Some Alternative Non-Linear Methods

Here we describe two ad hoc methods of combining single-dish and mosaic images. To some extent, these are of historical interest only - the linear and non-linear methods mentioned before are to be preferred.

These alternative methods are not implemented as tasks within Miriad, so the steps to implement them are more hands-on. For example, if needed, you will want to correct the flux calibration factor manually before using either of these (e.g. immerge can do this for you). Some examples of using these approaches are given in the paper by Snezana Stanimirovic (ASP Conf. Series, vol. 278, p. 375).

• An alternative combination technique using mosmem is to use the single-dish image as the default image (default keyword) rather than as a separate $\chi^2$ constraint. That is, you use just the mosaic image and beam with the map and beam parameters, and set the single-dish image as the default image. Strictly the default image should be in units of Jy/pixel. However mosmem is smart enough to divide the pixel values by the beam volume (where necessary) to perform some sort of crude conversion from Jy/beam to Jy/pixel. You can constrain the total flux of the resultant model to agree with the single-dish total flux, by using options=doflux. Task mosmem is smart enough to use the default image as an flux estimate when the doflux option is used and when the flux is left unset.

• You could form an image which is a simple weighted average of the mosaic and single-dish dirty images (i.e. before any deconvolution), and then pass this to mosmem as if it was a normal mosaic. In doing this, you are trying to fool mosmem to some extent, so there are a number of fudges that you should do to make the data better approximate what mosmem expects.

  Before adding the mosaic and single-dish images, you should apply the residual primary beam attenuation present in the mosaic to the single-dish data. Task mossen will generate an image of the residual primary beam attenuation (set the gain parameter). Having generated a the residual primary beam attenuation, you can then apply this to the single-dish image using maths.

  The weights used to add the mosaic and single-dish images should sum to 1, but the actual way of determining the relative weights is not well defined. One good choice, advocated by Stanimirovic et al., is to weight the images in inverse proportional to the beam volumes. When the difference in resolution between the mosaic and single-dish observations is appreciable (i.e. the normal situation), this weighting reduces to nearly the same as that performed by immerge. Normalisation the weights to add to 1 gives

  $$\begin{eqnarray*} w_{\rm mos} &=& \frac{\Omega_{\rm SD}}{\Omega_{\rm mos}+\Omega... ...D} &=& \frac{\Omega_{\rm mos}}{\Omega_{\rm mos}+\Omega_{\rm SD}} \end{eqnarray*}$$
  
  
  
  
  (where $\Omega_{\rm mos}$ and $\Omega_{\rm SD}$ are the beam volumes of the synthesised mosaic beam and the single-dish beam respectively).

  To generate the effective beam dataset, you will also want to add a component to the mosaic beam. Because of the process that mosmem believes was used to generate a mosaic, the component you add is not simply the single-dish beam - the mosaicing process narrows down the width of the beams from that of individual pointings. If both the interferometer primary beam and the single-dish beam are gaussian forms with widths $\theta_{\rm int}$ and $\theta_{\rm SD}$ respectively, then you will want to add a gaussian to the mosaic beam dataset which has a width
  

  $$\theta = \sqrt{\frac{2\theta_{\rm int}^2\theta_{\rm SD}^2}{2\theta^2_{\rm int}-\theta^2_{\rm SD}}}.$$
  
  
  This gaussian will be somewhat broader than the single-dish resolution (for an ATCA and Parkes combination, it will be about 7% broader than the Parkes beam). You will want to add this gaussian and the mosaic beam with the same weights that were used for the single-dish and mosaic images. Note that the mosaic beam dataset consists of one image per pointing, and that you will want to add a gaussian to each plane. Task imgen can be used to do this, although it warns you that its handling of cubes is crude! With imgen, you can set the weighting by appropriately setting the scaling factor (factor parameter) and the amplitude of the gaussian (spar parameter).

  Note that this technique ignores some edge effects. Also it uses the theoretical noise level derived for the mosaic observation. This will be close to what you want if the noise levels for all pointings are about the same.