Weighting

Weighting in the imaging process is important in order to improve the dynamic range and the fedility of a final aperture synthesized image. The ATNF's Users Guide gives a thorough discussion and description in both sampling density weighting and sensitivity weighting. The description of the weighting methods is relevant to the SMA users. For the SMA users convenience, we duplicate their discussion interleaving with some comments for the SMA specifications.

Each visibility sample is given a weight in the imaging step. This weighting can be used to account for differences in the density of sampling in different parts of the u-v plane, or to account for different noise variances in different samples, or to improve sensitivity to extended objects, etc. Here we briefly review some weighting schemes:

Natural weighting

This gives constant weights to all visibilities (or, more strictly, inversely proportional to the noise variance of a visibility). This weighting gives optimum point-source sensitivity in an image. However the synthesised beam-shape and sidelobe levels are usually poor.

Uniform weighting

This gives a weight inversely proportional to the sampling density function. This form of weighting minimizes the sidelobe level. However the noise level can be a factor of 2 worse than natural weighting.

Super- and sub-uniform weighting

Uniform weighting computes the sampling density function on a grid that is the same size as the gridded u-v plane. This results in the synthesised beam sidelobes being minimized over the same field-of-view as the region being image. Surprisingly, making the field-of-view very large (bigger than the primary beam size) or very small (comparable to the synthesised beam) both cause uniform weighting to reduce to natural weighting. Super- and sub-uniform weighting decouple the weighting from the field size being imaged. Instead, the sidelobes in the synthesised image are minimized over some arbitrary field size, with this field being either smaller or larger than the field being imaged (for super- or sub-uniform weights respectively).

Robust weighting

Uniform weighting (including super- and sub-uniform weighting) minimizing sidelobes, whereas natural weighting minimizes the noise level. Robust weighting provides a compromise between the two, doing so in an optimal sense (similar to Wiener optimisation). See Dan Briggs' thesis for more information (Dan Briggs' dissertations).

Tapering

In signal processing theory, the optimum way to detect a signal of known form, which is buried in noise, is to convolve that signal with a ``matched filter''. This filter has an impulse response which is just the reverse of the form of the signal that is being detected. Applying this principle to detecting sources in radio interferometry, the optimum weighting for detecting a Gaussian source is to weight the visibility data by a Gaussian. This is often called `tapering'. Using a Gaussian weight will significantly increase the detectability of an extended source. However it also degrades the resolution. Gaussian weighting can be combined with any of the above weighting schemes to achieve some form of balance between sidelobes and sensitivity.

Miriad gives good control over the visibility weighting schemes, via three parameters and one option.

• fwhm controls tapering of the data. Unlike AIPS, this taper is specified in the image domain, in arcseconds. If you are interested in features of a particular angular size, then the signal-to-noise ratio in the resultant dirty image is optimised for these features if the taper FWHM is the same as the source FWHM (or source scale size).

• sup is used to control super- and sub-uniform weighting. The sup parameter indicates the region of the dirty beam (centred on the beam centre) where sidelobes are to be suppressed or minimized. Like the fwhm parameter, the sup parameter is given in arcseconds. The weights that invert calculates are optimal, or near optimal, in a least-squares sense to minimizing the sidelobes in the specified region of the beam. Although the sidelobe suppression region is not a direct control of resolution and signal-to-noise ratio, it does have an effect on these characteristics.

  Suppressing sidelobes over the entire field of the dirty beam corresponds to uniform weighting - that is, we get uniform weighting if sup is set to the field size of the dirty beam; this is the default. Alternately making no attempt to suppress sidelobes (sup=0) corresponds to natural weighting.

  Increasing sup from 0 to the field size results initially in an improvement in resolution until the value of sup is approximately equal to the best resolution. Increasing sup beyond this results in a slow degradation in resolution. The noise level varies in a less regular way with sup. Apart from saying that sup=0 (natural weighting) gives the optimum signal-to-noise ratio, it is not possible to generalise. However the noise level will be no worse than a factor of a few from the optimum.

  The default value for sup is the field size (i.e. uniform weighting).

• robust: In Miriad, robust weighting is parameterized by a ``robustness'' parameter. Values less than about -2 correspond essentially to minimizing sidelobe levels only, whereas values greater than about +2 just minimizing noise. A value of about 0.5 gives nearly the same sensitivity as natural weighting, but with a significantly better beam.

• options=systemp: The basic weight of a visibility (the weight used for natural weighting, or the weight used for a point in determining the local sampling density function) is ideally $1/\sigma^2$, where $\sigma^2$ is the noise variance of a visibility. For no Tsys information given, invert normally assumes the noise variance is inversely proportional to the integration time ($\Delta t$) and the bandwidth ($\Delta \nu$) of a visibility. However SMA data loaded with Miriad task smalod contains antenna-based Tsys measurements and Miriad can calculate $\sigma^2$ using the formula below:

  
  

  $$\sigma^2(i,j)= JyperK^2 \left[ Ts(i)Ts(j) \over 2\Delta\nu \Delta t\right]$$
  
  
  where $JyperK = {2k/\eta A}$ and A is antenna area, $\eta=\eta_c\eta_a$ is an efficiency, k is Boltzmans constant.

  The options = systemp can instruct invert to compute the basic weights using $\sigma^2$. The calculation assumes that the parameters A, $\eta_c$ and $\eta_a$ are constant across the array and the sky, i.e. $JyperK$ is assumed to be constant. For optimizing the sensitivity of the final synthesized image, the relative weights derived from options = systemp need to be applied.