This page provides information on the continuum, spectral line, and mosaic sensitivity calculators. These can be used to assist in developing appropriately motivated sensitivity calculations and time requests that must be included as part of an observing proposal.

The calculators can be found at https://apps.sarao.ac.za/calculators/

Note that clicking on the ‘More information’ button on each tool will bring you to the appropriate section of this document.

Note: All MeerKAT bands are covered by the calculators and these calculators supersede the ipython notebooks released in a previous Call for Proposals. The calculators should be used for the preparation of any observation time request for MeerKAT.

Contents

Quick look tables of sensitivities

Table 1 below gives the theoretical thermal rms noise expected for some example integration times for continuum observations.

Table 1: Expected thermal noise for continuum observations in the L-band. We assume 58 antennas and consider only robust -0.5, which is a good default for continuum imaging, with no confusion noise estimates or Gaussian tapering. Confusion is not reflected in this table since it is dependent on declination and imaging parameters. Please use the continuum sensitivity tool for a more realistic calculation.

Open

Continuum sensitivity

This calculator takes into account expected beam size as a function of declination and imaging parameters to calculate confusion noise, in addition to scaled and tapered thermal noise. It does not take into account additional noise due to calibration errors and dynamic range limitations.

Using the calculator

Open

A screenshot of the continuum calculator, which consists of three input/output zones.

The top left area consists of a series of drop-down boxes for the observation and imaging parameters, and the planned integration time. To only apply Briggs weighting, select ‘Untapered’ in the tapering drop-down box.

Outputs are in the top right - details of calculations are described below.

The plot can be used to determine the optimal robust weighting for the proposed observations. Extraneous curves can be turned off by clicking on the legends:

Assumptions

L- and UHF bands

• A system-equivalent flux density (SEFD) of 425 Jy in L-band and 550 Jy in UHF band, per antenna, is assumed. Note that the SEFDs do have a slope as a function of frequency - full plots can be found here.

• The observatory minimum requirement for a science array is 58 antennas, though more are generally available. The calculator assumes 58.

• While careful flagging can, in some instances, yield some useful data in bands dominated by GNSS and GSM signals, these bands are generally discarded by continuum observers. This gives worst case effective bandwidths as follows:

S-band

Due to currently logistal issues, the sensitivity calculator assumes 54 antennas in operation at S-band. The sensitivity calculations are based on a limited set of measurements, and may be updated in future.

*No separate measurements were made in S1. The SEFD is inferred from the overlapping sub-bands.

Details of the calculations

The Stokes I point source sensitivity under natural weighting is given by the radiometer equation as follows:

Here is the number of antennas, is the total (unflagged) bandwidth available and is the total integration time on the source.

However, in practice, we don’t often use natural weighting, particularly for continuum point source observations where high resolution is important. The calculator therefore also offers standard imaging robust and tapering factors (see below).

Residual calibration errors do not scale with time, though a full uv track will improve the point spread function (PSF), reducing noise due to sidelobes from bright sources. For more information, please see our pages on dynamic range and direction-dependent effects.

Pure thermal noise will scale as a function of integration time. However after sufficient integration time, depending on the beam size, the field will become dominated by confusion noise. This represents a fundamental limit to continuum detections with MeerKAT.

Resolution

We estimate the major axis of the restoring beam, as fitted with the WSClean imager (Offringa et al. 2014).

The MeerKAT PSF at various weightings is shown below for context to highlight the flank structure seen at weightings closer to natural weighting.

As can be seen here the mixture of two distributions for the short and long spacings make the primary lobe of the PSF significantly non-Gaussian, therefore breaking the assumption of restoring routines implemented in CLEAN-based approaches. This PSF is synthesized for a transiting observation at declination -30. Higher declinations (equatorial and above) have substantial sidelobes which may impact the achievable dynamic range.

UV tapering and robust weighting

The sensitivity calculator has been extended to include basic point source sensitivity and confusion estimates based on a tabulated selection of cuts in the uv domain for both bands. This is implemented as a circular Tukey taper with no inner tapering applied. The cuts chosen form linear steps between 7.7 km and 0.9 km (between 23.11 and 2.83 kλ respectively at 900 MHz for L band, or between 15.41 and 1.89 kλ at 600 MHz for UHF band).

