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

$$I_\ensuremath{\mathrm{meas}}^\ensuremath{\mathrm{sd}} = B_\ensuremath{\mathrm{sd}} \star I_\ensuremath{\mathrm{source}} + N,$$ (5.6)

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

$$I_\ensuremath{\mathrm{meas}}^\ensuremath{\mathrm{int}} = B_... ...mathrm{primary}}.I_\ensuremath{\mathrm{source}} \right\}} + N,$$ (5.7)

i.e. the measured intensity ( $I_\ensuremath{\mathrm{meas}}^\ensuremath{\mathrm{int}}$) 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 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.