4 Theory

  In the first part of this section we show how Loop "A'' and Loop "B'' produce the same results only when the self calibration process is linear. In the second part we discuss the role played by the closure phase in linearizing self calibration.

4.1 Linear case

  Self calibration, based on the algorithm of Schwab (1980), estimates the errors affecting the observed phase of the visibility function in terms of antenna errors E by minimizing Eq. (1). We may express the observed visibility phase $\psi_{\rm o}$ on a baseline ik as a combination of the true phase $\psi_{\rm t}$ and the antenna errors E at the two telescopes as

 

$$\psi_{{ik}_{\rm o}}=\psi_{{ik}_{\rm t}}+(E_i-E_k).$$ (3)

We may characterize the difference $\epsilon$ between the true visibility phase $\psi_{\rm t}$ and the phase $\psi_{\rm m}$ predicted by a given source model as

 

$$\psi_{{ik}_{\rm t}}=\psi_{{ik}_{\rm m}}+\epsilon_{ik} .$$ (4)

By solving for antenna-based errors, the closure phase on a given triangle ijk is preserved, which implies the constraint:

 

< E_i -E_j>+ < E_j -E_k>- < E_i -E_k> =0 . (5)

We can see that by substituting Eq. (3) and Eq. (4) in Eq. (5), we have:

$$\begin{eqnarray} (\psi_{{ij}_{\rm o}}-\psi_{{ij}_{\rm m}}) & + & (\psi_{{jk}_{\r... ...\nonumber - ( \epsilon_{ij} & + & \epsilon_{jk}-\epsilon_{ik} ) =0\end{eqnarray}$$ (6)

(where hereafter we omit the baseline subscript for the visibility phases.)

By defining $\Phi_{\rm o}$ and $\Phi_{\rm m}$the observed and the model closure phases (Eq. 2), we have:

 

$$\epsilon_{ij}+\epsilon_{jk}-\epsilon_{ik} = \Phi_{\rm o}-\Phi_{\rm m} .$$ (7)

Where we omit the subscript "ijk'' for the closure phases.

In other words the estimation of the $\epsilon$'s depend on the closure phase. How the difference between the observed and model closure phases is distributed among the individual errors $\epsilon$it depends upon the weights attributed to the various baselines. Since we want here to illustrate in general the role played by the closure phase in the solution, we assume for simplicity equal weights, and estimations of the errors like:  

$$\epsilon =\epsilon_{ik}=\epsilon_{jk}=-\epsilon_{ij}= -(\Phi_{\rm o}-\Phi_{\rm m}).$$ (8)

Now we can see the impact of the closure phase in the self calibration solution. Let us examine the case of a point-like source as input model. In this case the estimation in Eq. (8) of the error $\epsilon$ depends entirely on the observed closure phase, since in this case $\Phi_{\rm m}=0$. Let us assume that the observed closure phase is a small quantity, $\Phi_{\rm o} \simeq 0$, as it is in our simulated u-v data set with the intermediate baseline deleted. The error estimation then becomes a small quantity, that is

 

$$\Phi_{\rm o} \simeq 0 ~~~ \Longrightarrow ~~~ \epsilon = - \Phi_{\rm o} \simeq 0 .$$ (9)

In other words a small closure phase gives rise to an erroneously low value for the estimated error in the model. This wrong estimation of the error in the model propagates itself into a wrong estimation of the antenna error,

 

$$E_i-E_k = \psi_{\rm o} -\psi_{\rm m} - \epsilon ,$$ (10)

which becomes $E_i-E_k\simeq\psi_{\rm o}$ since $\psi_{\rm m}$ and $\epsilon$ both vanish. As a result the corrected observed phase $\psi^\prime_{\rm o}$ is equal to

 

$$\psi^\prime_{\rm o} = \psi_{\rm o} - (E_i-E_k)\simeq 0 .$$ (11)

In conclusion, small closure phase values mean that self calibration is unable to correct an erroneous initial point model. Only low estimations of $\epsilon$ are possible, i.e., the model is taken as a good approximation of source structure, and large antenna corrections are applied to the observed phase to force it to follow the model. The result is that an input model with small closure phase, characteristic of any symmetric source, will produce corrected phases $\psi^\prime_{\rm o}$ set to zero as well. This will produce the observed symmetrization: part of the flux density is displaced to make the source symmetric.

Let us now assume that the model phase determined by the set of clean components, produces, due to the use of proper CLEAN boxes, an $\epsilon$ smaller than before and more comparable to the small closure phase. In this case self calibration should converge toward the proper solution, i.e., each iteration should remove the bias of the preceding one. Why, then, does this not happen? In a linear process such as described in Eq. (10), the solution derived from a given iteration of self calibration (sc) is simply added to the solution from the previous iteration, that is

$$\begin{eqnarray} (E_i-E_k) _{\rm Loop-A} & = & (E_i-E_k)_{\rm sc-1} + (E_i-E_k)_... ...mber & =& \psi_{\rm o}- \psi_{\rm cc}+(\Phi_{\rm o}-\Phi_{\rm cc})\end{eqnarray}$$ (12)

where $\psi_{\rm cc}$ and $\Phi_{\rm cc}$ are the model phase and the model closure phase determined by the set of clean components.

