Because the basic interpolated calibration procedure does not perfectly determine the antenna gains at each time stamp, the quality of the resultant image suffers. The technique of self calibration is often used to make additional corrections to the gains as a function of time. It is very similar to the basic calibration in which the model of the calibrator was a point source at the phase centre. In self calibration, a model of the source that you are interested in imaging, is used to refine the antenna gains. Like basic calibration, self calibration can only correct antenna based errors.
If the source is a point source, then, again, a point-source model could be used, although the restriction that it be located at the phase centre is lifted. The model for a more complicated source is more sophisticated. Generally, it is a set of CLEAN components. Recall that this is a list of delta functions at specified locations in the image. Alternatively, you might use the model provided by some other deconvolution technique such as maximum entropy. A useful trick is used in self calibration to reduce the problem to one very similar to the basic calibration, as already discussed in § 3. The model is first Fourier transformed to the visibility domain, and then the data are divided by the Fourier transform of the model. This reduces the data to a pseudo point source, and the gains are easily determined from the equations given in § 3.
The self-calibration procedure is applied iteratively, each time with a better model, until finally the sequence converges and no more improvement in the image quality can be made.
Self calibration is not a technique that should be applied blindly. This is especially true for ATCA data, because the problem is only slightly overdetermined, compared with the VLA. This is because the ATCA has, at most, 15 baselines to determine the gains for 6 antennas, whereas the VLA has, at most, 351 baselines to determine the gains for 27 antennas. Because we can set the phase of one antenna to zero (see § 3.1), the problem reduces to finding 11 real numbers from 30 real equations. With the 5-antenna compact array, we must find 9 real numbers from 20 real equations. This problem is exacerbated when one antenna is absent from the data for a period of time, or you have flagged it out because of poor data quality.
There are other problems to do with the east-west nature of the ATCA and I refer you to the document by Bob Sault (see references) which discusses some of these. One important point to keep in mind is that self calibration with the ATCA depends crucially on the initial model that you start with -- much more so than with the VLA where you can start with quite a poor model but arrive at the correct result after just a few iterations.
Self calibration, like basic calibration, requires that the signal-to-noise ratio on each baseline be of the order of at least 5 or so. For weak sources, this may require a long solution interval, within which the gain offsets for each antenna are assumed to be constant. If, in reality, the gains are changing on a time scale significantly shorter than your solution interval, and these changes are degrading your image quality, then you will be unable to improve the image quality. The time scale on which you will often find it necessary to correct the gains is approximately one minute. Thus, weak sources often cannot be self calibrated.
Note that it is not the receivers that cause the gains to be unsteady with time, but the atmosphere. In a similar way to the degradation of an optical image by the atmosphere, a radio image is defocused by the phase (and amplitude) noise that atmospheric cells induce into the wavefront. Self calibration can be thought of as an off-line mimicry of adaptive optics in optical astronomy.