# The Measurement Equation & Calibration

The visibilities measured by an interferometer must be calibrated before formation of an image. This is because the wavefronts received and processed by the observational hardware have been corrupted by a variety of effects. These include (but are not exclusive to): the effects of transmission through the atmosphere, the imperfect details amplified electronic (digital) signal and transmission through the signal processing system, and the effects of formation of the cross-power spectra by a correlator. Calibration is the process of reversing these effects to arrive at corrected visibilities which resemble as closely as possible the visibilities that would have been measured in vacuum by a perfect system. The subject of this chapter is the determination of these effects by using the visibility data itself.

# The HBS Measurement Equation

The relationship between the observed and ideal (desired) visibilities on the baseline between antennas i and j may be expressed by the Hamaker-Bregman-Sault Measurement Equation Hamaker, Bregman, & Sault (1996) [1] and Sault, Hamaker, Bregman (1996) [2] .

Citation Number 1 Hamaker, J.P., Bregman, J.D. & Sault, R.J. 1996, A&AS, 117, 137 (ADS).
Citation Number 2 Sault, R. J.; Hamaker, J. P.; Bregman, J. D. 1996, A&AS, 117, 149 (ADS)

\begin{eqnarray}
\vec{V}_{ij}~=~J_{ij}~\vec{V}_{ij}^{\mathrm{~IDEAL}}
\end{eqnarray}

where$\vec{V}_{ij}$ represents the observed visibility, a complex number representing the amplitude and phase of the correlated data from a pair of antennas in each sample time, per spectral channel. $\vec{V}_{ij}^{\mathrm{~IDEAL}}$ represents the corresponding ideal visibilities, and $J_{ij}$ represents the accumulation of all corruptions affecting baseline $ij$. The visibilities are indicated as vectors spanning the four correlation combinations which can be formed from dual-polarization signals. These four correlations are related directly to the Stokes parameters which fully describe the radiation. The $J_{ij}$ term is therefore a 4$\times$4 matrix.

Most of the effects contained in $J_{ij}$ (indeed, the most important of them) are antenna-based, i.e., they arise from measurable physical properties of (or above) individual antenna elements in a synthesis array. Thus, adequate calibration of an array of $N_{ant}$ antennas forming $N_{ant} (N_{ant}-1)/2$ baseline visibilities is usually achieved through the determination of only $N_{ant}$ factors, such that $J_{ij} = J_i \otimes J_j^{*}$.  For the rest of this chapter, we will usually assume that $J_{ij}$ is factorable in this way, unless otherwise noted.

As implied above, $J_{ij}$ may also be factored into the sequence of specific corrupting effects, each having their own particular (relative) importance and physical origin, which determines their unique algebra. Including the most commonly considered effects, the Measurement Equation can be written:

\begin{eqnarray}
\vec{V}_{ij}~=~M_{ij}~B_{ij}~G_{ij}~D_{ij}~E_{ij}~P_{ij}~T_{ij}~\vec{V}_{ij}^{\mathrm{~IDEAL}}
\end{eqnarray}

where:

• $T_{ij}~=~$ Polarization-independent multiplicative effects introduced by the troposphere, such as opacity and path-length variation.
• $P_{ij}~=~$ Parallactic angle, which describes the orientation of the polarization coordinates on the plane of the sky. This term varies according to the type of the antenna mount.
• $E_{ij}~=~$ Effects introduced by properties of the optical components of the telescopes, such as the collecting area's dependence on elevation.
• $D_{ij}~=~$ Instrumental polarization response. "D-terms" describe the polarization leakage between feeds (e.g. how much the R-polarized feed picked up L-polarized emission, and vice versa).
• $G_{ij}~=~$ Electronic gain response due to components in the signal path between the feed and the correlator. This complex gain term $G_{ij}$ includes the scale factor for absolute flux density calibration, and may include phase and amplitude corrections due to changes in the atmosphere (in lieu of $T_{ij}$). These gains are polarization-dependent.
• $B_{ij}~=~$ Bandpass (frequency-dependent) response, such as that introduced by spectral filters in the electronic transmission system
• $M_{ij}~=~$ Baseline-based correlator (non-closing) errors. By definition, these are not factorable into antenna-based parts.

Note that the terms are listed in the order in which they affect the incoming wavefront ($G$ and $B$ represent an arbitrary sequence of such terms depending upon the details of the particular electronic system). Note that $M$ differs from all of the rest in that it is not antenna-based, and thus not factorable into terms for each antenna.

As written above, the measurement equation is very general; not all observations will require treatment of all effects, depending upon the desired dynamic range. E.g., instrumental polarization calibration can usually be omitted when observing (only) total intensity using circular feeds. Ultimately, however, each of these effects occurs at some level, and a complete treatment will yield the most accurate calibration. Modern high-sensitivity instruments such as ALMA and JVLA will likely require a more general calibration treatment for similar observations with older arrays in order to reach  the advertised dynamic ranges on strong sources.

