3 Modifications to the self-calibration algorithm

3.1 Different size antennas

  I now turn to the problem of processing images generated from an array having antennas with different HPBWs (101.8 and 108.8 arcmin at 21 cm in the case of the DRAO ST). Measurements of the separate antenna patterns made by Dougherty (unpublished DRAO technical memos) show that at 21 cm the maximum difference in response is about 5% at a distance of 64 arcmin from the field centre (the primary beam response of the 9 m antennas is 32.0% of the peak and that of the 8 m antennas is 37.1% of the peak at this distance; the weighted mean attenuation is 36.0%). If one happens to detect a strong source at a fairly large distance from the field centre, this discrepancy will limit both the dynamic range attainable through standard self-calibration procedures and how well one can clean the associated image. The synthesized antenna pattern (PSF) used to compute the response to clean components is derived solely from the UV locations sampled by the array as it tracks the field. In reality, however, each source in a DRAO field will have a slightly different PSF that depends on its distance from the field centre because of the different sensitivities of the individual antenna pair combinations that sample the UV plane. Also, if one has a strong source at a large distance from the field centre and weak sources elsewhere, the contribution from the strong source will dominate the gain solutions derived from self-calibration causing the gain solutions to be incorrect for sources elsewhere in the field.

This effect is quite noticeable if one observes a strong source situated at a distance from the field centre where the beams are significantly discrepant. Figure 1 shows the results of a 12 hour test observation at 21 cm wavelength in which the ST field centre was placed at a distance of exactly 64 arcmin from the strong source 3C 147. There should be a maximum discrepancy between the primary beams of the large and small antennas at this distance. The image shows the results of doing standard cleaning and phase-only self-calibration under the assumption that the antennas all have the same HPBW. Residual grating rings are present because of the discrepancy between the actual and assumed PSF response.

  

Figure 1: A DRAO Synthesis Telescope 21 cm (1420 MHz) image of an area containing the strong source 3C 147 at a distance of 64 arcmin to the south of the field centre. The image has a width of 2.8 degrees. It shows the results of doing a standard phase-only self-calibration and clean of the field. Residual rings having amplitudes in the range 6 to 30 mJy (0.09 to 0.44% of the peak signal) surround 3C 147, which has an attenuated flux density of 6.7 Jy

  

Figure 2: The 3C 147 field after first subtracting off model visibilities corresponding to clean components in the UV plane and then adding these clean components back into the image when the clean restoration is done. The rms noise in the vicinity of 3C 147 is reduced to 1.3 mJy, or 0.02% of the peak signal. The image half-tone scale is similar to that of Fig. 1

How does one proceed to solve this problem? When doing self-calibration one normally obtains antenna-based amplitude and phase corrections by feeding the gain solver a collection of normalized baseline data, D / M, where D is the observed visibility and M is the model visibility. In the present case one must compute 3 separate sets of model visibility data corresponding to three separate model skys for antenna pair combinations L $\times$ L, L $\times$ B and B $\times$ B starting from an image which represents a weighted mean of these combinations. (Here, L = little antenna, B = big antenna).

In order to obtain the model visibilities one does the following steps:

1) Clean the combined image using the Cotton-Schwab algorithm (see Cotton 1989, and Cornwell & Braun 1989). Then for each clean component with derived signal S from the combined image determine its distance from the field centre.

2) Calculate its value on the sky before attenuation by the mean synthesized beam (i.e. correct the clean component for the mean primary beam attenuation A)

S' = S / A

where A is the mean primary beam attenuation.

3) Then compute the signal S'' which would be seen by a particular antenna pair combination

$$S'' = S' \sqrt{A_1A_2}$$ (4)

where A₁ is the attenuation factor for the first antenna of the baseline antenna pair and A₂ is the attenuation factor for the second member of the pair.

4) Compute the model visibilities M for the baseline associated with the antenna pair by taking the Fourier Transform of the `image' of all the S'' clean components and divide the observed visibilities, D, by M to get the normalized signal D/M for the baseline.

5) Repeat this procedure for each baseline in the array, and feed the normalized visibilities into the self-calibration solution solver, to get antenna-based amplitude and phase corrections that should be independent of the positions of the clean components in the field.

6) Apply the resulting antenna based amplitude and phase corrections to the observed visibilities in the usual way to self-calibrate the data.

Then one would usually make a new image from the corrected visibility data and clean the resulting image. If the previous six steps have gone according to plan the new image should have an improved dynamic range, and one may go further with another round of self-calibration by repeating steps 1) to 6) again. This cycle may take place several times until little or no improvement is seen in the final image.

The resulting restored image should have a higher dynamic range and indeed it does for the 3C 147 field as can be seen in Fig. 2.

The Cotton-Schwab algorithm works because the model visibilities that are computed in steps 3) and 4) above contain adjustments for the separate antenna attenuations and thus agree with the true instrumental response to the incoming signal. Since these visibilities are subtracted off in the UV plane, which is then inverted back to the image plane for the next clean cycle, one avoids large discrepancies between the actual and assumed PSF response.

