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The Measurement Equation for a Generic Interferometer

I will restate the HBS measurement equation as described by Noordam but with a few additions and simplifications to clarify the physical content of the equation. In addition, I wish to correct what I believe to be mistakes and omissions in his formulation.

A Generic Interferometer measures cross correlations between two channels per feed, leading to a four-term cross-correlation that we write as a vector, $\vec{V}\,$, subscripted by the pair of feeds i, j:

$$\vec{V}_{\rm ij}^{} \left(\vphantom{ \begin{array}{c} {V}_{\rm pp}\\ {V}_{\rm pq}\\ {V}_{\rm qp}\\ {V}_{\rm qq} \end{array}}\right. \begin{array}{c} {V}_{\rm pp}\\ {V}_{\rm pq}\\ {V}_{\rm qp}\\ {V}_{\rm qq} \end{array} \left.\vphantom{ \begin{array}{c} {V}_{\rm pp}\\ {V}_{\rm pq}\\ {V}_{\rm qp}\\ {V}_{\rm qq} \end{array}}\right)_{ij}^{}$$ (1)

This basic measureable can be decomposed as follows. The measured cross-correlation $\vec{V}\,$ is given by:

$$\vec{V}_{\rm ij}^{} \left[\vphantom{{{G}_i\otimes{G}^*_j}}\right. \otimes \left.\vphantom{{{G}_i\otimes{G}^*_j}}\right] \left[\vphantom{{{D}_i\otimes{D}^*_j}}\right. \otimes \left.\vphantom{{{D}_i\otimes{D}^*_j}}\right] \left[\vphantom{{{C}_i\otimes{C}^*_j}}\right. \otimes \left.\vphantom{{{C}_i\otimes{C}^*_j}}\right] \vec{\cal V}_{\rm ij}^{}$$ (2)

where G_i, D_i, C_i are 2 by 2 matrices representing specific feed-based effects. The operator $\otimes$ represents a direct matrix product yielding in this case 4 by 4 matrices. G_i represents the complex gain of the i'th feed, D_i the leakage, and C_i is a fixed matrix representing the nominal feed configuration. Noordam gives examples of these terms. The terms $\left[\vphantom{{{G}_i\otimes{G}^*_j}}\right.$G_i $\otimes$ G^*_j$\left.\vphantom{{{G}_i\otimes{G}^*_j}}\right]$, $\left[\vphantom{{{D}_i\otimes{D}^*_j}}\right.$D_i $\otimes$ D^*_j$\left.\vphantom{{{D}_i\otimes{D}^*_j}}\right]$, and $\left[\vphantom{{{C}_i\otimes{C}^*_j}}\right.$C_i $\otimes$ C^*_j$\left.\vphantom{{{C}_i\otimes{C}^*_j}}\right]$ are 4 by 4 matrices, and thus the formulation is 4-dimensional. Physically the matrices G_i, D_i, C_i represent coupling between the two polarization states for each feed and so the terms $\left[\vphantom{{{G}_i\otimes{G}^*_j}}\right.$G_i $\otimes$ G^*_j$\left.\vphantom{{{G}_i\otimes{G}^*_j}}\right]$, $\left[\vphantom{{{D}_i\otimes{D}^*_j}}\right.$D_i $\otimes$ D^*_j$\left.\vphantom{{{D}_i\otimes{D}^*_j}}\right]$, $\left[\vphantom{{{C}_i\otimes{C}^*_j}}\right.$C_i $\otimes$ C^*_j$\left.\vphantom{{{C}_i\otimes{C}^*_j}}\right]$ represent coupling between the four polarization correlation states. In appendix A, I give a catalog of various forms for the calibration matrices.

A hint on reading these and succeeding equations: to get the more usual equations, just convert matrices to scalars, ignore the transposes, and convert the direct matrix product to a simple product. The equations derived below should then look quite familiar.

The vector $\vec{\cal V}\,$ represents the visibility that would be measured in the absence of the visibility-domain calibration effects. ¹

$$\vec{\cal V}_{\rm ij}^{} \left(\vphantom{ \begin{array}{c} {\cal V}_{\rm pp}\\ {\cal V}_{\rm pq}\\ {\cal V}_{\rm qp}\\ {\cal V}_{\rm qq} \end{array}}\right. \begin{array}{c} {\cal V}_{\rm pp}\\ {\cal V}_{\rm pq}\\ {\cal V}_{\rm qp}\\ {\cal V}_{\rm qq} \end{array} \left.\vphantom{ \begin{array}{c} {\cal V}_{\rm pp}\\ {\cal ... ... pq}\\ {\cal V}_{\rm qp}\\ {\cal V}_{\rm qq} \end{array}}\right)_{ij}^{}$$ (3)

This includes only image-domain calibration effects. Thus, the feed-based effects must now be a function of direction, $\underline{\rho}$.