Users may select tapering settings up to core tapering. The latter sufficiently drives down the sidelobe contribution of the distribution of long baselines to remove the flank structure highlighted above. We also provide fitted beam sizes for a range of declinations [SCP, -60, -30, -10, +10 and +30]. Users should select the declination closest to their target coordinates. In all instances, the PSF is synthesized for a moderately long track (8 hours for all but +30 declination - the latter only allows for UV coverage of around 6 hours). The major-axis is then used to estimate the confusion noise at either 1.28 GHz or 816 MHz (L- and UHF-band MFS weighted averages).

The taper values were chosen to keep instrument resolution close to constant at all imaging sub-bands, which is typical for in-band spectral index estimation for compact emission. Note that we do not apply any inner tapers in these estimates – these are necessitated only when measuring spectra for extended emission and will degrade sensitivity at largest scales.

For a chosen tapering and robust weighting (and resulting fitted instrument resolution) we show the drop in unpolarized point source sensitivity. For example

Note that, as explained above, we do not show fitted resolutions where the MeerKAT PSF main lobe is significantly non-Gaussian – this would break the assumption underpinning the restoring operation in CLEAN-based approaches which are widely implemented in commonly used imagers. As such the indicated naturally weighted thermal point source sensitivity may not be achievable and is given only to quantitatively show the expected drop in sensitivity due to tapering.

It is possible to turn off tapering by leaving the selector on the blank option. In this case only Briggs robust weights are applied. This choice of tapering will consequently ensure maximum resolution at highest usable frequencies of the individual sub-bands. Values of +1 and above on the robust scale will asymptotically approach the indicated theoretical natural unpolarized point source sensitivity if no tapering is selected.

Estimating the confusion noise

Figure 15 of Mauch et al. (2020) shows that at a confusion signal-to-noise ratio of q = 5, the minimum number (β_min) of beam solid angles (Ω_b) per reliably detectable source is roughly β_min = 25 for a power-law differential source count n(S) ∝ S^-γ when 1.4 ≤ γ ≤ 2.0. The counts measured by Matthews et al. (2021) from the MeerKAT DEEP2 image show that γ lies within this range for 1.25uJy < S_1.4GHz < 1 Jy.

Using the Matthews et al. (2021) measurements for S < 25 Jy, we derive the rms confusion noise σ_c from the cumulative source count

We calculate the flux density S_min at which

for a circular Gaussian restoring beam FWHM θ with solid angle Ω_b =πθ²/(4 ln(2)) and compute σ_c = S_min/5. When computing the integral above we scale the 1.4 GHz NVSS (Condon et al. 1998) counts at S > 2.5 mJy from Matthews et al. (2021) to the DEEP2 frequency of 1.266 GHz assuming a spectral index α = -0.7.

Brightness noise

MeerKAT has such good surface-brightness sensitivity that it easily approaches the confusion limit.  We have added a calculation of the rms brightness noise to aid the design of observations of low-brightness extended continuum sources, as follows:

The spectral line sensitivity calculator

This sensitivity calculator uses the radiometer equation and a correction factor based on the desired beam size, at a specified target declination, to provide a theoretical lower bound on the expected noise floor of spectral line observations. 

Using the calculator

Channel width

The channel width refers to the channel width of the imaged data cube, in kHz, which would either be the native channel resolution of the mode or the binned channel width. A dropdown menu gives a quick look-up of the channel widths of various correlator modes, or you can enter your own value depending on your binning strategy. The resulting velocity width at your observation frequency will be calculated.

The table below provides the channel widths for each mode in kHz, and and indication of the velocity width (which is dependent on the frequency of interest). Note that narrowband modes are currently not available for the UHF or S bands.

The observation frequency is the red-shifted or sky frequency of your target, not the line rest frequency.

Assumptions

• 58 antennas (the minimum observatory operation requirement) are assumed. Generally more antennas are available.

• We assume continuum subtraction has been done, so confusion noise is not included.

