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
phase^{5.15} for a point source at
sky position be , where we have explicitly put
in a time dependence. Then the observed visibility on a baseline
would be

If it also the case that the ionospheric and instrumental gains are changing slowly, then they can be calibrated in the following manner. Suppose that close to the source of interest, there is a calibration source whose true visibility is known. Then one could intersperse observations of the target source with observations of the calibrator. For the calibrator, dividing the observed visibility by the (known) true visibility, one can measure the factor . This can then be applied as a correction to the visibilities of the target source. For slightly better corrections, one could interpolate in time between calibrator observations. This is the basis of what is sometimes called `ordinary' calibration. The calibrator source is usually an isolated point source, although this is not, strictly speaking, necessary. It is sufficient to know the true visibilities . Note that if the calibrators absolute flux is also known, then this calibration procedure will also calibrate the amplitude scale of the target source

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
errors^{5.18},
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
,
and
. If we call the true visibility phase
, the raw visibility phase
and the sum of the instrumental
and ionospheric phases , then we have

i.e. over any triangle of baselines the sum of the phases of the raw visibilities is the true source visibility. This is called

This is called

- Choose a suitable starting model for the brightness distribution. Compute the model visibilities.
- For this model, solve for the antenna gains, subject to the closure constraints.
- Apply these gain corrections to the visibility data, use the corrected data to make a fresh model of the brightness distribution.

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 hours^{5.20},
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 sources^{5.21}For 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
converge^{5.22}.

- ...
phase
^{5.15} - by which we mean the phase difference over what would have been obtained in the absence of the ionosphere
- ... beam
^{5.16} - i.e. we have set the factor to 1.
- ...
source
^{5.17} - provided, as we will discuss in more detail later, that the system temperature does not differ for the target source and the calibrator
- ...
errors
^{5.18} - which is often a good assumption for digital correlators
- ...
point
^{5.19} - Actually strictly speaking one means the signal to noise ratio over an interval for which the ionospheric phase can be assumed to be constant
- ... hours
^{5.20} - Or change in a predictable manner with changing azimuth and elevation of the antennas
- ... sources
^{5.21} - see Noordam, J. E.
& de Bruyn A. G., 1982, Nature
**299**, 597. - ...
converge
^{5.22} - See Subrahmanya, C. R., (in `Radio Astronomical Seeing', J. E. Baldwin & Wang Shouguan eds.) for more details