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 $$ I_\mathrm{meas}^\mathrm{sd} = B_\mathrm{sd} \star I_\mathrm{source} + N $$ i.e. the measured intensity ( $I_{\mathrm{meas}}^{\mathrm{sd}}$ ) is the convolution of the source intensity distribution ( $I_{\mathrm{source}}$ ) by the single-dish beam ( $B_{\mathrm{sd}}$ ) plus some thermal noise, while the measurement equation of an interferometer can be rewritten as $$ I_\mathrm{meas}^\mathrm{id} = B_\mathrm{dirty} \star \left(B_\mathrm{primary}.I_\mathrm{source}\right) + N, $$ i.e. the measured intensity ( $I_{\mathrm{meas}}^{\mathrm{id}}$ ) is the convolution of the source intensity distribution times the primary beam ( $B_{\mathrm{primary}}.I_{\mathrm{source}}$ ) by the dirty beam ( $B_{\mathrm{dirty}}$ ) plus some thermal noise. $B_{\mathrm{sd}}$ has very similar properties than $B_{\mathrm{primary}}$ and very different properties than $B_{\mathrm{dirty}}$ . In radioastronomy, $B_{\mathrm{sd}}$ and $B_{\mathrm{primary}}$ 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 $B_{\mathrm{sd}}$ and $B_{\mathrm{primary}}$ have similar full width at half maximum. Now, $B_{\mathrm{dirty}}$ 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 $B_{\mathrm{dirty}}$ has a full width at half maximum close to the interferometer resolution, i.e. much smaller than the FWHM of $B_{\mathrm{sd}}$ .

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.

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