4 The MODCAL procedure

4.1 Theory

 

Once the primary beam attenuates a signal to about 5% of its intrinsic value, the attenuation factor usually has quite a large error and consequently a calculation of the actual value becomes meaningless. Other instrumental effects such as bandwidth smearing and time smearing also distort the image at large distances. Consequently, for example, CGPS 21 cm continuum mosaics cut off the field of view of an individual field being joined to the mosaic at a radius of 100 arcmin. This distance corresponds to a primary beam attenuation of 6.6% of the intrinsic value of the signal.

I mentioned above that because the DRAO ST has a wide field of view and a narrow bandwidth there is a good chance that any single field is contaminated by responses to one of Cyg A, Cas A, or Tau A. A typical field in which Cyg A is detected is shown in Fig. 7. Other contaminating signals come from man-made interference and solar emission. In all these cases, if one assumes a purely geometic relationship between the source position and the telescope baseline (see Eqs. (A2) and (A3) of the Appendix), computation of model visibilities from clean components, followed by subtraction of the model visibilities from those observed, usually does not lead to successful removal of the contaminating source. However I shall now describe a method which has been very successful at removing the effects of such sources from DRAO fields, even when the exact instrumental response (such as a sidelobe gain) to the incoming signal is unknown.

  

Figure 7: A DRAO Synthesis Telescope 74 cm (408 MHz) image centred on the CTB 80 region. The total field of view is about 19.7 degrees. Cyg 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 centre. Low level grating artifacts having amplitudes of 50 to 100 mJy prevent our studying the emission regions at the field centre. These grating artifacts cannot be successfully removed by conventional clean or self-calibration procedures

Usually, the initial step is the derivation of model visibilities for the contaminating source. Then, for each baseline in the array one compares the model visibilities with those observed and calculates time dependent complex gain factors. These gain factors are used to degrade the model visibilities toward those actually observed, and the degraded model visibilities are then subtracted from the observed visibilities. The algorithm to determine these gain factors is called MODCAL (after MODel CALibration). It is derived from a procedure originally developed by Lloyd Higgs to improve the dynamic range of the four antenna version of the ST that was in service before the current seven antenna system came on-line in 1991. An initial description of this algorithm was given in Willis & Higgs (1996).

In order to remove a contaminating source by using the MODCAL algorithm one proceeds in detail as follows:

1) Compute model visibilities M for this contaminating source and model visibilities E for the remaining sources in the field.

2) Delete the model visibilities E corresponding to all other sources in the field from the observed visibilities D to obtain visibilities D'.

 

D' = D - E. (9)

It is desirable that the signal E contains a very good representation of the signal from all other sources in the field. This may not always be possible in practice, and some iteration is usually required (see Sects. 4.2).

D' should now represent the observed response to the contaminating source. At some particular instant in time the relationship between the observed visibility, D', and the contaminating source model visibility, M, can be described by a simple equation of the form

D' = GM + C

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. If E accurately represented the contribution to the observed visibilities from other sources in the field, C should have a minimal value. All quantities are complex.

3) If G and C are assumed to be approximately constant over some time interval, then one can now solve for G and C in the time interval by simple least-squares fitting techniques. If the combined real and imaginary residuals are to be minimized one obtains the two complex equations

$$\sum_{i=1}^N D_i^{'} M_i^* = G \sum_{i=1}^N M_iM_i^* + C \sum_{i=1}^N M_i^*$$ (11)

and

$$\sum_{i=1}^N D_i^{'} = G \sum_{i=1}^N M_i + C\sum_{i=1}^N 1$$ (12)

where M^* is the complex conjugate of the model visibility and N is the total number of visibility samples in the time interval. One can easily solve for the complex G and C terms by standard matrix inversion techniques.

At DRAO the above least squares analysis is usually done on one hour's worth of visibility data. Since a standard observation with this telescope requires 12 hours to completely sample the UV domain, one ends up with 12 values of G and C for each baseline. To obtain the MODCAL solutions for G and C one must assume that one is solving for a constant value of C over each time interval. This assumption is usually incorrect, and may affect the solution for G, especially for any baselines where C represents a significant contribution to the total visibility. As I will show in the next subsection, one can get around this problem by an iterative procedure.

4) Once the least-squares values of G have been obtained, one fits sequences of orthogonal polynomials to their amplitude and phase equivalents using the method of Peterson (1979). These best fitting polynomial sequences are then used to calculate a complex gain adjustment, $G_{\rm adj}$, at each original sample point of a given baseline in the UV plane. Multiplication of the model visibility at this location, M, by $G_{\rm adj}$gives one a best-fit contaminating visibility. Subtraction of this contaminating visibility from the original visibility D' yields a data visibility, D'', which should not have any signal due to the interfering source.

$$D'' = D' - G_{\rm adj}M.$$ (13)

5) Finally, to get a set of visibilities which represent the observed field without the contaminating source, add visibilities E back into the data.

