1. Anisa Bajkova Pulkovo Observatory (St. Petersburg) Multi-frequency study of AGN using generalized maximum entropy method The XI Russian-Finnish Radio Astronomy Symposium, Pushchino, 18-22 October2010

2. VLBI mapping is used for study of: Structure of AGN and its evolution Kinematics of AGN jets Polarization Spectral index distribution over the source

3. Image reconstruction in VLBI includes the following two basic operations: Reconstruction of phase of the visibility function, which is Fourier transform of source brightness distribution Deconvolution by synthesized “dirty” beam

4. 1. The phase problem traditionally is solved iteratively using relations for closure phases in selfcalibration or hybrid mapping loops if number of interferometer elements N>=3. Number of closure phases = (N-1)(N-2)/2. Closure amplitudes are available if N>=4. Number of closure amplitudes = (N-2)(N-3)/2. 2. Deconvolution is a well-known ill-posed problem and traditionally is solved using non-linear procedures such as CLEAN (Hogbom, 1974) or MEM (Frieden,1972). 3. The quality of image reconstruction depends on number of interferometer elements N, density of UV-coverage, signal-to-noise ratio of visibility measurements.

5. We stress attention on the following two problems: Kinematical study of AGN jets using multi- epoch multi-frequency data Spectral index mapping using multi- frequency data obtained simultaneously or quasi-simultaneously

6. Standard approaches: Kinematical analysis using multi-epoch data: Selfcalibration + CLEAN + model fitting + component identification Spectral index mapping using two-frequency data: - obtaining CLEAN maps at each frequency - convolution of the CLEAN maps obtained by the same clean beam which size is equal to the central lobe of the dirty beam for the lowest frequency - determining the spectral index distribution using the following relation:

7. Standard software packages: AIPS (Astronomical Image Processing System) (NRAO) -- calibration, imaging and analysis of radio interferometric data Difmap (CalTech) – fast, flexible, editing and mapping program

8. Problems of mapping arises in the following cases: Small-element interferometer sparse UV-coverage, no (two-element interferometer) or small number of closure relations problems with accurate deconvolution and phase reconstruction High-Orbit Space–Ground Radio Interferometer degeneracy to two- element interferometer

9. Russian VLBI systems Three-element Russian “Quasar” VLBI network (Svetloye, Zelenchukskaya, Badary) Future Space-Ground high-orbit “Radioastron” mission In both cases we have sparse UV- coverage, insufficient for imaging radio sources with complicated structure

10. Space-Ground Radio Interferometer “Radioastron”

11. “Quasar” “Radioastron”

12. Ways for solving the mapping problems: Multi-frequency synthesis (fast aperture synthesis) Reconstruction of visibility phase directly from visibility amplitude (well-known “phase” problem)

13. 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

14. Now CLEAN is practically only widely used deconvolution algorithm in VLBI mapping (AIPS, DIFMAP). In other fields more popular is MEM. Our aim is to study MEM-based algorithms in order to realize their advantages for solving modern VLBI mapping problems.

15. Maximum entropy image deconvolution principle ( Bob Sault ): 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.

16. -- MEM ensures maximally smoothed solution subject to data constraints (Frieden 1972). -- MEM possesses super-resolution effect due to nonlinearity which is attained due to positiveness of the solution. These properties of MEM can be valuable for increasing accuracy of kinematic analysis of AGN jets.

17. Standard MEM

18. Discrete form of MEM

19. Lacks of the standard MEM due to positiveness of the solution: 1)MEM gives biased solution: mean noise of the image is not equal to zero. This lack can lead to large nonlinear distortions of images (artifacts) due to input data errors. 2)Difference mapping is not possible due to overestimation of brightness peaks. It is necessary to generalize MEM to obtain sign- variable solutions and save nonlinearity (super- resolution) property.

