The use of the visibility weights
\((1/\sigma^2)\)
in the definition of the
sampling function is called natural weighting as it is natural to weight
each visibility by the inverse of noise variance. Natural weighting is also
the way to maximize the point source sensitivity in the final image.
However, the exact scaling of the sampling function is an additional degree
of freedom in the imaging process. In particular, the user may change this
scaling to give more or less weight to the long or short spatial frequencies.
We can thus introduce a weighting function
\(W(u,v)\)
in the definitions of
\(B_{\mathrm{dirty}}\)
and
\(I_{\mathrm{dirty}}\)
$$ B_\mathrm{dirty} = \mathrm{FT}^{-1} \left(W.S\right) $$
and
$$ I_\mathrm{dirty} = \mathrm{FT}^{-1} \left(W.S.V\right) $$
There are two main categories of weighting functions
- Robust weighting
- In this case,
\(W\)
is computed to enhance the
contribution of the large spatial frequencies. This is done by first
computing the natural weight in each cell of the uv plane. Then
\(W\)
is
derived so that
- The product
\(W.S\)
in a uv cell is set to a constant if the
natural weight is larger that a given threshold;
-
\(W = 1\)
(i.e. natural weighting) otherwise.
This decreases the weight of the well measured uv cells (i.e. very
low noise cells) while it keeps natural weighting of the noisy cells. It
happens that the cells of the outer uv plane (corresponding to the
large interferometer configurations) are often noisier than the cells of
the inner uv plane (just because there are less cells in the inner
uv plane). Robust weighting thus increases the spatial resolution by
emphasizing the large spatial frequencies at moderate cost in sensitivity
for point sources (but with a larger loss for extended sources, see below).
- Tapering
- is the apodization of the uv coverage by simple
multiplication by a Gaussian
$$ W = \mathrm{exp}\left(-\frac{\left({u^2+v^2}\right)}{t^2}\right), $$
where
\(t\)
is the tapering distance. This multiplication in the uv
plane translates into a convolution by a Gaussian in the image plane,
i.e. a smoothing of the result. The only purpose of this is to increase
the sensitivity to extended structure. Tapering should never be
used alone as this somehow implies that you throw away large spatial
frequencies measured by the interferometer. It is only a way to extract
the most information from the given data set. If you need more sensitivity
to extended structures, use compact configuration of the arrays
rather than extended configurations and tapering.
For more details on the whole imaging process the interested reader is
referred to ().
