Transcript МНОГОЧАСТОТНЫЙ СИНТЕЗ И СПЕКТРАЛЬНАЯ КО
Multi-frequency study of AGN using generalized maximum entropy method
Anisa Bajkova Pulkovo Observatory (St. Petersburg) The XI Russian-Finnish Radio Astronomy Symposium, Pushchino, 18-22 October2010
VLBI mapping is used for study of:
Structure of AGN and its evolution Kinematics of AGN jets Polarization Spectral index distribution over the source
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
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.
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
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: I
( |
x
,
y
)
I
( 0 |
x
,
y
) 0 (
x
,
y
)
Standard software packages:
AIPS (Astronomical Image Processing System) (NRAO) - calibration, imaging and analysis of radio interferometric data Difmap (CalTech) – and mapping fast, flexible, editing program
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
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
Space Ground Radio Interferometer “Radioastron”
“Quasar” “Radioastron”
Ways for solving the mapping problems:
Multi-frequency synthesis (fast aperture synthesis) Reconstruction of visibility phase directly from visibility amplitude (well-known “phase” problem)
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
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.
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.
-- 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.
Standard MEM
max
E
I
(
x
) ln[ 1 /
I
(
x
)]
dx
I
(
x
) ln[
I
(
x
)]
dx
Discrete form of MEM
min
l m I
(
l
,
m
) ln
I
(
l
,
m
)
k
(
k re
) 2
k
2 (
k im
) 2
l m
l m I
(
l
,
m k
)
a lm I
(
l
,
m k
)
b lm I
(
l
,
m
) 0
k re
k im
A k
B k
( 2 ) ( 3 ) ( 4 ) ( 1 )
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.
Let us define entropy of an arbitrary real function I(x) as the entropy of the modulus of this function
E
I
(
x
) ln
I
(
x
)
dx
( 5 )
E
I
(
x
) ln
I
(
x
)
dx
Let be
I
(
x
)
I
(
x
)
I
(
x
),
where I
(
x
)
I
(
x
),
I
(
x
)
I
(
x
),
if if I I
(
x
) 0
I
(
x
) 0 (
x
),
I
( ( 6 7 ) ) (
x
) 0 ,
I
(
x
)
I
(
x
) 0 ( 8 )
When the conditions (6)-(7) or (8) are fulfilled, expression (5) takes the standard form:
E
(
I
(
x
) ln(
I
(
x
))
I
(
x
) ln(
I
(
x
)))
dx
We determine generalized entropy variable function as follows: of real, sign-
E
(
I
(
x
) ln(
I
(
x
))
I
(
x
) ln(
I
(
x
)))
dx
, 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 conditions (6)-(7) or (8).
I
(
x
)
I
(
x
) α controls the accuracy of fulfillment of exp( 2 2 ln ).
Multi-frequency synthesis
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.
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
.
2006,
49
, 499.
Radioph. & Quantum Electr
., are based on CLEAN deconvolution algorithm for spectral correction of images (double-deconvolution [1,2,4,5], vector-relaxation algorithm [6]).
algorithm
Improving UV coverage (“Quasar”) (a) (b) Four element radio Interferometer: Svetloe, Zelenchukskaya, Badary, Matera (a) single frequency synthesis (b) multi-frequency synthesis
Improving UV coverage (“Radioastron”)
Spectral variation of brightness distribution
I
( )
I
( 0 ) 0
I
( )
I q
I
0
I
0 (
Q q
1 1
I q
0 0 1 ) [ (
q
q
1 )]
q
!
I
(
l
,
m
)
I q
(
l
,
m
)
I
0 (
l
,
m
)
Q q
1 1
I q
(
l
,
m
) 0 0
q I
0 (
l
,
m
) (
l
,
m
)[ (
l
,
m
) 1 ]
q
!
[ (
l
,
m
) (
q
1 )] ( 9 ) ( 10 ) ( 11 )
I
1 (
l
,
m
)
I
0 (
l
,
m
) (
l
,
m
) (
l
,
m
)
I
1 (
l
,
m
) /
I
0 (
l
,
m
) ( 12 )
GMEM functional to be minimized
E
{
l
,
m I
0 (
l
,
m
) ln[
I
0 (
l
,
m
)]
q
1
Q
l m
|
I q
(
l
,
m
) | ln[ |
I q
(
l
,
m
) |]}
u
,
v
(
u
re
,
v
) 2
u
2 (
u
im
,
v
,
v
) 2
I q
(
l
,
m
)
I q
(
l
,
m
)
I q
(
l
,
m
)
E
l
,
m I
0 (
l
,
m
) ln[
I
0 (
l
,
m
)]
q
1
Q
l m q
1
Q
l m I q
(
l
,
m
) ln[
a I q
(
l
,
m
)]
I q
(
l
,
m
) ln[
a I q
(
l
,
m
)]
u
,
v
(
u
re
,
v
) 2
u
2 (
u
im
,
v
,
v
) 2
Restrictions derived from visibility data
V u
,
v
F
{
I
(
l
,
m
)}
D u
,
v
1
Q q
0
F
I q
(
l
,
m
) 0 0
q
D u
,
v
q
1
Q
m I q
(
l
,
m
)
lm a u
,
v
0 0
q
u
re
,
v
A u
,
v
1
Q q
m I q
(
l
,
m
)
b u
lm
,
v
0 0
q
im u
,
v
B u
,
v
Modeling 3C120
Model of 3C120 SFS MFS
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
Problem with aligning VLBI images
There exists frequency-dependent core shift phenomenon, which must be taken into account before MFS.
Some examples of MFS with aligning VLBI images at different frequencies
(in collaboration with A. Pushkarev) Source Frequency band Reference frequency GHz Shift mas 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.
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)
J0336+3218 (Two-frequency synthesis) 2.3 GHz 8.6 GHz MFS at 5.5 GHz (jet aligning)
J1419+5423 (Three-frequency sytnthesis) 5 GHz 8.4 GHz 15.3 GHz MFS at 8.4 GHz (jet aligning) high resolution low resolution
J0958+6533 (1997 Apr 06) synthesis) (Four-frequency 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 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
MFS-maps:
MEM-solution MEM-solution MEM-solution (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 RMS=0.12 mJy/beam RMS=0.13 mJy/beam Map peak: 0.319 Jy/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 %
Spectral index distributions over the source KUXC synthesis UXC synthesis XC synthesis
KUXC_C KUXC_X KUXC_U KUXC_K
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
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.)
Simulation of the phase-less method
Model sources Reconstructed intermediate Images reconstructed from zero-phase images spectrum of intermediate images
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
Phase-less mapping of 3С120
Images of 3C120 obtained from VLBA + observations in 2002 at 8.4 GHz
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.
Model Dirty image Clean MEM Difference GMEM
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)
Multi-epoch GMEM solutions from VLBA observations at 5 GHz
G
MEM-solutions convolved by clean beam 05x0.5 mas
GMEM-solutions convolved by clean beam related to real system
Kinematics of the jet components Relative R.A.
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
epoch: 2004.58
2004.66
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
Kinematics of the jet components С B A
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.