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Basic Calibration Fundamentals

Calibration and imaging responsibilities are divided between the calibrater and imager tools. The calibrater tool handles visibility-plane calibration while the imager tool deals with image formation, deconvolution and image-plane calibration.

In a typical synthesis observation, a number of calibrators of known structure (usually point sources) and at least one of known flux density are observed in addition to the scientific target(s) in order to obtain a measure of the systematic properties of the instrument and, if relevant, the corrupting media above it. The effects of these systematic properties are then removed from the data for the scientific target. This technique is often referred to as cross-calibration and its success depends implicitly on the relationship between: 1) the true brightness distribution of the observed radio sources (what we are after); 2) the formal response of the instrument and its variety of imperfections of varying importance (what we are calibrating); 3) and the data that is actually recorded (what we have measured).

The AIPS++ Measurement Equation, originally defined by Hamaker, Bregman, & Sault (1996a, 1996b), is an explicit description of this relationship. The Measurement Equation (in the visibility plane) for a spectral line polarization calibration is given by

$$\vec{{V}}_{{ij}}^{} \vec{{V}}_{{ij}}^{{\mathrm{~IDEAL}}}$$ (1.1)

where:

$\vec{{V}}_{{ij}}^{}$  =  cross-correlations between two polarizations for each of two feeds (i, j) which characterize an individual baseline (e.g. the measured visibilities).

$\vec{{V}}_{{ij}}^{{\mathrm{~IDEAL}}}$  =  visibilities measured by an ideal interferometer (e.g. no instrumental errors or calibration effects). $\vec{{V}}_{{ij}}^{{\mathrm{~IDEAL}}}$  =  is directly related to the Fourier Transform of the true polarized sky brightness, $\vec{{I}}\,$ = (I, Q, U, V).

T_ij  =  Complex gain effects which are polarization-independent, e.g. tropospheric effects, high-frequency opacity corrections, antenna gain as a function of elevation, baseline corrections.

P_ij  =  Parallactic angle.

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 leiu of T_ij).

B_ij  =  Bandpass response.

Note that the Measurement Equation is a matrix equation. $\vec{{V}}_{{ij}}^{}$ and $\vec{{V}}_{{ij}}^{{\mathrm{~IDEAL}}}$ are 4-vectors with elements for each correlator product, and the T_ij, etc., are 4x4 matrices that `corrupt' the correlator products according to the properties of the specific term. The order of the corrupting terms from right-to-left, for the most part, is the order in which the errors affect the incoming wavefront. AIPS++ hardwires the Measurement Equation order and all of the specific algebra associated with the individual terms. As such, it is usually possible to ignore the fact that the Measurement Equation is a matrix equation, and treat the different terms as labeled `components' (black boxes) along the signal path.

In practical calibration, it is often best to begin the calibration process by determining solutions for those terms which affect the data most. Thus, the user would normally determine gain solutions (G) first (applying pre-computed parallactic angle corrections P if doing polarization calibration), then bandpass solutions (B) and/or polarization solutions (D). Errors which are polarization-independent (T) can be determined at any point in the calibration process. T has nearly identical form to G but it located at a different point in the Measurement Equation. A practical advantage to this factorization is the ability to store polarization-independent effects which are characterized by a certain timescale or a particular parametrization (e.g., tropospheric opacity) as a separate calibration factor, thus isolating them from electronic gain effects which may vary on a different timescale and/or with different systematics. Although most data reduction sequences will probably proceed in this order, the user has complete freedom and flexibility to determine the order in which calibration effects are solved for. Self-calibration, the process by which an improved model of the target source is used to obtain improved solutions for the calibration, is also not limited to just the electronic gain (G) corrections, as has been traditionally the case. Improved solutions for, say, the instrumental polarization (D) may be obtained by self-calibration as well. In general, decision-making along the calibration path is a process of determining which effects are dominating the errors in the data and deciding how best to converge upon the optimal set of calibration solutions (for all relevant components) and source models which best describe the observed data.

Once the calibration solutions have been determined, the AIPS++ software applies the solutions according to the order defined by the Measurement Equation. The individual calibration components represented in the Measurement Equation are described in detail in the Synthesis Calibration chapter of Getting Results. The generalization of the Measurement Equation to image plane effects is also described there.

The observed data, $\vec{{V}}_{{ij}}^{}$, are stored in the DATA column in the MAIN table of the MS; these are the raw data as loaded by the filler tool or imported from a UVFITS file. Associated with the DATA column are related columns to hold the most recent version of the calibrated data (CORRECTED_DATA), and the latest visibility plane representation of the source or field model, $\vec{{V}}_{{ij}}^{{IDEAL}}$, (MODEL_DATA). The latter two columns are filled in by the calibrater and imager tools, respectively. The actual calibration information is stored in separate calibration tables. The observed DATA column does not change during reduction, but the related columns do. When plotting the data using a tool such as msplot, the data sub-types (observed, corrected, and model) can be selected individually.