ADASS 2003 Conference Proceedings

In VLBI, generalized Linear Multi-Frequency Imaging (MFI) consists of multi-frequency synthesis (MFS) and multi-frequency analysis (MFA) of the VLBI data obtained from observations on various frequencies. A set of linear deconvolution MFI algorithms is described. The algorithms make it possible to obtain high quality images interpolated on any given frequency inside any given bandwidth, and to derive reliable estimates of spectral indexes for radio sources with continuum spectrum.

1. Statement of the problem

Let us consider a linear model for intensity $I_{kpq}=I(x_{p},y_{q},\nu _{k})$ of the radio source in a point $(x_{p},y_{q})$ on the observational frequency $\nu _{k}$ :

$\begin{displaymath} \begin{array}{l} I_{kpq}\approx \left( {I_{0}}\right) _{pq}+... ...nu _{k}}{\nu _{0}}-1,\quad k=1,\;2,\;\ldots \;,\;K, \end{array}\end{displaymath}$

where $\nu _0$ is reference frequency corresponding to the intensity $\left( {I_0 } \right)_{pq}$ .

$\begin{displaymath} I_{kpq}=\left( {I_{0}}\right) _{pq}\cdot \left( {\frac{\nu _{k}}{\nu _{0}}}%\right) ^{\alpha _{pq}}, \end{displaymath}$

$\begin{displaymath} I_{kpq}=\left( {I_{0}}\right) _{pq}e^{\xi _{k}\alpha _{pq}}\... ...{0}}\right) _{pq}\cdot \left( {1+\xi _{k}\alpha _{pq}}\right) \end{displaymath}$

and thus the spectral indexes $\alpha _{pq}=\alpha \left( {%x_{p},y_{q}}\right)$ can be obtained as

$\begin{displaymath} \left( {I_{1}}\right) _{pq}=\alpha _{pq}\cdot \left( {I_{0}}\right) _{pq} \end{displaymath}$

$\begin{displaymath} \rho =\sum\limits_{k=1}^{K}{\sum\limits_{n=0}^{M-1}{\sum\lim... ...m}\cdot \left\vert {V_{knm}-\hat{V}_{knm}}\right\vert ^{2}}}}, \end{displaymath}$

where, $w_{knm}=w(u_{n},v_{m},\nu _{k})\geq 0$ are weights, ${ V_{knm}}$ , ${\hat{V}_{knm}}$ is a measured and a model visibility function respectively,

$\begin{displaymath} {\hat{V}_{knm}=A}_{k}\cdot \sum\limits_{p,q=o}^{M-1}\left[ \... ...\{ -2\pi i\cdot \left( u_{n}x_{p}+v_{m}y_{q}\right) \right\} , \end{displaymath}$

$\begin{displaymath} \left( \widehat{I_{l}}\right) _{pq}=\Delta ^{2}\varphi _{pq}... ..._{l}\right) _{pq}\left( 1-x_{p}^{2}-y_{q}^{2}\right) ^{-0.5}, \end{displaymath}$

The problem of the optimization can be presented as a solution of the following system of linear equations:

$\begin{displaymath} \left( {D_0 } \right)_{pq} = 0,\;\ldots ,\;\left( {D_{N - 1} } \right)_{pq} = 0 \end{displaymath}$

for a vector of intensity $\left( \widehat{\mathbf{I}}\right) _{rt}=\left( {\left( \widehat{\mathbf{I}}{_{... ...{rt},\;\ldots ,\;\left( \widehat{\mathbf{I }}{_{N-1}}\right) _{rt}}\right) ^{T}$ , where the

-th residual map $\left( {D_{m}}\right) _{pq}$ can be defined as:

