## Appendix A: Hybrid Double Mappping (HDM)
For this analysis it is convenient to indentify 3 contributions to the visibility phase: where -
is due to source structure, evaluated w.r.t. a reference position for the source. -
is due to any offset of the true source position from the reference position. Both and are functions of the resolution coordinates,*u*and*v*, at time*t*. -
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, and , 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.)
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: However, the position offset term, , 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 , ; the "absolute" position information is lost: 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 coordinates are also essentially the same for both sources, for each baseline and time. The visibility sum, , can then be re-written: 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
, ,
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
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.
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 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. © European Southern Observatory (ESO) 2000 Online publication: March 9, 2000 |