10.4.2 Estimate of the atmospheric decorrelation factor f

Details about the origin of f are given in M.Bremer lecture 9. I will discuss here the practical implementation of the atmospheric phase correction done in real-time and in CLIC. More details are given in the IRAM report ``Practical implementation of the atmospheric phase correction for the PdBI'' by R.Lucas.

The atmospheric phase fluctuations are due to different time varying water vapor content in the line-of-sight of each antenna through the atmosphere. Between antenna i and j, this introduces a decorrelation factor $f\sim e^{-(\phi_i-\phi_j)^2/2}$ on the visibility V_ij. This term, non-linear, cannot be factorized by antenna. Moreover due to the physical properties of the atmosphere, there are several timescales. One can correct partially some, but not all, of them.

At Bure the basic integration time is 1 second and the scan duration is usually 60 seconds. The radiometric correction works then on timescales of a few seconds to one minute. It corrects only the amplitude: the phase is never changed because phase jumps between individual scans are dominated by instrumental limitations (mainly the receiver stability on a few minutes + ground pickup variations). The implications on the image quality are developed in the lecture by S.Guilloteau 16. Longer atmospheric timescales of about 2-8 hours are removed by the spline functions fitted inside the phase and the amplitude.

Intermediate timescales fluctuations from about one minute (the scan duration) to 1 hour are not removed. The resulting rms phase are measured by the fit of the splines in the phase. These timescales are not suppressed by the radiometric correction, and they contribute to the decorrelation factor f (see Eq.10.16). as the main component.

10.4.2.0.1 The method

The differences in water vapor content are measurable by monitoring the variations of the sky emissivity T_sky. A monitoring of the total power in front of each antenna will then lead to a monitoring of the phase fluctuations. At Bure, we monitor the total power P with the 1.3mm receivers (note that P is also called M_atm in the first part of this lecture). The variation of T_sky, $\Delta T_{sky}$ (equal to $\Delta T_{emi}$) is linked to the total power by

$$\frac{\Delta P}{P} = \frac{(\Delta T_{emi} + \Delta T_{loss})}{T_{sys}}$$ (10.18)

The monitoring of the atmospheric phase fluctuation works only when $\Delta T_{loss}$ due to the instrumentation is negligible on the time scales at which the phase correction is calculated and applied (typically a few seconds to one minute). Slow drifts on scale of hours have no effects.

With standard atmospheric conditions and following [Thompson et al 1986] (their Eq.13.20), the variation of the path length through the atmosphere at zenith is approximated by:

$$\Delta L= 6.3 \delta w$$ (10.19)

were $\delta w$ is the variation of water vapor content. $\Delta L$ is related to the phase fluctuation $\psi_i$ above the antenna i by

$$\psi_{i}(t) = \frac{2\pi}{\lambda}\Delta L(t)$$ (10.20)

For example, under standard conditions (see fig.10.1 or 10.2), a variation $\delta w =0.1$ mm corresponds to $\Delta L \simeq 630$ $\mu$m, $\Delta T_{sky}\simeq 1.5$ K and $\psi_i \simeq 250^o$ at 1.3mm. This value is enormous and would not allow to produce images of good quality.

To reduce the phase fluctuation to a reasonable value having a negligible impact on the image quality e.g. $\psi_i\sim 25^o$, one needs to get $\Delta T_{loss}+ \Delta T_{sky}\sim 0.15$ K corresponding to a global path length variation of $\sim 60 \mu m$. For a typical $T_{sys} \sim 150$ K (DSB in the antenna plane, not SSB outside the atmosphere as for astronomical use), the instrumental stability required ( $\Delta T_{loss} / T_{sys}$) must then be of order of $\sim 5 \cdot 10^{-4}$.

At Bure, on time scales of a few minutes, $\Delta T_{loss}$ is dominated by the stability of the receivers which must be carefully tuned to get the best stability. The 1.3mm receivers are systematically tuned to get a stability of a few 10^-4; the stability is checked by doing autocorrelations of 60 seconds on the hot load. Achieving the required stability may prove impossible at some frequencies.

10.4.2.0.2 Practical implementation

Ideally one would like to use T_emi measured each second on each antenna to compute $\psi_i(t)$ and correct the measured baseline phases. Practically, it is not so simple because $\psi_i(t)$ can do many turns and instrumental effects affect the measured T_emi.

Instead we use a differential procedure: once the antenna tracks a given source, one calibrates the atmosphere to calculate T_sys(t₀), $\Delta L(t_0)$ and $\Delta L/d T_{sky}(t_0)$. Phase corrections are then referenced to t₀.

$$\Delta\psi_i = \frac{2\pi}{\lambda}\frac{d\Delta L}{d T_{emi}}\frac {T_{sys}(t_0)}{P(t_0)}(P(t)-P(\mathrm{Ref}))$$ (10.21)

where P(Ref) is chosen in order to minimize as much as possible all the slow effects contributing to $\Delta T_{loss}$. A long term atmospheric effect can also be included in P(Ref) because these effects are not removed by the radiometric phase correction but by the traditional phase referencing on a nearby calibrator. The main steps are the following:

1.

The total power P is continuously monitored on calibrators and on sources (each one second).

2.

Using the standard calibration method (see first part of the lecture) P and T_emi (measured each second) are used to compute T_sky and w.

3.

The atmospheric model has also been upgraded to compute the path length $\Delta L$ and its derivative $d \Delta L/d T_{emi}$. $\Delta L$ is computed by integrating the refractive index of the wet air along the line of sight across the atmosphere.

4.

Inside the 60 seconds scan, the new phase (Eq.10.21) is computed and the correction applied to the amplitude.

10.4.2.0.3 Quasi-real Time Calibration

For the quasi-real time correction,

• the default value for P(Ref) is the measured atmospheric emission at the time of the last calibration, i.e. P(Ref)=P(t₀).

10.4.2.0.4 Calibrating using CLIC

The CLIC command ``MONITOR delta-time'' allows to re-compute all the parameters. This command is useful when you want to select a better value for P(Ref).

• This command is used to prepare the atmospheric radiometric phase correction. It processes the calibration scans to compute the correction factors (i.e. the change of path length for a given change in emission temperature of the atmosphere at the atmospheric monitor frequency (normally 1.3mm)).

• The scans in the current index are grouped in intervals of maximum duration delta-time (in seconds); source changes will also be used to separate intervals. In each interval a straight line is fitted in the variation of atmospheric emission as a function of time; this line will be the reference value for the atmospheric correction, i.e. the correction at time t is proportional to the difference between the atmospheric emission at time t and the reference at time t. This scheme is used to avoid contaminating the correction with total power drifts of non-atmospheric origin (changes in receiver noise and gain, and changes in ground noise).

• MONITOR 0 will use for each scan the average of the atmospheric emission as the reference value (i.e. P(Ref)=<P(t)>_scan). This will cause the correction to average to zero in one scan: the average phase is not changed, only the coherence is restored leading to an improved amplitude.

The automatic calibration procedure uses the command MONITOR 0.

  

Figure 10.4: The amplitude and phase versus time on baselines B12 and B13 with (black) and without (red and blue) the radiometric phase correction. The phase remains unchanged but the amplitude is significantly improved.