Imaging (UV_MAP and RUN MAKE_MOSAIC)

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 fields are noisy measurement 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

$$\displaystyle % M(\alpha,\delta) = \frac{\displaystyle\s... ...style\sum\nolimits_i \frac{B_i(\alpha,\delta)^2}{\sigma_i^2}},$$ (5.3)

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

$$N(\alpha,\delta) = \frac{\sum\nolimits_i \frac{B_i(\alpha... ...)} {\sum\nolimits_i \frac{B_i(\alpha,\delta)^2}{\sigma_i^2}},$$ (5.4)

and

$$\sigma(\alpha,\delta) = % \frac{\sqrt{\sum\nolimits_i \f... ...sqrt{\sum\nolimits_i \frac{B_i(\alpha,\delta)^2}{\sigma_i^2}}}$$ (5.5)

Not only, the noise strongly increases near the edges of the mosaic field-of-view. But also, the center of each field is contaminated by increased noise level coming from the external regions of the neighboring fields. Indeed, the noise corrected for the primary beam attenuation is largely increasing where the primary beam is going to zero. To limit these effects, both the primary beams used in the above formula and the resulting mosaic are truncated.