Wide-Field Imaging

W-term, Primary Beams (models, pbcor, A-Projection)

Widefield imaging in CASA is unverified - please use at own discretion.

 

Wide-field imaging typically refers to fields of view over which the basic 2D Fourier transform assumption of interferometric imaging does not apply and where standard on-axis calibration will not suffice.

 

The non-coplanar baseline effect: W-term

For wide-field imaging, sky curvature and non-coplanar baselines result in a non-zero w-term. Standard 2D imaging applied to such data will produce artifacts around sources away from the phase center. CASA has two methods to correct the w-term effect.

 

Faceting

In this method, visibilities are gridded multiple times onto the same uv-grid, each time with a different phase-reference center. One single dirty/residual image is constructed from the resulting grid and deconvolved using a single PSF (picked from the first facet). This deconvolution is not affected by emission that crosses facet boundaries, unlike in image-domain faceting, which is an older approach where small facet images are deconvolved separately before being stitched together. [1]

Citation Number 1
Citation Text Sault et al, 1999

In tclean, faceting is available via gridder='widefield' where you can specify the number of desired facets on a side.  It can be used along with W-Projection as well, for very large fields of view.

 

W-projection

In this method, visibilities with non-zero w-values are gridded using using a gridding convolution function (GCF) given by the Fourier transform of the Fresnel EM-wave propagator across a distance of w wavelengths. In practice, GCFs are computed for a finite set of w-values (wprojplanes) and applied during gridding. W-projection is roughly an order of magnitude faster than faceted imaging because it grids each visibility only once [2].

Citation Number 2
Citation Text Cornwell et al, 2008

In tclean, w-projection is available via gridder='widefield' or 'wproject' or 'awproject'.  In all cases, the 'wprojplanes' parameter must be set. It represents the number of discrete w-values to be used to quantize the range of w-values present in the dataset being imaged. An appropriate value of wprojplanes depends on whether there is a bright source far from the phase center, the desired dynamic range of an image in the presence of a bright far out source, the maximum w-value in the measurements, and the desired trade off between accuracy and computing cost. As a (rough) guide, VLA L-Band D-config may require a value of 128 for a source 30arcmin away from the phase center. A-config may require 1024 or more. To converge to an appropriate value, try starting with 128 and then increasing it if artifacts persist. W-term artifacts (for the VLA) typically look like arc-shaped smears in a synthesis image or a shift in source position between images made at different times. These artifacts are more pronounced the further the source is from the phase center. There is no harm in simply always choosing a large value (say, 1024) but there will be a significant performance cost to doing so, especially for gridder='awproject' where it is combined with A-Projection. wprojplanes=-1 may be used with gridder='widefield' or 'wproject' to automatically compute the number of planes.

The formula that CASA uses to calculate the number of plans when wprojplanes=-1 is:

$N_\mathrm{wprojplanes} = 0.5\times \frac{W_\mathrm{max}}{\lambda} \times \frac{\mathrm{imsize}}{\mathrm{(radians)}}$

where $W_\mathrm{max}$ is the maximum $w$ in your $uvw$ data and imsize is the largest linear size of your image. This formula is somewhat conservative and it is possible to achieve good results by using a smaller number of planes, which can also save on speed and memory.

 

 

Antenna Voltage/Power Patterns: Primary-Beam

The aperture-illumination-function (AIF) of each antenna results in a direction-dependent complex gain that can vary with time and is usually different for each antenna. The resulting antenna power pattern is called the primary beam. There are two methods to correct for the effect of the primary beam.  

Image-domain PB-correction

A simple method of correcting the effect of the primary beam is a post-deconvolution image-domain division of the model image by an estimate of the average primary beam or some other model. This method ignores primary-beam variations across baselines and time, and is therefore approximate, limiting the imaging dynamic-range even within the main lobe of the beam.  This approach also cannot handle heterogenous arrays.

In tclean, this option is available by setting pbcor=True.  When used with gridder='standard' or 'widefield' or 'wproject' which do not internally use any primary beam models, it will compute a model PB at the reference frequency per image channel, and divide it out of the output restored image.   If used with gridder='mosaic' or 'awproject', it will use a weighted average of the primary beam models used by the gridders per baseline and timestep.

Primary Beam correction for wide bandwidth observations is discussed in the Wideband Imaging section.

 

A-Projection

Time and baseline-dependent corrections are applied during gridding, by computing GCFs for each baseline as the convolution of the complex conjugates of two antenna aperture illumination functions. An additional image-domain normalization step is required, and can result in the image being "flat-sky" ( the image represents only the sky intensity) or "flat-noise" (the image represents the sky multiplied by the primary beam). The advantage of this method is that known time and baseline variability can be accounted for, both during gridding as well as de-gridding [3].

