wavelets for b/g error covariance modelling
DESCRIPTION
Wavelets for b/g error covariance modelling. Ross Bannister, Data Assimilation Research Centre, Reading, UK. Introductory remarks:. 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’. - PowerPoint PPT PresentationTRANSCRIPT
B/g error modelling with wavelets - Ross Bannister - Monday January 10th 2005 - Page 1
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:
B/g error modelling with wavelets - Ross Bannister - Monday January 10th 2005 - Page 2
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
B/g error modelling with wavelets - Ross Bannister - Monday January 10th 2005 - Page 3
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
B/g error modelling with wavelets - Ross Bannister - Monday January 10th 2005 - Page 4
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
B/g error modelling with wavelets - Ross Bannister - Monday January 10th 2005 - Page 5
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
B/g error modelling with wavelets - Ross Bannister - Monday January 10th 2005 - Page 6
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
B/g error modelling with wavelets - Ross Bannister - Monday January 10th 2005 - Page 7
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
~
B/g error modelling with wavelets - Ross Bannister - Monday January 10th 2005 - Page 8
The bandpass functions
B/g error modelling with wavelets - Ross Bannister - Monday January 10th 2005 - Page 9
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.
B/g error modelling with wavelets - Ross Bannister - Monday January 10th 2005 - Page 10
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)
B/g error modelling with wavelets - Ross Bannister - Monday January 10th 2005 - Page 11
5. This study: WS results
Diagnostics from the explicit B-matrix (control)
Diagnostics from the WS B-matrix model (1 band)
Real-space T variances Vertical T corrs (fn. of posn.) Vertical T corrs (fn. of scale)
B/g error modelling with wavelets - Ross Bannister - Monday January 10th 2005 - Page 12
5. This study: WS results
Diagnostics from the explicit B-matrix (control)
Diagnostics from the WS B-matrix model (2 bands)
Real-space T variances Vertical T corrs (fn. of posn.) Vertical T corrs (fn. of scale)
B/g error modelling with wavelets - Ross Bannister - Monday January 10th 2005 - Page 13
5. This study: WS results
Diagnostics from the explicit B-matrix (control)
Diagnostics from the WS B-matrix model (3 bands)
Real-space T variances Vertical T corrs (fn. of posn.) Vertical T corrs (fn. of scale)
B/g error modelling with wavelets - Ross Bannister - Monday January 10th 2005 - Page 14
5. This study: WS results
Diagnostics from the explicit B-matrix (control)
Diagnostics from the WS B-matrix model (4 bands)
Real-space T variances Vertical T corrs (fn. of posn.) Vertical T corrs (fn. of scale)
B/g error modelling with wavelets - Ross Bannister - Monday January 10th 2005 - Page 15
5. This study: WS results
Diagnostics from the explicit B-matrix (control)
Diagnostics from the WS B-matrix model (5 bands)
Real-space T variances Vertical T corrs (fn. of posn.) Vertical T corrs (fn. of scale)
B/g error modelling with wavelets - Ross Bannister - Monday January 10th 2005 - Page 16
5. This study: WS results
Diagnostics from the explicit B-matrix (control)
Diagnostics from the WS B-matrix model (6 bands)
Real-space T variances Vertical T corrs (fn. of posn.) Vertical T corrs (fn. of scale)
B/g error modelling with wavelets - Ross Bannister - Monday January 10th 2005 - Page 17
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