Weighting

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 of the imaging process. In particular, the user may change this scaling to give more or less weight to the large/short spatial frequencies.

We can thus introduce a weighting function $W(u,v)$ in the definitions of $B_\ensuremath{\mathrm{dirty}}$ and $I_\ensuremath{\mathrm{dirty}}$

$$B_\ensuremath{\mathrm{dirty}} = \mbox{FT}^{-1} \ensuremath{\displaystyle\left\{ W.S \right\}}$$ (4.4)

and

$$I_\ensuremath{\mathrm{dirty}} = \mbox{FT}^{-1} \ensuremath{\displaystyle\left\{ W.S.V \right\}}.$$ (4.5)

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 that the cells of the inner $uv$ plane (just because there are less cells in the inner $uv$ plane). Robust weighting thus increase the spatial resolution by emphasizing the large spatial frequencies at the cost of a worst sensitivity to point sources...

Tapering

is the apodization of the $uv$ coverage by simple multiplication of a Gaussian

$$W = \exp\ensuremath{\displaystyle\left\{ -\frac{\ensuremath{\displaystyle\left( u^2+v^2 \right) }}{t^2} \right\}},$$ (4.6)

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 almost never be used as this somehow implies that you throw away large spatial frequencies measured by the interferometer... In other words, use compact configuration of the arrays and not tapering to increase sensitivity to extended structures in your source.

For more details on the whole imaging process the interested reader is referred to Guilloteau (2000).