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2003
-12-
0120
04
SM
HI-
OH
Wavelets as a framework for describing
inhomogenity and anisotropy
Alex Deckmyn
Tomas Landelius
Anders Höglund
2003
-12-
0120
04
SM
HI-
OH
Variational data assimilation
Minimization of a cost function
N
jjjj
Tjjj
bT
b
xHyRxHy
xxBxxxJ
0
1
01
00
)()(
)( )()(
2003
-12-
0120
04
SM
HI-
OH
The importance of the B matrix
B contains the background error covariances:
• Weights background state against observations.
• Helps to impose balance to the assimilated fields.
• Smoothes out the observational information.
Ttbtb xxxxEB ))((
2003
-12-
0120
04
SM
HI-
OH
Problems with the B matrix
It is HUGE:
• Impossible to store explicitly.
• Impossible to fully determine.
142 10)()dim( zyx nnnB
2003
-12-
0120
04
SM
HI-
OH
Optimal interpolation
Direct local solution to 3D-VAR
Local models of B that differ for
different locations.
1
0
)(
)( )()(
RHBHBHK
HxyKixixTT
bib
2003
-12-
0120
04
SM
HI-
OH
Need a global model for B
Variable substitution
Fourier/Wavelet transform
)(
/ , /
)( )( )()(
**
1**
1
bo
b
obT
b
xuTJuu
ITBTuTxx
xJxxBxxxJ
*21* FDFBFDT
2003
-12-
0120
04
SM
HI-
OH
The Fourier transform
+ Fast implementation O(n log(n)).
- Homogenous; same for all locations.
- Boundary problems; periodic.
2003
-12-
0120
04
SM
HI-
OH
Discrete wavelet transform
Some problems with the DWT
- Oscillations near singularities.
- Shift variance.
- Sensitive to processing.
- Lack of directionality.
2003
-12-
0120
04
SM
HI-
OH
The wavelet transform
+ Very fast implementation O(n).
+ Some anisotropy and inhomogenity.
+ Boundary problem can be avoided.
2003
-12-
0120
04
SM
HI-
OH
Estimating D
Orthogonal T
Non-orthogonal T
iTii txxtEd *
2*
*
iiij
iT
ii
ttA
txxtEb
bAd