Fringe-fitting

Frequency ranges strongly affected by RFI are flagged (not included) before anything else. These ranges are chosen manually.

First we bandpass-correct the phases. For the bandpass we use either the result from a previous iteration, or we vector-average per baseline all times. The mean is subtracted from the bandpass phases to avoid absorbing phase calibration into bandpass calibration. This mean phase is determined from the vector-average of the bandpass over all good channels.

Per time block (e.g. individual 1-sec integrations, 10 sec or 60 sec blocks) are then fitted per baseline and polarisation. The model always includes a phase for the reference frequency freq0 and a (non-dispersive) group delay across the band. We can also add a curvature (sum of dispersive and non-dispersive delay with zero mean slope) and a delay-rate for variations with time.

For a non-dispersive delay, the absolute delay and group delay are equal with equal signs. For a dispersive delay the signs are the opposite. The phase is given by the absolute delay multiplied by 2*pi*freq. With this we can get the following equations:

nondisp = group + curv
disp    = curv
group   = nondisp - disp
phi0    = (nondisp + disp) * 2pi*freq0
        = (group + 2*curv) * 2pi*freq0
disp    = ( phi0/(2pi*freq0)-group ) / 2
nondisp = ( phi0/(2pi*freq0)+group ) / 2

Besides explicitly fitting the curvature to distinguish between dispersive and non-dispersive delays, we can thus combine the group delay with the phase to achieve the same result, actually with higher accuracy. Note that we can have arbitrary constants per baseline in each of the parameters (with the exception of phase), because they can be partly absorbed into the bandpass. Before applying the equations one should thus generally subtract the mean.

As a test for this concept we use the dataset (see the list) 1603186576, which is one hour on PKS 1934-638 in 4k mode in band S3. We fringe-fit over 60-sec blocks with curvature and delay rate. As example we use the baseline m045-m060, because it has the strongest dispersive delay variations.

Let us first plot phase against group delay (both mean-subtracted) for all data blocks:

We see almost a 1:1 correspondence, which means that most of the delay (variation) is non-dispersive. However, there are deviations, which may be due to the ionosphere.

Now we have a look at the two types of delays determined in the two possible ways:

Phase and group delay are fitted directly, and they are very similar as described above. For the curvature we have the direct fitting result (dotted) and the one from phase and group delay (solid). The two versions are consistent with each other, but the latter is much more accurate and therefore much smoother. In the bottom panel we combine the group delay and phase in two ways to determine the dispersive and non-dispersive delays indirectly without using the explicit curvature fit. The dispersive delay is very smooth, the non-dispersive delay shows real fluctuations.

In conclusion we can say that the non-dispersive delay dominates and that we do not need to fit dispersive delay or curvature explicitly, but can determine it much better from the group delay and the phase.

Here we show this for the same data set, this time over 10-sec blocks and without including curvature:

The dispersive delay stays smooth, while the non-dispersive delay shows even more (real) variations.

All these solutions can be reduced to station-based ones. For the phases, one has to be careful to use consistent total turns. We do this by first separating the complex value, then we use unwrapped functions for the stations and correct them by full turns to become consistent with the unwrapped functions on the baselines.

We absorb time-averaged closure errors into the baseline-based bandpasses before trying to separate them into station-based functions. See Bandpass part. It turns out that remaining closure errors (as small as they are) can be minimised even more by excluding short baselines. For PKS 1934-63 a baseline limit (measured in antenna XY coordinates or UV) of a few hundred metres works well and keeps a sufficient number of baselines for the fits.

As basis for this we use the station-based solutions, because of their better SNR. Whenever needed, we can re-derive the baseline-based values from them.

All effects are largest on the longest baselines and thus on the outer stations. Here we show solutions for non-dispersive and dispersive delays for four stations representing the corners of the array. Both polarisations are combined, because they are consistent with each other.

non-dispersive delays (troposphere+instrumental):

dispersive delays (ionosphere):

In the dispersive delays we notice that the large fluctuations in the first half are very similar but with opposite signs in the left and right panel. The signature of the second half has opposite signs in the top and bottom plots and is weaker in the bottom. We interpret this as follows: The ionosphere causes mostly large-scale gradients over the entire array. In the first half the direction of this gradient is east-west, in the second half it is more north-south. This is consistent with a large scale of ionospheric variations and with a high speed at which they cross the array. This means we do not see individual effects at each station, but are really dominated by one gradient at each time.

We do not see a similar behaviour for the non-dispersive delays. Their fluctuations are much faster and we know that they cannot cross the array faster than the wind speed. This means that we really see individual signatures and not just a large-scale gradient.

We can test the gradient-hypothesis by fitting such a gradient (independent for each time) to the data and check how well it can reproduce the observed signal. We can do this per station or per baseline. The former has the advantage that we have to check less plots.

Here are results for the antenna-based dispersive delay:

We see that the gradient fit (green) fits the observations (red) almost exactly for all stations. For the innermost stations (left) the noise dominates the signal, but the systematic variations still agree very well. This means the ionosphere really acts mostly as a large-scale time-varying gradient.

Here is the result for the non-dispersive delay:

For the outer stations (right) the agreement is still pretty good, but this is because we have only a few outer stations, which thus have a strong lever arm and dominate the solution. In contrast to the dispersive delay, this solutions does not fit the inner stations (left) well. Fluctuations there are dominated by small-scale and more local variations.

Here are PDF files of all these solutions for completeness:

gradient fit for antenna-based dispersive delay
gradient fit for antenna-based non-dispersive delay
gradient fit for baseline-based dispersive delay
gradient fit for baseline-based non-dispersive delay