Tpower = Tsys + Trms + Telev * cos(Elev) + Tatm * Phi -- (1)where Tsys is the receiver noise power, Trms represents the receiver gain instability and other random power fluctuations, Telev is the total power from ground radiation, and Tatm is the total power from atmospheric fluctuations. The receiver gain instabilities and atmospheric fluctuations are represented by Gaussian distributions. All units are expressed as antenna temperatures. Typical values at millimeter wavelengths are Tsys = 500 K, Trms = 0.05 K, (corresponding to a gain stability of 10-4 for Tsys=500 K), Telev = 50 K, and Tatm = 0.5 K/radian. The receiver noise power, Tsys, also generates additive Gaussian noise which corrupts the data. The additive noise fluctuations from the other, smaller, terms in equation 1 are not modeled. The atmospheric power fluctuations are modeled to be perfectly correlated with the atmospheric phase fluctuations, Phi.
For a point source there is no phase from source structure, so the interferometer phase is the phase difference between the antennas. In Figure 1 we plot the interferometer phase versus the total power difference for each baseline. Since all antennas are modeled to have the same elevation dependence, then the Telev term cancels perfectly. In the absence of receiver noise fluctuations, the correlation between phase and total power fluctuations is quite clear if Tatm * Phi > Trms, as shown in Figure 1. With added receiver noise fluctuations the correlation is still clear if Tatm * Phi > { Trms + Tsys/sqrt(BT) }, as shown in Figure 2. The correlation becomes poor if the receiver gain instabilities are larger than the atmospheric fluctuations. Figure 3 shows the situation for Tatm * Phi = Trms. This corresponds to a receiver gain stability of 10-3 for Tsys = 500 K. Clearly a receiver gain stability better than 10-3 is required to successfully predict phase fluctuations from total power measurements.
Figure 1 - 3
If we work with total power differences, then there is a correlation (in the model assumption) with the measured interferometer phases which are phase differences between antennas. The correlation is assumed to independent of baseline length, so we can reduce the number of equations by computing both the phase and the total power differences with respect to the same antenna. The gain and phase instabilities in this antenna are reflected in all the data, so that a stable antenna should be selected as the reference antenna. This method provides us with an algorithm for correcting the data. The first estimate of antenna gains is made from the measured total power differences. This can be followed by self-calibration if the signal to noise is sufficient.
The quality of the gain estimate depends upon the relative values of the atmospheric fluctuations, and the total power fluctuations due to gain instability, ground radiation etc. In the model calculation the elevation dependence is the same for all antennas and cancels perfectly, so that Trms includes variations in the total power from ground radiation received at each antenna. For a gain stability of 10-4 (Trms=0.05 and Tsys=500), and 40 degrees of phase noise, then the total power fluctuations are well correlated with atmospheric phase fluctuations and the data is corrected by applying the gains derived form total power differences. The rms noise was reduced from 0.047 to 0.019 Jy, close to the value without any phase noise. For a gain stability of 10-3 (Trms=0.5) the derived gains have large errors due to the gain instability and the maps degrade to an rms=0.063 Jy when the gains are applied to the data. As an intermediate case, for a gain stability of 4 x 10-4 (Trms=0.2, Tsys=500), and 40 degrees of phase noise, the rms is improved from 0.047 to 0.027 after applying the gains derived from the total powers.