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RM-synthesis with Wavelets
D. Sokolo (1), P. Frick (2), R. Stepanov (2), T. Ensslin (3), S. Duscha (3)
(1) University of Moscow, (2) Institute of Continuous Media Mechanics, (3) Max-Planck-Institute for Astrophysics

25

th

of March 2009

D. Sokolo (1), P. Frick (2), R. Stepanov (2), T. Ensslin (3), with Wavelets ((1) University of Moscow, (2) Institute1of Con RM-synthesis S. Duscha (3) 25th of March 2009 /8


Table of contents
1

RM-synthesis using Fourier Transform

2

Diculties with Fourier Transform RM-synthesis

3

Wavelet theory

4

RM-synthesis with Wavelets

5

Comparison of FT and Wavelet reconstructions

D. Sokolo (1), P. Frick (2), R. Stepanov (2), T. Ensslin (3), with Wavelets ((1) University of Moscow, (2) Institute2of Con RM-synthesis S. Duscha (3) 25th of March 2009 /8


RM-synthesis using Fourier Transform
Frequency space IFT Dec FT RA f RA Dec Faraday space

Physics of RM-synthesis and Fourier Transform approach Weighted p olarized intensities

~ P ~ F

=

W


F

2

P



2

Fourier Transform into Fourier Space

+

() =

()

R

() =

K
-

~ P



2

2 e -2i d

2

Deconvolute with Rotation Measure Spread Function (RMSF)

R

() =

+ - W



2

d

2

-1

+ - W



2

2 e -2i d



2

D. Sokolo (1), P. Frick (2), R. Stepanov (2), T. Ensslin (3), with Wavelets ((1) University of Moscow, (2) Institute3of Con RM-synthesis S. Duscha (3) 25th of March 2009 /8


Diculties with FT RM-synthesis
Fourier Transform only uses Sin and Cos functions to reconstruct the shap e of the original function. Paramters to cho ose are only the frequency comp onents, thus... Fourier transform only lo calized in frequency Wavelet transform is lo calized in frequency AND time Diculties in handling sparse sampling, esp ecially gaps Can lead to ghost features in reconstructed signal Choice of Mother Wavelet to adapt to physical situation

D. Sokolo (1), P. Frick (2), R. Stepanov (2), T. Ensslin (3), with Wavelets ((1) University of Moscow, (2) Institute4of Con RM-synthesis S. Duscha (3) 25th of March 2009 /8


Wavelet theory

Meyer

Morlet

Mexican Hat

The Mother Wavelet is scaled by a factor of a and translated by a factor of b to give Child Wavelets:

a,b =

1

t -b a a

The reconstructed signal is then the scalar pro duct of the WT co ecients and the Child Wavelets (discrete form):

x (t

)=

m

Z

n

Z x

, a,b m,n (t )

D. Sokolo (1), P. Frick (2), R. Stepanov (2), T. Ensslin (3), with Wavelets ((1) University of Moscow, (2) Institute5of Con RM-synthesis S. Duscha (3) 25th of March 2009 /8


RM-synthesis with Wavelets
The wavelet transform of the (partially) observed p olarization P is dened as



2

wP (a, b

)= 1 a

+ - P



2





2 -

a

b d

2

and implies the following inverse Wavelet Transform

P

(2 ) = C1 =
1 2

C

2

- 2 ^ | (k )| - |k |
0





2 -

a

b

w (a , b

) dadb a2

dk <

Reconstruction of the signal through Rotation Measure Wavelet

Synthesis (RMWS) is given by F
1 () = C




0

-

^ (2a) wP (a, b)

dadb e -2ib a

D. Sokolo (1), P. Frick (2), R. Stepanov (2), T. Ensslin (3), with Wavelets ((1) University of Moscow, (2) Institute6of Con RM-synthesis S. Duscha (3) 25th of March 2009 /8


Comparison of FT and Wavelet reconstructions
Wavelet Transform is a smo othed version of the Fourier Transform Greater accuracy in reconstructing multiple Faraday emission cases

Faraday emission simulation

Reconstruction with FTand WT

D. Sokolo (1), P. Frick (2), R. Stepanov (2), T. Ensslin (3), with Wavelets ((1) University of Moscow, (2) Institute7of Con RM-synthesis S. Duscha (3) 25th of March 2009 /8


Conclusions

Currently implemented as a Mathematica noteb o ok Use of Wavelet library for C++ implementation

blitzwave Wave++ others...

Part of the RM-synthesis library, librm, exchangeable algorithms Computationally demanding (dep ends on range of WT co ecients space a and b)

D. Sokolo (1), P. Frick (2), R. Stepanov (2), T. Ensslin (3), with Wavelets ((1) University of Moscow, (2) Institute8of Con RM-synthesis S. Duscha (3) 25th of March 2009 /8