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Next: The Design Up: The Generic Instrument: III Design of Calibration and Previous: Goals

Subsections

The Measurement Equation

The measurement equation is:

$\displaystyle \vec{V}_{{\sf i}{\sf j}}$ = X$\scriptstyle \sf i$$\scriptstyle \sf j$$\displaystyle \left(\vphantom{{\sf M}_{{\sf i}{\sf j}}{\left[{{\sf J^{vis}}_{{\... ...right)}^\ast \right]} {\sf S}\ {\vec{I}}_k + {\vec{A}}_{{\sf i}{\sf j}}}\right.$$\displaystyle \sf M_{{\sf i}{\sf j}}^{}$$\displaystyle \left[\vphantom{{{\sf J^{vis}}_{{\sf i}}}\otimes{{\sf J^{vis}}_{{\sf j}}}^\ast }\right.$$\displaystyle \sf J^{vis}_{{\sf i}}^{}$ $\displaystyle \otimes$ $\displaystyle \sf J^{vis}_{{\sf j}}^{\ast}$$\displaystyle \left.\vphantom{{{\sf J^{vis}}_{{\sf i}}}\otimes{{\sf J^{vis}}_{{\sf j}}}^\ast }\right]$$\displaystyle \sum_{k}^{}$$\displaystyle \left[\vphantom{{{\sf J^{sky}}_{{\sf i}}\left({\vec{\rho}}_k\right)}\otimes{{\sf J^{sky}}_{{\sf j}}\left({\vec{\rho}}_k\right)}^\ast }\right.$$\displaystyle \sf J^{sky}_{{\sf i}}^{}$$\displaystyle \left(\vphantom{{\vec{\rho}}_k}\right.$$\displaystyle \vec{\rho}_{k}$$\displaystyle \left.\vphantom{{\vec{\rho}}_k}\right)$ $\displaystyle \otimes$ $\displaystyle \sf J^{sky}_{{\sf j}}^{}$$\displaystyle \left(\vphantom{{\vec{\rho}}_k}\right.$$\displaystyle \vec{\rho}_{k}$$\displaystyle \left.\vphantom{{\vec{\rho}}_k}\right)^{\ast}$$\displaystyle \left.\vphantom{{{\sf J^{sky}}_{{\sf i}}\left({\vec{\rho}}_k\right)}\otimes{{\sf J^{sky}}_{{\sf j}}\left({\vec{\rho}}_k\right)}^\ast }\right]$$\displaystyle \sf S$ $\displaystyle \vec{I}_{k}$ + $\displaystyle \vec{A}_{{\sf i}{\sf j}}$$\displaystyle \left.\vphantom{{\sf M}_{{\sf i}{\sf j}}{\left[{{\sf J^{vis}}_{{\... ...right)}^\ast \right]} {\sf S}\ {\vec{I}}_k + {\vec{A}}_{{\sf i}{\sf j}}}\right)$ (1)

We need to be able to do the following things:

In previous memos, we described how these are to be accomplished. Here we recapitulate that description, with some minor changes.

Prediction of coherences

This is quite straightforward, except that efficiency requires that FFTs be used whereever possible.

Correction of coherences

The Jones Matrices can be usefully split into two classes: those that are sky-position dependent, $ \sf J^{sky}$, and those that are not, $ \sf J^{vis}$. The latter can be corrected trivially in the visibility domain whereas the former cannot be corrected without going through an imaging process.

To ensure compatibility, we have chosen to impose a standard set and ordering of matrices. These split as follows. The SkyJones matrices are: 


  

 
$ \sf F$($ \vec{r}_{{\sf i}}$,$ \vec{\rho}\,$) ionospheric Faraday rotation  

 
$ \sf T$($ \vec{r}_{{\sf i}}$,$ \vec{\rho}\,$) atmospheric gain 

 
$ \sf E_{{\sf i}}^{}$($ \vec{\rho}\,$$\scriptstyle \prime$) primary beam

The VisJones matrices are: 


  

 
$ \sf P_{{\sf i}}^{}$
receptor position angle 

 
$ \sf C_{{\sf i}}^{}$
configuration of receptors 

 
$ \sf D_{{\sf i}}^{}$
receptor cross-leakage 

 
$ \sf G_{{\sf i}}^{}$
electronic gain (antenna-based only) 

 
$ \sf K_{{\sf i}}^{}$($ \vec{\rho}_{0}$.$ \vec{r}_{{\sf i}}$) factorised `FT' phase forphase center

The order is:

$\displaystyle \sf J^{vis}$ = $\displaystyle \sf K$ $\displaystyle \sf G$ $\displaystyle \sf D$ $\displaystyle \sf C$ $\displaystyle \sf P$ (2)

$\displaystyle \sf J^{sky}$ = $\displaystyle \sf E$ $\displaystyle \sf T$ $\displaystyle \sf F$ (3)

The conventional meaning of these matrices is given by Noordam (1995) with one important exception: we have changed the K-matrix to be a single phase rotation for a coherence rather than something that is position-dependent.

