Representation of complex visibilities

• Background: For the storage of complex visibilities or triple products, the question arises whether to use the amplitude/phase or real/imaginary part representation.
• Pro amp/phase arguments: some interferometers measure only the phase of triple products, i.e. the closure phase; amplitude calibrations can be more uncertain than phase calibrations, in which case the error ellipses are more elongated along the complex vector.
• Pro real/imaginary arguments: in low SNR cases (for example measurements near the first Null), amplitude/phase representations are biased. This is discussed below.

High SNR case

Let's look at this question first for high SNR data. (The examples shown here use NPOI data.) Below are 200 ms coherent integrations of the visibility of FKV0803 (a calibrator) on the EW baseline (2002-09-15). A few color coded scans have been plotted in the complex plane. It can be seen that amplitude/phase representation would be the better choice here for the average complex visibility for each scan since the scatter is inside an arc/ellipse with the axes along and perpendicular to the average complex vector. The integration time would be chosen as a trade off between photon noise which decreases with time, and systematic errors wich increase with integration time. The width in phase is larger than the width in amplitude due to atmospheric phase noise. Amplitude calibration uncertainties would increase the axis of the ellipse along the average complex vector.

The two plots below are other examples of complex visibilities for the high SNR case (though not as high as shown in the example discussed above). This time though we show triple product data which only has instrumental/photon noise rather than also atmospheric phase noise. The data are of FKV0509 (a bright calibrator) observed on 2002-04-03, 200 ms averages, for a triple which has one of the baselines on a different spectrometer/detector. Several color coded scans are shown on the left and right. One can see that the noise distribution is more or less circular, in which case the amp/phase and real/imag. representations are equivalent. (Note that the amplitudes are not normalized.)

FKV0509 triple product (3 scans)	FKV0509 triple product (3 more scans)

Comparison to low SNR case

What about low SNR data? In this case, the distribution of the complex visibility samples in the complex plane used for an average overlaps the origin. We show below data taken on FKV0752 (large angular diameter star) with NPOI in channel 8 (680 nm) with the station/baseline triple CEW. The total integration time for each data point is 1 second, but made up of 500 2ms samples of the complex triple product below in the left column of plots, and made up of 10 samples of coherently integrated visibilities with an integration time of 50 ms below in the right column. (The delay and fringe phase estimates for the low-SNR EW baseline were derived from the differences of these quantities between the high-SNR CW and CE baselines. This is the phase bootstrapping technique.)

When comparing the scatter of the samples from which an average complex triple product would be derived, i.e. comparing the lower SNR case in the left column to the higher SNR case in the right column, it can be seen that in the former case, biases in the phase and amplitude distributions are present. This means that assigning uncertainty estimates for the complex triple product in the amp/phase representation would be impossible.

Vector averages of 500 2ms samples.	Average of 10 50ms coherent integrations
Real/imag. scatter plot
Closure phase plot (in radians)
Triple product amplitude plot (not normalized)

Recommendation

We recommend using the amplitude/phase representation for the complex visibilities, including the error of the visibility amplitude, but instead of the phase error use the phase error times the visibility amplitude. This quantity is equal to the width of the error ellipse orthogonal to the visibility vector. In the high signal-to-noise case, dividing this width by the visibility amplitude gives the phase error. In the low signal-to-noise case, one would not divide by the amplitude but rather use this width and the amplitude error as the errors of X and Y (if the errors are not correlated, i.e. if the error ellipse is circular.)

The alternative would have been to allow both amp/phase and X/Y representations, but our proposal would merge the two and still be able to handle all conceivable cases.