Calibration

We have assumed till now that we have been working with calibrated visibilities, i.e. free from all instrumental effects (apart from some additive noise component). In reality, the correlator output is different from the true astronomical visibility for a variety of reasons, to do with both instrumental effects as well as propagation effects in the earth's atmosphere and ionosphere.

At low frequencies, it is the effect of the ionosphere that is most dominant. As is discussed in more detail in Chapter 16, density irregularities cause phase irregularities in the wavefront of the incoming radio waves. One would expect therefore that the image of the source would be distorted in the same way that atmospheric turbulence (`seeing') distorts stellar images at optical wavelengths. To first order this is true, but for the ionosphere the `seeing disk' is generally smaller than the diffraction limit of typical interferometers. There are two other effects however which are more troublesome. The first is `scintillation', where because of diffractive effects the flux density of the source changes rapidly - the flux density modulation could approach 100%. The other is that slowly varying, large scale refractive index gradients cause the apparent source position to wander. At low frequencies, the source position could often wander by several arc minutes, i.e. considerably more than the synthesized beam. As we shall see below, provided the time scale of this wander is slow enough, it can be corrected for.

Let us take the case where the effect of the ionosphere is simply to produce an excess path length, i.e. for an antenna $i$ let the excess phase^5.15 for a point source at sky position $(l,m)$ be $\phi_i(l,m,t)$, where we have explicitly put in a time dependence. Then the observed visibility on a baseline $(i,j)$ would be

$$\widetilde{\mathcal{V}}_{ij}(t) = G_{ij}(t)\int e^{-i\left... ...t)-\phi_j(l,m,t)\right)} I(l,m)e^{-i2\pi (lu_{ij} + mv_{ij})}$$ (5.2.14)

where $I(l,m)$ is the sky brightness distribution and we have ignored the primary beam^5.16. $G_{ij}(t)$ is `instrumental phase', i.e. the phase produced by the amplifiers, transmission lines, or other instrumentation along the signal path. If $\phi_i(l,m,t)$ were some general, unknown function of $(l,m,t)$ it would not be possible to reconstruct the true visibility from the measured one. However, since the size scale of ionospheric disturbances is $\sim$ a few hundred kilometers, it is often the case that $\phi_i(l,m,t)$ is constant over the entire primary beam, i.e. there is no $(l,m)$ dependence. The source is then said to lie within a single iso-planatic patch. In such situations, the ionospheric phase can be taken out of the integral, and eqn(5.2.14) reduces to:

$$\widetilde{\mathcal{V}}_{ij}(t) = G_{ij}(t) e^{-i\left(\ph... ...)-\phi_j(t)\right)} \int I(l,m)e^{-i2\pi (lu_{ij} + mv_{ij})}$$ (5.2.15)

If it also the case that the ionospheric and instrumental gains are changing slowly, then they can be calibrated in the following manner. Suppose that close to the source of interest, there is a calibration source whose true visibility $\mathcal{V}^c_{ij}$ is known. Then one could intersperse observations of the target source with observations of the calibrator. For the calibrator, dividing the observed visibility $\widetilde{\mathcal{V}}^c_{ij}(t)$ by the (known) true visibility, $\mathcal{V}^c_{ij}(t)$ one can measure the factor $G_{ij}(t) e^{-i\left(\phi_i(t)-\phi_j(t)\right)}$. This can then be applied as a correction to the visibilities of the target source. For slightly better corrections, one could interpolate in time between calibrator observations. This is the basis of what is sometimes called `ordinary' calibration. The calibrator source is usually an isolated point source, although this is not, strictly speaking, necessary. It is sufficient to know the true visibilities $\mathcal{V}^c_{ij}(t)$. Note that if the calibrators absolute flux is also known, then this calibration procedure will also calibrate the amplitude scale of the target source^5.17.

In the approach outlined above, in order to calibrate the data one needs to solve for an unknown complex number per baseline, (i.e. N(N-1)/2 complex numbers for an N element interferometer). If we assume that the correlator itself does not produce any errors^5.18, i.e. that all the instrumental errors occur in the antennas or the transmission lines, then the instrumental gain can be written out as antenna based terms, i.e.

$$G_{ij}(t) = g_i(t)g^*_j(t)$$ (5.2.16)

where $g_i(t)$ and $g_j(t)$ are the complex gains along the signal paths from antennas $1$ and $2$. But the ionospheric phase can also be decomposed into antenna based quantities (see eqn 5.2.15), and can hence be lumped together with the instrumental phase. Consequently the total unknown complex gains that have to be solved for reduces from N(N-1)/2 to N, which can be a dramatic reduction for large N. (For the GMRT it is a reduction from 435 unknowns to 30 unknowns).

