3.10.1 Weighting

When forming the dirty map different weighting schemes may be employed to optimize the resolution and sensitivity of the map. Three criteria are used to decide the weight of each visibility point representing a single integration on a single baseline.

(i)

Weighting inversely proportional to the error. This can be found `blindly' from the number of visibilities or the scatter of visibility amplitudes over time intervals too short for any astrophysical change. However the high resolution of MERLIN means source structure can cause rapid amplitude variations, and its widely-spaced asymmetric $uv$ coverage means antenna weights strongly affect the synthesised beam-shape. For well-edited, well-calibrated data the time-variation due to noise is assumed to be negligible in comparison with the differences between the signal received at each antenna and this form of weighting is not advised for MERLIN data.

(ii)

It is usual to weight the data from each MERLIN antenna in proportion to the antenna sensitivity (see Table 4.3) which is a function of the relative collecting area and efficiency (and bandwidth, if different) at the relevant frequency. The appropriate antenna weights are applied to the $uv$ data in AIPS after all calibration but before the final mapping. Occasionally it is necessary to give antennas different weights during calibration also, and this can be done such that only a set of solutions is affected, not the data itself. Weighting individual antennas during calibration should be done with caution as increasing the contribution of more sensitive antennas may apparently improve the solutions but the results for the less sensitive antennas will be poorly constrained.

(iii)

Weighting according to distance in the $uv$ plane. This is most useful in initial exploratory maps. If confusion is suspected the inner portion of the $uv$ plane can be given a low or zero weight to start producing a model of a compact source. Conversely an initial point model will give better solutions for a weak and slightly resolved source if the outer regions of the $uv$ plane are given lower weight, known as tapering, which coarsens the resolution. This also allows a larger pixel size to be used, useful for searching a large area for confusion. If possible the weighting is incrementally brought back to unity as confusing sources are subtracted and/or a better model developed.

(iv)

Weighting according to the density of samples in the $uv$ plane. If each visibility point has the same weight then the density of MERLIN visibilities (per cell in the gridded data) is greatest towards the centre of the $(u,v)$ plane. This is known as natural weighting and produces a synthesised beam about 25% larger than that described in §3.2.2. At the other extreme, uniform weighting divides the weight of each data point by the number of visibilities in that $uv$ grid cell. This means each part of the $uv$ plane has equal weight, giving a smaller synthesised beam.

Each of these weighting schemes affects the map noise as well as resolution. Weighting up the more sensitive antennas as in (ii) reduces the noise but if the Lovell antenna is in the array the contributions of the shorter baselines are increased, giving a larger synthesised beam (e.g. Fig 3.4). If Cambridge is the most sensitive antenna present the resolution is increased but the side-lobe level may be worse. At 5 GHz Defford is usually given a low weight. Deviations from the values given in Table 4.2 may be necessary, and the effects of this weighting are changed by the other forms of weighting.

Weighting as described in (iii) and (iv) (apart from pure natural weighting) inevitably increases the $\sigma_{\rm rms}$ noise. However for circumpolar sources, using uniform weighting to interpolate the data more evenly across the $uv$ plane can reduce the beam sidelobe level (Table D.3). The AIPS task IMAGR allows a huge range of combinations of weighting, including a spectrum covering all stages between completely natural (ROBUST 5) and completely uniform (ROBUST -5) weighting. The effects on a given data set depend on the $uv$ coverage and the scales on which your target has structure. Fig. D.5 shows the MERLIN beam for sources at various $\delta$ and natural, intermediate and uniform weighting.

For many sources partial uniform weighting (in addition to weighting antennas according to individual sensitivity) often gives a more regular beam shape, making maps easier to interpret, with negligible increase in noise levels. More extreme uniform weighting can give very high resolution but extended structure will be lost and artefacts can appear. It can be useful to show bright detailed structure qualititively, but can be self-defeating for determination of positions of bright compact components, as model-fitting accuracy depends on the signal-to-noise ratio (Appendix G). It is usually difficult to use weighting other than type (ii) to improve maps of targets which are too faint to self-calibrate.

In any situation involving sparsely sampled data (MERLIN, VLBI, VLA snapshots) it must be remembered that the resolution and sensitivity do not have a linear relationship with changes in weighting. For example, increments in the proportion of uniform to natural weighting will produce sudden jumps in resolution and noise, and in some parts of the $uv$ plane a small change in $uv$ distance can abruptly include/exclude all baselines to the Cambridge antenna.

If $uv$ data from different arrays is being combined then the relative sensitivities of each array must be considered, see §3.7.1 and Appendix G.