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PS reprint
A. G. Willis, L. A. Higgs
Dominion Radio Astrophysical Observatory, P. O. Box 248,
Penticton, BC, Canada V2A 6K3
Assume that we detect a source such as Cygnus A at the edge of a field and that we have a good model for this source. We can describe the relationship between an observed visibility V, and the visibility, M, which the model predicts, by a simple equation of the form
where G is a gain factor and C is a quantity which represents the contribution to the visibility from the radio emission in the rest of the field.
Here all quantities are complex. When we observe a celestial source for a specific length of time, we obtain a sequence of observed visibilities as a function of time. If we compute visibilities predicted from the model source at the same times and assume G and C to be approximately constant over the time interval, then we can solve for G and C by simple least-squares fitting techniques. If we wish to minimize the combined real and imaginary residuals we obtain the four linear equations
where N is the total number of visibility samples, and the r and i subscripts indicate the real and imaginary components of the complex quantities.
We can easily solve for the unknown gains and constants 
, 
, 
,
by standard matrix inversion techniques.
At DRAO we presently do the above least squares analysis on one hour's worth of visibility data. Since a standard observation with our telescope requires 12 hours to completely sample the UV domain, we end up with 12 sets of gains and constants for each baseline.
To obtain these solutions we had to assume that we are solving for
constant values of 
and 
over each time interval. This is usually incorrect, and
contaminates the solutions for 
and 
, especially
for those baselines where C represents a significant contribution
to the total visibility. As we shall see below, we can get around this problem
by an iterative procedure.
Once we have obtained our 12 samples of 
and 
we do a least squares fit of
a polynomial to the 12 samples. We then use
this best fitting polynomial to determine a 
and 
at
each original sample point of a given baseline in the UV plane.
Complex multiplication of the model visibility at each sample point
by the corresponding gain 
gives us a
best fit contaminating visibility which we subtract from the original
visibility V.
We now have a final data visibility, 
from which we have subtracted
our best guess as to the visibility of the contaminating source.
This procedure is the exact opposite of the standard self-calibration method. In self-calibration, the observed visibility data is iteratively corrected toward source model visibilities (see, for example, Cornwell 1985); in MODCAL, source model visibilities are corrected toward observed visibilities.
Old Cambridge One-Mile Telescope 21cm images of Cygnus A, Cass A and Taurus A serve as models for these sources. These images contain Fourier components well beyond the maximum required by the DRAO Synthesis Telescope's longest baseline of approximately 600 meters.
To get the best results we proceed in an iterative manner. We do an initial run of MODCAL to get a preliminary subtraction of the interfering source. This subtraction will probably not be perfect, and may introduce some minor artifacts near the field center. These artifacts occur because of our intermingling of the G and C terms in the least squares fit. As we might expect, the artifacts tend to have low spatial frequencies---indicative of poor solutions on the short spacings, where the contribution of C will be dominant.
Once we have a preliminary removal of the interfering source, we can usually get a good model of the emission at the field center by a combination of CLEANing and self-calibration. We can now proceed to improve our MODCAL solution by inverting this model into the UV plane and subtracting its visibilities from the original observed visibilities. The majority of the remaining signal will be due to the contaminating source; the C component of the observed visibilities should be minimal.
We then proceed to do a second MODCAL on these altered visibilities. Now we will get a very good least squares estimate of the gain G regardless of baseline, and consequently, a very good subtraction of the contaminating source. Finally, we add back to the second set of modified visibilities the visibilities generated by our CLEAN component model of the emission at the field center.
We now have a set of visibilities from which we can generate images of the field undisturbed by the contaminating source. An example of an image corrected by the procedure is shown in Figure 1.
Figure 1: A DRAO Synthesis Telescope 74 cm (408 MHz) image of the CTB 80 region.
In the left image, Cygnus A is detected with a peak flux density of
approximately 36 Jy at a distance of 8.2 degrees to the north of the field
center. Low level grating artifacts having amplitudes of 50 to 100 mJy
prevent our studying the emission regions at the field center. These grating
artifacts cannot be successfully removed by conventional CLEAN or
self-calibration procedures. The right image shows the field after
removal of Cygnus A with the MODCAL procedure. All that is left of
Cygnus A are a few faint spokes and stripes centered at its position
with amplitudes in the range 10 to 50 mJy. The astronomer can
now study the radio emitting regions at the field center to a level
below 25 mJy per beam.
Figure 1: (left) PS 527 Kb,
Figure 1: (right) PS 527 Kb
We have found other innovative uses for MODCAL. The DRAO Synthesis Telescope is now routinely used to make polarization maps at 21 cm. Significant instrumental polarization is present at large distances from the field center and grating rings from these features may contaminate the central region of the image. By using a good quality total intensity image and the MODCAL procedure we can successfully remove the instrumental polarization effects. An example result is shown in Figure 2.
Figure 2: A DRAO Synthesis Telescope 21cm (1420 MHz) image of polarized
emission (Stokes U) from an area near W3 and 3C58.
In the left image we see strong apparent U emission from
W3 (at the left edge of the image) due to instrumental polarization
and from 3C58 near the top, along with
associated grating rings. The second image shows the field after application
of the MODCAL technique. The effects of the strong sources have been
almost completely removed so that we can more easily study the real U emission
from the central part of the field.
Figure 2: (left) PS 531 Kb,
Figure 2: (right) PS 531 Kb