21. Let us define entropy of an arbitrary real function I(x) as the entropy of the modulus of this function

22. Let be

23. When the conditions (6)-(7) or (8) are fulfilled, expression (5) takes the standard form:

24. We determine generalized entropy of real, sign- variable function as follows: where parameter α controls the accuracy of fulfillment of conditions (6)-(7) or (8), because the solutions for I(x) are connected by equation:where parameter α controls the accuracy of fulfillment of conditions (6)-(7) or (8).

25. MFS in VLBI assumes 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 the 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. Multi-frequency synthesis

26. Th e most important works on MFS 1.Conway, J.E. Proc. IAU Coll. 131, ASP Conf. Ser., 1991, 19, 171. 2.Conway, J.E., Cornwell T.J., Wilkinson P.N. MNRAS, 1990, 246, 490. 3.Cornwell, T.J. VLB Array Memo 324, 1984, NRAO, Socorro, NM. 4.Sault, R.J., Wieringa, M.H. A & A, Suppl. Ser., 1994, 108, 585. 5.Sault, R.J., Oosterloo, T.A. astro-ph/0701171v1, 2007. 6.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]).

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

28. Improving UV-coverage (“Radioastron”)

29. Spectral variation of brightness distribution

30. GMEM functional to be minimized

31. Restrictions derived from visibility data

32. Modeling 3C120 Model of 3C120 SFS MFS

33. Modeling J0958+6533 C-band model image X-band model image U-band model image K-band model image X-band model image convolved by beam SFS image Model spectral map Model spectral index map MFS-image at 8 GHz MFS image convolved by beam reconstructed spectral map reconstructed spectral index map

34. Problem with aligning VLBI images

35. There exists frequency-dependent core shift phenomenon, which must be taken into account before MFS.

36. Some examples of MFS with aligning VLBI images at different frequencies (in collaboration with A. Pushkarev) Source Frequency band Reference frequency GHz Shift mas We applied the MFS algorithm for synthesis images and spectral index maps for three radio sources J2202+4216, J0336+3218, J1419+5423 and J0958+6533 with aligning images obtained at different frequencies.

37. J2202+4216 (Two-frequency synthesis) 2.3 GHz 8.6 GHz MFS at 5.5 GHz (core aligning) MFS at 5.5 GHz (without aligning) MFS at 5.5 GHz (jet aligning)

38. J0336+3218 (Two-frequency synthesis) 2.3 GHz 8.6 GHz MFS at 5.5 GHz (jet aligning)

39. J1419+5423 (Three-frequency sytnthesis) 5 GHz 8.4 GHz 15.3 GHz MFS at 8.4 GHz (jet aligning) high resolution low resolution

40. Map peak: 0.166 Jy/beam Map peak: 0.219 Jy/beam Map peak: 0.311 Jy/beam Map peak: 0.359 Jy/bea Contours: 0.7, 1.4, 2.8, 5.6, 11.2, Contours: 0.25, 0.5, 1, 2, 4, 8, 16, Contours: 0.1, 0.2, 0.4, 0.8, 1.6, Contours: 0.2, 0.4, 0.8, 1.6, 3.2, 6.4, 22.5,45, 90 % 32, 64, 90 % 3.2,6.4,12.8, 25.6,51.2,90 % 12.8, 25.6,51.2,90 % Beam FWHM: 0.577x0.433 (mas) Beam FWHM: 0.812x0.592 (mas) Beam FWHM: 1.44x1.05 (mas) Beam FWHM:2.5x1.72 (mas) at -21.6 degr at -16 degr at -16.2 degr at -17.3 degr J0958+6533 (1997 Apr 06) (Four-frequency synthesis) K (22.2 GHZ) U (15.4 GHz) X (8.4 GHz) C (5 GHz) MEM-solution MEM-solution MEM-solution MEM-solution Map peak: 0.092Jy/pixel Map peak: 0.078 Jy/pixel Map peak: 0.116 Jy/pixel Map peak: 0.133 Jy/pixel

