4 Matrix self-alignment

   
4.1 Self-alignment

The argument of the preceding section carries over in its entirety to the matrix domain. We may follow through the same steps, simply replacing all scalar gains by Jones matrices and visibilities by coherency matrices. We must now solve the matrix equivalent of Eq. (10):

$$\begin{eqnarray*}\vec{V}(\vec{r}_{jkt \vphantom{'}}) = \vec{J}'_{jt \vphantom{'}... ...r}_{jkt \vphantom{'}})\ \vec{J}'_{kt \vphantom{'}}{}^{\dagger}. \end{eqnarray*}$$

The solution is the analogue of Eq. (13)

 

$$\vec{B}'(\vec{l}) = \vec{X}\, \vec{B}(\vec{l})\, \vec{X}^{\dagger}.$$ (14)

This equation is known as a congruence transformation. I shall follow mathematicians in referring to Eq. (14) as "the congruence transformation $\vec{X}$'', i.e. using the name of $\vec{X}$ as a synonym for the transformation that it effects: The source model $\vec{B}'$ is related to the true source $\vec{B}$ by an unknown congruence transformation $\vec{X}$.

Like Eq. (13) for scalar selfcal and with the same proviso, this is a basic relation that any matrix self-alignment solution must satisfy. And as for the scalar case, the appearance of this indeterminacy is fundamental and unavoidable. I call it the poldistortion. Although this result is formally the same as for scalar selfcal, it is worth some extra thought. We see that all the, probably time-varying, errors in the observation have given way to a single poldistortion representing a set of unknown errors that is constant over the observation. A similar effect occurs, e.g., in quasi-scalar interferometry, where scalar selfcal on the "parallel'' channels takes away the temporal gain/phase variations and leaves a single, constant XY or LR phase difference in their place. In matrix self-alignment, the time-varying errors to be converted into unknown constants include not only phases and gains, but also any other variations, e.g. in feed parameters (mainly the "leakage'' or "D'' terms in the quasi-scalar jargon), in parallactic angle (if one were not to correct for it beforehand) and in ionospheric Faraday rotation. Moreover, all this is true not only for a single observation contiguous in time, but also for a set of observations spaced over a time interval in which the source does not change.

A schematic of the combined self-alignment and poldistortion elimination procedure is shown in Fig. 1. As in scalar selfcal, we may first correct for the errors that we know of. Note that our argument does not require this; however, the actual selfcal algorithm ([Thompson et al. 1986]; [Perley et al. 1994]) starts with an image, to be made from the raw observations, that must be good enough to extract a reasonable initial source model from it. If such a model can be obtained otherwise, e.g. from prior knowledge about the source, the initial corrections may just as well be omitted: they will automatically be subsumed in the corrections to be derived in self-alignment.

Whether or not we apply the prior corrections is likely to affect the poldistortion in the final solution, but not our ignorance about its value. No matter how we arrive at a self-aligned image, we must assume that it contains an unknown poldistortion and undertake to eliminate it. This problem will be discussed below.

As an aside, I observe that Fourier transforming Eq. (3) for a single dish, one obtains an equation of the same form as Eq. (14): the self-aligned array is equivalent to a single dish with unkown Jones matrix. This result was derived in Paper II by a more qualitative argument.

Figure 1: Flow diagram of the self-alignment and poldistortion calibration procedure. The crux of the matter resides in the self-alignment: it effects a split between the true coherencies and the time-variable part of the observing errors. The price to be paid is the insertion of an unknown constant error artefact, the poldistortion, in the estimated coherencies and its inverse in the estimated Jones matrices. To eliminate it, we must compare these estimates with a priori instrumental and astronomical information

   
4.2 Scattering dynamic range and polarimetric fidelity

The solutions Eqs. (13) and (14) of the selfcal/self-alignment problem represent in-place transformations of the brightness: a simple uniform scaling in the scalar case, a uniform poldistortion in the matrix case. Their common characteristic is that they do not scatter radiation out of any point of the source into any other position in the image. In the scalar case, the scaling being the same for the entire image means that our image is a faithful (although scaled) replica of the source: dynamic range is conserved, - and it is this feature that makes selfcal such an important and valuable tool.

In the matrix case, image fidelity and dynamic range are no longer synonymous. Self-alignment suppresses spatial scattering in the image; the residual scattering defines to what extent weak structures remain recognisable in the presence of strong features elsewhere in the source. The concept of dynamic range is appropriate to describe this effect, but its proper quantitative definition is not as obvious as in the scalar case.

On top of any residual effect of scattering comes the poldistortion that is independent of it. Even if we were to produce a truly scatter-free image it would still misrepresent the source in an unknown way: as a complement to dynamic range we must consider the question of the polarimetric fidelity of the image.

We have no need for a formal definition of either dynamic range or polarimetric fidelity. What matters is that we are dealing with two quite different and mutually independent types of error in a self-aligned image.

   
4.3 Example: Faraday rotation

The concepts developed above and the benefits of the matrix approach can be neatly illustrated on the example of ionospheric Faraday rotation. This rotation changes the observed position angle of linear polarization. Over a synthesis observation, it varies and this results in scattering of linearly polarized brightness in the final image.

Corrections for the effect are based on external data: Ionosphere models and ground- and satellite-based measurements ([Thompson et al. 1986]). Often the results leave much to be desired. In some cases, the external correction can be improved upon by noting the apparent rotation of linear polarization during the observation (A.G. de Bruyn, private communication). One might call this Faraday self-aligment. In a similar vein, Sakurai & Spangler (1994) used a linearly polarized source to monitor temporal fine structure in the Faraday rotation. Obviously, one can only measure and eliminate variations in the rotation; to find the true position angle of linear polarization one must determine its zero point by other means. In the matrix approach, this adjustment of Faraday rotation is no longer a distinct operation: it is subsumed in the overall self-alignment process. A scatter-free image results directly. The absolute rotation and position angle of linear polarization remain undetermined: this is now recognised as part of the poldistortion that is the unavoidable by-product of self-alignment. Obtaining a good dynamic range and correct rendition of the polarization appear as two distinct problems that must be independently addressed.