Anisa T . Bajkova Central (Pulkovo) Astronomical Observatory of RAS

1. MULTI-FREQUENCY SYNTHESIS TECHNIQUE IN RADIO INTERFEROMETRIC IMAGING USING GENERALIZED MAXIMUM ENTROPY METHOD Anisa T. Bajkova Central (Pulkovo) Astronomical Observatory of RAS

2. MFS in VLBIassumes mapping at several observing radio frequencies simultaneously to improve UV-coverage, so MFS is a tool of rapid aperture synthesis. MFS is possible due to measurement of UV-coordinates of visibility function in wavelengths. The main problem of MFS is spectral dependence of a source brightness distribution and in order to avoid possible artifacts in the image it is necessary to fulfill spectral correction during the deconvolution stage of the image formation (CLEAN or MEM).

3. The most important works on MFS • Conway, J.E., Cornwell T.J., Wilkinson P.N. MNRAS, 1990, 246, 490. • Conway, J.E. Proc. IAU Coll. 131, ASP Conf. Ser., 1991, 19, 171. • Cornwell, T.J. VLB Array Memo 324, 1984, NRAO, Socorro, NM. • Sault, R.J., Wieringa, M.H. A&A, Suppl. Ser., 1994, 108, 585. • Sault, R.J., Oosterloo, T.A. astro-ph/0701171v1, 2007. • Likhachev, S.F., Ladygin, V.A., Guirin, I.A. Radioph. & Quantum Electr., 2006, 49, 499. are based on CLEAN deconvolution algorithm for spectral correction of images (double-deconvolution algorithm [1,2,4,5], vector-relaxation algorithm [6]).

4. The aim of this work: Development and investigation of a new MFS deconvolution algorithm based on maximum entropy method for effective solving spectral variation problem in broad-band frequency region and estimation of a spectral index distribution over a source.

5. CLEAN or MEM ? Bob Sault The answer is image dependent: • “High quality” data, extended emission, large images  Maximum entropy • “Poor quality” data, confused fields, point sources CLEAN

6. Importance of MFS for Russian Radio Astronomy • Three-element Russian “Quasar” VLBI network (Svetloye, Zelenchukskaya, Badary) • Future Space-Ground high-orbit“Radioastron” mission In both cases we have sparse UV- coverages, insufficient for imaging radio sources with complicated structure

7. Improving UV-coverage (a) (b) Four element radio Interferometer:Svetloe, Zelenchukskaya, Badary, Matera (a) single frequency synthesis (b) multi-frequency synthesis

8. Space-Ground Radio Interferometer “Radioastron”

10. Spectral variation

11. Bob Sault Maximum entropy image deconvolution principle: Of all the possible images consistent with the observed data, the one that has the maximum entropy is most likely to be the correct one.

12. Maximum Entropy Method Discrete form of practical MEM:

13. Visibility function constraints:

14. Reconstruction using Generalized Maximum Entropy Method(GMEM)

16. The Lagrange method:

17. Solution:

18. Unconditional optimization problem:

19. Simulation results 30% SFS UV-planes а б 90% 60% в г Fig.1

20. abc Fig.2 Model distributions of the source (a), first -order spectral map (b) and spectral index (c) (0<α(x,y)<0.8), size of maps 128x128 Contour levels: 0.0625,0.125,0.25,0.5,1,2,4,8,16,32,64,99 %

21. а б в Fig.3 Reconstructed images using (a) SFS; (b) МFS (30%),α(x,y)=0; (c) MFS α(x,y)≠0 Contour levels: 0.0625,0.125,0.25,0.5,1,2,4,8,16,32,64,99 %

22. Io(x,y) I1(x,y) α(x,y) 2 а bc 3 def Fig.4 (Frequency band30%)

23. 2 а bc 3 def 4 Fig.5(60%) ghi

24. 2 а bc 3 def 4 Fig.6 (90%) ghi

25. а bc Fig.7 MFS (90%), 27 frequences

26. SFSwithout spectral correction 2 3 а b cd 4 5 I1(x,y) α(x,y) ef hi Рис.8 MFS (90%, 9 frequencies) significant noise in data (visibility function) Contour levels: 0.25,0.5,1,2,4,8,16,32,64,99%

27. Modelling 3C120 Model of 3C120 at 8.2 GHz Spectral index distribution

28. Reconstructed images: -2.1 <α(x,y)<0.8, frequency bandwidth=30%, nonlinear spectral correction with N=4

29. Reconstructed images: -2.1 <α(x,y)<0.8, frequency bandwidth=60%, nonlinear spectral correction with N=4

30. Modelling Radioastron mission

31. SFS MFS (bandwidth=±30%) MFS (bandwidth=±60%) UV-coverage for Radioastron mission (U,V in 108 wavelengths)

32. Image synthesis by Radioastron Model SFS MFS with spectral correction MFS without spectral correction

33. Image synthesis by Radioastron Source model SFS MFS(±30%)(α(x,y)=0) MFS(±30%)( α(x,y)≠0) MFS(±30%)( α(x,y)≠0) MFS(±60%)( α(x,y)≠0) (without spectral correction) (linear spectral correction N=2) (nonlinear spectral correction N=4)

34. CONCLUSION We proposed and investigated simple and effective MFS-deconvolution technique based on the Generalized Maximum Entropy Method which allows to provide accurate spectral correction of images in wide frequency band and reconstruct both source brightness and spectral index distributions. The results obtained will be published in Astronomy Reports (2008), v.85, N 12.

35. THANK YOU FOR ATTENTION!