Let us define the sky coherence to be:

$\displaystyle \vec{V}^{\sf sky}_{{\sf i}{\sf j}}$ = $\displaystyle \sum_{k}^{}$$\displaystyle \left[\vphantom{{{\sf J^{sky}}_{{\sf i}}\left({\vec{\rho}}_k\right)}\otimes{{\sf J^{sky}}_{{\sf j}}\left({\vec{\rho}}_k\right)}^\ast }\right.$$\displaystyle \sf J^{sky}_{{\sf i}}^{}$$\displaystyle \left(\vphantom{{\vec{\rho}}_k}\right.$$\displaystyle \vec{\rho}_{k}$$\displaystyle \left.\vphantom{{\vec{\rho}}_k}\right)$ $\displaystyle \otimes$ $\displaystyle \sf J^{sky}_{{\sf j}}^{}$$\displaystyle \left(\vphantom{{\vec{\rho}}_k}\right.$$\displaystyle \vec{\rho}_{k}$$\displaystyle \left.\vphantom{{\vec{\rho}}_k}\right)^{\ast}$$\displaystyle \left.\vphantom{{{\sf J^{sky}}_{{\sf i}}\left({\vec{\rho}}_k\right)}\otimes{{\sf J^{sky}}_{{\sf j}}\left({\vec{\rho}}_k\right)}^\ast }\right]$$\displaystyle \sf S$ $\displaystyle \vec{I}_{k}$ (4)

We then have that:

$\displaystyle \vec{V}_{{\sf i}{\sf j}}$ = X$\scriptstyle \sf i$$\scriptstyle \sf j$$\displaystyle \left(\vphantom{{\sf M}_{{\sf i}{\sf j}}{\left[{{\sf J^{vis}}_{{\... ...ght]} {\vec{V}^{\sf sky}}_{{\sf i}{\sf j}}+ {\vec{A}}_{{\sf i}{\sf j}}}\right.$$\displaystyle \sf M_{{\sf i}{\sf j}}^{}$$\displaystyle \left[\vphantom{{{\sf J^{vis}}_{{\sf i}}}\otimes{{\sf J^{vis}}_{{\sf j}}}^\ast }\right.$$\displaystyle \sf J^{vis}_{{\sf i}}^{}$ $\displaystyle \otimes$ $\displaystyle \sf J^{vis}_{{\sf j}}^{\ast}$$\displaystyle \left.\vphantom{{{\sf J^{vis}}_{{\sf i}}}\otimes{{\sf J^{vis}}_{{\sf j}}}^\ast }\right]$$\displaystyle \vec{V}^{\sf sky}_{{\sf i}{\sf j}}$ + $\displaystyle \vec{A}_{{\sf i}{\sf j}}$$\displaystyle \left.\vphantom{{\sf M}_{{\sf i}{\sf j}}{\left[{{\sf J^{vis}}_{{\... ...ght]} {\vec{V}^{\sf sky}}_{{\sf i}{\sf j}}+ {\vec{A}}_{{\sf i}{\sf j}}}\right)$ (5)

And then a natural definition of a corrected (visibility) coherence would be:

$\displaystyle \vec{V}^{\sf cal}_{{\sf i}{\sf j}}$ = $\displaystyle \left[\vphantom{{{\sf J^{vis}}_{{\sf i}}}\otimes{{\sf J^{vis}}_{{\sf j}}}^\ast }\right.$$\displaystyle \sf J^{vis}_{{\sf i}}^{}$ $\displaystyle \otimes$ $\displaystyle \sf J^{vis}_{{\sf j}}^{\ast}$$\displaystyle \left.\vphantom{{{\sf J^{vis}}_{{\sf i}}}\otimes{{\sf J^{vis}}_{{\sf j}}}^\ast }\right]^{-1}$$\displaystyle \sf M_{{\sf i}{\sf j}}^{-1}$$\displaystyle \left(\vphantom{{{X}}_{{\sf i}{\sf j}}^{-1}\left({\vec{V}}_{{\sf i}{\sf j}}\right) -{\vec{A}}_{{\sf i}{\sf j}}}\right.$X$\scriptstyle \sf i$$\scriptstyle \sf j$-1$\displaystyle \left(\vphantom{{\vec{V}}_{{\sf i}{\sf j}}}\right.$$\displaystyle \vec{V}_{{\sf i}{\sf j}}$$\displaystyle \left.\vphantom{{\vec{V}}_{{\sf i}{\sf j}}}\right)$ - $\displaystyle \vec{A}_{{\sf i}{\sf j}}$$\displaystyle \left.\vphantom{{{X}}_{{\sf i}{\sf j}}^{-1}\left({\vec{V}}_{{\sf i}{\sf j}}\right) -{\vec{A}}_{{\sf i}{\sf j}}}\right)$ (6)

Full correction for sky-plane-based calibration terms is not possible without constructing an image. This we consider next.