However to appreciate the real power of this decomposition into antenna based gains, consider the following quantities. First let us look at the sum of the phases of the raw visibilities $\widetilde{\mathcal{V}}_{12}$, $\widetilde{\mathcal{V}}_{23}$ and $\widetilde{\mathcal{V}}_{31}$. If we call the true visibility phase $\psi_{\mathcal{V}_{ij}}$, the raw visibility phase $\psi_{\widetilde{\mathcal{V}}_{ij}}$ and the sum of the instrumental and ionospheric phases $\chi_i$, then we have

$$\psi_{\widetilde{\mathcal{V}}_{12}} + \psi_{\widetilde{\mathcal{V}}_{23}} + \psi_{\widetilde{\mathcal{V}}_{31}} = \chi_1 - \chi_2 + \psi_{\mathcal{V}_{12}} + \chi_2 - \chi_3 + \psi_{\mathcal{V}_{12}} + \chi_3 - \chi_1 + \psi_{\mathcal{V}_{31}}$$

$$= \psi_{\mathcal{V}_{12}} + \psi_{\mathcal{V}_{23}} + \psi_{\mathcal{V}_{31}}$$ (5.2.17)

i.e. over any triangle of baselines the sum of the phases of the raw visibilities is the true source visibility. This is called phase closure. Similarly it is easy to show that for any baselines 1,2,3,4, the ratio of the raw visibilities will be the same as the true visibilities, i.e.

$${\vert\widetilde{\mathcal{V}}_{12}\vert\vert\widetilde{\math... ...4}\vert \over \vert\mathcal{V}_{23}\vert\mathcal{V}_{41}\vert}$$ (5.2.18)

This is called amplitude closure. For an N element interferometer, we have $1/2 N (N-1) - (N-1)$ constraints on the phase and $1/2 N (N-1) - N$ constraints on the amplitude. For large N, this is considerably more than the N unknown gains that one is solving for. The large number of available constraints means that the following iterative scheme would work.

1. Choose a suitable starting model for the brightness distribution. Compute the model visibilities.
2. For this model, solve for the antenna gains, subject to the closure constraints.
3. Apply these gain corrections to the visibility data, use the corrected data to make a fresh model of the brightness distribution.

For arrays with sufficient number of antennas, convergence is usually rapid. Note however, for this to work, the signal to noise ratio per visibility point^5.19 has to be reasonable, i.e. 2-3. This is often the case at low frequencies, and this technique of determining antenna gains (which is called self calibration) is usually highly successful.

Note that if one adds a phase $\chi_i = 2\pi(l_0 u_i + m_0 v_i)$ to each antenna (where $l_0,\ m_0$ are arbitrary and $(u_i,v_i)$ are the (u,v) co-ordinates of the $i$th antenna), the phase closure constraints (eqn 5.2.17) continue to be satisfied. That means that in self calibration the phases can be solved only upto a constant phase gradient across the uv plane, i.e. the absolute source position is lost. Similarly, it is easy to see that the amplitude closure constraints will be satisfied even if one multiplies all the gains by a constant number, i.e. in self calibration one loses information on the absolute source flux density . The only way to determine the absolute source flux density is to look at a calibrator of known flux. Since antenna gains and system temperatures are usually stable over several hours^5.20, it is usually sufficient to do this calibration only once during an observing run. A more serious problem at low frequencies is that the Galactic background (whose strength varies with location on the sky) makes a significant contribution to the system temperature. Hence, when attempting to measure the source flux density, it is important to correct for the fact that the system temperature is different for the calibrator source as compared to the target source. The system temperature can typically be measured on rapid time scales by injecting a noise source of known strength at the front end amplifier.

Another related way (to selfcal) of solving for the system gains is the following. Suppose that the visibility on baselines $(i,j)$ and $(k,l)$ are identical. Then the ratio of the measured visibilities is directly related to the ratio of the complex instrumental gains of antennas $i,j,k\ \&\ l$. If there are enough number of such `redundant' baselines, one could imagine solving for the instrumental gains. Some arrays, like the WSRT have equispaced antennas, giving rise to a very large number of redundant baselines, and this technique has been successfuly used to calibrate complex sources^5.21For a simple source, like a point source, all possible baselines are redundant, and this technique reduces essentially to self-calibration.

At the very lowest frequencies ($\nu < 200$ MHz, roughly for the GMRT) the assumption that the source lies within the iso-planatic patch probably begins to break down. The simple self calibration scheme outlined above will stop working in that regime. A possible solution then, is to solve (roughly speaking) for the phase changes produced by each iso-planatic patch. Often the primary beams of several antennas will pass through the same iso-planatic patches, so the extra number of degrees of freedom introduced will not be substantial, and an iterative approach to solving for the unknowns will probably converge^5.22.

Footnotes

... phase^5.15

by which we mean the phase difference over what would have been obtained in the absence of the ionosphere

... beam^5.16

i.e. we have set the factor $B(l,m)/\sqrt{1 - l^2 -m^2}$ to 1.

... source^5.17

provided, as we will discuss in more detail later, that the system temperature does not differ for the target source and the calibrator

... errors^5.18

which is often a good assumption for digital correlators

... point^5.19

Actually strictly speaking one means the signal to noise ratio over an interval for which the ionospheric phase can be assumed to be constant

... hours^5.20

Or change in a predictable manner with changing azimuth and elevation of the antennas

... sources^5.21

see Noordam, J. E. & de Bruyn A. G., 1982, Nature 299, 597.

... converge^5.22

See Subrahmanya, C. R., (in `Radio Astronomical Seeing', J. E. Baldwin & Wang Shouguan eds.) for more details