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====== 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. ^ parameter ^ meaning ^ ^ phi0 | phase at freq0 | ^ curv | curvature delay | ^ group | group delay | ^ nondisp | total non-dispersive delay (clocks, atmosphere) | ^ disp | total dispersive delay (ionosphere), measured at freq0 | 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: <code> 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 </code> 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 [[projects:mkat_sband:pub:imobs_list|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: \\ {{:projects:mkat_sband:pub:phase_vs_delay.png?direct&400}} 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:\\ {{:projects:mkat_sband:pub:delays_direct_indirect.png?direct&500}} 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: \\ {{:projects:mkat_sband:pub:delays_indirect.png?direct&500}} 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 [[projects:mkat_sband:pub:bandpass|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 in order to be time-independent, not in UV) of 800 metres works well and keeps a sufficient number of baselines for the fits.