3.8.1 Overview

Correction of the $uv$ data for instrumental and atmospheric effects is done using observations of calibrator sources, usually scheduled by the operators. Following Fomalont & Perley, p. 79 in Taylor et al. (1999) we may express the relationship between the observed visibilities $\tilde{V}$ and the true visibilities $V$ for the baseline between antennas $i$ and $j$ as

$$\tilde{V}_{ij} = G_i G^{\ast}_j G_{ij} V_{ij}$$ (3.6)

Calibration is the determination of the complex quantities $G$ which are functions of both time and frequency. The subscripts $i$ and $j$ refer to the two antennas which form the baseline and $G_i$ is the complex gain of an individual antenna. The complex gains are best thought of in terms of their modulus and argument, known as the telescope amplitudes and phases, since variations in amplitude and phase have different physical origins and timescales. Variations in telescope phases are mainly caused by variations in the refractive index of the atmosphere and ionosphere along the path to the source. Timescales are often as short as a few minutes or less. Variations in telescope amplitude are mainly due to slow drifts in receiver sensitivity and have a typical timescale of hours. In addition there are predictable variations in amplitude with elevation (see §3.8.3). The term $G_{ij}$ in Eq. 3.6 is a small (typically within 1% of unity) baseline-dependent gain (e.g. an offset introduced in the correlator).

For an array of $N$ antennas there are $N(N-1)/2$ equations which may be used with model values of $V$ to solve for the $N$ complex telescope gains. For an unresolved i.e. point source, $V_{ij} = S$ for all baselines where $S$ is the flux density. Hence, observations of a point source may be used to determine the telescope gains and correlator offsets. Resolved sources may also be used, in which case an initial estimate of $V_{i,j}$ is derived by Fourier transforming the source visibilities to make a crude map. The errors in these estimates are usually factorized into errors associated with individual antennas, making this a robust technique for the determination of telescope gains: this is the basis of self-calibration (see § 3.9.3). In contrast the baseline-dependent correlator gains are so small (rarely above 0.5% in amplitude) and there is no redundancy in eq. 3.6 for the determination of the correlator gains, these are almost always determined using a point source.