EVLA Advanced Topics 3C391
Continuum Observations Data Reduction Tutorial: 3C 391---Advanced Topics
In this document, we discuss various "advanced topics" for further reduction of the 3C 391 continuum data. This tutorial assumes that the reader already has some familiarity with basic continuum data reduction, such as should have been obtained [Continuum Data Reduction Tutorial] on the first day of the NRAO Synthesis Imaging Workshop data reduction tutorials. If one did not participate in the EVLA Continuum Data Reduction Tutorial, one could use the script extractor to generate a CASA reduction script and process the data to form an initial image. Current experience on a standard desktop computer suggests that such a data set could be processed in 20 min. or less.
Polarization Imaging
In the previous data reduction tutorial, a full polarization imaging cube of 3C 391 was constructed. This cube has 3 dimensions, the standard two angular dimensions (right ascension, declination) and a third dimension containing the polarization information. Considering the image cube as a matrix, [math]\displaystyle{ Image[l,m,p] }[/math], the [math]\displaystyle{ l }[/math] and [math]\displaystyle{ m }[/math] axis describe the sky brightness or intensity for the given [math]\displaystyle{ p }[/math] axis. If one opens the viewer and loads the 3C 391 continuum image, the default view contains an "animator" or pane with movie controls. One can step through the polarization axis, displaying the images for the different polarizations.
As [constructed], the image contains four polarizations, for the four Stokes parameters, I, Q, U, and V. Recalling the lectures, Q and U describe the linear polarization and V describes the circular polarization. Specifically, Q describes the amount of linear polarization aligned with a given axis, and U describes the amount of linear polarization at a 45 deg angle to that axis. The V parameter describes the amount of circular polarization, with the sign (positive or negative) describing the sense of the circular polarization (right- or left-hand circularly polarized).
In general, few celestial sources are expected to show circular polarization, with the notable exception of masers, while terrestrial and satellite sources are often highly circularly polarized. The V image is therefore often worth forming because any V emission could be indicative of unflagged RFI within the data (or problems with the calibration!).
Because the Q and U images both describe the amount of linear polarization, it is more common to work with a linear polarization intensity image, [math]\displaystyle{ L = \sqrt{Q^2 +U^2} }[/math]. ([math]\displaystyle{ L }[/math] can also be denoted by [math]\displaystyle{ P }[/math].) Also important can be the polarization position angle [math]\displaystyle{ tan 2\chi = U/Q }[/math].
The relevant task is [immath], with specific examples for processing of polarization images given in [Polarization Manipulation]. The steps are the following.
1. Extract the Q and U planes from the full Stokes image cube, forming separate Q and U images.
2. Combine the Q and U images using the mode='poli' option of immath to form the linear polarization image.
3. If desired, combine the Q and U images using the mode='pola' option of immath to form the polarization position angle image. Because the polarization position angle is derived from the tangent function, the order in which the Q and U images are specified is important.
4. If desired, form the fractional linear polarization image, defined as L/I.
One can then view these various images using viewer.
Self-Calibration
Recalling the lectures, even after the initial calibration using the amplitude calibrator and the phase calibrator, there are likely to be residual phase and/or amplitude errors in the data. Self-calibration is the process of using an existing model, often constructed by imaging the data itself. Provided that sufficient visibility data have been obtained, and this is essentially always the case with the EVLA (and often the VLBA, and should be with ALMA), the system of equations is wildly over-constrained for the number of unknowns.
More specifically, the observed visibility data on the [math]\displaystyle{ i }[/math]-[math]\displaystyle{ j }[/math] baseline can be modeled as
[math]\displaystyle{ V'_{ij} = G_i G^*_j V_{ij} }[/math]
where [math]\displaystyle{ G_i }[/math] is the complex gain for the [math]\displaystyle{ i^{\mathrm{th}} }[/math] antenna and [math]\displaystyle{ V_{ij} }[/math] is the "true" visibility. For an array of [math]\displaystyle{ N }[/math] antennas, at any given instant, there are [math]\displaystyle{ N(N-1)/2 }[/math] visibility data, but only [math]\displaystyle{ N }[/math] gain factors. For an array with a reasonable number of antennas, [math]\displaystyle{ N \gt ~ 8 }[/math], solutions to this set of coupled equations converge quickly.
