In this family, most of the processing is done on the interferometric data
alone. Indeed, the interferometric data is deconvolved and corrected for
the primary beam contribution to obtain
$$
I_\mathrm{sky}^\mathrm{id} = B_\mathrm{clean} \star I_\mathrm{source} + N',
$$
where
\(B_{\mathrm{clean}}\)
is a Gaussian of FWHM equal to the interferometer
resolution and
\(N'\)
is some thermal noise corrected for the primary beam
contribution. Two main facts are hidden in this formulation: 1) the
field-of-view of the observation is obviously limited to the observed
portion of the sky and 2) more importantly, the lack of short-spacings has
not yet been overcome and a better formulation would be
$$
I_\mathrm{sky}^\mathrm{id} = \mbox{Highpass-filter} \left(B_\mathrm{clean} \star I_\mathrm{source}\right) + N'.
$$
We shall keep the simpler formulation as notation, but bear in mind the
hidden filtering in the discussions. The hybridization method
consists in combining two images (
\(I_{\mathrm{meas}}^{\mathrm{sd}}\)
and
\(I_{\mathrm{clean}}^{\mathrm{id}}\)
) in the uv plane.
- Both images are first spatially regridded on the same fine grid.
- The FFT of those two images are computed, and linearly combined by
selecting the low spatial frequencies from FFT(
\(I_{\mathrm{meas}}^{\mathrm{sd}}\)
)
and the high spatial frequencies from FFT(
\(I_{\mathrm{sky}}^{\mathrm{id}}\)
).
$$ \mathrm{FFT}(uv) = f(uv) \mathrm{FFT}(I_{\mathrm{meas}}^{\mathrm{sd}}) + (1-f(uv)) \mathrm{FFT}(I_{\mathrm{sky}}^{\mathrm{id}}) $$
The transition
\(f(uv)\)
between low and high spatial frequency is selected
to use the best regions of the uv plane in both images.
- The result is FFTed back to the image plane to produce a final,unique
image, which takes into account both single-dish and interferometric
information.
The method has the following free parameters: the transition radius and the
detailed shape of that transition. To avoid discontinuity, the transition
shape is chosen to be reasonably smooth. When the low resolution image
is provided by a single-dish, the best signal-to-noise combination
is obtained using a function
\(f(uv)\)
that is Fourier transform
of the single-dish beam. However, that is only optimal if the noise
in this image is small enough and no other instrumental effect
(such as pointing errors or baseline ripples) affect the data. So,
\(f(uv)\)
can also be chosen arbitrarily. The spatial frequency of
transition is selected close to the smallest spatial frequency
reliably measured by the interferometer (e.g. about 18 m for NOEMA),
and/or the largest spatial frequency measured by the low
resolution image (e.g. about 20 m for data taken with the IRAM 30-m
telescope). For combining ACA and ALMA data, this would be 15 m,
and of ACA and 12-m single dish, about 9 m.
