In practice, however, the antenna based amplitude ( ) and phase () 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 gains, these can be written as complex gains where . For an unresolved source at the phase tracking center, variations in this amplitude will be indistinguishable from a variations in the ratio of and .
In terms of s, we can write Equation D.1 as
For an unresolved source at the phase tracking center, all terms in the exponent of are exactly zero. in this case would be proportional to the flux density of the source.
Assuming that the antenna dependent complex gains are independent, with a gaussian probability density function (this implies that the real and imaginary parts are independently gaussian random processes), one can estimate s by minimizing, with respect to s, the function given by
Dividing the above equation by (the source model, which is presumed to be known - it is trivially known for an unresolved source), and writing , we get
Expanding Equation D.5, we get
This can also be derived by equating the partial derivatives of with respect to real and imaginary parts of as shown in Section D.3.
Since the antenna dependent complex gains also appear on the right-hand side of Equation D.8, it has to be solved iteratively starting with some initial guess for s or initializing them all to 1.
Equation D.8 can be written in the iterative form as:
(the change in from one iteration to another) where is the tolerance limit.
is a product of two complex numbers, namely and , which we wish to determine. is itself derived from the measured quantity . Numerically speaking, each term in the summation of the numerator of Equation D.8 will involve (via ) and the multiplication of with would give an effective weight of . 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 over all correlations with antenna .
In the very first iteration, when , the normalization would be incorrect since the numeric value of , as it appears inside would be different from that used in the denominator of Equation D.8. However, as the estimates of s improve with iterations, the equation would progressively approach a true weighted average equation. The speed of convergence will depend upon the value of and the convergence would be defined by the constraint in Equation D.10. In the ideal case when the true value of all s is known, right hand side of Equation D.8 also reduces of .
Estimating 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, would be present in N-1 measurements (all the ) and the best estimate of would be the weighted average of all these measurements. Proper weight for , buried in each of the products , can be found heuristically as follows. , estimated from the measurements of a given baseline, must obviously be weighted by the signal-to-noise ratio on that baseline. This is in the above equations. It must also be weighted by the amplitude gain of the other antenna making the baseline, namely , to account for variation in antenna sensitivities and . The total weight for would then be , the sum of which appears in the denominator of Equation D.8. Knowing that ideally , each of the must be multiplied by (to apply the the above mentioned weights to ), before being summed for all values of and normalized by the sum of weights to form the weighted average of . One thus arrives at Equation D.8 using these heuristic arguments.
All contributions to , which cannot be factored into antenna dependent gains, will result in the reduction of . remaining constant, this will be indistinguishable from an increase in the effective system temperature. Since majority of later processing of interferometry data for mapping (primary calibration, bandpass calibration, SelfCal, etc.) is done by treating the visibility as a product of two antenna based numbers, this is the effective system temperature which will determine the noise in the final map (though, as a final step in the mapping process, baseline based calibration can possibly improve the noise in the map).
In the normal case of no significant baseline based terms ( ) in , the system temperature as measured by the above method will be equivalent to any other determination of .
can also be determined by recording interferometric data for a strong point source with and without an independent noise source of known temperature at each antenna. In this case
Expanding Equation D.5, ignoring s and writing it in terms of real and imaginary parts we get