W-term Correction

The term $w{\sqrt{1-l^2-m^2}}$ in eq. 2.3 is referred as w-term. It arises due to the fact that fringe rotation effectively phases the array for a point source in the sky referred to as the phase center direction. A wavefront away from the phase tracking center but on the celestial sphere ( $l^2+m^2+n^2 =1$), will introduce an 'extra' phase equal to $2\pi w\sqrt{1-l^2-m^2} = \pi w \theta^2$. The phase error $\phi \alpha B_{max} \lambda / D^2$, where $B_{max}$ is the length of the maximum baseline and D is the antenna diameter, and $\lambda$ the wavelength.

In earth rotation aperture synthesis, the baselines are not co-planer, since the array geometry as viewed from the source changes with time. And for the larger primary field of view, the map contains around 10 to 20 confusing sources or more, the side-lobes from these sources can contribute to image noise. If such sources are away from the phase center, they produce phase errors which increases with the square of distance from the phase center. Thus, non-coplanar baselines and spherical curvature of the sky causes the apparent shift in source position which varies with time causing smearing which results in limiting the dynamic range of the map and distortion across the field of view.

There are techniques for reducing non-coplanarity problem which is called w-term correction and the technique called 'faceted transform' of the image is used. To reduce the w-term effect, a 2-D approximation can be done by shifting the phase center to different parts of the image correcting the w-term and initially a small field around the shifted phase center, which is nearly a tangent plane, called 'facets'. The many facets that form the field of view are 'CLEANed' simultaneously and the CLEANed images combined and give the larger field. To reconcile multiple fields made from the single image, an interpolation is done on a common tangent plane with respect to the pointing position of the observation.