THE FULL MEASUREMENT EQUATION

The formalism discussed hitherto has been called the uv-domain Measurement Equation because it ignores image-plane effects, i.e. instrumental effects that depend on the position in the field. This is only valid for a dominating compact source in the centre of the field, and then only in the case of axially symmetric systems. Only then does the matrix $\vec{B_i}\,$ , which describes effects like primary beam and instrumental polarisation, reduce to the unit matrix.

In this section, the formalism will be generalised to the full Measurement Equation, which describes the behaviour of a real instrument observing an arbitrary brightness distribution, and thus includes image-plane effects. For k (point) sources within the primary beam, we get:

$\displaystyle \vec{v}_{ij}^{}$ = $\displaystyle \left(\vphantom{\begin{array}{c} v_{\rm pp}\\ v_{\rm pq}\\ v_{\rm qp}\\ v_{\rm qq} \end{array}}\right.$ $\displaystyle \begin{array}{c} v_{\rm pp}\\ v_{\rm pq}\\ v_{\rm qp}\\ v_{\rm qq} \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{c} v_{\rm pp}\\ v_{\rm pq}\\ v_{\rm qp}\\ v_{\rm qq} \end{array}}\right)_{ij}^{}$ = ( $\displaystyle \vec{G}^{\null}_{i}$ $\displaystyle \otimes$ $\displaystyle \vec{G}^{\ast}_{j}$ ) ( $\displaystyle \vec{D}^{\null}_{i}$ $\displaystyle \otimes$ $\displaystyle \vec{D}^{\ast}_{j}$ ) ( $\displaystyle \vec{C}^{\null}_{i}$ $\displaystyle \otimes$ $\displaystyle \vec{C}^{\ast}_{j}$ ) $\displaystyle \Sigma$

(18)

in which the matrices $\vec{G}\,$ (complex gain), $\vec{D}\,$ (leakage) and $\vec{C}\,$ (nominal feed configuration) are typical uv-domain effects in the sense that they affect all the sources in the same way. The matrix $\Sigma$ describes image-plane effects, which depend on the position $\rho_{k}^{}$ in the field, and therefore have to be applied to each source independently:

$\displaystyle \Sigma$ = $\displaystyle \sum_{k}^{}$ $\displaystyle \sum_{t}^{}$ $\displaystyle \sum_{f}^{}$ ( $\displaystyle \vec{B}^{\null}_{i}$ $\displaystyle \otimes$ $\displaystyle \vec{B}^{\ast}_{j}$ ) ( $\displaystyle \vec{P}^{\null}_{i}$ $\displaystyle \otimes$ $\displaystyle \vec{P}^{\ast}_{j}$ ) ( $\displaystyle \vec{F}^{\null}_{i}$ $\displaystyle \otimes$ $\displaystyle \vec{F}^{\ast}_{j}$ ) $\displaystyle \vec{S}\,$ $\displaystyle \left(\vphantom{\begin{array}{c} I\\ Q\\ U\\ V \end{array}}\right.$ $\displaystyle \begin{array}{c} I\\ Q\\ U\\ V \end{array}$ $\displaystyle \left.\vphantom{\begin{array}{c} I\\ Q\\ U\\ V \end{array}}\right)_{k}^{}$ exp^-i .

(19)

in which the sum is taken over all k sources in the field. The sums over time t and frequency f take care of the effects of finite integration time and bandwidth. The effects of integrating over a finite antenna size are indistinguishable from the primary beam, and are thus part of $\vec{B}\,$ (see below).

Note that the matrices $\vec{P}\,$ for parallactic angle and $\vec{F}\,$ for ionospheric Faraday rotation are now behind the sum, even though they are (usually) the same for all sources in the field. This is because they do not commute with $\vec{B}\,$ . The same is true for tropospheric gain effects (e.g. extinction and refraction), which should not really be lumped with the receiver gain $\vec{G}\,$ . It is clear that current calibration practices could do with some critical scrutiny!

The matrix $\vec{B}_{i}^{}$ describes the primary beam response (including pointing errors!) of antenna i. The matrix elements are functions of the difference vector $\Delta$ $\vec{\rho_{k}}\,$ between the direction vector $\vec{\rho}_{i}^{}$ of the optical axis of antenna i, and the direction vector $\vec{\rho_{k}}\,$ of a point source k,

$\displaystyle \vec{B}_{i}^{}$ = $\displaystyle \vec{B}_{i}^{}$ ( $\displaystyle \vec{\rho}_{i}^{}$ - $\displaystyle \vec{\rho}_{k}^{}$ ) = ^...

(20)

Instrumental polarisation is an artifact caused by asymmetries in $\vec{B}_{i}^{}$ , and thus in $\vec{B}^{\null}_{i}$ $\otimes$ $\vec{B}^{\ast}_{j}$ . Ionospheric Faraday rotation does not affect it, of course. There is a growing consensus that at least two effects are involved: the linear shape of the dipoles, which cause the familiar `clover-leaf' patterns in Q and U, and a standing wave in the front-end support legs. The latter effect is strongly frequency-dependent. NB: Expressions for the elements of $\vec{B}_{i}^{}$ for the WSRT will emerge over the summer.

**Figure 1:** Label: matform File: matform.eps
$\begin{figure}\begin{center} \epsfxsize=15truecm \epsfysize=15truecm \leavevm... ...subset of parameters, given sufficient constraints. }\end{center} \end{figure}$