The current DRAO software package does not have a direct equivalent of the Cotton-Schwab variant of clean, which is implemented in the AIPS task MX. However, a strategy of cleaning an image to a level to somewhat above about 2% of the initial peak signal (and thus avoiding the error levels shown in Fig. 1), subtracting off the corresponding visibilities in the UV plane, generating a new image from the residual UV data, and then cleaning this residual image, etc., allows one to obtain a reasonable approximation to the Cotton-Schwab algorithm. Before generating the final restored image all clean components obtained in the various steps are added together.

The strategy of doing self-cal with big and little antennas together is necessitated by the fact that there is a total of only 21 correlated baselines in the DRAO ST array. If one had an array with large numbers of both big and little antennas then one could investigate algorithms where data from sub arrays of big antennas was processed separately from data obtained with the small antennas.

3.2 Variable source removal  

  

Figure 3: A DRAO Synthesis Telescope 21 cm (1420 MHz) image of an area containing PSR1930+20. The image, 2.8 degrees across, shows the field as seen before any processing. The pulsar itself has a mean flux of about 189 mJy. The brightest source in the field has a signal of 2 Jy

As I mentioned earlier in Sect. 2, a source whose signal varies significantly over the time period of a synthesis observation violates the assumption that the sky is invariant and therefore the standard clean algorithm will fail to remove the effects of such a source. Figure 3 shows a field surrounding pulsar PSR 1939+20 before any image processing has been done. Figure 4 shows the same area after standard cleaning and self-calibration has been done.

If the variable source is sufficiently strong one can remove its effects by using a variant of the standard self-calibration procedure. In standard self calibration (see the beginning of Sect. 2) by performing the operation of dividing the data by the calculated gain (D/G) one effectively "adjusts" the data toward the model and hopefully improves the data.

However, there is nothing to stop the astronomer from taking an opposite approach with a variable source. Use an imperfect model, the time averaged signal (represented by the clean components found in the 12 day averaged image, or the flux density measured in the averaged image), and move this imperfect model toward the visibility data (which one can consider as perfect since the visibility data contain the time varying information). One can then obtain the time varying gain corrections which should be applied to the model to "degrade" the model toward the data.

  

Figure 4: The pulsar field after standard cleaning and self-calibration has been done. Because of time variations in the pulsar signal, residual grating rings and spokes having amplitudes of up to 3% and 2% respectively of the mean flux surround the pulsar

  

Figure 5: This image shows the PSR1930+20 field after application of the inverted self-calibration procedure. The remaining spokes due to the pulsar have a level of about 0.5% (1 mJy) of the pulsar's mean signal. Some residual rings with amplitudes 1 to 2% of the source peaks are visible around several sources in the field

The algorithm contains the following steps:

1) Obtain a visibility data set D' in which one has deleted the signal E from all other sources in the field as best one can

D' = D - E.

Since D' is used to compute gain solutions for the variable source, it is critical that the signal E contains a very good estimate of the flux from all other sources in the field.

2) Take visibilities M computed from the imperfect (time averaged) model of the variable source and, using D' and M, calculate the least-squares antenna-based gain corrections as a function of time using a standard self-cal procedure; then obtain the corresponding baseline-based gain corrections G from Eq. (1) and Eq. (2).

3) After obtaining the gain corrections G, instead of dividing the visibility data D' by G as was done in Sect. 2, multiply the model visibilities by the gain corrections and subtract the resulting visibilities from D'

D'' = D' - GM.

One should end up with a visibility set D'' which is essentially the noise during the observation.

4) Finally, to get a set of visibilities which represent the proper time-averaged field add visibilities for E and M back into the data.

D''' = D'' + E + M.

Images made from visibilities D''' should give a very good time-averaged image for the field. Figure 5 shows the pulsar field after use of the above algorithm. Most of the effects due to the pulsar have been removed. (The ability of this method to remove very short time fluctuations will obviously depend on the intrinsic signal-to-noise ratio of the source. The weaker the source, the longer the time interval over which we must average the data in order to get acceptable solutions for the gains (Cornwell & Fomalont 1989).)

  

Figure 6: This image shows the PSR1930+20 field after applying a baseline-based averaging procedure to the individual baseline data. The remaining spokes due to the pulsar have a level of about 0.2% of the pulsar's mean signal. Residual rings around other sources in the field were removed by the MODCAL algorithm described in Sects. 4.1 and 4.5. The RMS noise in this field is about 0.3 mJy

As an alternative to step 2) above, one can directly start with a baseline-based approach in which one computes an average gain correction G for each baseline

G = D' / M

over some time interval. Figure 6 shows the results of using this method. The baseline-based approach (Fig. 6) seems to give slightly better results as far as subtracting off a time variable source is concerned. However some distortion of the noise in the area close to the pulsar occurs because the gain correction, G, is derived from simple baseline averaging, rather than from an antenna-based least-squares fit.

Note that residual grating rings visible around some of the sources in Fig. 5 are no longer visible in Fig. 6. They were removed by the MODCAL algorithm described in Sects. 4.1 and 4.5.