Calibration of the visibility

Calibration involves solving for the possibly unknown feed-based matrices such as G_i, D_i from calibration observations². Imaging means estimating $\vec{\cal I}\,$ from data corrected for the calibration matrices. Note that parametrization of the free variables is probably wise. For calibration, this means that, for example, G_i is diagonal with parameters a_p, a_q, $\phi_{p}^{}$ and $\phi_{q}^{}$ describing the amplitude and phase of the gain in the two polarization channels. For imaging, the parameters could be the pixels of an image or the parameters of a number of Gaussian components.

If the image parameters are known for some subset of the measurements, then one can solve for the calibration parameters and thus calibrate. Then from another subset one can solve for image parameters of another object.

Suppose that we have a number of measurements $\vec{V}\,$ of the visibility function, and that furthermore we have estimates of the visibility function $\vec{V}\,$ that would be measured in the absence of visibility-based calibration effects. We can then estimate the calibration matrices by a least squares fit whereby the matrices are adjusted to fit the data in a least-squares sense. Define an error norm $\chi^{2}_{}$ :

$\displaystyle \chi^{2}_{}$ = $\displaystyle \sum_{ij}^{}$ $\displaystyle \Delta$ $\displaystyle \vec{V}^{*T}_{\rm ij}$ W_ij $\displaystyle \Delta$ $\displaystyle \vec{V}_{\rm ij}^{}$

(10)

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

(11)

$\displaystyle \widehat{\vec{V}}$ = $\displaystyle \left[\vphantom{{{G}_i\otimes{G}^*_j}}\right.$ G_i $\displaystyle \otimes$ G^*_j $\displaystyle \left.\vphantom{{{G}_i\otimes{G}^*_j}}\right]$ $\displaystyle \left[\vphantom{{{D}_i\otimes{D}^*_j}}\right.$ D_i $\displaystyle \otimes$ D^*_j $\displaystyle \left.\vphantom{{{D}_i\otimes{D}^*_j}}\right]$ $\displaystyle \left[\vphantom{{{C}_i\otimes{C}^*_j}}\right.$ C_i $\displaystyle \otimes$ C^*_j $\displaystyle \left.\vphantom{{{C}_i\otimes{C}^*_j}}\right]$ $\displaystyle \widehat{\vec{\cal V}}_{\rm ij}^{}$

(12)

and W_ij is a (4 by 4) weight matrix. For natural weighting, W will usually be diagonal with elements given by the inverse of the corresponding variance. However, it is more strictly the inverse of the covariance matrix of the errors. For uniform weighting, the usual correction for the local density of samples will be applied.

Choosing the calibration matrices that minimize $\chi^{2}_{}$ is a non-linear least squares problem. To solve it, we will need the first and second derivatives of $\chi^{2}_{}$ with respect to the calibration matrices. Since this gets quite complicated, I've deferred complete exposition to another memo. For the moment, I say only that to check these formulas, I generated the scalar update equation for the case where only the gain matrix need be calculated. I do indeed obtain the update formula used in the usual feed-based gain solution algorithms.

This is all good as far as it goes, but we probably will prefer to parametrize the calibration matrices by a small number of parameters. We will then require derivatives with respect to these parameters, instead of the elements of the matrices directly. By applying the chain rule, this is quite straightforward, if tedious.

$\displaystyle \vec{{\cal V}}_{\rm ij}^{}$ = $\displaystyle \left[\vphantom{{{C}_i\otimes{C}^*_j}}\right.$ C_i $\displaystyle \otimes$ C^*_j $\displaystyle \left.\vphantom{{{C}_i\otimes{C}^*_j}}\right]^{-1}_{}$ $\displaystyle \left[\vphantom{{{D}_i\otimes{D}^*_j}}\right.$ D_i $\displaystyle \otimes$ D^*_j $\displaystyle \left.\vphantom{{{D}_i\otimes{D}^*_j}}\right]^{-1}_{}$ $\displaystyle \left[\vphantom{{{G}_i\otimes{G}^*_j}}\right.$ G_i $\displaystyle \otimes$ G^*_j $\displaystyle \left.\vphantom{{{G}_i\otimes{G}^*_j}}\right]^{-1}_{}$ $\displaystyle \vec{V}_{\rm ij}^{}$

(13)

The other calibration matrices, E_i $\left(\vphantom{\underline{\rho}_k}\right.$ $\underline{\rho}_{k}^{}$ $\left.\vphantom{\underline{\rho}_k}\right)$ , P_i $\left(\vphantom{\underline{\rho}_k}\right.$ $\underline{\rho}_{k}^{}$ $\left.\vphantom{\underline{\rho}_k}\right)$ , F_i $\left(\vphantom{\underline{\rho}_k}\right.$ $\underline{\rho}_{k}^{}$ $\left.\vphantom{\underline{\rho}_k}\right)$ , cannot be corrected in the visibility domain. Instead, it is necessary to make an image. This I describe next.