Weighting and Tapering

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 $B_{\ensuremath{\mathrm{dirty}}}$ and $I_{\ensuremath{\mathrm{dirty}}}$

  $$B_\ensuremath{\mathrm{dirty}} = \mbox{FT}^{-1} \ensuremath{\displaystyle\left\{ W.S \right\}}$$ (4)

and

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

  $$W = \exp\ensuremath{\displaystyle\left\{ -\frac{\ensuremath{\displaystyle\left( u^2+v^2 \right) }}{t^2} \right\}},$$ (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 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 Guilloteau (2000).