$\displaystyle \left( {D_{m}}\right) _{pq}$	$\textstyle =$	$\displaystyle \sum\limits_{k=1}^{K}{\beta _{k}^{m}\cdot } \left\{ {D_{kpq}}-{\s... ...{\left( \widehat{\mathbf{I}}{ _{n}}\right) _{il}\cdot \beta _{k}^{n}}\right\} ,$	(1)
$\displaystyle m$	$\textstyle =$	$\displaystyle 0,\;1,\;\ldots ,\;N-1,$	(2)

where, ${D_{kpq}=}\sum\limits_{n,m=o}^{M-1}w_{knm}\cdot V_{knm}\cdot \exp \left\{ 2\pi i\left( u_{n}x_{p}+v_{m}y_{q}\right) \right\}$ is a k-th "dirty" map at the point $\left( x_{p},y_{q}\right)$ ,

$B_{k,p-i,q-l}=\sum\limits_{n,m=o}^{M-1}w_{knm}\exp \left\{ 2\pi i\left[ u_{n}\left( x_{p}-x_{i}\right) +v_{m}\left( y_{q}-y_{l}\right) \right] \right\}$ is a k-th "dirty" beam at the point $\left( x_{p}-x_{i},y_{q}-y_{l}\right)$ .

2. Solution of the problem

Let us choose the following initial conditions: ${\left( \widehat{ \mathbf{I}}{_{m}}\right) _{il}^{\left( 0\right) }=0}$ for all

and form

initial arrays $\left( {D_{m}}\right) _{pq}^{\left( 0\right) },m=0,\;1,\;\ldots ,\;N-1,$ and $\left( \widehat{B}{_{m}}\right) _{pq}=\sum\limits_{k=1}^{K}A_{k}^{2}\cdot \left( {\beta _{k}}\right) ^{m}\cdot B_{kpq}$ ,

Calculation of the next s-th step ( $s=1,\;2,\;\ldots )$ begins from the choice of the point $(x_{p},y_{q})$ , of the map maximum

$\begin{displaymath} \varepsilon ^{(s-1)}=\max_{x_{r}^{2}+y_{t}^{2}<\;1;\quad 0\;... ...left\vert {\left( {D_{m}}\right) _{rt}^{(s-1)}}\right\vert }. \end{displaymath}$

$\begin{displaymath} \left( \widehat{\mathbf{I}}\right) _{pq}^{(s)}=\left( \wideh... ...a \mathbf{E}^{-1}\cdot \left( \mathbf{D}\right) _{pq}^{(s-1)}, \end{displaymath}$

and the residual maps $\left( \mathbf{D}\right) _{rt}=\left\{ {\left( {D_{0}} \right) _{rt},\left( {D_{1}}\right) _{rt},...,\left( {D_{N-1}}\right) _{rt}} \right\} ^{T}$ :

$\begin{displaymath} \left( \mathbf{D}\right) _{rt}^{(s)}=\left( \mathbf{D}\right... ...(s)}-\left( \widehat{\mathbf{I}}\right) _{pq}^{(s-1)}\right] . \end{displaymath}$

Here $\mathbf{E=}\left( E_{ij}\right)$ is a positive defined matrix of maximum values of weighted "dirty" beams, $E_{ij}=\left( \widehat{B}_{i+j}\right) _{0,0},i,j=0,...,N-1$ ; $\gamma$ is a loop gain. The process of the iteration can be completed if $\varepsilon ^{\left( s-1\right) }<\varepsilon$ , where $\varepsilon$ is a given accuracy. Otherwise it is necessary to suppose

and to calculate the next $\ \varepsilon ^{\left( s\right) }.$ Conditions of the convergence of the algorithm above is $0<\gamma <2$ , $1\leq N\leq K$ .

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 $\left( u,v\right)$ -coverage.

3. Implementation of the linear deconvolution algorithm

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).

**Figure 1:** 3C84 observations at 11 (interpolated) and 15 GHz
$\begin{figure} \plottwo{P6-1f1.eps}{P6-1f2.eps} %\plottwo{P6-1_1.eps}{P6-1_2.eps} \end{figure}$

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.

4. Acknowledgments

The authors thank Jon Romney (NRAO) for providing the observational data used in testing the new MFI algorithms.

References

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.

Generalized Linear Multi-Frequency Imaging in VLBI

Abstract:

1. Statement of the problem

2. Solution of the problem

3. Implementation of the linear deconvolution algorithm

4. Acknowledgments

References