Citation Number 3
Citation Text Bhatnagar et al, 2008

Different primary beam effects cause artifacts at different levels in the image [4]. Depending on the available sensitivity of an observation or desired dynamic range, one can choose to leave out some corrections and save on computing time.  In general, the varying dish size in a heterogenous array is the dominant source of errors causing a dynamic range limit of a few 100. Next come large pointing offsets (such as beam squint or illumination offsets) and at a higher dynamic ranges ($10^4$ and beyond) are other factors such as the details about feed leg structures. On its own, parallactic angle rotation causes artifacts only at a dynamic range of around $10^5$ but if any of the other large effects (pointing offset or illumination pattern errors) are not azimuthally symmetric, then parallactic angle rotation will have an effect at much lower dynamic ranges.

Citation Number 4
Citation Text Kundert et al 2016

gridder = 'awproject'

In tclean, gridder='awproject' applies the full A-Projection algorithm and uses baseline, frequency and time dependent primary beams. They are azimuthally asymmetric to account for feed leg structures. They also include beam squint, which is corrected during gridding by applying an appropriate phase gradient across the GCFs to cancel out the polarization dependent pointing offset.  The frequency dependence of the primary beam within the data being imaged is included in the calculations and can optionally also be corrected for during gridding (see Wideband Imaging section for details). 

Full Mueller matrix support is being added into the system for full-polarization widefield imaging.  Currently, heterogenous arrays like ALMA are not supported, but it will be suitable for VLA widefield imaging.  The computing cost of full A-Projection can be very large, and there are a number of parameters to apply approximations that can reduce the computing load. The beam models used here are derived from ray-traced aperture illumination functions and currently no external beam models can be fed in.

 

gridder='mosaic'

In tclean, gridder='mosaic' applies an approximation of the A-Projection algorithm where it uses azimuthally symmetric beam models that can be different per baseline. It includes the diagonal of the Mueller matrix for multi-Stokes images, but ignores off-diagonals. The frequency dependence of the primary beam is accounted for but is not eliminated during gridding. Since time dependence is not supported by default, the computational cost is lower than A-Projection.   Since ALMA imaging typically involves small fractional bandwidths, includes data with multiple dish sizes, and needs to operate on very large cubes with many channels, this option is suitable for ALMA.  It is also possible to supply external beam models to this gridder, by setting up the vpmanager tool, and one can in principle assign beams separately for each antenna as a function of time, if needed. Note that gridder='mosaic' can be used even on a single pointing, especially to account for effects due to a heterogenous array. 

 

Mosaics

Data from multiple pointings can be combined during gridding to form one single large image. Details are are described in the Mosaicing page.  In a Linear Mosaic, data from multiple pointings are imaged (and optionally deconvolved too) before being stitched together. A Joint Mosaic is a simple extension of A-Projection in which phase gradients are applied to the gridding convolution functions to map data from each pointing to a different position on the sky.  In tclean, gridder='mosaic' and 'awproject' will both create joint mosaics if data from multiple pointings are selected as the input.

  

Primary Beam Models

gridder='standard', 'wproject', 'widefield', 'mosaic'

Default PB models :

VLA: PB polynomial fit model (Napier and Rots, 1982) [5]

Citation Number 5
Citation Text Napier and Rots, 1982

EVLA: New EVLA beam models (Perley 2016) [6]

Citation Number 6
Citation Text Perley 2016

ALMA : Airy disks for a 10.7m dish (for 12m dishes) and  6.25m dish (for 7m dishes) each with 0.75m blockages (Hunter/Brogan 2011). Joint mosaic imaging supports heterogeneous arrays for ALMA  (Hunter/Brogan 2011)

These are all azimuthally symmetric beams. For EVLA, these models  limit the dynamic range to 10^5 due to  beam squint with rotation and the presence of feed leg structures.  For ALMA, these models accounting only for differences in dish size, but not in any feed-leg structural differences between the different types of antennas.

 

Adding other PB models

Use the vpmanager tool, save its state, and supply as input to tclean's vptable parameter

Example : For ALMA and gridder='mosaic', ray-traced (TICRA) beams are also available via the vpmanager tool. To use them, call the following before the tclean run: 

vp.setpbimage(telescope="ALMA", compleximage='/home/casa/data/trunk/alma/responses/ALMA_0_DV__0_0_360_0_45_90_348.5_373_373_GHz_ticra2007_VP.im',  antnames=['DV'+'%02d'%k for k in range(25)])
vp.saveastable('mypb.tab')


Then, supply vptable='mypb.tab' to tclean.

 

gridder = 'awproject'

VLA / EVLA : Uses ray traced models (VLA and EVLA) including feed leg and subreflector shadows, off-axis feed location (for beam squint and other polarization effects), and a Gaussian fit for the feed beams [7]

Citation Number 7
Citation Text Brisken 2009

ALMA : Similar ray-traced model as above (but the correctness of its polarization properties remains un-verified).

These models do not yet support heterogenous arrays (although the version of CASA's AWProjection used by LOFAR's LWImager does have fully heterogenous support). Full-polarization support is currently being worked on.