5 The poldistortion

   
5.1 Polrotation and polconversion

An arbitrary square matrix $\vec{X}$ can be subjected to a polar decomposition (Appendix B.6)

$$\begin{eqnarray*}\vec{X}= x \vec{H}\vec{Y}= x \vec{Y}\vec{H}' \end{eqnarray*}$$

where

• x is a complex constant;
• $\vec{Y}$ is a unimodular unitary matrix (i.e. $\vec{Y}\vec{Y}^{\dagger}= \mathbf{I}$ and $\det \vec{Y}= 1$);
• $\vec{H}$ is a unimodular positive hermitian matrix (i.e. $\vec{H}= \vec{H}^{\dagger}$, $\det \vec{H}= 1$ and $\mbox{Tr}\ \vec{H}>0$); so is $\vec{H}'$.

Applying the polar decomposition to Eq. (14) we get

 

$$\vec{B}' = xx^* \,\vec{H}\ (\vec{Y}\vec{B}\vec{Y}^{\dagger})\ \vec{H}^{\dagger}.$$ (15)

The positive scaling factor is the same as in scalar selfcal and I shall further ignore it. Apart from it, $\vec{B}'$ is derived from $\vec{B}$ through a succession of two specific transformations.

The first is the unitary transformation $\vec{Y}$. Its effect is to leave the intensity unchanged (naturally, since $\vec{Y}\mathbf{I}\vec{Y}^{\dagger}= \mathbf{I}$) and to rotate the polvector in its three-dimensional space (Appendix C.1). In quaternion notation

$$\begin{eqnarray*}[\, I' + \vec{p}' \,]= [\, I + \vec{R}\,\vec{p}] \end{eqnarray*}$$

where $\vec{R}$ is a rotation operator. I call $\vec{Y}$ the polrotation (transformation). The rotation and $\vec{Y}$ can be characterised by a vector in polvector space, the Gibbs vector (Appendix B.1)

$$\begin{eqnarray*}\vec{1}_{\vec{y}} \sin\eta, \end{eqnarray*}$$

where $\vec{y}$ is the direction of the rotation axis and $2\eta$ the rotation angle. In Appendix C.1 I derive the relation between the Gibbs vector and $\vec{Y}$. Obviously, any multiple of the Gibbs vector is an eigenvector of the transformation: $\vec{R}\,\vec{1}_{\vec{y}} = \vec{1}_{\vec{y}}$.

Like the polrotation, the positive hermitian transformation $\vec{H}$ is characterised by a vector

$$\begin{eqnarray*}\vec{1}_{\vec{h}} \sinh\gamma \end{eqnarray*}$$

that may also be called a Gibbs vector (Appendix C.4). The transformation exchanges brightness between the intensity and that component of the polvector that is parallel to $\vec{h}$; any perpendicular component is an eigenvector. I call $\vec{H}$ the polconversion. There is no mutual conversion between polvector components: The transformation is rotation-free. We may summarise the above by stating that the self-aligned source model $\vec{B}'$ is related to the true brighness $\vec{B}$ by a transformation that is the product of a polrotation, a polconversion and a positive scale factor. The poldistortion $\vec{X}$ is far more complicated than a simple scale error. Polrotation and polconversion each contain three unknown parameters (the cartesian components of their Gibbs vectors) which, together with the scale factor, make a total of seven. This is the same number that Paper II arrived at through a more elementary analysis.

$\vec{H}$ and $\vec{Y}$ being unimodular

 

$$\det \vec{B}\equiv \mathbf{I}^2 - \vec{p}^2$$ (16)

is, apart from the scaling of the entire image, an invariant of the poldistortion transformation.

   
5.2 Controlling the poldistortion

With self-alignment only the first half of the calibration job is done. To eliminate the poldistortion, we must bring other information to bear on our problem. It may take the form of either prior knowledge or additional measurements; both may be used either to constrain the self-alignment algorithm so as to (partly) suppress the poldistortion, or to remove the poldistortion $\vec{X}$ afterwards. For example, if we have reason to believe that our source field is unpolarized, we can impose this condition on our image and source model, as in Sect. 6.1 below. Alternatively, we may allow self-alignment to produce a polarized image and determine and remove the polconversion afterwards, on the basis that in the proper image certain sources must be unpolarized. Apart from the image $\vec{B}'$, self-alignment also yields estimates $\vec{J}'_{j \vphantom{'}}= \vec{J}_{j \vphantom{'}} \vec{X}^{-1}$ of the antenna Jones matrices. Comparing these with what we know about the true values gives us another handle on $\vec{X}$.