wavelets for b/g error covariance modelling

B/g error modelling with wavelets - Ross Bannister - Monday January 10th 2005 -

Wavelets for b/g error covariance modelling

Ross Bannister, Data Assimilation Research Centre, Reading, UK

• In D.A., need to estimate the P.D.F. of the a-priori (forecast) error.

• Assuming errors are normally distributed leads to ‘B-matrix’.

• B (as an explicit matrix) is too large.

• Model B approximately as a factorization of sparse matrices.

• Actually deal with the ‘square-root’ of B:

½space

½multi.v.

½ BBB

Introductory remarks:


Why the square-root?The square-root is interpreted as a transformation between control variables and model variables.

Why this particular factorization?½space

½multi.v.

½ BBB

The ‘parameter’ transform. Deals with multivariate aspects of covariances. The spatial transform (‘vertical’ and

‘horizontal’) deals with the spatial aspects of covariances (within each variable).

½BU

U

vx


Coming up …

1. What spatial covariance features are desirable to capture?

2. How do we assess the covariance model performance (without doing data assimilation)?

3. What is ‘wrong’ with the current Met Office model of ?

4. The new ‘waveband summation’ approach.

5. My simplified study.

6. Summary and references.

½spaceB


1. Aspects of spatial covariancesUninspiring/impossible to look at matrix operators and so instead plot

diagnostics derived from the operators.

Two parts of covariance: (i) variance and (ii) correlation

Each can be plotted in ‘real’-space or in ‘spectral’-space.

Real space Spectral space

Variances Variances

Vertical correlations Vertical correlations

Horizontal correlations Horizontal correlations

Examples given in the literature …MetO: Ingleby N.B., Q.J.R.Meteor.Soc. 127, 209-231 (2001).ECMWF: Derber J. & Bouttier, Tellus 51A, 195-221 (1999).

CB

ShR xx

F RhS xx 1F


1. The real-space structure functions:

allow us to compute variances,

position dependent vertical correlations,

and lengthscales) in real-space,

2. (Similar formulae exist for spectral-space.)

2. Diagnostics

),,(),,( 000 zzxR B

),,(),,(var 000000 zxz R

),,(cov),,(cov

),,(),,,(cor

100000

1001000 zz

zxzz R

½

0002

2

000 ),,(cor),,( Eg.

zzLR

T½½BBB


3. The Met Office operational spatial error covariance model

1. GOOD: The structure functions are non-separable. association between vertical and horizontal scales.

2. BAD: The transform cannot represent fully the position-dependent variances.

hvUUB ½space

Horizontal transform(isotropic and homogeneous correlation model)

Vertical transform

Need an alternative that achieves (1) and (2) simultaneously.

vU~

BAD

GOOD can

nearly

little


4.The waveband summation (WS) covariance model

GOOD: This transform will allow position dependencies (through design of ) and scale dependencies (through presence of ).

COMPROMISE: The transform cannot represent position and scale dependencies perfectly (c.f. Heisenberg uncertainty principle).

How does this transform compare to the Met Office transform?

Horizontal transform(same design as before)

Vertical transform(now band-dependent and redesigned to be position dependent)

J

j

jhvjb

0

2~ UUSpectral bandpass fn

Standard deviation field(diagonal matrix)

2jvjU

~

½spaceB hv UU

~


The bandpass functions


5. This studyA simplified set-up:

• 2d only (lat/ht).• Reduced resolution.• Only one variable (temperature).• Small number of diagnostics:

Real-space variances. Position dependent vertical

correlations. Scale dependent vertical

correlations. Lengthscales.

• Study the models:

• For each model:

Explicit covariance matrix. Implied diagnostics from

the MetO cov model. Implied diagnostics from

the WS cov model (vary No. of bands).

1. Perform the calibration (determine numbers to used in transforms) by examining f/c differences.

2. Compute diagnostics.


5. This study: MetO results

Diagnostics from the explicit B-matrix (control)

Diagnostics from the MetO B-matrix model

Real-space T variances Vertical T corrs (fn. of posn.) Vertical T corrs (fn. of scale)


5. This study: WS results


Diagnostics from the WS B-matrix model (1 band)





Diagnostics from the WS B-matrix model (2 bands)



6. Summary and References The explicit B-matrix has many properties.

- Revealed in diagnostics.

- Cannot use explicit B-matrix in operational DA.

- Need a managable ‘B-matrix-model’ that replicates the essential features of B.

- (Model square-root of B as a control variable transform.) Concentrate here on the spatial aspects of B. The MetO operational spatial B-model:

- It is cable of capturing non-separable aspects,

- It cannot represent position and scale dependencies simultaneously. The new WS spatial B-model:

- It is cable of capturing non-separable aspects,

- It can represent position and scale dependencies simultaneously,

- Involves a trade-off between resolution in real- and spectral-spaces.

- Some properties can be investigated analytically. References

- file:///home/mm0200/frxb/public_html/WS/Waveband.html (MetO intranet)

- www.met.rdg.ac.uk/~ross/DARC/WS/Waveband.html

wavelets for b/g error covariance modelling

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