In practice, it is usually far too difficult to adequately measure most calibration effects absolutely (as if in the laboratory) for use in calibration. The effects are usually far too changeable. Instead, the calibration is achieved by making observations of calibrator sources on the appropriate timescales for the relevant effects, and solving the measurement equation for them using the fact that we have $N_{ant}(N_{ant}-1)/2$ measurements and only $N_{ant}$ factors to determine (except for $M$ which is only sparingly used). Note: By partitioning the calibration factors into a series of consecutive effects, it might appear that the number of free parameters is some multiple of $N_{ant}$, but the relative algebra and timescales of the different effects, as well as the  multiplicity of observed polarizations and channels compensate, and it can be shown that the problem remains  well-determined until, perhaps, the effects are direction-dependent within the field of view. Limited solvers for such effects are under study; the calibrater tool currently only handles effects which may be assumed constant within the field of view. Corrections for the primary beam are handled in the imager tool.  Once determined, these terms are used to correct the visibilities measured for the scientific target. This procedure is known as cross-calibration (when only phase is considered, it is called phase-referencing).

The best calibrators are point sources at the phase center (constant visibility amplitude, zero phase), with sufficient flux density to determine the calibration factors with adequate SNR on the relevant timescale. The primary gain calibrator must be sufficiently close to the target on the sky so that its observations sample the same atmospheric effects. A bandpass calibrator usually must be sufficiently strong (or observed with sufficient duration) to provide adequate per-channel sensitivity for a useful calibration. In practice, several calibrators are usually observed, each with properties suitable for one or more of the required calibrations.

Synthesis calibration is inherently a bootstrapping process. First, the dominant calibration term is determined, and then, using this result, more subtle effects are solved for, until the full set of required calibration terms is available for application to the target field. The solutions for each successive term are relative to the previous terms. Occasionally, when the several calibration terms are not sufficiently orthogonal, it is useful to re-solve for earlier types using the results for later types, in effect, reducing the effect of the later terms on the solution for earlier ones, and thus better isolating them. This idea is a generalization of the traditional concept of self-calibration, where initial imaging of the target source supplies the visibility model for a re-solve of the gain calibration ($G$ or $T$). Iteration tends toward convergence to a statistically optimal image. In general, the quality of each calibration and of the source model are mutually dependent. In principle, as long as the solution for any calibration component (or the source model itself) is likely to improve substantially through the use of new information (provided by other improved solutions), it is worthwhile to continue this process.

In practice, these concepts motivate certain patterns of calibration for different types of observation, and the calibrater tool in CASA is designed to accommodate these patterns in a general and flexible manner. For a spectral line total intensity observation, the pattern is usually:

1. Solve for $G$ on the bandpass calibrator
2. Solve for $B$ on the bandpass calibrator, using $G$
3. Solve for $G$ on the primary gain (near-target) and flux density calibrators, using $B$ solutions just obtained
4. Scale $G$ solutions for the primary gain calibrator according to the flux density calibrator solutions
5. Apply $G$ and $B$ solutions to the target data
6. Image the calibrated target data

If opacity and gain curve information are relevant and available, these types are incorporated in each of the steps (in future, an actual solve for opacity from appropriate data may be folded into this process):

1. Solve for $G$ on the bandpass calibrator, using $T$ (opacity) and $E$ (gain curve) solutions already derived.
2. Solve for $B$ on the bandpass calibrator, using $G$, $T$ (opacity), and $E$ (gain curve) solutions.
3. Solve for $G$ on primary gain (near-target) and flux density calibrators, using $B$, $T$ (opacity), and $E$ (gain curve) solutions.
4. Scale $G$ solutions for the primary gain calibrator according to the flux density calibrator solutions
5. Apply $T$ (opacity), $E$ (gain curve), $G$, and $B$ solutions to the target data
6. Image the calibrated target data

For continuum polarimetry, the typical pattern is:

1. Solve for $G$ on the polarization calibrator, using (analytical) $P$ solutions.
2. Solve for $D$ on the polarization calibrator, using $P$ and $G$ solutions.
3. Solve for $G$ on primary gain and flux density calibrators, using $P$ and $D$ solutions.
4. Scale $G$ solutions for the primary gain calibrator according to the flux density calibrator solutions.
5. Apply $P$, $D$, and $G$ solutions to target data.
6. Image the calibrated target data.

For a spectro-polarimetry observation, these two examples would be folded together.

In all cases the calibrator model must be adequate at each solve step. At high dynamic range and/or high resolution, many calibrators which are nominally assumed to be point sources become slightly resolved. If this has biased the calibration solutions, the offending calibrator may be imaged at any point in the process and the resulting model used to improve the calibration. Finally, if sufficiently strong, the target may be self-calibrated as well.

# General Calibrater Mechanics

The calibrater tasks/tool are designed to solve and apply solutions for all of the solution types listed above (and more are in the works). This leads to a single basic sequence of execution for all solves, regardless of type:

1. Set the calibrator model visibilities
2. Select the visibility data which will be used to solve for a calibration type
3. Arrange to apply any already-known calibration types (the first time through, none may yet be available)
4. Arrange to solve for a specific calibration type, including specification of the solution timescale and other specifics
5. Execute the solve process
6. Repeat 1-4 for all required types, using each result, as it becomes available, in step 3, and perhaps repeating for some types to improve the solutions

By itself, this sequence doesn't guarantee success; the data provided for the solve must have sufficient SNR on the appropriate timescale, and must provide sufficient leverage for the solution (e.g., D solutions require data taken over a sufficient range of parallactic angle in order to separate the source polarization contribution from the instrumental polarization).