Problem definition

The normalized cross-correlation function (the correlator output), measured by an interferometer using two antennas, antenna $i$ and antenna $j$, in the limit $I \ll T_{sys_i}/\eta_i$, can be written as:

$$\begin{split}\rho_{ij}^{Obs}=\rho^{Obs}(u_{ij},v_{ij},w_{ij})... ... &{dl dm\over\sqrt{(1-l^2-m^2)}} + \epsilon_{ij}\end{split}$$ (1)

where $I(l,m)$ is the sky surface brightness, $\eta_i$ is the sensitivity and $T_{sys_i}$ the system temperature of the antenna $i$ in units of Kelvin/Jy and Kelvin respectively, $\epsilon_{ij}$ is the additive noise on the baseline $i$-$j$, and $\phi_i$ is the antenna based phase of the signal. The rest of the symbols have the usual meaning.

In practice however, the antenna based amplitude ( $\sqrt{{\eta_i}/{T_{sys_i}}}$) and phase ($\phi_i$) are potentially time varying quantities. This could be due to changes in the ionosphere, temperature variations, ground pick up, antenna blockage, noise pick up by various electronic components, background temperature, etc. Treating the quantities under the square root in the above equation as the antenna dependent amplitude gain, these antenna dependent quantities can be written as complex gains $g_i=a_ie^{-\iota\phi_i}$ where $a_i=\sqrt{\eta_i/T_{sys_i}}$. For an unresolved source at the phase tracking center, variations in this amplitude will be indistinguishable from a variations in the ratio of $\eta$ and $T_{sys}$.

In terms of $g_i$s, we can write Eq. 1 as

$$\rho_{ij}^{Obs} = g_i g^\star_j \rho^\circ_{ij} + \epsilon_{ij}$$ (2)

where

$$\rho_{ij}^\circ=\int\limits_{-\infty}^{+\infty}\int\limits_{-\inf... ...pi\iota(u_{ij}l+v_{ij}m+w_{ij}\sqrt{1-l^2-m^2})} {dl dm\over\sqrt{(1-l^2-m^2)}}$$ (3)

The use of the word ``antenna based gains'' for $g_i$s result into confusion for many and needs some clarifications. $g_i$s are called complex ``gains'' since they multiply with the complex quantity $\rho_{ij}$. For an unresolved source, $\vert g_i\vert$ represents the fraction of correlated signal and $arg(g_i)$ represents the phase of the correlated part of the signal from the antenna with respect to the phase reference (usually the reference antenna). It is in this sense that it is referred to as ``antenna based'' gains. $g_i$s are antenna based but a function of direction in the sky since, as defined here, they include $T_{sys}$ which in turn includes the sky background temperature. However, here we assume that the angular scale over which $g_i$s vary is larger than the antenna primary beam (isoplanatic case).

For an unresolved source at the phase tracking center, all terms in the exponent of $\rho_{ij}^\circ$ are exactly zero. $\rho_{ij}^\circ$ in this case would be proportional to the flux density of the source.

Given $\rho_{ij}^{Obs}$ and knowing $\rho_{ij}^\circ$ the goal is to determine the antenna dependent complex gains $g_i$s.