interferometer-based effects

Until now, we have assumed that all instrumental effects could be factored into feed-based contributions, i.e. we have ignored any interferometer-based effects. This is justified for a well-designed system, provided that the signal-to-noise ratio is large enough (thermal noise causes interferometer-based errors, albeit with a an average of zero). However, if systematic errors do occur, they can be modelled:

$\displaystyle \vec{{V}}_{{{{\sf i}{\sf j}}}}^{{{'}}}$ = $\displaystyle \sf X_{{{\sf i}{\sf j}}}^{}$ ( $\displaystyle \vec{{A}}_{{{{\sf i}{\sf j}}}}^{{}}$ + $\displaystyle \sf M_{{{\sf i}{\sf j}}}^{}$ $\displaystyle \vec{{V}}_{{{{\sf i}{\sf j}}}}^{{}}$ )

(9)

The 4 x 4 diagonal matrix $\sf X$ , the `Correlator matrix', represents interferometer-based corrections that are applied to the uv-data in software by the on-line system. Examples are the Van Vleck correction. In the newest correlators, it approaches a constant ( $\sf x$ ).

$\displaystyle \sf X_{{{\sf i}{\sf j}}}^{}$ = $\displaystyle \left(\vphantom{\begin{array}{rrrr} {\sf x}_{{\sf i}{\sf p}\,{\s... ...\\ 0 & 0 & 0 & {\sf x}_{{\sf i}{\sf q}\,{\sf j}{\sf q}} \end{array}}\right.$ $\displaystyle \begin{array}{rrrr} {\sf x}_{{\sf i}{\sf p}\,{\sf j}{\sf p}}& 0 ... ...f p}}& 0\\ 0 & 0 & 0 & {\sf x}_{{\sf i}{\sf q}\,{\sf j}{\sf q}} \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{rrrr} {\sf x}_{{\sf i}{\sf p}\,{\s... ...\\ 0 & 0 & 0 & {\sf x}_{{\sf i}{\sf q}\,{\sf j}{\sf q}} \end{array}}\right)$ $\displaystyle \approx$ $\displaystyle \sf x$ $\displaystyle \cal {U}$

(10)

The 4 x 4 diagonal matrix $\sf M$ represents multiplicative interferometer-based effects.

$\displaystyle \sf M_{{{\sf i}{\sf j}}}^{}$ = $\displaystyle \left(\vphantom{\begin{array}{rrrr} {\sf m}_{{\sf i}{\sf p}\,{\s... ...\\ 0 & 0 & 0 & {\sf m}_{{\sf i}{\sf q}\,{\sf j}{\sf q}} \end{array}}\right.$ $\displaystyle \begin{array}{rrrr} {\sf m}_{{\sf i}{\sf p}\,{\sf j}{\sf p}}& 0 ... ...f p}}& 0\\ 0 & 0 & 0 & {\sf m}_{{\sf i}{\sf q}\,{\sf j}{\sf q}} \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{rrrr} {\sf m}_{{\sf i}{\sf p}\,{\s... ...\\ 0 & 0 & 0 & {\sf m}_{{\sf i}{\sf q}\,{\sf j}{\sf q}} \end{array}}\right)$ $\displaystyle \approx$ $\displaystyle \cal {U}$

(11)

The 4-element vector $\vec{{A}}_{{{{{\sf i}{\sf j}}}}}^{{{}}}$ represents additive interferometer-based effects. Examples are receiver noise, and correlator offsets.

$\displaystyle \vec{{A}}_{{{{\sf i}{\sf j}}}}^{{}}$ = $\displaystyle \left(\vphantom{\begin{array}{c} {\sf a}_{{\sf i}{\sf p}\,{\sf j... ...f j}{\sf p}}\\ {\sf a}_{{\sf i}{\sf q}\,{\sf j}{\sf q}} \end{array}}\right.$ $\displaystyle \begin{array}{c} {\sf a}_{{\sf i}{\sf p}\,{\sf j}{\sf p}}\\ {... ... q}\,{\sf j}{\sf p}}\\ {\sf a}_{{\sf i}{\sf q}\,{\sf j}{\sf q}} \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{c} {\sf a}_{{\sf i}{\sf p}\,{\sf j... ...f j}{\sf p}}\\ {\sf a}_{{\sf i}{\sf q}\,{\sf j}{\sf q}} \end{array}}\right)$ $\displaystyle \approx$ $\displaystyle \vec{{0}}\,$

(12)

In some cases, interferometer-based effects can be calibrated, e.g. when they appear to be constant in time. It will be interesting to see how many of them will disappear as a result of better modelling with the Measurement Equation. In any case, it is desirable that the cause of interferometer-based effects is properly understood (simulation!).