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 30 min. or less.

Image Analysis and Manipulation

This topic is perhaps not "advanced," but it appears to fit more naturally here. It is assumed that an image 3c391_ctm_spw0_IQUV.image, resulting from the [3C391 Continuum Tutorial] exists.

The three most basic analyses are to determine the peak brightness, the flux density, and the image noise level. These are useful measures of how well one's imaging efforts are in approaching the thermal noise limit or in reproducing what is already known about a source. Additional discussion of image analysis and manipulation, including the combination of multiple images, mathematical operations on images, and much more can be found in [Image Analysis] in the CASA Reference Book.

The most straightforward statistic is the peak brightness, which is determined by imstat.

imstat(imagename='3c391_ctm_spw0_IQUV.image',stokes='')

• stokes=' ' : This example determines the peak brightness in the entire image, which has all four Stokes planes. If one wanted to determine the peak brightness in just, say, the Stokes V image, one would set stokes='V'.

The other two statistics require slightly more care. The flux density of a source is determined by integrating its brightness or intensity over some solid angle, i.e., [math]\displaystyle{ S = \int d\Omega I }[/math], where [math]\displaystyle{ I }[/math] is the intensity (measured in units of Jy/beam), [math]\displaystyle{ \Omega }[/math] is the solid angle of the source (e.g., number of synthesized beams), and [math]\displaystyle{ S }[/math] is the flux density (measured in units of Jy). In general, if the noise is well-behaved in one's image, when averaged over a reasonable solid angle, the noise contribution should approach 0 Jy. If that is the case, then the flux density of the source is also reported by imstat. However, there are many cases for which a noise contribution of 0 Jy may not be a safe assumption. If one's source is in a complicated region (e.g., a star formation region, the Galactic center, near the edge of a galaxy), a better estimate of the source's flux density will be obtained by limiting carefully the solid angle over which the integration is performed.

Open viewer and use it to display an image, such as 3c391_ctm_spw0_IQUV.image.

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 }[/math] >~ 8, solutions to this set of coupled equations converge quickly.

There is a small amount of discussion in the CASA Reference Manual on [self calibration]. In self-calibrating data, it is useful to keep in mind the structure of a Measurement Set: there are three columns of interest for an MS, the DATA column, the MODEL column, and the CORRECTED_DATA column. In normal usage, as part of the initial split, the CORRECTED_DATA column is set equal to the DATA column. The self-calibration procedure is then

• Produce an image (clean) using the CORRECTED_DATA column.
• Derive a series of gain corrections (gaincal) by comparing the DATA columns and the Fourier transform of the image, which is stored in the MODEL column. These corrections are stored in an external table.
• Apply these corrections (applycal) to the DATA column, to form a new CORRECTED_DATA column.

The following example begins with the standard data set, 3c391_ctm_mosaic_spw0.ms, resulting from the [3C391 Continuum Tutorial]. A model image is generated (3c391_ctm_spw0_IQUV.image), this model is used to generate a series of gain corrections (stored in 3C391_ctm_mosaic_spw0.G2), those gain corrections are applied to the data to form a set of self-calibrated data, and new image is then formed (3c391_ctm_spw0_IQUV_G2.image).

#In CASA
clean(vis='3c391_ctm_mosaic_spw0.ms',imagename='3c391_ctm_spw0_IQUV',
      field='',spw='',
      mode='mfs',
      niter=5000,
      gain=0.1,threshold='0.0mJy',
      psfmode='clarkstokes',
      imagermode='mosaic',ftmachine='mosaic',
      multiscale=[0, 6, 18, 54],smallscalebias=0.9,
      interactive=True,
      imsize=[576,576],cell=['2.5arcsec','2.5arcsec'],
      stokes='IQUV',
      weighting='briggs',robust=0.0,
      calready=True)

gaincal(vis='3c391_ctm_mosaic_spw0.ms',caltable='3C391_ctm_mosaic_spw0.G2',
        field='',spw='',selectdata=False,
        solint='30s',refant='ea21',minblperant=4,minsnr=3,
        solnorm=True,gaintype='G',calmode='p',append=False)

applycal(vis='3c391_ctm_mosaic_spw0.ms',
         field='',spw='',selectdata=False,
         gaintable= ['3c391_ctm_mosaic_spw0.G2'],gainfield=[''],interp=['nearest'])

clean(vis='3c391_ctm_mosaic_spw0.ms',imagename='3c391_ctm_spw0_IQUV_G2',
      field='',spw='',
      mode='mfs',
      niter=5000,
      gain=0.1,threshold='0.0mJy',
      psfmode='clarkstokes',
      imagermode='mosaic',ftmachine='mosaic',
      multiscale=[0, 6, 18, 54],smallscalebias=0.9,
      interactive=True,
      imsize=[576,576],cell=['2.5arcsec','2.5arcsec'],
      stokes='IQUV',
      weighting='briggs',robust=0.0,
      calready=True)

Commonly, this procedure is applied multiple times. The number of iterations is determined by a combination of the data quality and number of antennas in the array, the structure of the source, the extent to which the original self-calibration assumptions are valid, and the user's patience. With reference to the original self-calibration equation above, if the observed visibility data cannot be modeled well by this equation, no amount of self-calibration will help. A not-uncommon limitation for moderately high dynamic range imaging is that there may be baseline-based factors that modify the "true" visibility. If the corruptions to the "true" visibility cannot be modeled as antenna-based, as they are above, self-calibration won't help.

Self-calibration requires experimentation. Do not be afraid to dump an image, or even a set of gain corrections, change something and try again. Having said that, here are several general comments or guidelines:

• Bookkeeping is important! Suppose one conducts 9 iterations of self-calibration. Will it be possible to remember one month later (or maybe even one week later!) which set of gain corrections and images are which? In the example above, the descriptor 'G2' is attached to various files to help keep straight which is what. 'G2' is used because the original calibration already included a gain calibration, in 'G1'.

• A common metric for whether self-calibration is whether the image dynamic range (= max/rms) has improved. An improvement of 10% is quite acceptable.

• Be careful when making images and setting CLEAN regions or masks. Self-calibration assumes that the model is "perfect." If one CLEANs a noise bump, self-calibration will quite happily try to adjust the gains so that the CORRECTED_DATA describe a source at the location of the noise bump. As the author demonstrated to himself during the writing of his thesis, it is quite possible to take completely noisy data and manufacture a source. It is far better to exclude some feature of a source or a weak source from initial CLEANing and conduct another round of self-calibration than to create an artificial source. If a real source is excluded from initial CLEANing, it will continue to be present in subsequent iterations of self-calibration; if it's not a real source, one probably isn't interested in it anyway.

• Start self-calibration with phase-only solutions (calmode='p' in gaincal). As [Rick Perley] has discussed in previous summer school lectures, a phase error of 20 deg is as bad as an amplitude error of 10%.

• In initial rounds of self-calibration, consider solution intervals longer than the nominal sampling time (solint in gaincal) and/or lower signal-to-noise ratio thresholds (minsnr in gaincal). Depending upon the frequency and configuration and fidelity of the model image, it can be quite reasonable to start with solint='30s' or solint='60s' and/or minsnr=3 (or even lower). One might also want to consider specifying a uvrange, if, for example, the field has structure on large scales (small [math]\displaystyle{ u }[/math]-[math]\displaystyle{ v }[/math]) that is not well represented by the current image.

• One can track the agreement between the DATA, CORRECTED_DATA, and MODEL in plotms. The options in 'Axes' allows one to select which column is to be plotted. If the MODEL agrees well with the CORRECTED_DATA, one can use shorter solint and/or higher minsnr values.