# Presentation: 080329

## Tools for Anisotropic Covariances at NCEPR. James Purser, Manuel de Pondeca, David F. Parrish and John C. Derber

- A Fast Algorithm for anisotropic covariances in 2D and 3D has been developed based on geometrical properties of regular lattices.
- Extensions to 4D are possible (e.g., for proper treatment of the model-error term in weak-constraint 4D Var).
- Some preliminary testing of adaptive algorithms is being carried out at NCEP.

- THORPEX meeting, Jan. 18 2006, NCEP.

**NOAA/NWS/NCEP/EMC, Camp Springs, MD 20746-4304)**

## Covariances in Statistical Objective Analysis

- Construction of an increment field in 3DVar involves the convolution of data forcing terms with a spatial covariance.
- Many iterations of this operation occur in a typical solution algorithm.
- There is a need to accommodate anisotropic, inhomogeneous covariances, especially as we move to mesoscale assimilation.

**Properties of a covariance:**

- (Necessary, Desirable, Undesirable, Questionable)

- Positive definite

- Self-adjoint

- Smooth

- Computationally efficient to apply

- Locally isotropic

- Spatially homogeneous

- Gaussian?

- A Gaussian isotropic homogeneous covariance is efficient, smooth, symmetric and
- self-adjoint, - but limited! Several methods that approximate such a covariance exist:

**Direct evaluation by rational functions (Gaspari and Cohn)**

**Simulated diffusion (Derber and Rosati)**

**Cartesian product of one-dimensional filters (Purser, Wu, Parrish and Roberts)**

**Other methods: Multigrid; Spectral, etc.**

## The Basic 2D Triad Method

- The normalized central second-moment of a Gaussian covariance is a quantity we
- refer to as the “**ASPECT TENSOR**”. The crucial property of aspect tensors that we exploit in our filtering algorithms, regardless of the number of dimensions, is that:

**ASPECT  TENSORS  OF  SEQUENTIAL HOMOGENEOUS FILTERS  ADD.**

- Note: The aspect tensor of any line filter is of rank ONE.

- Therefore, given any “triad” of generalized grid lines, a given aspect tensor resolves uniquely, by “linear projection”, into three “**SPREAD**” components associated with these three grid lines. (“Spread” refers to a 1D normalized central second moment.)

**BUT ONLY **_**ONE**_** MINIMAL TRIAD GIVES +ve SPREAD COMPONENTS!**

**Minimal Triads**

**In a minimal triad, the determinant of the matrix formed by any two of the lines’ generators (shortest grid steps) has a determinant of +1 or -1.**
**The aspect vectors of a unit-spread filter in each of these directions, in units measured by the generators, form a BASIS for any aspect vector.**

## Geometry of Minimal Triads

**Each potential triad basis has its aspect vector on the surface of the “aspect cone”. The triads of such vectors subdivide the interior of the aspect cone in a sort of  (triangular) “honeycomb”.  Aspect space has a natural hyperbolic**
**geometry in which this partitioning of the space becomes regular. The interior of each triad’s region of aspect space are the vectors expressible as **_**positive**_** linear combinations of the 3 basis vectors.**

- A Poincare representation of the hyperbolic “plane” of normalized
- aspect vectors and the honeycomb
- of triads by which they are sorted

**The Basic Triad Algorithm:**

**1. Take any triad**
**2. Resolve the aspect tensor into this triad’s spread**
**    ****components (line second-moments).**
**3. If all three spreads are nonnegative, STOP.**
**4. If one spread is negative, exchange that **
**    ****generator by the unique alternative that**
**    ****makes up a new valid triad, and repeat 2, etc.**

## The Basic Hexad Algorithm

**The 6-line “Hexad” method is the analogue in 3D of the 3-line Triad in 2D.**
**The lines are the 6 vertex-diameters of a minimal skewed grid-cuboctahedron (a 12-vertex polyhedron of 8 triangles alternating with 6 parallelograms).**

**Hexad transition rules when a vertex pair’s spread component is negative**

-  Remove the offending vertex-pair of the cuboctahedron.
-  Create the replacement pair by extending outwards by *2 the twin centers of the   two parallelograms that do _**not**_ contain the old disqualified vertices.
-  A new skewed cuboctahedron with grid point vertices is formed!

- See:
-  Purser, R. J., 2005: “A geometrical approach to the synthesis of smooth anisotropic
- covariance operators for data assimilation.” NOAA/NCEP Office Note 447. 60 pp.
- De Pondeca, M. S. F. V,  R. J. Purser,  D. F. Parrish and J. C. Derber, 2006: Comparison
- of strategies for the specification of anisotropies in the covariances of a three-dimensional
- atmospheric data assimilation. Extended Abstracts, Fourth WMO International
- Symposiumon Assimilation of Observations in Meteorology and Oceanography,
- Prague, 18-22 April 2005.

## The geometrical interpretation of a pair of neighboring hexads

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## Covariances based on the Kinematic Method

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## 6. Preliminary ResultsSynoptic Situation

- 4 March 2004: Storm system with associated cold front moved from the Southwest United States into New Mexico and Texas.
- Strong to severe thunderstorms along the front.
-  4 March 2004: Well developed cyclone with central pressure located in Eastern Iowa.
- Passage of system produced intense rainfall at times and damaging winds.
- Initial Time for our Case Study is 12Z 5 March 2004
- All Calculations performed within the Eta 3DVAR Data Assimilation System

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## Forecast MSE Skill Scores

## Forecast MSE Skill Score

## Forecast MSE Skill Score

## 24-Hour Acc. Rainfall Ending 12 Z 05 March 2004

## 24-h Accum. Rainfall and Accum. Rainfall Difference with respect to the Control Run

## 24-h Accum. Rainfall and Accum. Rainfall Difference with Respect to the Control Run

## Summary and Conclusions

- We have used the Eta 3DVAR system to test  three different methods  (and their variations) for prescribing flow dependent background error covariances.
- In all methods, the covariances are shaped in accordance with selected background field diagnostics.
- Results from a case study featuring a late winter storm system over the Central and Eastern US suggest that all methods hold promise.
- The Mean Square Error Forecast Skill Scores for the 500 hPa Geopotential height show a positive impact when the covariances are prescribed via the semigeostrophic+isentropic method, the Riishojgaard method using Ertel PV, and the kinematic method.
- Compared with the results from the control forecast  (isotropic model), all methods  led to improved 24-hour accumulated rainfall.
- Further testing is needed before firm conclusions can be drawn on the merits and weaknesses of each method. In particular, sensitivity studies to some of the parameters that enter the problem (eg. spatial correlation lengths, time interval in the kinematic method, etc.) is needed. In addition, these methods must be tested for different synoptic situations and seasons.
- We hope that the adaptive prescription of background error covariances will in the future allow us to attain some of the benefits of 4DVAR while keeping the computationally less expensive 3DVAR.