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Future Covariance Enhancements

R. J. PurserIMSG at NOAA/NCEP/EMC

Acknowledgements: Helpful and stimulating discussions with Dave Parrish,Manuel de Pondeca, Geoff DiMego, John Derber, Wan-Shu Wu and Daryl Kleist.

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Recent technical developments offer an opportunity to introduceenhancements in both capability and, most likely, in computational performance of GSI’s background error covariances.

This would apply to:

• RTMA• Meso GSI• Global GSI

The objective is to provide a more versatile covariance operator:

• Anisotropic• Adaptive to the day’s weather• Spanning a greater range of scales• Allowing greater control of profile shape

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“Wheel of Pain” (from Conan the Barbarian)

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The operators and (column) vector spaces of variational data assimilation:

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It is helpful to bear in mind the picture of six distinct “spaces” through which the vectors must progress in strict cyclic order.

Each iteration of the conjugate-gradient algorithm represents onerevolution of the “wheel of pain”, whether this is done in the “primal”form, or in the “dual” form of the assimilation.

In either case, the sequence of operators follows the same cyclicpattern – color-coding the operators (at least mentally) preventsaccidental errors.

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primal

dual

Defining the “primal” control variable, v, we seek to minimize:

Whether in primal or dual form, the operators always act in the same cyclic order:

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Our interest in the first part of this presentationcenters on the operator C and its adjoint, C , and specifically, those parts of these operators that do NOT referto the {u,v} {phi,chi} conversion, or the strong constraints.

Even restricting attention to the apparently very similar-lookingrecursive filtering operator, C, and its adjoint, we MUSTcontinue to recognize that C-adjoint carries vectors from the “red” space to a DIFFERENT “yellow” space, and C thencecarries the vectors to yet another DIFFERENT “green”space.

T

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The RTMA is already anisotropic, but not yet adaptive in the day-to-day sense.

The 3D GSI is not even statically anisotropic. Anisotropic filtering involvesmore computations, but evidence from an “enhancement” to the RTMAsuggests that the real bottleneck is communications. Dave Parrish implementedan alternative factorization of the line-recursive filters which should havehalved the computational cost (half the calculations) – but it made essentiallyno difference.

This suggests that it is the parallelization and all the associated communicationsthat need radical redesign.

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At present, the line filters of the anisotropic filtering scheme are gathered as longcontinuous strings and each string contained completely within one processor.This relieves the need for communication during each individual line-filtering procedure, but the gathering and redistribution must be done before and afterevery line-filtering stage – and there are very many of these in the anisotropicfilters.

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At present, the line filters of the anisotropic filtering scheme are gathered as longcontinuous strings and each string contained completely within one processor.This relieves the need for communication during each individual line-filtering procedure, but the gathering and redistribution must be done before and afterevery line-filtering stage – and there are very many of these in the anisotropicfilters.

An alternative is to take inputFrom the “red” space insidea relatively narrow rectangularsub-region. Each red regiongoes to a different processorand, collectively, they tilethe (regional) analysis domain.

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At present, the line filters of the anisotropic filtering scheme are gathered as longcontinuous strings and each string contained completely within one processor.This relieves the need for communication during each individual line-filtering procedure, but the gathering and redistribution must be done before and afterevery line-filtering stage – and there are very many of these in the anisotropicfilters.

An alternative is to take inputFrom the “red” space insidea relatively narrow rectangularsub-region. Each red regiongoes to a different processorand, collectively, they tilethe (regional) analysis domain.

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At present, the line filters of the anisotropic filtering scheme are gathered as longcontinuous strings and each string contained completely within one processor.This relieves the need for communication during each individual line-filtering procedure, but the gathering and redistribution must be done before and afterevery line-filtering stage – and there are very many of these in the anisotropicfilters.

An alternative is to take inputFrom the “red” space insidea relatively narrow rectangularsub-region. Each red regiongoes to a different processorand, collectively, they tilethe (regional) analysis domain.

Operator C-adjoint than smoothsThis windowed data over anExtended domain – the resultsAre added back to a referenceGrid in the “yellow space”.

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After adding together, on the “yellow” reference grid, all the overlapping contributions,we are ready for the action of operator, C, itself.

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After adding together, on the “yellow” reference grid, all the overlapping contributions,we are ready for the action of operator, C, itself.

Adjoint interpolate back fromthe reference grid and applyall the factors of C in the orderreversing the order in whichtheir adjoints appear in C-adjoint.

But the only part of the “green”space result we want, is thesmaller region.

The output in the smallerregion is copied to the “green”reference grid.

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How wide do the margins have to be?

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The contributing terms to C are quasi-Gaussians.

We can take a one-dimensional look at the magnitude of the error.

In the following idealizations, an actual gaussian smoother isapplied , with boundary conditions that mimic what we would beforced to use.

