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The Measurement Equation

All radio-astronomical synthesis data reduction packages have an implicit calibration formalism, which includes a set of calibration effects which may be applied or solved for, and a mathematical framework for how these provide a relationship between the observed and corrected data. Even if this mathematical relationship is not given explicilty, it is, of necessity, enshrined in the infrastructure code.

AIPS++ has explicitly adopted the measurement equation (ME) formalism (Hamaker, Bregman and Sault 1996; Sault, Hamaker and Bregman 1996; Hamaker and Bregman 1996, Hamaker 1998, Noordam 1995; and Cornwel 1995) as the underlying calibration framework. It meets the objectives described in the preceding paragraph, and is easily extended to full coverage of a complete set of image-plane calibration effects (see AIPS++ Note 191). The generic nature of the ME, and its independence on the polarization basis for the data or the calibration parametrization are particularly strong advantages.

A misperception of the measurement equation is that it seeks to describe the physical processes or environment affecting the observed data at a needless or semi-infinite level of detail. This is mistaken; the arbitrary parametrization of the individual calibration effects allows the level of detail to be set at an arbitrary level of coarseness, consistent with available data or even expediency.

It is useful to examine a mathematical representation of the calibration formalism, as it has been adopted. Before describing the measurement equation in this form we need to consider some preliminary definitions for this particular formalism. A generic interferometer is considered to measure a four-vector of cross-correlations $\vec{{V}}\,$ between two recorded polarizations (p, q) for each of two feeds (i, j) characterizing an individual baseline. This measured vector includes instrumental calibration effects and is denoted by:

$$\vec{{V}}_{{\rm ij}}^{} \left(\vphantom{ \begin{array}{c} {V}_{\rm pp}\\ {V}_{\rm pq}\\ {V}_{\rm qp}\\ {V}_{\rm qq} \end{array}}\right. \begin{array}{c} {V}_{\rm pp}\\ {V}_{\rm pq}\\ {V}_{\rm qp}\\ {V}_{\rm qq} \end{array} \left.\vphantom{ \begin{array}{c} {V}_{\rm pp}\\ {V}_{\rm pq}\\ {V}_{\rm qp}\\ {V}_{\rm qq} \end{array}}\right)_{{ij}}^{}$$ (1.1)

The ME relates the measured vector $\vec{{V}}\,$ to the true polarized sky brightness, $\vec{{\cal I}}\,$, which is expressed in a Stokes basis as:

$$\vec{{\cal I}}\, \left(\vphantom{ \begin{array}{c} I\\ Q\\ U\\ V\\ \end{array}}\right. \begin{array}{c} I\\ Q\\ U\\ V\\ \end{array} \left.\vphantom{ \begin{array}{c} I\\ Q\\ U\\ V\\ \end{array}}\right)$$ (1.2)

Different instrumental and propagation effects are modeled in the ME as calibration components and take the form of four-by-four matrices, acting on four-vectors of the type given above. The calibration components are sub-divided into two general categories based on whether they act in the image- or uv-plane, and they are expressed in the polarization basis (p, q) of the feed, which may typically be circular or linear. Calibration components which are feed-based are constituted as the direct matrix product of separate two-by-two Jones matrices for each feed on the baseline under consideration, as:

$$\left[\vphantom{{{J}_i\otimes{J}^*_j}}\right. \otimes \left.\vphantom{{{J}_i\otimes{J}^*_j}}\right]$$ (1.3)

For a particular calibration component (types listed below), the diagonal elements in the two-by-two Jones matrices describe response to like polarization, and the off-diagonal describe the response to opposite polarization. The direct matrix product of two matrices is defined as:

$$\left(\vphantom{ {\begin{array}{cc} a_{00} & a_{01} \\ a_{10} & a_{11} \\ \end{array}} }\right. \begin{array}{cc} a_{00} & a_{01} \\ a_{10} & a_{11} \\ \end{array} \left.\vphantom{ {\begin{array}{cc} a_{00} & a_{01} \\ a_{10} & a_{11} \\ \end{array}} }\right) \otimes \left(\vphantom{ {\begin{array}{cc} b_{00} & b_{01} \\ b_{10} & b_{11} \\ \end{array}} }\right. \begin{array}{cc} b_{00} & b_{01} \\ b_{10} & b_{11} \\ \end{array} \left.\vphantom{ {\begin{array}{cc} b_{00} & b_{01} \\ b_{10} & b_{11} \\ \end{array}} }\right) \left(\vphantom{ {\begin{array}{cccc} a_{00}b_{00} & a_{00}b_{0... ..._{10} & a_{10}b_{11} & a_{11}b_{10} & a_{11}b_{11} \\ \end{array}} }\right. \begin{array}{cccc} a_{00}b_{00} & a_{00}b_{01} & a_{01}b_{00} &... ... a_{10}b_{10} & a_{10}b_{11} & a_{11}b_{10} & a_{11}b_{11} \\ \end{array} \left.\vphantom{ {\begin{array}{cccc} a_{00}b_{00} & a_{00}b_{0... ..._{10} & a_{10}b_{11} & a_{11}b_{10} & a_{11}b_{11} \\ \end{array}} }\right)$$ (1.4)

The direct product of two Jones matrices thus describes the calibration component's impact on all possible correlation combinations between two feeds.

Provision is also made for multiplicative and additive baseline-based calibration components, which are not decomposed into feed-based terms. These take the form of four-by-four and four-by-one matrices, respectively.

