distinct line generators of the hexad, corresponding to diameters of the associated cuboctahedron, are each assigneda different color.

Thus, in the basic hexad smoothing algorithm, we may perform all the line smoothingoperations of one color before we move on to the next. In this way, computationalconflicts are avoided, even when operations are performed in parallel.

In an extension of the hexad method to a “blended” form, the aspect tensor is,in effect, mapped symmetrically to a spherical “cloud” (in aspect space) beforebeing resolved into hexad weights. This is done to ensure better smoothness in aninhomogeneous covariance model. Coded carefully, this can be shown to result ina configuration of smoothing operations that separate naturally into the 13 “colors”that are implied by the Galois field associated with a repetition of 3*3*3 blocks –the field, GF(27).

Repeated exponentiation of lambda is done modulo-3 and modulo- the primitive polynomial.

[GF(27)]

DISCUSSION AND CONCLUSIONDISCUSSION AND CONCLUSION The algorithm in 3D can be extended, in its basic (unblended) form, to a The algorithm in 3D can be extended, in its basic (unblended) form, to a

corresponding 4D generalization.corresponding 4D generalization. However, the “tiling” of the 10-dimensional space of aspect tensors However, the “tiling” of the 10-dimensional space of aspect tensors

cannot be accomplished using just one form of “tile” analogous to a cannot be accomplished using just one form of “tile” analogous to a generic hexad. generic hexad.

Instead, it requires two distinct shapes that translate into 4D grid space Instead, it requires two distinct shapes that translate into 4D grid space as two different kinds of configurations of lines. as two different kinds of configurations of lines.

Nevertheless, the algorithm itself is straightforward to apply and another Nevertheless, the algorithm itself is straightforward to apply and another Galois field, GF(16), supplies a convenient “coloring” byGalois field, GF(16), supplies a convenient “coloring” by

which the line-smoothing operations can be scheduled, with 15 colors.which the line-smoothing operations can be scheduled, with 15 colors. This could be of use in smoothing a model error term in a weak-constraint This could be of use in smoothing a model error term in a weak-constraint

4DVAR scheme.4DVAR scheme. Meanwhile, the 3DVAR algorithms can now benefit from adaptiveMeanwhile, the 3DVAR algorithms can now benefit from adaptive anisotropic covariance operators being synthesized by a reasonablyanisotropic covariance operators being synthesized by a reasonably efficient “fast” numerical algorithm that exploits abstract algebraic ideas.efficient “fast” numerical algorithm that exploits abstract algebraic ideas.