The self-calibration idea is based on the fact that the dominant error terms
are antenna-based, while source information is baseline-based.
With antennas, one gets at any time
visibility measurements,
but
amplitude gains, and only
error terms for the morphology of the source (phase gains).
The
number is because only relative phases count. The absolute flux
scale is a separate problem, and therefore also
relative amplitude gains count.
The measured visibilities
on baselines from antenna to antenna
at time
are,
from the simplified measurement equation:
![]() |
(17) |
Given a source model , one can derive the antenna gain
products at time
, based on the system:
![]() |
(18) |
For any (not a point-like) source, 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
be the ``observed'' visibilities at iteration
, with
the raw calibrated visibilities. Some
of the Clean components derived from
are used to define
''model' visibilities
. Then, solving for the antenna
gains, one obtains:
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,
Self-calibration is related to the ``closure'' relations. For any triplet
of antennas, the phase of the triple product
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.
(19) 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
,
there is noise amplification. It may even be the case that
is
zero (case of an extended, over-resolved emission), and thus some
(long) baselines will yield no direct constraint on the antenna gains
. 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
for an antenna are
close to zero, it implies
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.
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
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.
Next: Self-Calibration Implementation
Up: Self Calibration
Previous: Self-Calibration in a nutshell
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Gildas manager
2023-06-01