Some investigations of leakage into MeerKAT images from far-field sources.

I. M. Stewart.

I have run some simulations to investigate the power leakage into the field of view caused by sources outside of it.

Array configurations

Simulations were run for both the KAT7 and MeerKAT array layouts. The centre frequency for all runs was 1420 MHz. For KAT7 the frequency and time binning was as follows:

Channels:20 x 40 MHz
Time bins:120 x 300s

This binning scheme ensures that time- and bandwidth-smearing is kept below 10% for sources 0.5 degrees from the phase centre. To achieve the same criterion for the MeerKAT layout, the channel width must be <= 1 MHz and the time bin duration <= 10s. These smaller numbers meant that to achieve with the MeerKAT simulation runs the same total bandwidth and observation duration as for KAT7, a prohibitively large number of visibilities would have been necessary. Because of this, 3 different combinations of lesser bandwidth and duration were employed, as follows:

Scheme A:(1x1 MHz) x (3600x10s)
Scheme B:(60x1 MHz) x (60x10s)
Scheme C:(720x1 MHz) x (5x10s)

All hour angle ranges were symmetrical about the meridian. No thermal noise was added to the visibilities, and perfect calibration was assumed.

The primary beam

The attenuation of sources due to the shape of the primary beam was calculated as follows. All antennas in both arrays were assumed to have the same primary beam. All polarization considerations were ignored: ie a single, total-power primary beam in Stokes I was all that was treated, and all sources were considered to be unpolarized. All antennas were assumed to have alt-azimuth mounts with no feed rotation, giving rise to parallactic rotation of the primary beam pattern upon the sky. The total-power beam was considered to be the square of a pattern of pseudo-intensity which was constructed as the sum of radial and orthogonal components. The radial component was constructed from the Hankel transform of a function R(r)=1-(r/r_dish)^2 for r<r_dish, =0 else. The orthogonal components were obtained as the Fourier transform of a rectangular cutout (ie, a rectangular area within which the value =-1) having the width and projected radial length of the feed support leg. After obtaining the power, the beam was multipied by cosine of the angle from the optic axis.

The breakdown into radial and orthogonal components was done for the sake of computational speed, since this allows the primary beam attenuation at any place in the field to be calculated by interpolation of 3 one-dimensional functions, one of which (the radial one) only needs to be performed once per source. Because of these short cuts, the primary beam could not be an exact replica of the measured KAT7 primary beam, but it was designed to approximately preserve the principal features of same, in particular the approximate size of the first sidelobe, and the existence of extended sidelobes originating in the four feed support legs. It may be desirable to alter the software at a later stage to make use of a more exact model of the primary beam.

The primary beam was defined out to 90 degrees from the direction of point. Overspill behind the antenna was not considered. A log-scale plot of the (total-power) primary beam model is shown in the figure.

Image location and properties

The phase centre (the location in the sky from which signals from all antennas arrive at the correlator in phase) was set to be the same place as the pointing centre of the antennas. The phase centre chosen for the simulated observations was at RA=53.11667 deg, dec=-27.80833 deg, which is the location of the Chandra Deep Field South. Since this is close to the latitude of the KAT7/MeerKAT arrays, the rate of parallactic rotation becomes relatively rapid near culmination.

In all cases a 1x1 degree field was imaged. The KAT7 images had 512x512 pixels, the MeerKAT 2048x2048. No attempt was made to remove spherical aberration. Both natural and uniform weighting were utilized, as indicated for each figure below. In all but one case the gridding kernel was a simple single-pixel 'top-hat'. The exceptional case used an expsinc kernel (see AIPS task IMAGR). In no case has the dirty image been divided by the Fourier transform of the gridding kernel. No cleaning was done.

Location and nature of simulated radio sources

Visibilities were simulated for a list of sources. The sources were assumed to be point-like and thus the visibilities were constructed not, as is often done, by Fourier transforming a model sky image, but by sampling complex exponentials of the appropriate argument. This sampling was done via pre-calculation of a 'lookup table', the visibilities then being interpolated from this table. The w contribution was calculated correctly and the smearing due to finite channel and time-sample width was approximated to first order.