The “margin” is on one side of each of two abutting domains andeach is of width 50 units.

The characteristic widths of the gaussians tested are:• 40• 30• 20• 15

An initial impulse is placed just inside the right tile (at its extremeleft).

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The margin width needs to be about 2 or 3 times the width of the widestquasi-gaussian component on the grid.

If we adopt the MULTIGRID concept, and synthesize each gaussianon a grid of resolution JUST of sufficient resolution, we do not needsuch an unreasonable number of additional margin grid points andcorresponding computations.

Remember: the cost of filtering at double the scale is ¼ in 2D and 1/8 in3D !

But we can do even better is we exploit the fact that the resolutionwe need in the margin is not uniform. At the outer edge of the “yellow”region, only the largest scale of gaussian is “felt” and the grid resolutioncan be made correspondingly smaller. Approximately, dx can be madeproportional to distance outside the inner region.

Moreover, this resolution applies transversally as well as a normally,since the two go together for the gaussians dealt with.

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A “Fibonacci” constellation allows exactly this combined change of both normal andtransverse resolution.

(It is best in this context not to think of it as defining any single “grid” – it offerscurvilinear and smooth connections simultaneously in MANY different directions,which is exactly what the poly-triad and poly-hexad algorithms require.)

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A “Fibonacci” constellation allows exactly this combined change of both normal andtransverse resolution.

(It is best in this context not to think of it as defining any single “grid” – it offerscurvilinear and smooth connections simultaneously in MANY different directions,which is exactly what the poly-triad and poly-hexad algorithms require.)

The Fibonacci arrangement was first invented by Mother Nature

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A “Fibonacci” constellation allows exactly this combined change of both normal andtransverse resolution.

(It is best in this context not to think of it as defining any single “grid” – it offerscurvilinear and smooth connections simultaneously in MANY different directions,which is exactly what the poly-triad and poly-hexad algorithms require.)

The Fibonacci arrangement was first invented by Mother Nature

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A “Fibonacci” constellation allows exactly this combined change of both normal andtransverse resolution.

(It is best in this context not to think of it as defining any single “grid” – it offerscurvilinear and smooth connections simultaneously in MANY different directions,which is exactly what the poly-triad and poly-hexad algorithms require.)

The Fibonacci arrangement was first invented by Mother Nature

Its use as a grid for meteorological applications was suggested by R. Swinbank andR.J. Purser – see QJ, 2006: vol 132, 1769—1793.

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The “basis” vectors of thelocally useful grid generallyvary from place to place.

An analogue of the Fibonaccimagic can be found in threedimensions also.

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A locally quasi-isotropic, but variable resolution, Fibonacci constellation WITHOUT spirals!

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This optimally-self-avoiding property that characterizes the Fibonacci constellations is NOT a characteristic of randomorientations of the initial square grid relative to theorientation of the imposed deformation.

We can see this immediately when we actually run the same kind of simulation of deformation on a grid thatwe orient randomly:

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A randomly-oriented latticealmost alwayscollapses intouseless configurations, When subjectedTo the influenceof a constant deformation.

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The crucial property of the Fibonacci constellation is that it is optimally SELF-AVOIDING under appropriately oriented dilatations or deformations.

In the 3D context (Meso and Global GSI), the analogue of the “Fibonacci constellation” of points would allow the resolution of the resulting effective grid in the stratosphere to be made much more consistent with the actual scale of the covariance contributions there.

In other words, the horizontal resolution of the filtering grid would automatically become smoothly and progressively smaller on ascending through to the upper troposphere and on to the stratosphere. This could cut the requirement of filtering grid points to a fraction of what is presently needed.

In the case of the global GSI, we could use the Fibonacci constellation and its 3Danalogue to simplify the configuration of “patches” from the present 3-patcharrangement to a more elegant and symmetrical one with two congruent overlappingpatches, each extending as much as 60 degrees beyond the equator, if necessary.

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The implied “metric” used to infer preferred connectionsbetween points of the Fibonacci constellation need notbe isotropic or uniform throughout the domain.

According to this choice of metric we can thereforegenerate a wide variety of “grids” from the same constellation.

For example, the covariances in the stratospheretend to have very broad and flat covariancescompared to those of the troposphere which, especially low down, will tend to be much morevertically oriented:

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The large characteristic sizes of covariances in the stratosphereallows them to be adequately resolved with a much smallerallowance of grid points – BUT ONLY IF WE ADOPT THEFIBONACCI CONSTELLATIONS OF POINTS (or their 3Dgeneralization).

Let’s take a look at the 3D self-avoiding constellation.

In this case, our point of view is from a designated “central”member of the constellation and the imposed dilatation actsas a uniform abd homogeneous rate of deformation.