Algorithms to merge single-dish and interferometer information

The measurement equations of a single-dish and an interferometer are quite different from each other. Indeed, the measurement equation of a single-dish antenna is

  $$I_\ensuremath{\mathrm{meas}}^\ensuremath{\mathrm{sd}} = B_\ensuremath{\mathrm{sd}} \star I_\ensuremath{\mathrm{source}} + N,$$ (12)

i.e. the measured intensity ( $I_{\ensuremath{\mathrm{meas}}}^{\ensuremath{\mathrm{sd}}}$) is the convolution of the source intensity distribution ( $I_{\ensuremath{\mathrm{source}}}$) by the single-dish beam ( $B_{\ensuremath{\mathrm{sd}}}$) plus some thermal noise, while the measurement equation of an interferometer can be rewritten as

  $$I_\ensuremath{\mathrm{meas}}^\ensuremath{\mathrm{id}} = B_\ensure... ... B_\ensuremath{\mathrm{primary}}.I_\ensuremath{\mathrm{source}} \right\}} + N,$$ (13)

i.e. the measured intensity ( $I_{\ensuremath{\mathrm{meas}}}^{\ensuremath{\mathrm{id}}}$) is the convolution of the source intensity distribution times the primary beam ( $B_{\ensuremath{\mathrm{primary}}}.I_{\ensuremath{\mathrm{source}}}$) by the dirty beam ( $B_{\ensuremath{\mathrm{dirty}}}$) plus some thermal noise. $B_{\ensuremath{\mathrm{sd}}}$ has very similar properties than $B_{\ensuremath{\mathrm{primary}}}$ and very different properties than $B_{\ensuremath{\mathrm{dirty}}}$. In radioastronomy, $B_{\ensuremath{\mathrm{sd}}}$ and $B_{\ensuremath{\mathrm{primary}}}$ both have (approximately) Gaussian shapes. Moreover, the fact that we will use the single-dish information to produce the short-spacing information filtered out by the interferometer implies that $B_{\ensuremath{\mathrm{sd}}}$ and $B_{\ensuremath{\mathrm{primary}}}$ have similar full width at half maximum. Now, $B_{\ensuremath{\mathrm{dirty}}}$ is quite far from a Gaussian shape with the current generation of interferometer (in particular, it has large sidelobes) and the primary side lobe of $B_{\ensuremath{\mathrm{dirty}}}$ has a full width at half maximum close to the interferometer resolution, i.e. much smaller than the FWHM of $B_{\ensuremath{\mathrm{sd}}}$.

Merging both kinds of information obtained from such different measurement equations thus asks for a dedicated processing. There are mainly two families of short-spacing processing: the hybridization and the pseudo-visibility techniques.