General description

In this family, the single-dish information is heavily processed before merging with the interferometric information. The basic idea is to produce from the single-dish observations pseudo-visibilities similar to the ones that would be produced by the interferometer if they were not filtered out.

1. The Single-Dish measurements are re-gridded and then FFTed into the uv plane.
2. The data are deconvolved of the single-dish beam ( $B_{\ensuremath{\mathrm{sd}}}$) convolution by division by its Fourier Transform (truncated to the antenna diameter).
3. The data are FFTed back to the image plane and multiplied by the interferometer primary beam, $B_{\ensuremath{\mathrm{primary}}}$.
4. The result is FFTed again in the uv plane where the visibilities are sampled on a regular grid.
5. In the case of a mosaic, the two last operations are performed for each pointing center.

Using the properties of the Fourier transform, we can rewrite the measurement equation of an interferometer as

  $$V(u,v) = \ensuremath{\displaystyle\left\{ \mbox{FT}(B_\ensuremath... ...m{primary}}) \star \mbox{FT}(I_\ensuremath{\mathrm{source}}) \right\}}(u,v)+N.$$ (16)

This equation means that the visibility measured by an interferometer at the spatial frequency $(u,v)$ is the convolution of the Fourier transform of the source intensity distribution by the Fourier transform of the primary beam. Hence, to get pseudo-visibilities truly consistent with interferometric visibilities, we must be able to reliably compute the convolution by the Fourier transform of the primary beam. This implies that we can compute pseudo-visibilities only for spatial frequencies lower than D-d. The use of the IRAM-30m to produce the short-spacing information of the NOEMA is thus ideal as it enables to recover pseudo-visibilities up to 15 m (=30 m-15 m). Once the pseudo-visibilities have been computed, they are merged with the interferometric visibilities and standard imaging and deconvolution are then applied to the merged data set.