The sources were taken from the PKSCAT catalog. The flux densities were taken from the S1410 column, corresponding to 1410 MHz; hence only sources with a measured flux density at that frequency were used. There were 2375 of these in total. All sources were assumed to have a spectral index of zero. Sources were extinguished when below the horizon.

For some of the simulations, the subset of sources within 10 degrees of the phase centre were used. There were 13 sources which obeyed this criterion (see figure); they range in flux density from 0.5 to 5.3 Jy. Note that none of the PKSCAT sources fall within the 1x1 degree imaged field (just as well).

Software

The programs which construct the simulated visibilities are written in python, and run on my desktop. The gridder was written in fortran 90, but makes use of c++ libraries. I can only run the gridder at Leicester in the UK at present, since it is built to use the extensive libraries of the XMM-Newton SAS, which I have not yet built on any local machine. The (double-precision) visibilities were stored in a simplified format as FITS files.

I plan to make the python software available after I get a sensible versioning scheme going, and the f90 stuff after I put the bloated SAS on a severe diet and install a slimline version locally.

Results (dirty images)

Ideally I would have simulated a full MeerKAT observation, using all 2375 sources in the 1410 MHz subset of PKSCAT. However in point of processing time, this was thought to be impractical. Truncated bandwidths and durations were used as described above; also, only those 13 PKSCAT-1410 sources within 10 degrees of the phase centre were simulated. A further simplification was the use of single-precision arithmetic within the gridder. Attempts were made to test each of these simplifications and approximations to gain an idea of what the results would be on a 'full' MeerKAT simulation.

Firstly, I ran simulations for the KAT7 array, using only the 13 inner sources. Results for uniform and natural weighting are as follows.

The fringes in the uniformly-weighted image probably result from a grid cell which has only a single visibility within it, and which happens by chance to sample the rapidly-oscillating visibility function of a bright, far-field source near its maximum value. More sophisticated gridding kernels and weighting functions would be expected to remove such effects.

Next I ran a KAT7 simulation using all the PKSCAT-1410 sources:

Clearly the inclusion of the extra sources makes very little difference. Next using a double-precision gridder:

Again no discernable difference.

The MeerKAT images are presented in a table for compactness:

Note that the brightness scales differ from image to image.

The final check was to grid using a more sophisticated kernel. The only other which I currently implement in my software (although it hasn't been thoroughly exercised) is the expsinc kernel described in the AIPS task IMAGR, also in the 'Imaging' chapter of the NRAO interferometry summer school lectures. The result is as follows:

Comparing this to the top-hat version (middle row, left column of the table above) we see that the more sophisticated gridding kernel has erased the aliased sources but left the traces of far-field dirty beam sidelobes largely unchanged. This is to be expected from interferometry theory, since far sidelobes arise from edges in the density distribution of visibility samples. Such edge cells often contain input from few samples and so the higher spatial frequency visibility components are not as well averaged as in more densely populated areas.

Results (RMS values)

The RMS values for the images shown above are tabulated as follows:

Description: RMS (microJy):
KAT7, inner 13, natural 13.91
KAT7, inner 13, uniform 11.55
KAT7, all PKSCAT, natural 13.88
KAT7, all PKSCAT, uniform 11.52
MeerKAT A, natural 1.30
MeerKAT A, uniform 0.40
MeerKAT B, natural 10.17
MeerKAT B, uniform 1.61
MeerKAT C, natural 8.61
MeerKAT C, uniform 2.77
MeerKAT B, natural, expsinc10.32

Conclusions

To reduce the RMS level of dirty beam sidelobes of out-of-field sources we need to pay attention to the primary beam, which is an issue of correct weighting of visibilities. Weighting schemes exist which will reduce such far-field sidelobes. One might also be able to reduce the traces by a factor of 10 or so by subtraction of a sky model from the visibilities. So I would conclude that this effect is unlikely to be a show-stopper, regardless of the primary beam shape.