If the self calibration process were indeed linear, Eq. (12) would produce the same result as from a single iteration of Loop "B'':

 

$$(E_i-E_k)_{\rm Loop-B}= \psi_{\rm o}- \psi_{\rm cc}+(\Phi_{\rm o}-\Phi_{\rm cc}) .$$ (13)

But we have shown empirically that in the case of Test-2

 

$$(E_i-E_k)_{\rm Loop-A} \not= (E_i-E_k)_{\rm Loop-B} .$$ (14)

This implies a non-linear equation for $(E_i-E_k)_{\rm sc}$.

As we will show in Sect. 4.2 self calibration is in fact a non linear process. This explains why Loop "B'' works better than Loop "A''. At each iteration Loop "B'' discards the previous (E_i-E_k) estimation and simply starts again, approaching the proper solution for the antenna corrections as $\psi_{\rm m} \to \psi_{\rm t}$.On the contrary Loop "A'' accumulates previous, incorrect estimations and being unable to correct for them, biases the total antenna corrections even though $\psi_{\rm m} \to \psi_{\rm t}$.We have checked this surmise by setting CLEAN boxes to exclude the spurious features, forcing the CLEAN model to be the "true" one. Self calibration nevertheless determines corrections which still reproduce the artificial counter-jet.

However, if we accept that in general Loop "B'' works better than Loop "A'' since self calibration is non-linear, why does, in some cases, Loop "A'' converge to the same solution as Loop "B''? There must be some linearizing quantity in Test-1 that is lacking in Test-2. We remind the reader that Test-2 has the same u-v coverage as Test-1, except for the single intermediate baseline.

4.2 Condition for linearity

  In general as shown in Massi & Comoretto (1990) the minimization of Eq. (1), dealing with the difference of two complex quantities, can be reduced to the minimization of the difference of their phases, only when three conditions are satisfied: only phase errors are present, good SNR in the data, and small errors in the model. When these three conditions are satisfied the problem reduces to the linear case.

The presence of large errors in the model violates the condition of linearity; we will see below quantitatively what "large" means. Let us consider in Eq. (1) only phase errors, , that the gain terms g_i = g_k = 1 and neglect the amplitude and weight terms. Then, minimizing Eq. (1) with respect to the phase error terms gives  

$$\sum _{i\not= k} \sin \left[ \psi_{\rm o}-\psi_{\rm m}-(E_i-E_k) \right] = 0 .$$ (15)

We see as the presence of a large error $\epsilon\equiv\psi_{\rm o}-\psi_{\rm m}-(E_i-E_k)$prevents the linearization of Eq. (15). Let us therefore expand the sine function including the first non linear term, as $~\sin x = x-{x^3\over 3!}$.

Equation (15) then becomes $\epsilon_{ik}+(\Phi_{\rm o}-\Phi_{\rm m})- \sum _{i\not= k} {\epsilon^3\over 3!}=0$, or  

$$\epsilon_{ik}=-(\Phi_{\rm o}-\Phi_{\rm m})+ \sum _{i\not= k} {\epsilon^3\over 3!} \cdot$$ (16)

This implies that for the linear case, that is for Eq. (16) to be equal to the linear case expressed in Eq. (8), it must hold true that $\left\vert \left( \Phi_{\rm o}-\Phi_{\rm m} \right) \right\vert \gg \sum{\epsilon^3\over 3!}$.

Assuming an initial point-like source, the condition for linearity is further reduced to  

$$\left\vert \Phi_{\rm o} \right\vert \gg \sum {\epsilon^3\over 3!} \cdot$$ (17)

In conclusion, self calibration may reduce to a linear equation only if the errors $\epsilon$ of the model, that is the difference between the phases of the model and the true ones are such that the combination of their cube is much smaller (at least one order of magnitude lower) than the observed closure phase.

Since the errors $\epsilon$ for a point-like model are exactly the phase of our simulated source from FAKE, we can illustrate this result.

Figure 9a shows the closure phase available in Test-1 along with the combination of the errors. The closure phase is in this case more than one order of magnitude higher than the combination of the errors; these last appear in fact quite unsignificant in the plot. This implies that Eq. (17) is for Test-1 satisfied and as a consequence self calibration should reduce to the linear case, that is Eq. (12) should be satisfied as well. And in fact we have shown as Loop "A'' give same results as Loop "B'' for Test-1.

On the contrary we see in Fig. 9b that when the closure phase drops to a few degrees, as is the case for unbalanced triangles, the errors in the model, previous negligible, become a significant fraction of the closure phase. In other words the condition of Eq. (17) no longer applies and the non linear terms in self calibration cannot be neglected; we are therefore in the condition expressed in Eq. (14). A conclusion verified in Sect. 2 with test2 which gives different results for Test-2A and Test-2B.

Figure 9c illustrates that when the source has a complicated structure and a point source has been used as an initial model, the errors are so large to become an important fraction of the closure phase even for large closure phase values. In the case of Fig. 9d finally we show the worst combination of small closure phase values and large errors giving rise to a solution dominated by the non linear terms.

  

Figure 9: Closure phase (continuous line) and non-linear error terms (crosses) in the self calibration solution (see Sects. 4 and 5)