41. (KUXC) map convolved by K-band beam (UXC) map convolved by U-band beam (XC) map convolved by X-band beam Map peak: 0.1817 Jy/beam Map peak: 0.212 Jy/beam Map peak: 0.319 Jy/beam RMS=0.12 mJy/beam RMS=0.13 mJy/beam RMS=0.15 mJy/beam Beam FWHM: 0.577x0.433 (mas) at -21.6 o Beam FWHM: 0.812x0.592 (mas) at -16 o Beam FWHM: 1.44x1.05 (mas) at -16.2 Contours: 0.4, 0.8, 1.6, 3.2, 6.4,12.8, 25.6, 51.2, 90 % Contours: 0.4, 0.8, 1.6, 3.2, 6.4,12.8, 25.6, 51.2, 90 % Contours: 0.2,0.4, 0.8, 1.6, 3.2, 6.4,12.8, 25.6, 51.2, 90 % MEM-solution MFS-maps:

42. KUXC synthesis UXC synthesis XC synthesis Spectral index distributions over the source

43. KUXC_C KUXC_X KUXC_U KUXC_K

44. Core shift versus frequency for J0958+6533 using 22 GHz as the reference frequency: dr=A(fr^(-1/kr)-22^(-1/kr)), A=2.62, kr=2 m=2, n=3

45. Phase-less mapping Reconstruction of intermediate zero-phase, symmetric image from visibility amplitude using GMEM algorithm Reconstruction of the spectrum phase of sought for image from the spectrum amplitude of the intermediate image using Fienup’s algorithm or MEM-based phase-reconstruction algorithm (Bajkova, Astronomy Reports, Vol. 49, No. 12, 2005, pp. 973–983.)

46. Simulation of the phase-less method Model sources Reconstructed intermediate Images reconstructed from zero-phase images spectrum of intermediate images

47. Modeling of “phase-less” mapping for source 0716+714 Model “Dirty” image Reconstructed Reconstructed intermediate sought for image zero-phase image Contours: 0.0015, 0.0030, 0.00625, 0.0125, …% of peak UV-coverage

48. Phase-less mapping of 3С120 Images of 3C120 obtained from VLBA + observations in 2002 at 8.4 GHz

49. Difference-mapping method The difference-mapping method is based on the fundamental property of linearity of the Fourier transform. Bright components in the source that are reconstructed in the first stage are subtracted from the input spectrum, the remaining reconstruction is carried out for the residual spectral data, and the results of the two reconstructions are finally summed. Difference-mapping method requires the GMEM because the residual spectral data obtained after subtracting bright components reconstructed in previous stages of the algorithm can correspond to an image with negative values. The largest improvement from the difference-mapping method is obtained for compact structures embedded in a weak, extended base. This method was able to reconstruct both the compact and extended features with high accuracy.

51. ModelDirty imageClean MEM Difference GMEM

52. Kinematic study of the blazar S5 0716+714 during the active state in 2004 at frequencies 5 and 1.6 GHz (in collaboration with E. Rastorgueva and K. Wiik, Tuorla Observatory)

53. Multi-epoch GMEM solutions from VLBA observations at 5 GHz

54. G MEM-solutions convolved by clean beam 05x0.5 mas

55. GMEM-solutions convolved by clean beam related to real system

56. Relative R.A. Kinematics of the jet components

57. epoch: 2004.11 2004.22 2004.46 Multi-epoch GMEM solutions obtained from observations at 1.66 GHz, the solutions convolved by clean beam corresponding to the real resolution of the system and gaussian models

58. epoch: 2004.58 2004.66

59. Kinematics of the jet components С B A

60. CONCLUSIONS The GMEM can be used for reconstruction of signals of any type not only real non-negative ones as the standard MEM. The GMEM is free from such a lack of the MEM as biased solution and ensures much less nonlinear distortions. Only thanks to the generalization of MEM it became possible: -- to reconstruct correctly sign-variable spectral maps in the frame of MFS imaging technique, -- to reconstruct correctly sign-variable intermediate zero-phase images when solving phase problem, -- to fulfill difference-mapping method intended to increase dynamical range of images, especially in the case of compact structures embedded in a weak, extended base.

61. THANK YOU!