dtdwt

1. 2003-12-01
200
4
SMHI-
OH
Wavelets as a framework for describing
inhomogenity and anisotropy
Alex Deckmyn
Tomas Landelius
Anders Höglund

2. 2003-12-01
200
4
SMHI-
OH
Variational data assimilation
Minimization of a cost function
∑=
−
−
−−
+−−=
N
j
jjj
T
jjj
b
T
b
xHyRxHy
xxBxxxJ
0
1
0
1
00
)()(
)()()(

3. 2003-12-01
200
4
SMHI-
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.
{ }T
tbtb xxxxEB ))(( −−=

4. 2003-12-01
200
4
SMHI-
OH
Problems with the B matrix
It is HUGE:
• Impossible to store explicitly.
• Impossible to fully determine.
142
10)()dim( ≈= zyx nnnB

5. 2003-12-01
200
4
SMHI-
OH
Optimal interpolation
Direct local solution to 3D-VAR
Local models of B that differ for
different locations.
1
0
)(
)()()(
−
+=
−+=
RHBHBHK
HxyKixix
TT
bib

6. 2003-12-01
200
4
SMHI-
OH
Need a global model for B
Variable substitution
Fourier/Wavelet transform
)(
/,/
)()()()(
**
1**
1
bo
b
ob
T
b
xuTJuu
ITBTuTxx
xJxxBxxxJ
++=
==−=
+−−=
−
−
*21*
FDFBFDT =⇒=

7. 2003-12-01
200
4
SMHI-
OH
The Fourier transform
+ Fast implementation O(n log(n)).
- Homogenous; same for all locations.
- Boundary problems; periodic.

8. 2003-12-01
200
4
SMHI-
OH
Tiling the time-frequency plane
Gridpoint
STFT
Fourier
Wavelet

9. 2003-12-01
200
4
SMHI-
OH
Discrete wavelet transform
Some problems with the DWT
- Oscillations near singularities.
- Shift variance.
- Sensitive to processing.
- Lack of directionality.

10. 2003-12-01
200
4
SMHI-
OH
Dual tree complex wavelet transform

11. 2003-12-01
200
4
SMHI-
OH
The wavelet transform
+ Very fast implementation O(n).
+ Some anisotropy and inhomogenity.
+ Boundary problem can be avoided.

12. 2003-12-01
200
4
SMHI-
OH
Estimating D
Orthogonal T
Non-orthogonal T
{ }i
T
ii txxtEd *
=
{ }
2*
*
iiij
i
T
ii
ttA
txxtEb
bAd
=
=
=

13. 2003-12-01
200
4
SMHI-
OH
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