Appendix A: Computation of model visibilities

  Most of the procedures described in this paper require the computation of model visibilities, which are then compared with actual data. Accurate computation of these model visibilities is therefore important.

Most standard self-calibration techniques derive the model visibilities from Fourier inversion of one or more clean components (see e.g. Cornwell & Fomalont 1989). Clean components have the advantage that they represent a deconvolution of an observed image into a set of point-components. As such they usually, but not always, reflect exactly what was actually observed; if the observed visibility data are distorted by instrumental effects, the distribution of clean components will generally reflect this distortion. Consequently model visibilities generated from the clean components should appear similar to those observed.

However, if one wishes to compute model visibilities from objects other than clean components, then instrumental effects must usually be taken into consideration. Two common instrumental effects that cause distortion of images obtained by a radio interferometer are bandwidth smearing and time-averaging of raw visibility data. These effects cause the image of a point source to become increasingly distorted as its distance from the field centre increases.

An excellent introduction to the effects of bandwidth smearing and time averaging of visibility data on the resulting image is given in Sects. 6.3 and 6.4 of Thompson et al. (1986). In summary, bandwidth smearing will distort an image parallel to the direction to the field centre while time averaging distorts the image perpendicular to the field centre direction. These effects become increasingly important the larger the distance from the field centre becomes. When one deconvolves an image into clean components the distribution of the clean components reflects this distortion and their corresponding model visibilities will suffer the same attenuation as those observed (see e.g. Cotton 1989).

However, there are a number of cases where use of clean components is not the best way to derive model visibilities. Because the DRAO ST has a wide field of view and is predominantly used to observe that part of the galactic plane visible in the northern hemisphere, there is a good chance that any single field is contaminated by responses to one of Cyg A, Cas A, or Tau A.

These sources, which have angular sizes in the range 2 to 5 arcmin are usually only modestly resolved by the DRAO synthesized beam, which has an east-west size of 3 and 1 arcmin at 74 and 21 cm wavelength respectively. Decomposition of these sources into clean components and calculating visibilities based on clean components does not seem to work particularly well. There are several reasons for this; firstly clean components do not form a particularly good model for a slightly resolved source (Briggs & Cornwell 1993). Secondly, these sources are often detected at very large distances from the field centre, in the sidelobes of an antenna with unknown phase and amplitude response. Currently, visibility data on all ST baselines is averaged to a 90 second integration period. If a source is very far away from the field centre, not only does simple time smearing distort the sampled data, but aliasing due to under sampling of the fringes may occur. In such cases, clean components may represent a "Global Average" of the data, but produce relatively poor agreement between model visibilities and actual data on any given baseline.

Better model visibilities for these bright sources are generally obtained by taking a high resolution image of the source obtained by another array that includes spatial information out to wavenumbers much higher than can be sampled by the DRAO interferometer, and inverting this image back into the UV plane. Since the strong sources the ST detects are usually at quite large distances from the field centre, one must then explicitly correct the model visibilities produced by this method for both time and bandwidth smearing effects.

Assuming that the bandpass function is square, one can describe the bandwidth attenuation, A, at a given UV location D, for a source at a distance $\xi$ from the field centre, by a sinc function ($\sin(x)/x$) where, for bandwidth smearing,  

$$x = \pi D \xi B / c.$$ (A1)

Here B is the bandwidth and c is the velocity of light (see Thompson et al. 1986). (In Sect. 4.2 of this paper, I discuss the limitations of this assumption in the case of the DRAO ST.)

The attenuation due to time smearing is also a function of increasing distance from the field centre and of increasing baseline length (Eq. (6.67) of Thompson et al. 1986). One can model the time smearing effect by computing model visibilities in some sub increment of the actual time integration period used for the observations, and then averaging the computed model visibilities together over each actual integration period. At present the DRAO software package uses a sub increment length of 10 seconds, or 1/9 of the final integration period.

If one wishes to simply derive model visibilities for one source then the fastest way to generate them is to FFT the high resolution image into the Fourier domain and then interpolate from the UV grid to derive the visibility function at an observed UV location. Visibilities derived by this method can then be corrected for bandwidth and time smearing by application of the equations and methods outlined above.

This procedure already introduces one source of potential error: we must use some sort of interpolation function to obtain visibilities at observed UV locations from values computed at UV grid locations.

However, if an FFT inversion of a model image which contains components at widely differing distances and directions from the field centre is done and one wishes to properly correct the FFTd visibilities for bandwidth and time smearing, there is an additional problem: the degridded model visibilities already represent a summation of the visibilities due to the different sources of emission. At best, one can only compute some sort of weighted mean correction for time and bandwidth smearing.

More accurate model visibilities can be obtained if one starts from the actual definition of the phase term in the Fourier relationship between a celestial source and an observed visibility as seen by a radio interferometer  

$${\rm phase} = 2\pi(U l - V m)$$ (A2)

where U and V are the UV locations of the sampled data point, and l and m are the direction cosines of the source on the celestial sphere. For an east west interferometer, such as the ST, this equation reduces to the rather simple result  

$${\rm phase}= 2\pi R(\cos \delta \sin t - \cos \delta_{\rm o} \sin t_{\rm o})$$ (A3)

by substitution of the appropriate definitions for $U\!\!,~V\!\!,~l$ and m (see e.g. Brouw 1971, p. 25), here R is the actual baseline length in wavelengths, $\delta$ and t are the declination and hour angle of the source (or of each pixel in the image), and $\delta_{\rm o}$ and $t_{\rm o}$are the declination and hour angle of the phase centre of the array.

If one uses this direct Fourier transform relationship for each sky pixel one can compute correct model visibilities at each observed UV location and make a proper correction for bandwidth and time smearing. This approach also allows one to correct for other instrumental distortions that are a function of direction in the sky. The final model visibility is obtained by summing the visibilities corresponding to each image pixel. Although use of this method implies a considerable increase in computer time over that required for FFT methods, computation

of model visibilities is an excellent example of what computer scientists call an "embarrassingly parallel" application. The work can be easily spread out over a network of workstations.

Note that an image (as opposed to clean components) represents a convolution of emission from the sky with the point spread function (PSF) of the telescope that made the image. Therefore, when an image is inverted into the Fourier domain, the intrinsic visibilities will be multiplied by the Fourier transform of the originating instrument's PSF. Since the images inverted by this procedure normally have a considerably higher resolution than does the ST, in practice this means that there is an arbitrary amplitude gain ratio between model visibilities generated by this method and actual visibilities observed by the ST. The easiest way to determine the value of this ratio is to make an image from the model visibilities, and compare the maximum signal in this model image with that in an image derived from the actual observations. The model visibilities can then be scaled by the value found for this ratio.