We have assumed till now that we have been working with calibrated visibilities, i.e. free from all instrumental effects (apart from some additive noise component). In reality, the correlator output is different from the true astronomical visibility for a variety of reasons, to do with both instrumental effects as well as propagation effects in the earth's atmosphere and ionosphere.
At low frequencies, it is the effect of the ionosphere that is most dominant. As is discussed in more detail in Chapter 16, density irregularities cause phase irregularities in the wavefront of the incoming radio waves. One would expect therefore that the image of the source would be distorted in the same way that atmospheric turbulence (`seeing') distorts stellar images at optical wavelengths. To first order this is true, but for the ionosphere the `seeing disk' is generally smaller than the diffraction limit of typical interferometers. There are two other effects however which are more troublesome. The first is `scintillation', where because of diffractive effects the flux density of the source changes rapidly - the flux density modulation could approach 100%. The other is that slowly varying, large scale refractive index gradients cause the apparent source position to wander. At low frequencies, the source position could often wander by several arc minutes, i.e. considerably more than the synthesized beam. As we shall see below, provided the time scale of this wander is slow enough, it can be corrected for.
Let us take the case where the effect of the ionosphere is simply
to produce an excess path length, i.e. for an antenna let the excess
phase5.15 for a point source at
sky position be , where we have explicitly put
in a time dependence. Then the observed visibility on a baseline
In the approach outlined above, in order to calibrate the
data one needs to solve for an unknown complex number per baseline,
(i.e. N(N-1)/2 complex numbers for an N element interferometer). If we
assume that the correlator itself does not produce any
i.e. that all the instrumental errors occur in the antennas or the
transmission lines, then the instrumental gain can be written out as
antenna based terms, i.e.
However to appreciate the real power of this decomposition into
antenna based gains, consider the following quantities. First let us
look at the sum of the phases of the raw visibilities
. If we call the true visibility phase
, the raw visibility phase
and the sum of the instrumental
and ionospheric phases , then we have
Note that if one adds a phase to each antenna (where are arbitrary and are the (u,v) co-ordinates of the th antenna), the phase closure constraints (eqn 5.2.17) continue to be satisfied. That means that in self calibration the phases can be solved only upto a constant phase gradient across the uv plane, i.e. the absolute source position is lost. Similarly, it is easy to see that the amplitude closure constraints will be satisfied even if one multiplies all the gains by a constant number, i.e. in self calibration one loses information on the absolute source flux density . The only way to determine the absolute source flux density is to look at a calibrator of known flux. Since antenna gains and system temperatures are usually stable over several hours5.20, it is usually sufficient to do this calibration only once during an observing run. A more serious problem at low frequencies is that the Galactic background (whose strength varies with location on the sky) makes a significant contribution to the system temperature. Hence, when attempting to measure the source flux density, it is important to correct for the fact that the system temperature is different for the calibrator source as compared to the target source. The system temperature can typically be measured on rapid time scales by injecting a noise source of known strength at the front end amplifier.
Another related way (to selfcal) of solving for the system gains is the following. Suppose that the visibility on baselines and are identical. Then the ratio of the measured visibilities is directly related to the ratio of the complex instrumental gains of antennas . If there are enough number of such `redundant' baselines, one could imagine solving for the instrumental gains. Some arrays, like the WSRT have equispaced antennas, giving rise to a very large number of redundant baselines, and this technique has been successfuly used to calibrate complex sources5.21For a simple source, like a point source, all possible baselines are redundant, and this technique reduces essentially to self-calibration.
At the very lowest frequencies ( MHz, roughly for the GMRT) the assumption that the source lies within the iso-planatic patch probably begins to break down. The simple self calibration scheme outlined above will stop working in that regime. A possible solution then, is to solve (roughly speaking) for the phase changes produced by each iso-planatic patch. Often the primary beams of several antennas will pass through the same iso-planatic patches, so the extra number of degrees of freedom introduced will not be substantial, and an iterative approach to solving for the unknowns will probably converge5.22.