3.2.2 Synthesis array

The aim of synthesis imaging is to obtain an image of the source from the measurements of the visibility made by all the interferometers in the array. In the simplest case, the visibility is sampled densely out to some maximum radius $b_{\rm max}$ in the $(u,v)$ plane. The array thus produces exactly the same information as a single circular aperture of diameter $D = \lambda b_{\rm max}$, and an image with resolution $1/b_{\rm max}$ can be produced by simple Fourier inversion, usually implemented digitally using a fast Fourier transform. In this case the term `aperture synthesis' is strictly accurate. To produce an image using an FFT, $V(u,v)$ must be determined at regular intervals on a grid. If the grid contains $2^M$ samples separated by $\Delta u$ the resulting image will also have $2^M$ cells covering a total range $2^M \Delta l = 1 / \Delta u$, with cell size $\Delta l = 1/ 2^M \Delta u$. In words, the size of the image is the inverse of the cellsize in the $(u,v)$ plane, while the size of the $(u,v)$ grid is the inverse of the cellsize in the image.

Clearly the grid must be large enough to contain the maximum value of $u$ and its hermitian counterpart in the opposite direction, so $2^M \Delta u > 2 u_{\rm max}$, with the result that the cell size in the image must be smaller than the width of the interferometer PSF or synthesised beam, $\Theta_{\rm B} \sim 1 / u_{\rm max} > 2 \Delta l$. By the same token, the image must be large enough to contain the source with maximum end-to-end size $\theta_{\rm max}$, providing a maximum size for the $(u,v)$ grid spacing $\Delta u < 1 / \theta_{\rm max}$. The sampling theorem implies that this cell size is also the maximum distance over which we can safely interpolate the visibility function using linear methods, so that we need some measurements within $1 / \theta_{\rm max}$ of each grid point out to the maximum radius $b_{\rm max}$.

For MERLIN, the distribution of sampled points in the $(u,v)$ plane (the uv coverage) is far from regular (see Appendix D) and contains a number of large gaps, so that the method of direct Fourier inversion would be restricted to extremely small images, only a few beamwidths across. Instead, more powerful non-linear methods are used to derive the brightness distribution, such as CLEAN and Maximum Entropy (see §3.9.2). Using these methods, requirements on the $uv$ coverage for successful imaging are substantially loosened compared to that needed for direct Fourier inversion: in particular gaps in the coverage can be partially compensated for by denser coverage elsewhere, and in fields containing well separated regions of emission, it is the solid angle covered by the emission and not the total field of view which determines the tolerable extent of gaps in the coverage.