Image estimation

In Cornwell (1995), we described how to solve for the sky brightness. A generalized dirty image can be defined as:

$\displaystyle \vec{I}^{D}_{k}$ = - $\displaystyle \left[\vphantom{\frac {\partial^{2}{\chi^2}}{\partial{{\vec{I}}_k}\partial{{\vec{I}}^T_k}}}\right.$$\displaystyle {\frac{\partial^{2}{\chi^2}}{\partial{{\vec{I}}_k}\partial{{\vec{I}}^T_k}}}$ $\displaystyle \left.\vphantom{\frac {\partial^{2}{\chi^2}}{\partial{{\vec{I}}_k}\partial{{\vec{I}}^T_k}}}\right]^{-1}_{}$$\displaystyle {\frac{\partial \chi^2}{\partial {\vec{I}}_k}}$$\displaystyle \mid_{{\vec{I}}_k=0}^{}$ (7)

where $ \chi^{2}_{}$ is given by:

$\displaystyle \chi^{2}_{}$ = $\displaystyle \sum_{{\sf i}{\sf j}}^{}$$\displaystyle \Delta$$\displaystyle \vec{V}_{{\sf i}{\sf j}}^{*T}$$\displaystyle \Lambda_{{\sf i}{\sf j}}$$\displaystyle \Delta$$\displaystyle \vec{V}_{{\sf i}{\sf j}}$ (8)

where the residual is given by the difference between that observed and predicted using the Measurement Equation:

$\displaystyle \Delta$$\displaystyle \vec{V}\,$ = $\displaystyle \vec{V}\,$ - $\displaystyle \widehat{{\vec{V}}}$ (9)

The derivatives are given by:

$\displaystyle {\frac{\partial \chi^2}{\partial {\vec{I}}_k}}$ = - 2 $\displaystyle \Re$$\displaystyle \sum_{{\sf i}{\sf j}}^{}$$\displaystyle \left[\vphantom{{{\sf J}_{{\sf i}}\left({\vec{\rho}}_k\right)}\otimes{{\sf J}_{{\sf j}}\left({\vec{\rho}}_k\right)}^\ast }\right.$$\displaystyle \sf J_{{\sf i}}^{}$$\displaystyle \left(\vphantom{{\vec{\rho}}_k}\right.$$\displaystyle \vec{\rho}_{k}$$\displaystyle \left.\vphantom{{\vec{\rho}}_k}\right)$ $\displaystyle \otimes$ $\displaystyle \sf J_{{\sf j}}^{}$$\displaystyle \left(\vphantom{{\vec{\rho}}_k}\right.$$\displaystyle \vec{\rho}_{k}$$\displaystyle \left.\vphantom{{\vec{\rho}}_k}\right)^{\ast}$$\displaystyle \left.\vphantom{{{\sf J}_{{\sf i}}\left({\vec{\rho}}_k\right)}\otimes{{\sf J}_{{\sf j}}\left({\vec{\rho}}_k\right)}^\ast }\right]^{*T}$$\displaystyle \Lambda_{{\sf i}{\sf j}}$ $\displaystyle \Delta$$\displaystyle \vec{V}_{{\sf i}{\sf j}}$ (10)

