15.5 Weighing and Tapering

There is still a free parameter in the image construction process: the weighting function. At $uv$ table creation, the sampling function is defined as

$$S(u,v) = \frac{1}{\sigma^2(u,v)}$$ (15.25)

where the noise $\sigma$ is computed from system temperature, bandwidth, integration time, and system efficiency (including quantization and decorrelation).

$$\sigma(u,v) = \frac{\ensuremath{\mathcal{J}}_{I} T_{\mathrm{sys}}... ...suremath{\eta_\mathrm{\scriptscriptstyle Q}}\sqrt{2 \Delta \nu t_\mathrm{int}}}$$ (15.26)

$\ensuremath{\mathcal{J}}_{I}$ being the antenna temperature to flux density conversion factor:

$$\ensuremath{\mathcal{J}}_{I} = \frac{2 k}{\ensuremath{\eta_\mathrm{\scriptscriptstyle A}}A}$$ (15.27)

The weights $W(u,v)$ can be freely chosen. The selecting of weights is usually decomposed in two slightly different processes, called Weighting and Tapering.

Weighting deals with the local variations of weights for each grid cell after the gridding process. Since the original $uv$ coverage is an ensemble of ellipses, the gridding may leave a weight distribution with very large dispersion. Weighting can be applied to uniformize this distribution.

On the other hand, Tapering consists in apodizing the $uv$ coverage by $T(u,v) = \exp (-(u^2+v^2)/t^2))$ where $t$ is a tapering distance. This corresponds to smoothing the data in the map plane (by convolution with a gaussian).

The simplest possibility, called Natural Weighting, without taper is to keep the original spectral sensitivity function by setting $W(u,v) = 1$. This can be demonstrated to maximize sensitivity to point sources (i.e. sources smaller than the synthesized beam). Proper design (and use) of the array can ensure that the resulting synthesized beam is appropriate, in terms of size (angular resolution matched to the scientific goal) and shape (lowest possible sidelobes).

If the sources of interest are somewhat extended, tapering can be used to increase brightness sensitivity. Tapering may also have the advantage of lowering the sidelobes. This is usually true for limited tapering, which reduces the effect of the discontinuity at the outer edge of the $uv$ plane, but is not the case for strong tapering, where the result becomes critically dependent on the actual sampling of the inner part of the $uv$ plane. However, tapering is always throwing out some information, namely the long baselines part of the data set. Hence, it should be used either with moderate tapers, or as a complementary view on a data set. To increase brightness sensitivity, one should use preferentially compact arrays rather than tapering.

Uniform Weighting consists in selecting the weights $W(u,v)$ so that the sum of weights $\sum W \times S$ over a $uv$ cell is a constant function (or zero if no $uv$ data exists in that cell). The size (radius) of the $uv$ cell is an arbitrary parameter. It can be the cell size resulting from the gridding process, i.e. the inverse of the field of view, but any other choice is possible. Using half of the dish diameter is well justified, since the visibilities are convolution of Fourier transform of the sky brightness by the Fourier transform of the primary beam. Uniform Weighting gives more weight to long baselines than natural weighting (because you spend less time per $uv$ cell on long baselines than on short baselines for earth synthesis). Uniform Weighting produces smaller beam. Because it fills the $uv$ plane more regularly, Uniform weighting could be thought also to produce lower sidelobes. However, because of the discontinuity of the weights at the edge of the sampled portion of the $uv$ plane, the inner sidelobes tend to be increased, unless some tapering is combined with Uniform weighting.

Robust Weighting is a variant of uniform weighting which avoids to give too much weight to a $uv$ cell with low natural weight. There are several ways to implement such a scheme. Roughly speaking, if the sum of natural weights in a cell is less than a threshold, the weighting is unchanged, if it is more, the weight is set to this threshold. Let $S_n$ be the natural weight of a cell, and $S_t$ a threshold for such weight. Robust weighting could be implemented by selecting the weight W as

$$S_n < S_t \rightleftharpoons W = 1$$

$$S_n > S_t \rightleftharpoons W = S_t/S_n$$ (15.28)

or a more continuous formula like

$$W = \frac{1}{\sqrt{1 + S_n^2/S_t^2}}$$ (15.29)

Robust weighting combines the advantages of Natural and Uniform weighting, by increasing the resolution and lowering the sidelobes without degrading too much the sensitivity. By adjusting the threshold, it approaches either case (large threshold $\longleftrightarrow$ Natural, small threshold $\longleftrightarrow$ Uniform).

Weighting and Tapering reduce point source sensitivity by

$$\sqrt{\sum T^2 W^2} / ( \sum T W )$$ (15.30)