Dirty Map and Dirty Beam

But the real situation is much worse. With the advent of the Very Large Array (VLA), the majestic filling in of the $u-v$ plane with samples spaced at $D/2$ went out of style. If one divides the field of view into pixels of size $1/(2b_{max})$, then the total number of such pixels (resolution elements) would be significantly larger than the number of baselines actually measured in most cases. This is clearly seen in plots of $u-v$ coverage which have conspicuous holes in them. The inverse Fourier transform of the measured visibility is now hardly the true map because of the missing data. But it still has a name - the ``dirty map'' $I^D$. We define a sampling cum weighting function $W(u,v)$ which is zero where there are no measurements and in the simplest case (called uniform weighting) is just unity wherever there are measurements. So we can get our limited visibility coverage by taking the true visibilities and multiplying by $W(u,v)$. This multiplication becomes a convolution in the sky domain. The ``true'' map with full visibility coverage is therefore convolved by the inverse Fourier transform of $W$ which goes by the name of the ``dirty beam" $B^D(l,m)$.

$$I^D(l,m)=\int \int I(l',m')B^D(l-l',m-m')~dl'~dm'$$

where

$$B^D(l,m)\propto \sum W(u,v)\exp(+2\pi i (lu+mv)).$$

For a patchy $u-v$ coverage, which is typical of many synthesis observations, $B^D$ has strong sidelobes and other undesirable features. This makes the dirty map difficult to interpret. What one sees in one pixel has contributions from the sky brightness in neighbouring and even not so neighbouring pixels. For the case of $W=1$ within a disk of radius $b_{max}$ we get an Airy pattern as mentioned earlier. This is not such a dirty beam after all, and could be cleaned up further by making the weighting non-uniform, i.e. tapering the function $W$ down to zero near the edge $\vert(u,v)\vert=b_{max}$. For example, if this weighting is approximated by a Gaussian, then the sky gets convolved by its transform, another Gaussian. This dirty map is now related to the true one in a reasonable way. But, as Ables remarked, should one go to enormous expense to build and measure the longest baseline and then multiply it by zero?