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.
- The Single-Dish measurements are re-gridded and then FFTed into the
uv plane.
- The data are deconvolved of the single-dish beam (
\(B_{\mathrm{sd}}\)
)
convolution by division by its Fourier Transform (truncated to the
antenna diameter).
- The data are FFTed back to the image plane and multiplied by the
interferometer primary beam,
\(B_{\mathrm{primary}}\)
.
- The result is FFTed again in the uv plane where the visibilities
are sampled on a regular grid.
- 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
$$
V(u,v) = \left(\mbox{FT}(B_\mathrm{primary}) \star \mbox{FT}(I_\mathrm{source})\right)(u,v)+N.
$$
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.
