The most basic form of calibration that the user undertakes (there are many forms of calibration taken care of by the on-line system) is aimed at making corrections for changes in the visibilities induced by the atmosphere and electronics that the on-line system cannot deal with. Such changes may be on time scales from minutes to hours.
We determine the corrections by observing known sources. It is advantageous that their structure be as simple as possible. Ideally, we would like a strong unresolved source with a well known position. If such a source is observed at the phase centre, then regardless of the baseline, the visibility amplitude should be equal to the source flux density, and the visibility phase should be zero. The observed visibilities of such a source thus yield direct estimates of the desired corrections. The corrections that we determine are referred to as the complex (amplitude and phase) gains, and the calibration formula can be written simply
where t is the time of the observation, 
is the frequency, i
and j refer to a measurement associated with a pair of antennas
(i,j), V is the true visibility, 
is the observed
visibility, and G is the gain.
The visibilities and gains are complex quantities; they have an amplitude and a phase (or a real and imaginary part if you prefer to think that way). Each instantaneous visibility measures a Fourier component of the sky intensity distribution. The combination of these two-dimensional `corrugations' with the correct (as they should be after calibration) relative phases and amplitudes yields the sky intensity distribution convolved by the point response function of the synthesis array.
Since most of the data corruption occurs before correlation, we can
factor the baseline-based gain, 
into antenna
based gains, 
and 
so that the calibration equation becomes
If there are N antennas, then there are N complex antenna gains,
However, there are N(N-1)/2 baseline based gains, 
from which we must extract these N antenna gains. That is, we must
solve the set of equations
Because this set of equations remains true even if we subtract an
arbitrary phase from every 
, we can subtract the phase of one 
from all the 
, so that the phase of 
becomes zero. This is
known as the reference antenna. The number of real valued unknowns is
therefore 2N - 1. A non-linear least squares technique is usually
used to extract these from the N(N-1)/2 complex equations. In
practice, we deal with the amplitude and phase of the gains, rather than
the real and imaginary parts. This is because the primary effects of
the wave propagation are to rotate the phase and attenuate the
amplitude.