MultiMulti--frequency Imaging frequency Imaging in VLBI in VLBI

Sergey Likhachevin collaboration with Vladimir Ladygin

and Igor Guirin

Astro Space Center of P.N. Lebedev Physical Institute, Moscow, Russia

MultiMulti--frequency Imagingfrequency Imaging

Multi-frequency imaging (MFI) consists of:

multi-frequency synthesis (MFS): image synthesis and deconvolution procedures (CLEAN, etc);multi-frequency analysis (MFA): estimates of the image spectral parameters (spectral indexes, etc).

MFI Formalism.MFI Formalism.Linear Multi-frequency Deconvolution Algorithm

asda( )pq

00

kkpq pqI I

ανν

=

Let us minimize a metrics21 1

0 01

ˆK M M

knm knm knmn mk

w V Vρ− −

= === −∑ ∑ ∑

( ) ( ) ( )( )1 11

0 1, ,1 1 0 0

,D ( ) ( ) ( )K K M M

m m Nm Nkpq k k p i q l k kpq il ilk k i l

D B I Iβ β β− −

−−− −

= = = == − + +∑ ∑∑∑

) )K

( ),pq p qx yα α= is a spectral index

( , )p qx yin a pointIf brightness kpqI can be presented as

( ) ( ) 10 1 .pq pqpqI Iα

−

= ⋅

then a dirty map and the spectral index in a point (pq) can be presented as

MultiMulti--frequency Imagingfrequency ImagingThe developed Linear Multi-frequency Deconvolution Algorithm

(LMFDA) allows:

to synthesize and analyze of high-quality VLBI images directly from the visibility data measured on a few frequencies, without analyses of the images itself;

to synthesize an image of a radio source at any intermediatefrequency inside any given frequency band. Thus, spectral interpolation of the image is feasible;

it is possible to obtain an estimate of the spectral index for a given radio source;

to provide the highest angular resolution possible for any VLBI project due to its improved (u,v) -coverage.

MFI SimulationsMFI Simulations

Flux [Jy] Distance [as]Delta func.1 1 0Delta func. 2 0.5 9-6

Delta func. 3 0.5 1.2-5

Radioastron + VLBA (one hour observations)

MFI Simulations. No MFSMFI Simulations. No MFS

MFI Simulations. MFSMFI Simulations. MFS

Source 1733Source 1733--56. 56. MFS&MFAMFS&MFA

Courtesy of Bob Sault (ATNF)

12 frequencies [MHz]:

4418, 4451, 4562, 4637, 4706, 4815,

5072, 5223, 5308, 5393, 5478, 5804.

17331733--56. MFS56. MFS

17331733--56. MFA56. MFA

3C84. VLBA. 8GHz&15GHz3C84. VLBA. 8GHz&15GHzCourtesy of Jon Romney (NRAO)

3C84. VLBA. 8GHz&15GHz3C84. VLBA. 8GHz&15GHz

MF Synthesis. 3C84. 11GHzMF Synthesis. 3C84. 11GHz

MF Analysis. 3C84MF Analysis. 3C84

Jet Spectra

3C84. Spectral Index 3C84. Spectral Index Core

Jet

W49B. MFI AnalysisW49B. MFI Analysis327, 1452, 4885 MHz

W49B. MFI AnalysisW49B. MFI Analysis

OutcomesOutcomes

The problem is completely solved in the case of linear subtractive deconvoltion (MF CLEAN).Developed algorithms are implemented in the ASL for Windowsand tested on the various VLBI data sets.Future plans: MF MEM, MF Self –calibration.