Imaging

When combining together (dirty or clean) images, it is important to correct the primary beam attenuation to avoid modulation of the signal in the combined image. If we forget for the moment the dirty beam convolution, the images associated to each field are noisy measurements of the same quantity (the sky brightness distribution) weighted by the primary beam. The best estimation of the measured quantity is thus given by the least mean square formula $$ M(\alpha,\delta) = \frac{\displaystyle\sum\nolimits_i \frac{B_i(\alpha,\delta)}{\sigma_i^2}\,F_i(\alpha,\delta)} {\displaystyle\sum\nolimits_i \frac{B_i(\alpha,\delta)^2}{\sigma_i^2}}, $$ where \(M(\alpha,\delta)\) is the brightness of the dirty/cleaned mosaic image in the direction \((\alpha,\delta)\) , \(B_i\) are the response functions of the primary antenna beams in the tracking direction of field \(i\) , \(F_i\) are the brightness distributions of the individual dirty/cleaned maps, and \(\sigma_i\) are the corresponding noise values. As may be seen on this equation, the intensity distribution of the mosaic is corrected for primary beam attenuation. This implies that noise is inhomogeneous. Indeed, if \(N(\alpha,\delta)\) is the noise distribution and \(\sigma(\alpha,\beta)\) is its standard deviation in the direction \((\alpha,\beta)\) , we have $$ N(\alpha,\delta) = \frac{\sum\nolimits_i \frac{B_i(\alpha,\delta)}{\sigma_i^2}\,N_i(\alpha,\delta)} {\sum\nolimits_i \frac{B_i(\alpha,\delta)^2}{\sigma_i^2}} $$ and $$ \sigma(\alpha,\delta) = \frac{\sqrt{\sum\nolimits_i \frac{B_i(\alpha,\delta)}{\sigma_i^2}}} {\sum\nolimits_i \frac{B_i(\alpha,\delta)^2}{\sigma_i^2}} = \frac{1}{\sqrt{\sum\nolimits_i \frac{B_i(\alpha,\delta)^2}{\sigma_i^2}}} \simeq \frac{\sigma} {\sqrt{\sum\nolimits_i B_i(\alpha,\delta)^2}} $$

with the approximate formula assuming nearly equal noises \(\sigma_i \approx \sigma\) . Thus, the noise strongly increases near the edges of the mosaic field-of-view, but is also non-uniform in the central regions, especially if the pointings are not sufficiently packed.

To limit this, it is possible to truncate the primary beams (preferably by tapering them by a continuous function to avoid sharp edge effects). A better approach is to limit the deconvolved region in the deconvolution step.