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Astron. Astrophys. 355, 552-563 (2000)

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Appendix A: Hybrid Double Mappping (HDM)

A.1 Principle of Hybrid Double Mapping
The visibility function, V, measured at time t, on a baseline between antennas i and j, is represented by a complex function with amplitude A, phase P:

[EQUATION]

For this analysis it is convenient to indentify 3 contributions to the visibility phase:

[EQUATION]

where

  • [FORMULA] is due to source structure, evaluated w.r.t. a reference position for the source.

  • [FORMULA] is due to any offset of the true source position from the reference position.
    Both [FORMULA] and [FORMULA] are functions of the resolution coordinates, u and v, at time t.

  • [FORMULA] is due to inaccuracies in the correlator model calculation of the interferometer geometry and the signal propagation delays in the ionosphere, troposphere and receiving system; it is an unknown function of time.
    This term can be represented by the difference of two "antenna-based" phases, [FORMULA] and [FORMULA], since it can be related to the difference in signal arrival times at the two sites. (This analysis is a simplification which ignores possible "non-closing" instrumental baseline phase terms arising from e.g. un-matched bandpasses and polarisation impurities.)

[EQUATION]

In conventional hybrid mapping, an iterative procedure is used to separate out the antenna-based phase terms from the "source" terms; the latter must produce a consistent and physically plausible source structure after Fourier transformtion of the corrected visibility:

[EQUATION]

However, the position offset term, [FORMULA], can also be expressed as a difference in wavefront arrival times at the 2 antennas and so it is also "absorbed" in antenna phase terms [FORMULA], [FORMULA]; the "absolute" position information is lost:

[EQUATION]

In Hybrid Double Mapping (HDM), the visibility functions of two sources observed simultaneously are added. For a close source pair, we make the same assumption as for conventional phase-referencing - that the model error phase terms are essentially the same for both sources. We make a further assumption that the [FORMULA] coordinates are also essentially the same for both sources, for each baseline and time. The visibility sum, [FORMULA], can then be re-written:

[EQUATION]

[EQUATION]

This may be recognised as the visibility function of a "composite" source consisting of the sum of the brightness distributions of sources 1 and 2, with antenna-based phase error terms [FORMULA], [FORMULA], as before. The HDM method consists of performing the normal hybrid mapping procedure with the visibility sum, resulting in the separation of the antenna-based errors, and a physically plausible map of the sum of the two source brightness distrubutions. An important point is that, whereas the origin of the map of the composite source is arbitrary (as it depends on the position of the starting model), the separation of the two source brightness distributions within the composite map (determined by [FORMULA]) is fixed during the phase separation procedure, and is equal to the difference of the errors in the two source positions used for correlation. We call this the "residual separation".

A.2 Practical aspects

There are some practical aspects to be considered. If the source coordinates used in the correlator model are very precise, then the residual separation may be less than the interferometer beamwidth, and the two source distributions will lie on top of each other. In this case it is desireable to introduce an artificial position offset into one of the source visibility functions before forming the visibility sum, to ensure that the two source reference features are well separated in the HDM map. One should also arrange that the peak of one source does not lie on the sidelobe response of the other in the "dirty" map, as this may degrade the CLEAN deconvolution process in the mapping step.
Another important consideration is that the time-averaged samples of the summed visibility function contain equal contributions from both source visibility functions. When both sources are observed simultaneously this will normally be the case, except when different amounts of data are lost in the two separate correlator passes needed for the two source positions. It is important to edit the data sets carefully to fulfil this condition.
The range of validity of the assumption that the [FORMULA] coordinates for the two sources are the same depends on the "dilution factor", i.e. the reciprocal of the source separation, measured in radians. The [FORMULA] value assigned to the summed visibility will be incorrect for either source by roughly 1 part in the dilution factor (roughly 1 in 6000 for 1038+528A,B). This is equivalent to having source visibility phase errors of this order, and thus limits the size of an HDM map to be less than the beamwidth times the dilution factor; the residual separation should be much smaller than this value.
In the actual analysis used in this work, we first made a rough correction to the phase of the summed visibility of 1038+528 A + B, using the antenna phase and phase derivative errors from fringe-fitting 1038+528A using a point source model. However, there is no reason why one should not fringe-fit the summed visibility function directly.

A.3 Applications

The HDM method can in principle be applied whenever two (or more !) radio sources are observed simultaneously, but are correlated at separate field centres; however, they must be close enough so that the conditions of same [FORMULA] coverage and same correlator model errors apply. The method uses the structures of BOTH sources simultaneously to separate out the antenna phase errors, as opposed to a single source in simple hybrid mapping. If both sources are strong (as with 1038+528 A and B), constraining the (single) antenna phase solutions with two structures should lead to a more rigorous and robust separation between the source and antenna phase terms. One field of application is in high resolution VLBI imaging of gravitational lens systems with wide image separations (e.g. images A and B of QSO 0957+561 with 6.1 arcsec separation) where preserving the necessary wide field-of-view from a single correlation may result in inconveniently large data sets. When one source is very weak, however, there is probably little to be gained over normal hybrid mapping.
For relative astrometry studies (as described in this paper), the HDM method has some advantages over conventional phase-reference mapping and explicit phase-differencing methods. In phase-differencing astrometry, separate hybrid maps must be made of both sources to correct for source structural phase terms and the antenna phase errors are NOT constrained to be the same. Imperfect separation between source and antenna phase terms can increase the noise on the differenced phase, as well as lead to possible systematic errors. In phase-reference astrometry, only one of the sources is used to solve for the antenna phase terms; imperfect separation can lead to extra phase noise in the phase-referenced visibility of the "target" source. In HDM we use both source structures simultaneously to separate the (common) antenna error phases from that of a single "structure" in which the reference points of the two sources are spaced by the residual separation.
When the separation between the two sources of a pair exceeds the telescope primary beamwidths, astrometric and phase-reference observations must involve switching between the sources, and the visibility phase of at least one of the sources must normally be interpolated in the observing gap. The condition that must be fulfilled for HDM to work in this case is that an equal number of observations of both sources must be added to form an average visibility function for the length of the "solution interval" in the phase self-calibration step of HDM. This length is generally limited by the coherence time of the atmosphere, and would imply a very fast switching cycle in most cases. Another application for HDM could be in the the analysis of "cluster-cluster" VLBI (see e.g. Rioja et al. 1997b), in which two or more sources are observed simultaneously on VLBI baselines by using more than one telescope at each site.

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Online publication: March 9, 2000
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