Self Calibration

Prerequisites

• Measurement Equation

Short Topical Videos

• Self-Calibration for Radio Interferometers (by Aaron Parsons)

Reference Material

• Synthesis Imaging in Radio Astronomy II, ed. Taylor, Carilli, Perley, Ch. 10 (Cornwell & Fomalont)

1 Self-Calibration

Data obtained by a pair of antennas for a specific baseline is the observed visibility, or ${\tilde {V}}_{ij}$. The goal of calibrating data is to recover the true visibility $V_{ij}$ of the sky, which differs from the observed visibility for several reasons. In this section, the focus will be on self-calibration, which is a method of solving for the internal degrees of freedom of a telescope.

The gain of an antenna element is unique to each antenna, and its value needs to be calibrated against a target source in the sky. The basis of self-calibration is treating antenna gains as free parameters that can be iteratively adjusted to reconcile observed visibilities with a model of a target source.

A baseline-based complex gain $G_{ij}(t)$ can be approximated by the product of two antenna-based complex gains:

$G_{ij}(t)=g_{i}(t)g_{j}^{*}(t)\,\!$

The basic calibration formula is:

${\tilde {V}}_{ij}(t)=G_{ij}(t)V_{ij}(t)+\epsilon _{ij}(t)+\eta _{ij}(t)\,\!$

where $\epsilon _{ij}(t)$ is a complex offset and $\eta _{ij}(t)$ is a complex noise. By clever telescope design, $\epsilon _{ij}(t)=0$, and we’ll ignore noise for simplicity. Therefore:

${\tilde {V}}_{ij}(t)\sim G_{ij}(t)V_{ij}(t)=g_{i}(t)g_{j}^{*}(t)V_{ij}(t)\,\!$

If there is a calibration source near the region to be imaged, the antenna gains can be solved for. Solving for these gains is neat because you can simultaneously solve for their values while also increasing the accuracy of the model sources. The method essentially involves producing a model of the sky that when fourier-transformed and corrected by the gain factors, reproduces observed visibilities.

One way of obtaining gain factors is to minimize the sum of the square of the residuals:

$S\sim \sum _{k}\sum _{i,j}|{\tilde {V}}_{ij}(t_{k})-g_{i}(t_{k})g_{j}^{*}(t_{k})V_{ij}(t_{k})|^{2}\,\!$

An example of an iterative model that both solves for gains and bootstraps the sky model is the following:

1) Make an initial model of the source using constraints on the source structure

2) Use this model to solve for the gains using a least squares fit as mentioned above

3) Find the corrected visibility $V_{ij,cor}$

$V_{ij,cor}={\tilde {V}}_{ij}(g_{i}(t)g_{j}^{*}(t))^{-1}\,\!$

4) Form a new model using this corrected data

5) Go to step (2)

Alternatively, if there are multiple sources in the model, it is possible to first solve for the gain parameters using only 1 source and then using those gains to measure the 2nd source spectrum, and then in turn using that to refine the gains. Note that this iterative approach is possible because there are many visibilities to work with (many baselines), since there are $N$ antennas (and therefore gain factors) and $N(N-1)/2$ baselines (and therefore visibilities). The system is overdetermined, and the extra degrees of freedom go into the assumptions about the model source structure.

The main advantage of self-calibration is that fairly robust gain solutions are derived as a function of time. However, it requires a sufficiently bright source to calibrate to and the results depend on the model. If the model sources are not accurate, their interferences can leak into the gains ("coupling sidelobes of sources").

One other thing to note is that the complex gains can be broken into antenna-based amplitude corrections and a phase correction:

$G_{ij}(t)=a_{i}(t)a_{j}(t)e^{i(\phi _{i}(t)-\phi _{j}(t))}\,\!$

These two terms are discussed in more detail in the video link at the top of the page.