This page was provided by Christian Hummel, Tom Pauls, and Dave
Mozurkewich, and is a contribution to an ongoing discussion on the
oi-data mailing list.
# 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.

Christian Hummel, Tom Pauls, David Mozurkewich.