Let us consider a linear model for intensity of the radio source in a point on the observational frequency :
where is reference frequency corresponding to the intensity .
If the intensity in the point can be approximated by power law as
then we can present it as
and thus the spectral indexes can be obtained as
Let us consider a target function
are weights, ,
is a measured and a model visibility function
where, is a gain coefficient
for k-th antenna,
is a normalized beam, is a grid step.
The problem of the optimization can be presented as a solution of the following system of linear equations:
for a vector of intensity
, where the -th residual map
can be defined as:
where, is a k-th "dirty" map at the point ,
is a k-th "dirty" beam at the point .
Let us choose the following initial conditions: for all and form
initial arrays and , .
Calculation of the next s-th step ( begins from the choice of the point , of the map maximum
Now it's possible to specify a vector
and the residual maps
Here is a positive defined matrix of maximum values of weighted "dirty" beams, ; is a loop gain. The process of the iteration can be completed if , where is a given accuracy. Otherwise it is necessary to suppose and to calculate the next Conditions of the convergence of the algorithm above is , .
The developed algorithm is nothing other than the multi-frequency linear deconvolution, itself. this is described in more detail this procedure by Likhachev, et al. (2003). Notice that the developed algorithm 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. In case of multi-frequency linear deconvolution, it is possible to synthesize an image of a radio source at any intermediate frequency inside any given frequency band. Thus, spectral interpolation of the image is feasible. This part of the algorithm is carry out the synthesis of the image itself. However, the algorithm also makes it possible to obtain an estimate of the spectral index for a given radio source, i.e., it implements the analysis of the image. It is clear that multi-frequency imaging (MFI) will provide the highest angular resolution possible for any VLBI project due to its improved -coverage.
The algorithm described above was implemented in the software, Astro Space Locator (ASL) for Windows (http://platon.asc.rssi.ru/dpd/asl/asl.html). It was developed by the Laboratory for Mathematical Methods of the Astro Space Center (Likhachev, 2003).
Fig.1 shows two deconvolved images of 3C84 as observed on the VLBA at 11 and 15 GHz respectively. Due to the better (u,v)-coverage, the quality and angular resolution of the interpolated MFS-image at 11 GHz is much better than for the same source at 15 GHz.
The authors thank Jon Romney (NRAO) for providing the observational data used in testing the new MFI algorithms.
Likhachev, S. F., Ladygin, V. A., & Guirin, I. A. 2003, Lebedev Phys. Institute Preprint, 31, 30p.
Likhachev, S. F.,Multi-Frequency Imaging for VLBI, Future Directions in High Resolution Astronomy, in print.