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The self-calibration idea is based on the fact that the dominant error terms are antenna-based, while source information is baseline-based. With \(N\) antennas, one gets at any time \(N (N-1) /2\) visibility measurements, but \(N\) amplitude gains, and only \(N-1\) error terms for the morphology of the source (phase gains). The \(N-1\) number is because only relative phases count. The absolute flux scale is a separate problem, and therefore also \(N-1\) relative amplitude gains count.
The measured visibilities on baselines from antenna \(i\) to antenna \(j\) at time \(t\) are, from the simplified measurement equation: $$ V_{\rm obs}(i,j,t) = G(i,t) G^{{\star}}(j,t) V_{\rm true}(i,j) + Noise $$ where \(G(i,t)\) is the complex (phase and amplitude) gain for the antenna \(i\) at time \(t\) . The true visibility \(V_{\rm true}(i,j)\) only depends on the baseline \((i,j)\) , not on the time.
Given a source model \(V_{mod}(i,j)\) , one can derive the antenna gain products at time \(t\) , based on the system: $$ \frac{V_{\rm obs}(i,j,t)}{V_{\rm mod}(i,j)} = G(i,t) G^{{\star}}(j,t) $$ which is an over-constrained process, since there are \(N (N-1) /2\) constraints for \(N-1\) unknowns. Solving for this over-constrained problem is similar to deriving the amplitude and phase solution from a calibrator observation. In the calibrator case (i.e., an unresolved source like a distant bright quasar), \(V_{\rm mod(i,j)} = (1.0,0.0)\) (constant amplitude, zero phase), so there is no risk of noise amplification in the process.
For any (not a point-like) source, \(V_{\rm mod}\) must be guessed. Self-calibration will use your source to improve the calibration of the antenna-based (complex) gains as a function of time. The practice is to proceed iteratively, based on a preliminary deconvolution solution. Let \(V_{\rm obs}(k)\) be the “observed” visibilities at iteration \(k\) , with \(V_{\rm obs(k=0)} = V_{\rm obs} \) the raw calibrated visibilities. Some of the Clean components derived from \(V_{obs}(k)\) are used to define ”model' visibilities \(V_{\rm mod}(k)\) . Then, solving for the antenna gains, one obtains:
$$ V_{\rm obs}(k+1) = \frac {V_{obs}(k)}{(G_i G^{{\star}}_j)} $$
The model is thus progressively refined, and in the end, satisfies better the initial constraints on the source shape and on the antenna gains as a function of time provided by the measurements. Note that the absolute phase (and hence the position) can be lost in the self-calibration process and it should not be used for absolute astrometry.
There are two types of self-calibration: phase and amplitude self-calibration. The amplitude gain is a more complex problem than the phase gain. Amplitude gains can (and often do) vary with time, but from the measurement equation, a scale factor in the amplitude gain can be exchanged by a scale factor on the source flux. It is thus customary to re-normalize the gains so that the source flux is conserved in the process. An alternate (perhaps not strictly equivalent) solution is to ensure that the time averaged product of the amplitude gains is 1. The two approaches differ by the averaging process.
For any typical source, \(V_{\rm mod}\) is non zero and of magnitude smaller than 1 (using the total flux as a scale factor) since the source is partially resolved. So in computing \(V_{obs}/V_{\rm mod}\) , there is noise amplification. It may even be the case that \(V_{mod}\) is zero (case of an extended, over-resolved emission), and thus some (long) baselines will yield no direct constraint on the antenna gains \(G(i) G^{{\star}}(j)\) . But this should not matter too much for self-calibration, for two reasons. First, other (i.e., shorter) baselines may provide contraints on the gains. Second, if all \(V_{\rm obs}\) for an antenna are close to zero, it implies \(V_{\rm mod}\) must be close to zero too, so an error on the phase of those visibilities (as well as on its magnitude) is not so important.
Self-calibration is related to the “closure” relations. For any triplet of antennas, the phase of the triple product \(V_{ij} V_{jk} V_{ki}\) is independent of the antenna errors, and thus is (within the noise) a bias free constraint on the source. Similarly, for any quadruplet of antennas, the amplitude of the ratio \((V_{ij} V_{kl}) / (V_{ik} V_{jl})\) is independent of the antenna errors. But here, the noise amplification can be large because of the likelihood to have two small visibilities. For this reason, amplitude self-calibration requires in practice higher signal to noise ratios than phase calibration in the initial deconvolved data set used as a model.
Among the advantages of self-calibration, one may emphasize that antenna gains are derived at the correct time of the science object observation, while they must be interpolated in the classical calibration approach. Both atmospheric and electronic noises are supposed to vary with time, although with different timescales. Gains are also computed in the correct direction on the celestial sphere, while the calibrator-based approach introduces differences in the pointing direction with respect to the science object. The robustness of the approach increases with the number of baselines.
In order to implement self-calibration, it is however necessary that the signal to noise ratio be large enough (the process will require a sufficient bright source). Self-calibration can especially bring significant improvements to the calibration solution in the case of higher than expected background noise, or in the presence of convolutional artifacts around objects, especially point sources.