9 The general polconversion problem

As long as we can rely on an intensity calibration of some sort to suppress polconversion, both the quasi-scalar and matrix methods provide for elimination of the polrotation. The most difficult problem arises when no intensity calibrators are available. The preceding sections provide several leads on how this problem might be approached. I now consider it further.

   
9.1 Use of a priori receiver characteristics

In the absence of poldistortion, prior knowledge about the instrument can be used to optimise the polrotation. An obvious question now is whether that method can take care of polconversion as well. I have not succeeded in making a proper analysis. It is informative, however, to look again at the heterogeneous array, in which we have seen that prior knowledge about the feeds alone suffices to eliminate all polrotation, after intensity self-alignment had suppressed polconversion. In an attempt to eliminate the entire poldistortion without recourse to an intensity calibrator, I modified the least-squares minimisation method of Sect. 8.2 by removing the restriction that $\vec{X}$ be unitary. It appears that there is still a unique solution, but it is not the correct one: The algorithm transforms part of the polconversion factor $\vec{H}$ in the poldistortion $\vec{X}$ through the feeds into spurious amplitude gains in the receiver. It has the freedom to do so because the amplitude-gain matrices are also positive hermitian. To suppress this effect, we need additional prior knowledge, e.g. about those gains.

This example suffices to demonstrate that, if we want to control polconversion by applying prior instrumental knowledge, we need more of such knowledge than to control polrotation. Undoubtedly, observers will find ways to reproduce with full-fledged matrix self-alignment the polarimetric fidelity now obtainable with the linearised approximation. Whether matrix theory will alow us to progress any further remains to be seen.

   
9.2 Maximum entropy

A different approach might be to apply some statistical optimisation criterion representing prior assumptions on the source. It has been suggested that the maximum-entropy (ME) method can also be applied in polarimetric imaging ([Ponsonby 1973]; [Narayan & Nityananda 1986]; [Sault et al. 1999]). The quantity proposed for maximisation is the integral over the image of the product of the eigenvalues of the brightness matrix.

This product equals the value of the determinant (Eq. (16)), $\det \vec{B}= I^2 - \vec{p}^2$. Physically this makes some sense: indeed, maximising it is similar to maximising the unpolarized (and therefore most "disorderly'') brightness $I - \vert\vec{p}\vert$. However, we have seen that $\det \vec{B}$ is invariant under poldistortion. So the ME algorithm must be indifferent to it: out of many possible solutions it will arbitrarily select one, with an unknown poldistortion, -- just as self-alignment does. For the polrotation this is obvious, because the ME criterion provides no clue as to the orientation of the polvector. What my analysis shows is that it does not provide a handle on the polconversion either.

   

Table 2: Comparison of the properties of quasi-scalar intensity selfcal versus matrix self-alignment, both with inclusion of the subsequent poldistortion elimination step

Quasi-scalar	Matrix
Small-error/weak-polarization approximation.	Exact representation.
Homogeneous arrays required.	Arbitrary arrays allowed.
Fragmented stepwise self-alignment, treating successive error types in mutually isolated procedures. Coupling may be introduced through iteration.	Self-alignment includes all errors in one procedure.
Lack of overall perspective obscures view of what actually happens. Scattering and poldistortion intertwined.	Holistic perspective, individual effects clearly separated and identifiable: Alignment, dynamic range, scattering, poldistortion, polconversion, polrotation.
Faraday rotation to be externally calibrated prior to calibrating polrotation.	Faraday-rotation variations absorbed in self-alignment. One overall rotation to be calibrated externally.
Due to intermixing of various effects dynamic range cannot be clearly assessed.	Dynamic range strictly defined in self-alignment, should be comparable to that in scalar selfcal on unpolarized sources.
Second-order effects produce nasty artefacts: $(Q,U) \Rightarrow I$ "inverse leakage'' shows up as interferometer-based errors in the I,V selfcal ([Massi et al. 1996]).	All higher-order effects are properly accounted for.
Intensity selfcal measures leakage terms absolutely per interferometer.	Leakage terms absorbed in antenna Jones matrices derived in self-alignent.
Intensity calibration suppresses polconversion through determination of leakage terms.	Intensity self-alignment suppresses polconversion directly.
Two axes of polrotation suppressed through determination of leakage terms.	Two axes of polrotation suppressed through least-squares fit of feed errors.
Homogeneous array: Phase difference representing third axis of polrotation must be measured
Heterogeneous array: Intractable	Heterogeneous array: Phase difference does not exist as an independent term. It is coupled to the feed parameters and the feed-error fit takes care of it.