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Import standard modules:


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import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
from IPython.display import HTML 
HTML('../style/course.css') #apply general CSS

Import section specific modules:


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pass

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HTML('../style/code_toggle.html')

8.3 2GC Calibration: direction independent self-calibration

After performing 1GC (applying the antenna gains from the calibrator to the target field) we should be able to make a decent image of our target field. The dynamic range of this image can be improved even further by using the self-calibration framework. We formally present self-calibration and hybrid-mapping (a predecessor of self-calibration) in $\S$ 8.3.1 ⤵ and $\S$ 8.3.2 ⤵ respectively.

8.3.1 Self-Calibration

Self-calibration makes use of the observed field to calibrate the visibilities. In Fig. 8.3.1 ⤵ we have a block diagram depicting the self-calibration framework. We continuously switch between two domains; the image domain and the visibility domain. In the image domain we perform deconvolution and source finding, while calibration takes place in the visibility domain. Deconvolution and source finding are discussed in $\S$ 6.2 ➞ and $\S$ 6.5 ➞ respectively. It is important to realize that Fig. 8.3.1 ⤵ is a very rudimentary representation of the self-calibration framework. Self-calibration is performed by using the following procedure:

Self-calibration algorithm

1. We start by creating an incomplete initial sky-model of our target field (using a post-1GC image).
2. We use the initial/improved sky-model to calibrate our observed visibilities which are subsequently imaged.
3. We deconvolve the resulting image.
4. We run a source finder on the deconvolved image to construct a more accurate sky model.
5. We return to step 2, or terminate the algorithm if the we have reached the target dynamic range or if further improvement is not possible.

Figure 8.3.1: The self-calibration framework.

We now present some radio images which illustrate the usefulness of self-calibration. The images below all depict the NCP as observed by LOFAR. Fig. 8.3.2 ⤵ was created directly after the calibrator solutions were applied. Fig. 8.3.3 ⤵ is an image of the target field after self-calibration (and additional flagging). We produced Fig. 8.3.4 ⤵ by decreasing the colour-bar range of Fig. 8.3.3 ⤵ . We can make a few observations by comparing Fig. 8.3.2 ⤵ , Fig. 8.3.3 ⤵ and Fig. 8.3.4 ⤵ :

  • The radio galaxy 3C 220.3 only becomes visible after we perform self-calibration.
  • The radio galaxy 3C 61.1 becomes brighter after self-calibration (it now approximates its true apparent flux).
  • Some of the weaker sources are visible after self-calibration (see Fig. 8.3.4 ⤵).

In short, self-calibration greatly improves the quality of radio images.

Figure 8.3.2: The NCP before self-calibration (only 1GC calibration has been performed). This observation was made with the LOFAR telescope.

Figure 8.3.3: The NCP after self-calibration.

Figure 8.3.4: The NCP after self-calibration. The colour-bar range has been clipped so that the weaker sources are visible.

8.3.2 Hybrid-mapping

Up until this point we have not described the actual calibration algorithm one needs to use in the self-calibration framework. This is because the algorithm used is irrelevant in the context of the framework. However, it is useful to note that the (currently) most used algorithm is the least-squares approach discussed in $\S$ 8.1 ➞ . This was not always the case.

In the past self-calibration actually employed closure quantities (see A phase sensitive interferometer technique for the measurement of the Fourier transforms of spatial brightness distributions of small angular extent and The measurement of the angular diameter of radio stars). Back when closure quantities were used to implement the calibration sub-block in Fig. 8.3.1 ⤵ , self-calibration was known by a different name: hybrid-mapping.

Warning: Generally speaking, when the calibration framework in [Fig. 8.3.1 ⤵](#cal:fig:self_cal) employs a the least squares solver we use the term "self-calibration", while the name "hybrid-mapping" is used when the calibration framework in [Fig. 8.3.1 ⤵](#cal:fig:self_cal) relies on closure quantities instead.

We discussed closure quantities in $\S$ 8.2 ➞ . One of the best known hybrid-mapping approaches was developed by Readhead and Wilkinson (The mapping of compact radio sources from VLBI data). It requires the following steps:

Hybrid-mapping algorithm

1. If we have an $N$-element array, we obtain $N-1$ baseline phases from our initial/updated model visibilities.
2. The baseline phases are determined such that the closure-phases are minimised.
3. After imaging the corrected visibilities, deconvolution is performed.
4. We update our sky-model based on the deconvolved image.
5. Return to step 1 or terminate if convergence has been reached.


Note: *Redundant calibration* (see [High dynamic range mapping of strong radio sources, with application to 3C84 ⤴](http://adsabs.harvard.edu/abs/1982Natur.299..597N)) also requires the use of closure quantities, but does not use a sky-model. It exploits the redundancy in an array to accomplish this. As there are several measurements at the same $uv$-point, those measurements need simply be calibrated such that they agree.

Using a least-squares solver to calibrate was first proposed in Adaptive calibration of radio interferometer data and A new method for making maps with unstable radio interferometers. Another interesting concomitant of the least-squares approach is that it allows us to solve for individual antenna gains instead of baseline-based gains. Good literature reviews on self-calibration can be found in The almost serendipitous discovery of self-calibration and Image formation by self-calibration in radio astronomy.


Note: Self-calibration can be regarded as a variant of the *Gerchberg-Saxton* algorithm (see [A practical algorithm for the determination of phase from image and diffraction plane pictures ⤴](http://www.u.arizona.edu/~ppoon/GerchbergandSaxton1972.pdf)).

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