9.1 Definitions and formalism

As has been seen in the previous lectures, each interferometer baseline provides a measurement of the source visibility at a given point in the $u,v$ plane of spatial frequencies; the source brightness distribution can then be reconstructed by an appropriate Fourier Transform.

In reality things are not so simple. Interferometers are designed with a lot of care; however many electronic components will have variable gains both in amplitude and in phase; these variations will affect the results and have to be taken out. It is generally sometimes more efficient to have a slightly varying instrument response, and a more sensitive instrument, than a very stable one with less sensitivity, provided the varying terms in the response are slow and may be easily calibrated out. At millimeter wavelengths the atmospheric absorption and path length fluctuations will dominate the instrument imperfections in most cases.

For a given observation, if we interpret the correlator response (amplitude and phase) as the source visibility, ignoring any imperfections, we have an observed (apparent) visibility $\ensuremath{\widetilde{V}}_{ij}(t)$, where $i,j$ are antenna numbers, $\nu$ the frequency and $t$ is time. If the true source visibility is $\ensuremath{V}_{ij}(t)$, we may define :

$$\ensuremath{\widetilde{V}}_{ij}(t) = \ensuremath{\mathcal{G}}_{ij}(t) \ensuremath{V}_{ij}(t) + \epsilon_{ij}(t)+ \eta_{ij}(t)$$ (9.1)

where the $\ensuremath{\mathcal{G}}_{ij}(\nu,t)$ are the complex gains of each baseline. $\eta_{ij}(t)$ is a noise term resulting from thermal fluctuations in the receivers; $\epsilon_{ij}(t)$ is an offset term. This assumes that the system is linear. $\pi$ phase switching applied on the first local oscillators is a very efficient method of suppressing the offsets $\epsilon_{ij}(t)$; they are generally negligible and will not be considered any further.

9.1.1 Baseline based vs antenna based gains

Since amplitude and phase distortions have different physical origins it is generally useful to write

$$\ensuremath{\mathcal{G}}_{ij}(t) = g_i(t) g_j^*(t) = a_i(t) a_j(t) e^{i(\phi_i(t)-\phi_j(t))}$$ (9.2)

Here we have split the gains into antenna based factors. This is generally legitimate since the gains represent properties of the data acquisition chains which are in the analogue part of the system. The correlator itself is a digital machine and we assume it is perfectly working (including the clipping correction). This assumption is certainly valid when considering a single frequency and a single instant. When we start averaging in time or frequency, the average of the product may not be the product of averages, and we may have some baseline-based effects.

The baseline-based gains can be determined by observing a point source. This is usually a strong quasar. In that case the true visibilities $\ensuremath{V}_{ij}(t)$ should all be equal to the quasar flux density $S$. Then

$$\ensuremath{\mathcal{G}}_{ij}(t) = \frac{\ensuremath{\widetilde{V}}_{ij}(t)}{S}$$ (9.3)

The antenna gains $g_i(t)$ can also be deduced from the non-linear set of equations:

$$g_i(t) g_j^*(t) = \frac{\ensuremath{\widetilde{V}}_{ij}(t)}{S}$$ (9.4)

This is a system with $N$ complex unknowns and $N(N-1)/2$ equations. In terms of real quantities there are $N (N-1)$ measured values (amplitudes and phases; there are only $2 N-1$ unknowns since one may add a phase factor to all complex gains without affecting the baseline-based complex gains. When $N$ is larger than 2 the system is over determined and may be solved by a method of least squares.

If we note $\ensuremath{\widetilde{V}}_{ij} = \tilde{A}_{ij} e^{i \tilde{\varphi}_{ij}}$, the equations for phases are simply:

$$\phi_i - \phi_j = \tilde{\varphi}_{ij}$$ (9.5)

It can be shown that the least-squares solutions (when the same weight is given to all baselines, and if we impose the condition $\sum_{j=1,N} \phi_j = 0$), is given by:

$$\phi_i = \frac{1}{N} \sum_{j\neq i} \tilde{\varphi}_{ij}$$ (9.6)

For the amplitudes we can define in order to get a linear system:

$$\gamma_i = \log{g_i}, \tilde{\alpha}_{ij} = \log{\tilde{A}_{ij}}$$ (9.7)

$$\gamma_i + \gamma_j = \tilde{\alpha}_{ij}$$ (9.8)

This time the least square solution is, when the same weight is given to all baselines:

$$\gamma_i = \frac{1}{N-1}\sum_{j\neq i}\alpha_{ij} - \frac{1}{(N-1)(N-2)}\sum_{j\neq i}\sum_{k\neq i, >j}\alpha_{jk}$$ (9.9)

Obviously this antenna gain determination needs at least three antennas. For three antennas it reduces to the obvious result:

$$g_1 = \frac{\tilde{A}_{12}\tilde{A}_{13}}{\tilde{A}_{23}}$$ (9.10)

These formulas can be generalized to the cases where the baselines have different weights.

It can be seen in the above formulas that the precision to which the antenna phases and amplitudes is determined is improved by a factor $\sqrt{N}$ over the precision of the measurement of the baseline amplitudes and phases.

9.1.2 Gain corrections

The determination of antenna-based gains (amplitudes and phases) has an obvious advantage: the physical cause of the gain variations are truly antenna-based. One may solve for the gains at the time of the observations, and correct the occurring problems to improve the quality of the data. One may re-point or re-focus the antennas to correct for an amplitude loss, correct for an instrumental delay (affecting the frequency dependence of the phases) ...