Signal decorrelation

\ensuremath{J_\ensuremath{\mathrm{ant}}^\ensuremath{\mathrm{sd}}} characterizes the antenna hardware, i.e. it assumes perfect atmospheric conditions or the use of autocorrelations, as in single-dish measurements. In interferometric mode, the phase of the turbulent atmosphere above each antenna of a given baseline has a random part that causes an additional “attenuation” of the amplitude of the correlation. A point source of 1 Jy flux will appear as a source of \ensuremath{\eta_\ensuremath{\mathrm{atm}}} Jy flux if we only use the \ensuremath{J_\ensuremath{\mathrm{ant}}^\ensuremath{\mathrm{sd}}} factor. This is called atmospheric decorrelation and it depends on the weather during the observations.

However, the last calibration step of interferometric data is to measure a point source of known flux to deduce the actual conversion factor, \ensuremath{J_\ensuremath{\mathrm{ant}}^\ensuremath{\mathrm{int}}}, taking into account the atmospheric decorrelation that happens during the observations. By definition of \ensuremath{\eta_\ensuremath{\mathrm{atm}}}, we yield

  $\displaystyle \ensuremath{J_\ensuremath{\mathrm{ant}}^\ensuremath{\mathrm{sd}}}...
...rm{atm}}}\,\ensuremath{J_\ensuremath{\mathrm{ant}}^\ensuremath{\mathrm{int}}}.
$ (7)
It can be shown that \ensuremath{\eta_\ensuremath{\mathrm{atm}}} is related to the atmospheric rms phase noise ( \ensuremath{\phi_\ensuremath{\mathrm{rms}}}) through
  $\displaystyle \ensuremath{\eta_\ensuremath{\mathrm{atm}}}{} = e^{-\frac{\ensuremath{\phi_\ensuremath{\mathrm{rms}}}^2}{2}}.
$ (8)