$$\vec{\cal V}_{\rm ij}^{} \sum_{k}^{} \left[\vphantom{{{E}_i {\left(\underline{\rho}_k\right)} \otimes {E}^*_j{\left(\underline{\rho}_k\right)}}}\right. \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \otimes \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \left.\vphantom{{{E}_i {\left(\underline{\rho}_k\right)} \otimes {E}^*_j{\left(\underline{\rho}_k\right)}}}\right] \left[\vphantom{{{P}_i {\left(\underline{\rho}_k\right)} \otimes {P}^*_j{\left(\underline{\rho}_k\right)}}}\right. \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \otimes \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \left.\vphantom{{{P}_i {\left(\underline{\rho}_k\right)} \otimes {P}^*_j{\left(\underline{\rho}_k\right)}}}\right] \left[\vphantom{{{F}_i {\left(\underline{\rho}_k\right)} \otimes {F}^*_j{\left(\underline{\rho}_k\right)}}}\right. \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \otimes \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \left.\vphantom{{{F}_i {\left(\underline{\rho}_k\right)} \otimes {F}^*_j{\left(\underline{\rho}_k\right)}}}\right]$$

$$\vec{\cal I}_{k}^{} \pi \left(\vphantom{{\underline{r}_i-\underline{r}_j}}\right. \underline{r}_{i} \underline{r}_{j} \left.\vphantom{{\underline{r}_i-\underline{r}_j}}\right) \underline{\rho}_{k}$$ (4)

where E_i$\left(\vphantom{\underline{\rho}_k}\right.$$\underline{\rho}_{k}^{}$ $\left.\vphantom{\underline{\rho}_k}\right)$ represents the feed voltage receptivity pattern of the i'th feed (Noordam used B for this term), P_i$\left(\vphantom{\underline{\rho}_k}\right.$$\underline{\rho}_{k}^{}$ $\left.\vphantom{\underline{\rho}_k}\right)$ parallactic angle rotation, and F_i$\left(\vphantom{\underline{\rho}_k}\right.$$\underline{\rho}_{k}^{}$ $\left.\vphantom{\underline{\rho}_k}\right)$ atmospheric terms including tropospheric and ionospheric phase (Noordam used this term for Faraday rotation alone; here I note that Faraday rotation can also be accomodated in F). The term $\vec{\cal I}\,$ represents the polarized sky brightness. In the Stokes representation, this is given by:

$$\vec{\cal I}\, \left(\vphantom{ \begin{array}{c} I\\ Q\\ U\\ V\\ \end{array}}\right. \begin{array}{c} I\\ Q\\ U\\ V\\ \end{array} \left.\vphantom{ \begin{array}{c} I\\ Q\\ U\\ V\\ \end{array}}\right)$$ (5)

The matrix S converts from the representation used in $\vec{\cal I}\,$ to that most naturally used in describing the interferometer. HBS use linear polarization as the canonical representation, for which the S-matrix is:

$${1\over 2} \left(\vphantom{{ \begin{array}{cccc} 1&1&0&0\\ 0&0&1&i\\ 0&0&1&-i\\ 1&-1&0&0 \end{array} }}\right. \begin{array}{cccc} 1&1&0&0\\ 0&0&1&i\\ 0&0&1&-i\\ 1&-1&0&0 \end{array} \left.\vphantom{{ \begin{array}{cccc} 1&1&0&0\\ 0&0&1&i\\ 0&0&1&-i\\ 1&-1&0&0 \end{array} }}\right)$$ (6)

It is worth emphasizing that this is an arbitary choice and that one could, instead, use circular polarization, in which case the S-matrix would be:

$${1\over 2} \left(\vphantom{{ \begin{array}{cccc} 1&0&0&1\\ 0&1&i&0\\ 0&1&-i&0\\ 1&0&0&-1 \end{array} }}\right. \begin{array}{cccc} 1&0&0&1\\ 0&1&i&0\\ 0&1&-i&0\\ 1&0&0&-1 \end{array} \left.\vphantom{{ \begin{array}{cccc} 1&0&0&1\\ 0&1&i&0\\ 0&1&-i&0\\ 1&0&0&-1 \end{array} }}\right)$$ (7)

It may seem that C_i and S are redundant since one can choose S so that C_i is a unit matrix. However, I follow HBS and choose S to be the canonical S^lin. Consequently, C_i must be allowed to vary to suit the actual measurement scheme of the feed. This flexibility is required for a system of interferometers in which a mixture of linear and circular polarization is measured. Note that the calibration matrices will then inevitably be more complicated than would be the case if a single, natural representation was used.