D''' = D'' + E.

Images made from visibilities D''' should give a high quality view of the field.

One can do least-squares fitting on one-hour bins of DRAO data followed by further fitting of orthogonal polynomials because the DRAO antennas have equatorial mounts. Consequently most unknown instrumental responses to an incoming signal are expected to change slowly, if at all, as a function of time. This will not necessarily be the case for arrays of telescopes with altazimuth mounts, such as the VLA, and it is not clear that the MODCAL algorithm has any applicability to such arrays.

4.2 Removal of strong sources

  If one wishes to remove the effects of sources such as Cyg A, Cas A and Tau A from a field, Cambridge or VLA images of these sources are normally used to compute model visibilities by the procedure described in the Appendix. These images contain Fourier components well beyond the maximum required by the DRAO Synthesis Telescope's longest baseline of approximately 600 metres.

The accuracy with which model visibilities can be computed for the DRAO ST from images obtained with other telescopes depends on how well one can emulate the instrumental responses of the ST. For example, the effect of bandwidth smearing on model visibilities is currently calculated under the assumption that the bandpass function is square (Eq. (A1)). However this theoretical bandpass deviates in shape from what is actually found. Recent measurements by Thorsley (unpublished DRAO technical memo) show that the 21 cm passbands are skewed and asymmetrical in shape, and differ from baseline to baseline. Consequently the model visibilities will be in error. However, this error should be compensated to some extent, by a counteracting adjustment to the gain factors G that will occur during the least-squares fit for the determination of G. Therefore the complex visibility, GM, that one wants to subtract from the original visibility, D', should be relatively insensitive to errors in the modelling procedure.

(The asymmetric bandpasses are certainly one of the limiting factors in our ability to obtain high dynamic range in DRAO images by standard self-calibration procedures. Future enhancements to the modelling software will take this property into account.)

To get the best MODCAL results one proceeds in an iterative manner. Do an initial run of MODCAL on the original observed visibility data D to get a preliminary subtraction of the interfering source. This subtraction will not be perfect because the visibilities being fitted still contain the contribution E from the other sources in the field, and images produced from the MODCAL output visibilities may contain some minor artifacts near the field centre. These artifacts occur because of our intermingling of the G and C terms in the least squares fit. As one 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.

  

Figure 8: The CTB 80 field after removal of Cyg A with the MODCAL procedure. All that is left of Cyg A are a few faint spokes and stripes centred at its position with amplitudes in the range 10 to 50 mJy. The astronomer can now study the radio emitting regions at the field centre to a level below 25 mJy per beam

Once one has a preliminary removal of the interfering source, one can usually get a reasonable model of the emission at the field centre by a combination of cleaning and self-calibration. One can now proceed to improve the MODCAL solution by inverting this model into the UV plane and subtracting its visibilities (E) from the original observed visibilities D. The majority of the remaining signal D' will be due to the contaminating source; the C component of the observed visibilities should be minimal.

Then proceed to do a second MODCAL on these altered visibilities as described in Sect. 4.1. Now one 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, add back to the second set of modified visibilities D'' the visibilities E generated by the clean component model of the emission at the field centre.

One now has a set of visibilities D''' from which one can generate images of the field undisturbed by the contaminating source. An example of an image corrected by the procedure is shown in Fig. 8.

4.3 Removal of solar interference

At 74 cm, solar interference can be a problem as the Sun is often detected in antenna sidelobes; it then produces significant fringes on shorter baselines. One does not want to flag short spacing data contaminated by solar emission unnecessarily, since short spacings contain information about the broad structure of a field. A simple solution which allows the astronomer to remove most of the solar interference is to proceed by approximating the solar image as a uniform disk with a radius of 17.5 arcmin.

The Fourier transform of a uniform disk is a first order Bessel function (Bracewell 1965); so one can easily calculate the visibility amplitude at a given distance from the UV origin as

$$A = R J_1(2\pi RD) / D$$ (15)

where A is the visibility amplitude, R the solar radius in radians, D the distance from the UV origin in wavelengths, and J₁ is the first order Bessel function.

Using this equation for the amplitude and an ephemeris giving the Sun's position as a function of time, one computes model visibilities for those hour angles where the Sun was visible above the local DRAO horizon. The solar fringes can then be removed by application of the MODCAL algorithm.