$\displaystyle {\frac{\partial^{2}{\chi^2}}{\partial{{\vec{I}}_k}\partial{{\vec{I}}_k^T}}}$ = 2 $\displaystyle \Re$$\displaystyle \sum_{{\sf i}{\sf j}}^{}$$\displaystyle \sf S^{*T}_{}$$\displaystyle \left[\vphantom{{{\sf J}_{{\sf i}}\left({\vec{\rho}}_k\right)}\otimes{{\sf J}_{{\sf j}}\left({\vec{\rho}}_k\right)}^\ast }\right.$$\displaystyle \sf J_{{\sf i}}^{}$$\displaystyle \left(\vphantom{{\vec{\rho}}_k}\right.$$\displaystyle \vec{\rho}_{k}$$\displaystyle \left.\vphantom{{\vec{\rho}}_k}\right)$ $\displaystyle \otimes$ $\displaystyle \sf J_{{\sf j}}^{}$$\displaystyle \left(\vphantom{{\vec{\rho}}_k}\right.$$\displaystyle \vec{\rho}_{k}$$\displaystyle \left.\vphantom{{\vec{\rho}}_k}\right)^{\ast}$$\displaystyle \left.\vphantom{{{\sf J}_{{\sf i}}\left({\vec{\rho}}_k\right)}\otimes{{\sf J}_{{\sf j}}\left({\vec{\rho}}_k\right)}^\ast }\right]^{*T}$tex2html_image_mark>#tex2html_wrap_indisplay2797# $\displaystyle \left[\vphantom{{{\sf J}_{{\sf i}}\left({\vec{\rho}}_k\right)}\otimes{{\sf J}_{{\sf j}}\left({\vec{\rho}}_k\right)}^\ast }\right.$$\displaystyle \sf J_{{\sf i}}^{}$$\displaystyle \left(\vphantom{{\vec{\rho}}_k}\right.$$\displaystyle \vec{\rho}_{k}$$\displaystyle \left.\vphantom{{\vec{\rho}}_k}\right)$ $\displaystyle \otimes$ $\displaystyle \sf J_{{\sf j}}^{}$$\displaystyle \left(\vphantom{{\vec{\rho}}_k}\right.$$\displaystyle \vec{\rho}_{k}$$\displaystyle \left.\vphantom{{\vec{\rho}}_k}\right)^{\ast}$$\displaystyle \left.\vphantom{{{\sf J}_{{\sf i}}\left({\vec{\rho}}_k\right)}\otimes{{\sf J}_{{\sf j}}\left({\vec{\rho}}_k\right)}^\ast }\right]$ $\displaystyle \sf S$ (11)

where $ \Lambda$ is a weighting array.

The misfit, $ \chi^{2}_{}$, can be evaluated with respect to the corrected values, thus allowing imaging in the traditional two-step process. One caveat is that the Weight matrix must then be adjusted for the correction.

Calibration estimation

We allow least squares estimation of calibration effects by providing derivatives of $ \chi^{2}_{}$ with respect to the various gain matrices. Differentiating $ \chi^{2}_{}$ with respect to each element of a gain matrix in turn we can build up the gradient with respect to the overall matrix. For clarity, consider the case where only $ \sf G$ is active in $ \sf J^{vis}$. We then have that:

$\displaystyle {\frac{\partial \chi^2}{\partial {\sf G}_{{\sf i},p,q}}}$ = - 2 $\displaystyle \sum_{{\sf j}}^{}$ $\displaystyle \left[\vphantom{{U_{p,q}}\otimes{{\sf G}_{{\sf j}}}^\ast }\right.$Up, q $\displaystyle \otimes$ $\displaystyle \sf G_{{\sf j}}^{\ast}$$\displaystyle \left.\vphantom{{U_{p,q}}\otimes{{\sf G}_{{\sf j}}}^\ast }\right]^{*T}_{}$ $\displaystyle \Lambda_{{\sf i}{\sf j}}$ $\displaystyle \Delta$$\displaystyle \vec{V}_{{\sf i}{\sf j}}$ (12)

$\displaystyle {\frac{\partial^{2}{\chi^2}}{\partial{{\sf G}_{{\sf i},p,q}}\partial{{{\sf G}_{{\sf i},p,q}}^{*T}}}}$ = 2 $\displaystyle \sum_{{\sf j}}^{}$ $\displaystyle \left[\vphantom{{U_{p,q}}\otimes{{\sf G}_{{\sf j}}}^\ast }\right.$Up, q $\displaystyle \otimes$ $\displaystyle \sf G_{{\sf j}}^{\ast}$$\displaystyle \left.\vphantom{{U_{p,q}}\otimes{{\sf G}_{{\sf j}}}^\ast }\right]^{*T}_{}$ $\displaystyle \Lambda_{{\sf i}{\sf j}}$ $\displaystyle \left[\vphantom{{U_{p,q}}\otimes{{\sf G}_{{\sf j}}}^\ast }\right.$Up, q $\displaystyle \otimes$ $\displaystyle \sf G_{{\sf j}}^{\ast}$$\displaystyle \left.\vphantom{{U_{p,q}}\otimes{{\sf G}_{{\sf j}}}^\ast }\right]$ (13)

where the matrix Up, q is unity for element p, q and zero otherwise. This equation simplifies to the well-known scalar version and corresponds physically to just referencing errors to one antenna and summing over all baselines to that antenna (see Cornwell and Wilkinson, 1981 for a heuristic derivation of the scalar version).

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Next: The Design Up: The Generic Instrument: III Design of Calibration and Previous: Goals   Contents
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2006-10-15