To emphasize the true glory of the full measurement equation, here I give the whole expression:

$$\vec{V}_{\rm ij}^{} \left[\vphantom{{{G}_i\otimes{G}^*_j}}\right. \otimes \left.\vphantom{{{G}_i\otimes{G}^*_j}}\right] \left[\vphantom{{{D}_i\otimes{D}^*_j}}\right. \otimes \left.\vphantom{{{D}_i\otimes{D}^*_j}}\right] \left[\vphantom{{{C}_i\otimes{C}^*_j}}\right. \otimes \left.\vphantom{{{C}_i\otimes{C}^*_j}}\right]$$

$$\sum_{k}^{} \left[\vphantom{{{E}_i {\left(\underline{\rho}_k\right)} \otimes {E}^*_j{\left(\underline{\rho}_k\right)}}}\right. \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \otimes \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \left.\vphantom{{{E}_i {\left(\underline{\rho}_k\right)} \otimes {E}^*_j{\left(\underline{\rho}_k\right)}}}\right] \left[\vphantom{{{P}_i {\left(\underline{\rho}_k\right)} \otimes {P}^*_j{\left(\underline{\rho}_k\right)}}}\right. \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \otimes \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \left.\vphantom{{{P}_i {\left(\underline{\rho}_k\right)} \otimes {P}^*_j{\left(\underline{\rho}_k\right)}}}\right] \left[\vphantom{{{F}_i {\left(\underline{\rho}_k\right)} \otimes {F}^*_j{\left(\underline{\rho}_k\right)}}}\right. \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \otimes \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \left.\vphantom{{{F}_i {\left(\underline{\rho}_k\right)} \otimes {F}^*_j{\left(\underline{\rho}_k\right)}}}\right]$$

$$\vec{\cal I}_{k}^{} \pi \left(\vphantom{{\underline{r}_i-\underline{r}_j}}\right. \underline{r}_{i} \underline{r}_{j} \left.\vphantom{{\underline{r}_i-\underline{r}_j}}\right) \underline{\rho}_{k}$$ (8)

It is worth making a number of comments about this very general form of the measurement equation:

Ordering of terms

The ordering of terms follows the signal path, reading right to left in the equation.

Linearity

The ME is linear in the sky brightness $\vec{\cal I}\,$. It is almost always non-linear in various calibration parameters. This will have implications for solvers, as I discuss below.

Time and Frequency

The formalism as presented here ignores indexing by time and frequency. Both of these are trivial to add but obscure the notation. Averaging over both time and frequency is then accomodated easily.

No new physics

There is no new physics in this formulation. The equations are totally equivalent to those derived earlier (see e.g. Schwab, 1984). However, I expect that the formulation will suggest new forms of calibration and imaging.

Use of these equations

In general, one will want to determine one or both of the calibration parameters and the sky brightness. To determine the calibration, one will fix the sky brightness model, and solve for the calibration parameters, perhaps by a least squares approach. The determine the sky brightness, one will want to fix the calibration parameters and solve for the sky brightness, using a deconvolution algorithm. One could alternate between calibration and sky brightness estimation, as is done in most self-calibration procedures, or one could do a joint solution by some very powerful least squares type algorithm.

Analytical forms for $\vec{\cal I}\,$

In the calibration cycle, one will have to perform an integration over $\underline{\rho}$. This will in general be quite difficult and usually, the integration will have to be performed numerically. The ``art'' of programming this equation will principally be in finding quick methods of numerical integration using, for example, FFTs.

Non-invertibility

The ME is clearly not invertible for either the sky brightness or the calibration matrices. This is really a truism since it represents only one sample.

Self-cal assumption

The fundamental assumption yielding the direct products is that all these instrumental effects factorize per feed. It is hard to think of any exceptions to this, apart from problems with the correlation process itself. Note that failure of closure due to different bandpasses at the different feeds is actually accomodated by the summation over frequency.

Non-isoplanatism

The location of the atmospheric phase terms is actually ambiguous in most cases. Physically it belongs with the F term. In isoplanatic conditions, it cannot be distinguished easily from the electronic term G, and so it is usually written as belonging in G. However, if the atmosphere is non-isoplanatic then the atmospheric phase must be included in the F-term. F then actually decomposes a follows:

$$\left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right) \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right)^{troposphere}_{} \left(\vphantom{\underline{\rho}_k}\right. \underline{\rho}_{k}^{} \left.\vphantom{\underline{\rho}_k}\right)^{ionosphere}_{}$$ (9)

Assumption of perfect correlation

Note that the correlation process itself introduces errors that must be corrected, such as the van Vleck correction. These we ignore for the moment.

Completeness of information

It is assumed that the full 4-dimensional visibility is measured. In many cases, this is not so and, for example, only the parallel hands will be correlated. Full correction is not then possible but one expects that good approximations will be obtained in many cases simply by inserting zeroes in the appropriate 4-vectors. A similar but distinct difficulty arises in those interferometers that measure different hands at different times. Such wrinkles can be dealt with straightforwardly but further discussion is deferred to a specifications document.