An excellent example of solar removal by means of this method is seen in a CGPS survey field centred just to the southwest of the W3, W4, W5 complex. Figure 9 shows the image made from the original data. In addition to the objects of interest, a significant ripple is present in the field.

After application of the MODCAL removal procedure, one obtains the image shown in Fig. 10. The low level solar waves have completely disappeared, and the astronomer can proceed with cleaning, self-calibration etc.

  

Figure 9: A DRAO Synthesis Telescope 74 cm (408 MHz) image of an area to the southwest of the W3, W4, W5 complex. The field is 9.9 degrees across. The low level ripples with an amplitude of approximately 30 mJy that are present throughout the image are due to the Sun

  

Figure 10: The image of the W5 area after processing the visibilities with the solar removal procedure outlined in the article. Some low level systematic offsets with an amplitude of approximately 8 to 25 mJy remain. These irregularities are caused by a lack of short UV spacings in the synthesized image

4.4 Removal of system and man-made interference

  The previous two subsections described how to remove interfering objects by starting with some predefined model, either a high resolution image, clean components, or an abstract mathematical definition. However the MODCAL removal procedure usually works very well to remove features that do not have either a predefined structure or known origin. Figure 11 shows another 74 cm field near Cyg A after initial removal of Cyg A by MODCAL. A faint grating ring is visible crossing through the lower middle part of the image. Some detective work revealed that the cause of this artifact is a "ghost" Cyg A (see Fig. 12) located 12 h away in right ascension from Cyg A but at the same declination! The exact cause of this artifact is not known but it is probably caused by some error in the 408 MHz fringe stopping or delay system. I infer this because the phase of an object varies according to a $\sin(t)$ term (Eq. (A3)) where t is the hour angle. Since $\sin(t+\pi)$ equals $-\sin(t)$, I suspect that incorrect fringe stopping or delay setting is somehow injecting a small percentage of the Cyg A signal into a negative $\sin$ term.

  

Figure 11: 74 cm wide field image (23.9 degrees on one side) of a field after initial removal of Cyg A by the MODCAL algorithm. Cyg A has been removed reasonably well, but a faint low-level ring-like feature, having a systematic amplitude offset of 100 to 300 mJy, remains, stretching from the lower left to the lower right part of the image. In the original image Cyg A had an attenuated peak flux density of 2300 Jy. Diffuse emission from other objects in the field that is as faint as about 200 mJy can be seen

  

Figure 12: 74 cm image (6 degrees across) of a field centred at J2000 $\alpha$ 7 h 59 m 27.92 s, $\delta$ 40 $\deg$ 44' 0.7'', 12 hours away in right ascension from Cyg A but at the same declination. The "object" present in the image is responsible for the ring feature seen in Fig. 11. It has a maximum signal of 2 Jy. The RMS noise in the field is approximately 145 mJy

In any case one can remove the effects of this ghost signal from the region of interest quite easily. Simply use the image shown in Fig. 12 as the model, invert it into the UV plane, and apply MODCAL in the way described previously. The end result is shown in Fig. 13. The unwanted grating ring has disappeared.

The astronomer can remove most ground-based man-made interference in a similar manner. Since a ground-based source is not moving, an interferometer data collection system usually interprets its signal as coming from a fixed point on the sky, namely the celestial pole. So all that needs to be done is to make an image of the region surrounding the pole, and use this image as the MODCAL model.

  

Figure 13: A 74 cm image of the Cyg A area after use of MODCAL to remove artifacts due to the object seen in Fig. 12

4.5 Removal of residual grating rings

  It is not uncommon for low-level residual rings to remain in DRAO images after self-calibration, even when one adjusts for different antenna sizes. Such low level residual rings may be seen in Fig. 5. There are several potential causes of these residual effects. Firstly, holography measurements by Gray (in preparation) indicate that some dishes have significant surface irregularities. Secondly, as was discussed in Sect. 4.2 the correlator bandpasses have significant departures from the ideal rectangular shape. Both these effects are likely to introduce effects that either vary across the field (dish surface irregularities), or are baseline dependent (correlator bandpasses). In theory one could attempt to model these effects. In practice, it is easier to remove these effects using the MODCAL procedure.

All that is necessary is for each source that has residual rings:

1) Subtract out model visibilities corresponding to the other sources in the field.

2) Create model visibilities for the offending source and run the MODCAL algorithm to remove the baseline dependent effects of the offending source.

3) Add the model visibilities created in step 2 to the adjusted raw data visibilities and continue on to the next source. In Fig. 6 the background sources have had their residual rings removed by this method.