9.3 Phase calibration

We now turn to the more difficult problem of correcting for the time dependence of the complex gains, contained in the functions $\mbox{\ensuremath{g_\mathrm{\scriptscriptstyle C}}}_{i}(t)$. Variations occur in both amplitude and phase. Let us summarize the various effects to be calibrated:

• Interferometer geometry:
  Small errors in the baseline determination will cause slow phase drifts (period 24 hours); unfortunately these errors are dependent on the source direction so they cannot be properly calibrated out by phase referencing on a calibrator, only reduced by a factor of the order of the source to calibrator distance expressed in radians.

• Atmosphere
  The atmosphere introduces phase fluctuations on time scales 1s to a few hours, depending on baseline length and atmospheric conditions (see Chapter 11). The effect of fluctuations on short time scales is to cause loss of amplitude by decorrelation, while on the long term the phase fluctuations can be mistaken for structure in the source itself. The critical time there is the time it takes for the projected baseline vector to move by half the diameter of one antenna:

  $$\Delta t_C = \frac{\cal D}{2} \frac{86400}{2 \pi b} = 224 \frac{\cal D}{15} \frac{450}{b}$$ (9.21)

  
  
  This time ranges from 4 minutes to 1 hour for Plateau de Bure, depending on the baseline length. The phase fluctuations on short time scales may be corrected by applying the radiometric phase correction method, if it is available.

• Antennas
  The antennas may cause variations in amplitude gains due to degradation in the pointing and in the focusing due to thermal deformations in the antennas. These can be corrected to first order by amplitude calibration but it is much better to keep the errors low by proper frequent monitoring of the pointing and focus, since these errors affect differently sources at the center and at the edges of the field. Focus errors also cause strong phase errors due to the additional path length which is twice the sub reflector motion in the axial direction.

• Electronics
  Phase and amplitude drifts in the electronics are kept low by efficient design (see Chapter 5 and Chapter 7). The electronics phase drifts are generally slow and of low amplitude, except for hardware problems.

A detailed analysis can be found in [Lay 1997].

9.3.1 Phase referencing by a nearby point source

This is the standard, traditional way to calibrate the phases with current interferometers. A point source calibrator is typically observed for $\ensuremath{T_\mathrm{\scriptscriptstyle I}}$ (a few minutes) every $\ensuremath{T_\mathrm{\scriptscriptstyle C}}$ (20-30 minutes). One fits a gain curve to the data observed on the calibrator, this gain curve is an estimate of the actual gain curve $\mbox{\ensuremath{g_\mathrm{\scriptscriptstyle C}}}_i(t)$. This enables removing most long-term phase drifts from the observation of the target source.

9.3.1.0.1 Decorrelation

It can be shown that the decorrelation factor for a given baseline is approximated by:

$$f \sim 1-\frac{1}{2}\int_{-log{2.5 \ensuremath{T_\mathrm{\scriptscriptstyle I}}}}^\infty{\nu P_\phi(\nu) d(\log{\nu})}$$ (9.22)

The decorrelation is fundamentally a baseline-based quantity: it cannot generally be expressed as a product of antenna-based factors. Both the target source and the calibrator are affected, so amplitude referencing will correct for decorrelation. However, the amount of decorrelation will vary from an integration to the next, so that the amplitude uncertainty is increased.

9.3.1.0.2 Phase referencing

The slow component of $\mbox{\ensuremath{g_\mathrm{\scriptscriptstyle C}}}_i(t)$ to be calibrated out is sampled at intervals $\ensuremath{T_\mathrm{\scriptscriptstyle C}}$ so that only variations with periods longer than $2\ensuremath{T_\mathrm{\scriptscriptstyle C}}$ can be followed. However one fits a slow component into the data points so one is sensitive to errors due to the presence of the fast component: the fast component is aliased into a slow component. It is essential to fit a curve that does not go through the points.

9.3.1.0.3 Fast phase referencing

One may reduce $\ensuremath{T_\mathrm{\scriptscriptstyle C}}$ as much as possible to remove a larger part of the atmospheric fluctuation spectrum. Time scales of the order of 10s may be used, at the expense of:

• the time efficiency is decreased (relatively more time is spent on the calibrator resulting in a larger overhead)
• caution must be taken that the time $T_\mathrm{\scriptscriptstyle C}$ may become comparable to the time it takes to water eddies to drift along the apparent distance between source and calibrator.

9.3.1.0.4 Water vapor radiometry

A radiometry system may be used to monitor the emission of water vapor at a suitable frequency in front of all antennas (dedicated instrument or astronomy receiver; water line or quasi continuum). The fluctuations in the path length can be of a few % of the total path length due to water vapor.

The fluctuations of the water emission are converted into fluctuations of path length by using an atmospheric model (see lecture by M. Bremer, Chapter 11). In principle one could hope to correct for all phase fluctuations this way. However limitations due to receiver stability, to variations in emission from the ground, and to uncertainty in the determination of the emission to path length conversion factor have the consequence that it is not yet possible to consistently correct for the variation of path length between the source and the calibrator.

So far this method at IRAM/Plateau de Bure is used only to correct for on-source fluctuations. Its main effect is to remove the decorrelation effect due to short-term phase fluctuations, improving the precision of amplitude determination.

9.3.2 Phase referencing by a point source in the primary beam

We now consider the simple case where the field contains a strong point source: it can be a continuum source (quasar) or a line source (maser). In that case all phase fluctuations with period longer than $\sim 2 \ensuremath{T_\mathrm{\scriptscriptstyle I}}$ are removed, where $T_\mathrm{\scriptscriptstyle I}$ is the integration time. However statistical errors may be mistaken for true atmospheric phase fluctuations, causing additional decorrelation.

This method gives good results, but for very specific projects which can be observed in very poor atmospheric conditions (e.g. observation of radio emitting quasars, of stars with strong maser lines).

9.3.3 Phase referencing using another band or another frequency

It is generally easier to measure the path lengths fluctuations at a lower frequency (even though the phase scales like frequency), due to both better receiver sensitivity and larger flux of the referencing source. Moreover, in marginal weather conditions, if the rms phase fluctuations at 100 GHz is $\sim 40^\circ$, then at 230 GHz they are of $\sim 100^\circ$, and the phase becomes impossible to track directly due to $2 \pi$ ambiguities.

If two receivers are available simultaneously, one may subtract to the high frequency phase the phase measured at the low frequency. The atmospheric fluctuations are cancelled and only a slow instrumental drift remains. The gain curve at the high frequency is then determined as the sum of two terms: the low frequency gain curve (including the slow atmospheric terms) plus that slow instrumental drift (which represents any phase fluctuation affecting one of the signal paths of the two receivers).

This method is currently used at Plateau de Bure.