NDPPP has an option msin.autoweights that derives weights for cross-correlations from the autocorrelations of the corresponding antennas. The equations are based on Gaussian noise even for RFI. These errors should be correct in the low S/N regime, when the (normalised) correlation coefficients are < < 1. There is another contribution from the cross-correlation (wave noise of the correlated signal) that is not taken into account by NDPPP. This only affects very bright sources (of the order of the SEFD).

The weights are scaled such that the real and imaginary parts of V*sqrt(weight) both have an rms scatter of approximately 1/sqrt(2), and the mean |V|^2*weight is approximately unity.

Here are test results from the data set L2012_50631 (3C196), using only subband 016. [The plots and numbers were produced with astro_wm/lofar_obs_wm/test_weights.py, in particular with the version in astro_wm/lofar_obs_wm/L2012_50631/test_weights/.] I only show plots for (all) baselines to FR606LBA.

data normalised with noise rms:

First: Using unflagged data. The formal rms for real and imaginary parts are 0.7348361 and 7.348214e-01, mean |V|^2*weight is 1.079947. The Gaussian curves have 0.7283392,0.7283548,1.060979, derived from mean |V|*sqrt(weight). Second: Using flagged data. The corresponding numbers are 0.7212914,7.213106,1.040550 and 0.7222662,0.7222830,1.043361. [ Horizontal axis is value in units of rms, vertical is the histogram. ]

We see that after normalisation the data have a nearly Gaussian distribution. Flagging cuts a bit too much of the wings, but this is expected.

data not normalised with noise rms:

First: unflagged, second: flagged

The unnormalised data are obviously not Gaussian distributed. This is a result of varying weights.

Conclusions: The weights derived from autocorrelations are (at least in this case) very reliable. They are normalised in such a way that the weighted variance is very close to unity, about 1.04. The convention used in e.g. difmap is different. The weights should be scaled by 2/1.04 to achieve a chi^2 of unity. With the current convention, we get about 1.04/2. This scaling has to be taken into account when calculating theoretical noise limits.

The 4 per cent excess is probably (partly) due to the wave noise of the correlated signal.