wavelets as a framework for describing inhomogenity and anisotropy
DESCRIPTION
Wavelets as a framework for describing inhomogenity and anisotropy. Alex Deckmyn Tomas Landelius Anders Höglund. Variational data assimilation. Minimization of a cost function. The importance of the B matrix. B contains the background error covariances: - PowerPoint PPT PresentationTRANSCRIPT
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Wavelets as a framework for describing
inhomogenity and anisotropy
Alex Deckmyn
Tomas Landelius
Anders Höglund
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Variational data assimilation
Minimization of a cost function
N
jjjj
Tjjj
bT
b
xHyRxHy
xxBxxxJ
0
1
01
00
)()(
)( )()(
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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 ))((
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Problems with the B matrix
It is HUGE:
• Impossible to store explicitly.
• Impossible to fully determine.
142 10)()dim( zyx nnnB
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Optimal interpolation
Direct local solution to 3D-VAR
Local models of B that differ for
different locations.
1
0
)(
)( )()(
RHBHBHK
HxyKixixTT
bib
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Need a global model for B
Variable substitution
Fourier/Wavelet transform
)(
/ , /
)( )( )()(
**
1**
1
bo
b
obT
b
xuTJuu
ITBTuTxx
xJxxBxxxJ
*21* FDFBFDT
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The Fourier transform
+ Fast implementation O(n log(n)).
- Homogenous; same for all locations.
- Boundary problems; periodic.
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Tiling the time-frequency plane
Gridpoint
STFT
Fourier
Wavelet
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Discrete wavelet transform
Some problems with the DWT
- Oscillations near singularities.
- Shift variance.
- Sensitive to processing.
- Lack of directionality.
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Dual tree complex wavelet transform
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The wavelet transform
+ Very fast implementation O(n).
+ Some anisotropy and inhomogenity.
+ Boundary problem can be avoided.
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Estimating D
Orthogonal T
Non-orthogonal T
iTii txxtEd *
2*
*
iiij
iT
ii
ttA
txxtEb
bAd
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Work status and issues
ALADIN implementation
- Wavelet transform v0.1 (cy30t1_wavbec)
Software for estimation of D
- FESTATWAV assumes orthogonal T
- Treatment of vertical correlations
Design of boundary wavelets