The full ME including image-plane and uv-plane calibration effects can be defined in terms of the quantities described above as:

$$\vec{{V}}_{{ij}}^{} \prod \left[\vphantom{{{J^{vis}}_i\otimes{J^{vis}}^*_j}}\right. \otimes \left.\vphantom{{{J^{vis}}_i\otimes{J^{vis}}^*_j}}\right]$$ (1.5)

$$\sum_{k}^{} \prod \left[\vphantom{{{J^{sky}}_i {\left(\underline{\rho}_k\right)} \otimes {J^{sky}}^*_j{\left(\underline{\rho}_k\right)}}}\right. \left(\vphantom{\underline{\rho}_k}\right. \underline{{\rho}}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \otimes \left(\vphantom{\underline{\rho}_k}\right. \underline{{\rho}}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \left.\vphantom{{{J^{sky}}_i {\left(\underline{\rho}_k\right)} \otimes {J^{sky}}^*_j{\left(\underline{\rho}_k\right)}}}\right]$$

$$\vec{{\cal I}}_{k}^{} \pi \left(\vphantom{{\underline{r}_i-\underline{r}_j}}\right. \underline{{r}}_{i} \underline{{r}}_{j} \left.\vphantom{{\underline{r}_i-\underline{r}_j}}\right) \underline{{\rho}}_{k}$$

where:

• $\left[\vphantom{{{J^{vis}}_i\otimes{J^{vis}}^*_j}}\right.$J^vis_i $\otimes$ J^vis*_j$\left.\vphantom{{{J^{vis}}_i\otimes{J^{vis}}^*_j}}\right]$ and $\left[\vphantom{{{J^{sky}}_i {\left(\underline{\rho}_k\right)} \otimes {J^{sky}}^*_j{\left(\underline{\rho}_k\right)}}}\right.$J^sky_i$\left(\vphantom{\underline{\rho}_k}\right.$$\underline{{\rho}}_{k}^{}$$\left.\vphantom{\underline{\rho}_k}\right)$ $\otimes$ J^sky*_j$\left(\vphantom{\underline{\rho}_k}\right.$$\underline{{\rho}}_{k}^{}$$\left.\vphantom{\underline{\rho}_k}\right)$$\left.\vphantom{{{J^{sky}}_i {\left(\underline{\rho}_k\right)} \otimes {J^{sky}}^*_j{\left(\underline{\rho}_k\right)}}}\right]$ are individual uv-plane and image-plane calibration components respectively.

• $\underline{{\rho}}_{k}^{}$ denotes direction, subscript k indexes directions, and ($\underline{{r}}_{i}^{}$ - $\underline{{r}}_{j}^{}$) denotes the interferometer baseline vector.

• The four-by-four matrix S converts the sky brightness Stokes polarization basis to the polarization basis used in the interferometer itself.

• M_ij and A_ij denote multiplicative and additive baseline-based instrumental effects respectively.

• $\prod$ and $\sum$ denote product and summation, respectively.

The ME thus reflects the standard Fourier transform relationship between the uv- and image-planes, and allows for generic calibration in both domains. Time and frequency averaging are implicit in the ME form given here, and are assumed to be over ranges shorter than the timescale for variation of any of the terms in the equation.

In the case of fields-of-view narrow enough that the J_i$\left(\vphantom{\underline{\rho}_k}\right.$$\underline{{\rho}}_{k}^{}$$\left.\vphantom{\underline{\rho}_k}\right)$ are constant with $\underline{{\rho}}_{k}^{}$, these terms may be factored out of the summation and treated uv-plane effects J_i, and the measurement equation may be simplified to (ignoring baseline-based effects):

$$\vec{{V}}_{{ij}}^{} \prod \left[\vphantom{{{J^{vis}}_i\otimes{J^{vis}}^*_j}}\right. \otimes \left.\vphantom{{{J^{vis}}_i\otimes{J^{vis}}^*_j}}\right] \vec{{V}}_{{ij}}^{{ideal}}$$ (1.6)

$\vec{{V}}_{{ij}}^{{ideal}}$ represents the visibilities measured by an ideal interferometer, and implicitly contains the Fourier transform of the sky.

The individual calibration effects (in either the uv- or image-plane) are denoted by single letter abbreviations, each representing the two-by-two Jones matrix for the ith feed. These include:

G_i

- the composite complex electronic gain of all components located after the feed in the signal path, but excluding the correlator.

T_i

- atmospheric tranmission effects which are independent of polarization (i.e., not including Faraday rotation). Useful for modelling tropospheric gain losses.

F_i

- propagation effects due to Faraday rotation.

D_i

- instrumental polarization response.

E_i$\left(\vphantom{\underline{\rho}_k}\right.$$\underline{{\rho}}_{k}^{}$$\left.\vphantom{\underline{\rho}_k}\right)$

- electric field pattern describing the primary beam response, usually normalized with respect to G_i.

B_i

- bandpass response, possibly normalized relative to G_i.

C_i

- feed configuration matrix. This describes the nominal feed configuration, including rotation or conversion of linear to circular polarization. This term is known a priori, and handled implicitly.

P_i

- parallactic angle. This is known analytically.

Since these matrix components may not commute, the order in which they appear in the measurement equation is important. In general, the correct order is given by the order in which the terms affect the incoming wavefront. Thus, a spectral line polarization observation would be described (ignoring primary beam and direction-dependent Faraday rotation) by the measurement equation as (recalling Equation 1.3):

$$\vec{{V}}_{{ij}}^{} \vec{{V}}_{{ij}}^{{ideal}}$$ (1.7)

The measurement equation is implemented in AIPS++ by sub-dividing the machinery dealing with image-plane and uv-plane calibration between imager and calibrater respectively. The remainder of this chapter treats the case illustrated in equation 1.7, i.e., uv-plane effects which are handled by the calibrater tool.