The phase correction

The atmospheric phase affecting the observations is given by:

$$\phi(t) = \frac{2 \pi}{\lambda} l(t)$$

Ideally one would like use $T_{\rm EM}$ measured every second for each antenna, to compute the corresponding $\phi(t)$, and to correct the measured baseline phases.

Practically this is not feasible, since $\phi(t)$ amounts to many turns, and instrumental effects affect the measured $T_{\rm EM}$. The receiver gains, the forward efficiencies vary with the source elevation. So when the antennas are moved in elevation by more that a few degrees, like when switching between observed sources, these effects spoil the measured $T_{\rm EM}$ and prevent the use of the derived pathlength values.

Instead we use a differential procedure: once the antennas track a given source, one calibrates the atmosphere to calculate $\mbox{$T_{\rm EM}$}(t_0)$, $l(t_0)$, and ${\partial l}/{\partial \mbox{$T_{\rm EM}$}}(t_0)$.

The relative change in total power is:

$$\Delta \mbox{$T_{\rm EM}$}(t) = \mbox{$T_{\rm SYS}$}\frac{\Delta P(t)}{P} + \Delta\mbox{$T_{\rm LOSS}$}(t)$$

The phase correction applied is then:

$$\Delta \phi(t) = \frac{2 \pi}{\lambda} \frac{\partial l}{\pa... ...}\frac{\Delta P(t)}{P} + \Delta\mbox{$T_{\rm LOSS}$}(t) \right)$$

which we may rewrite as

$$\Delta \phi(t) = \frac{2 \pi}{\lambda} \frac{\partial l}{\pa... ...m SYS}$}(t_0)}{P(t_0)}\left(P(t)-\mbox{$P_{\rm REF}(t)$}\right)$$

The choice of $P_{\rm REF}(t)$ will be made in order to include as much as possible all the slow effects that contribute to $\Delta\mbox{$T_{\rm LOSS}$}(t)$. It is not a problem if long term atmospheric effects are also included in $P_{\rm REF}(t)$ ; these effects will not be removed by the radiometric phase correction, but by the traditional phase referencing on a nearby calibrator.

Several choices of $P_{\rm REF}(t)$ may be used:

1. Use for a given time interval (e.g. a scan of $1-4$ min. duration) the average of $P(t)$ in the same interval. This ``minimal'' choice has the advantage of correcting the amplitude of the decorrelation effect as much as possible (of course decorrelation effects occurring inside an elementary sampling interval, one second, will not be corrected; but this may usually be neglected). The average phase should not be affected. This is the scheme we plan to apply in quasi-real time in the correlators, to the spectral line data.
2. Use, for a longer time scale (e.g. the on-source time between two observations of the phase calibrator), a linear fit to the $P(t)$ data as the $P_{\rm REF}(t)$ reference. This is basically the same choice, but additionnaly allows for a linear drift of $\Delta \mbox{$T_{\rm LOSS}$}$ during the observation. This choice is currently implemented in CLIC (see below).
3. Use a smooth curve approximation to $P(t)$ on a longer time scale (one or several hours). This approximation should be different for the source data and the phase calibrator data. This has not been really tried yet.
4. Use a realistic model of the system noise as a function of azimut and elevation, and other parameters: although some effects should be easily calibrated out (e.g. by monitoring the temperature of the receiver cabin), such a detailed knowledge of the ground noise properties of the PdB antennas is difficult to reach, and not yet available.