Data representation and the calibration mechanism

AIPS++ stores synthesis uv-data in a MeasurementSet (MS), and it is important to understand how observed and calibrated data are treated in this framework. The observed data are stored in the DATA column in the MAIN MS table; additional columns are added during processing for CORRECTED_DATA and MODEL_DATA. The CORRECTED_DATA column contains the data which result when applying all desired visibility-plane calibration corrections, moving from left to right across the uv-plane portion of the ME. This provides corrected data suitable for imaging.

The MODEL_DATA column, in contrast, is formed by propagating initial source models (perhaps for trial imaging) from right to left through the ME, applying the supported image-plane corrections, and Fourier transforming to the uv-plane. This realizes a best guess for the $\vec{{V}}_{{ij}}^{{ideal}}$ term in equation 1.7. The point at which these two columns meet in the ME allows a residual to be computed and thus an estimate of $\chi^{2}_{}$ . The calibration component(s) at the intersection point in the ME are solved for by computing gradients of $\chi^{2}_{}$ with respect to the Jones matrices, and performing a standard non-linear least-squares minimization^1.1.

As an example of the calibration mechanism, we recast equation 1.7 to the implicit form used when solving for D, given an existing solution for G and knowledge of P. We ignore the B calibration component, either because the data is from a continuum observation (only one channel), or because we assume that the bandpass shape is unimportant, at least for this stage of the calibration. Since G and P appear on either side of D in the ME, they are applied to the observed data ( $\vec{{V}}_{{ij}}^{}$ ) and model data ( $\vec{{V}}_{{ij}}^{{ideal}}$ ) terms, respectively:

G_ij^-1 $\displaystyle \vec{{V}}_{{ij}}^{}$ = $\displaystyle \left(\vphantom{{G}_{ij}^{-1}{G}_{ij}}\right.$ G_ij^-1G_ij $\displaystyle \left.\vphantom{{G}_{ij}^{-1}{G}_{ij}}\right)$ D_ij $\displaystyle \left(\vphantom{{P}_{ij} \vec{V}_{ij}^{ideal}}\right.$ P_ij $\displaystyle \vec{{V}}_{{ij}}^{{ideal}}$ $\displaystyle \left.\vphantom{{P}_{ij} \vec{V}_{ij}^{ideal}}\right)$

(1.8)

D_ij = G_ij^-1 $\displaystyle \vec{{V}}_{{ij}}^{}$ $\displaystyle \left(\vphantom{{P}_{ij} \vec{V}_{ij}^{ideal}}\right.$ P_ij $\displaystyle \vec{{V}}_{{ij}}^{{ideal}}$ $\displaystyle \left.\vphantom{{P}_{ij} \vec{V}_{ij}^{ideal}}\right)^{{-1}}_{}$

(1.9)

$\displaystyle \left[\vphantom{{{D^{vis}}_i\otimes{D^{vis}}^*_j}}\right.$ D^vis_i $\displaystyle \otimes$ D^vis*_j $\displaystyle \left.\vphantom{{{D^{vis}}_i\otimes{D^{vis}}^*_j}}\right]$ = G_ij^-1 $\displaystyle \vec{{V}}_{{ij}}^{}$ $\displaystyle \left(\vphantom{{P}_{ij} \vec{V}_{ij}^{ideal}}\right.$ P_ij $\displaystyle \vec{{V}}_{{ij}}^{{ideal}}$ $\displaystyle \left.\vphantom{{P}_{ij} \vec{V}_{ij}^{ideal}}\right)^{{-1}}_{}$

(1.10)

In equation 1.10, the D calibration is realized as the feed-based (two-by-two) Jones matrices which are solved for in the $\chi^{2}_{}$ minimization. It is straightforward to see how this algebraic manipulation can be extended to any of the calibration components, and the flexible implementation of the calibrater tool is clearly evident.