Interpretation of the equation

Equation D.8 offers itself for some intuitive understanding in the following way.

$X_{ij}$ is a product of two complex numbers, namely $g_i$ and $g_j^\star$, which we wish to determine. $X_{ij}$ is itself derived from the measured quantity $V^{Obs}_{ij}$. Numerically speaking, each term in the summation of the numerator of Equation D.8 will involve $g_i$ (via $X_{ij}$) and the multiplication of $X_{ij}$ with $g_j w_{ij}$ would give $g_i$ an effective weight of $\left\vert g_j\right\vert^2 w_{ij}$. Since the denominator is the sum of this effective weight, the right-hand side of Equation D.8 can be interpreted as the weighted average of $g_i$ over all correlations with antenna $i$.

In the very first iteration, when $g_j=(1,0)$, the normalization would be incorrect since the numeric value of $g_j$, as it appears inside $X_{ij}$ would be different from that used in the denominator of Equation D.8. However, as the estimates of $g_j$s improve with iterations, the equation would progressively approach a true weighted average equation. The speed of convergence will depend upon the value of $\alpha$ and the convergence would be defined by the constraint in Equation D.10. In the ideal case when the true value of all $g_i$s is known, right hand side of Equation D.8 also reduces of $g_i$.

Estimating $g_i$ for an antenna, by averaging over the measurements from all baselines in which it participates (for a unresolved source) makes sense since for an N element array, $g_i$ would be present in N-1 measurements (all the $\left. X_{ij}\right\vert _{j=1,N; j \ne i}$) and the best estimate of $g_i$ would be the weighted average of all these measurements. Proper weight for $g_i$, buried in each of the products $X_{ij}$, can be found heuristically as follows. $g_i$, estimated from the measurements of a given baseline, must obviously be weighted by the signal-to-noise ratio on that baseline. This is $w_{ij}$ in the above equations. It must also be weighted by the amplitude gain of the other antenna making the baseline, namely $g_j$, to account for variation in antenna sensitivities and $T^s$. The total weight for $g_i$ would then be $\left\vert g_j\right\vert^2 w_{ij}$, the sum of which appears in the denominator of Equation D.8. Knowing that ideally $X_{ij} = g_i g_j^\star$, each of the $\left. X_{ij}\right\vert _{j=1,N}$ must be multiplied by $g_j w_{ij}$ (to apply the the above mentioned weights to $g_i$), before being summed for all values of $j$ and normalized by the sum of weights to form the weighted average of $g_i$. One thus arrives at Equation D.8 using these heuristic arguments.