Time Average Smearing

As discussed before, the $u$ and $v$ co-ordinates of an antenna are a function of time and continuously change as earth rotates generating the $uv$-coverage. To improve the signal-to-noise ratio as well as reduce the data volume, the visibility function $V(u,v)$ is recorded after finite integration in time (typically 10-20s for imaging projects) and the average value of the real and imaginary parts of $V$ are used for average values of $u$ and $v$ over the integration time. Effectively then, the assigned values of $u$ and $v$ for each visibility point is evaluated for a time which is wrong from the correct (instantaneous) time by a maximum of $\tau/2$ where $\tau$ is the integration time.

In the map domain, the resulting effect can be visualized by treating the resulting map from the time average visibilities as the average for a number of maps made from the instantaneous (un-averaged) visibilities. The baseline vectors in the $uv$-domain follow the loci of the $uv$-tracks (which are parabolic tracks) and rotate at an angular velocity equal to the that of earth, $\omega_e$. Since a rotation of one domain results into a rotation by an equal amount in the conjugate domain in a Fourier transform relation, the effect in the map domain is that the instantaneous maps also are rotated with respect to each other, at the rate of $\omega_e$. Hence, a point source located at ($l,m$) away from the center of the map would get smeared in the azimuthal direction. This effect is same as the smearing effect due to finite bandwidth of observations, but in an orthogonal direction.