Tl

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

Tiling the time-frequency plane

Gridpoint

STFT

Fourier

Wavelet

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

Dual tree complex wavelet transform

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

2003

-12-

0120

04

SM

HI-

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