Details of the calculations

The correction factor or "Tapering Factor"  is calculated by doing simulations in wsclean for a target where the robust weighting factor is varied between -2 and +2 (natural weighting). 

This results in several datasets with differing beam sizes and effective numbers of visibilities used in imaging.  This is repeated for several declinations, which allows an interpolation between declination, frequency scaled beam size to the effective number of visibilities and hence a correction factor.  

There is also a warning if there is known persistent RFI at the frequency of the observation.  This just checks whether there is RFI present within the bandwidth range used in the calculation and does not take into account the amount of the bandwidth that is affected - just that it is present. Users should check the RFI documentation on the wiki.  The HI Column Density calculation uses the velocity width, which takes into account the redshift of the HI line as determined by the observation frequency.   

The mosaic calculator

This calculator does not calculate the rms noise of a single pointing. This needs to first be calculated by using the continuum or spectral line calculators discussed above and entered into the appropriate field of the calculator. The mosaic calculator will take into account beam overlap to produce a plot of the combined sensitivity across the region of interest.

Recommended workflow

1. Use either the continuum or spectral line sensitivity calculator to calculate the rms noise for a single pointing, according to your planned observation and imaging parameters.

2. Determine your pointing grid, either from your own calculations or using the ‘Targets’ tab on the mosaic calculator.

3. Determine the resulting rms noise distribution across the map area.

Using the calculator

If you are not certain of the exact pointing centres to be used, but you have a boundary of the area that you wish to cover, the first tab of the calculator can be used to generate an optimal set of pointings. If you are uncertain of what the separation should be, it can be calculated for you based on the frequency to be optimised for. Note that MeerKAT’s large bandwidth implies almost a factor of 2 difference between the highest and lowest frequencies. General practice is to space the pointings by FWHM/sqrt(2), however, you may choose your own separation or pointing centres to concentrate on areas of particular interest.

Calculate pointing centres

1. Decide on the area that you wish to image. Use your favorite image viewer or published image to calculate the nodes of a polygon to encapsulate the area of interest.

2. Decide on the separation that you wish to use. If uncertain the tool can compute it for you, based on the frequency of interest. For spectral line work, this would be at the observed line frequency. For continuum observations, you may wish to optimise at the highest observed frequency to ensure optimal signal-to-noise across the entire band.

3. Enter or upload polygon nodes. An example csv file is shown below:

The result is as follows:

You can set the border and rotation of pointings fields to 0 to start with. Now click ‘Calculate’.

4. Refine positions of points by tweaking either the border of the polygon, or rotate the line-up of the points. The calculated grid can be tilted by the rotation angle, which may be needed to get optimal filling of the polygon.

The ‘border’ parameter sets a border around the polygon. If the border is positive, no grid point can have a distance lower than border towards the input polygon. If the border is negative, grid points outside the polygon are allowed if their distance to a side of the polygon is less than the absolute value of border.

5. Save the target pointings as csv (you will need it later to set up your observation). And click the ‘fast forward’ button to copy the target pointings to the next tab. Now you’re all set to see whether you hit your target rms noise across the area of your mosaic.

Generating a sensitivity map

Click on the second tab, labelled Sensitivity. You can either carry over the pointings generated from the first tab, enter a few points manually or upload a csv file with RA/Dec coordinates.

The map produced will have grey scale of the expected noise and contours.  Statistics are calculated from this map as the average within 5 contours on the map. For the mosaic to be appropriate for your science case, your science target should fall entirely within the contour that has an average sensitivity appropriate for your science question.   

Assumptions

The Stokes I primary beam shape is assumed to be a tapered cosine (Mauch et al 2020).

Details of the calculation

The script will first calculate a weight map (shown in greyscale), and then generates a noise map (shown by contours). The beam attenuation patterns are added in quadrature.

This calculation is repeated for a sampling* of frequencies contained within the bandwidth centered on the observation frequency. The resulting noises maps are then combined. Generally when the bandwidth is much smaller than the observation frequency there will be little effect from the change of the primary beam size over that bandwidth. However  this will become more significantly as the bandwidth gets larger. In the case of  a spectral line observation, where the bandwidth == channel width this is a negligible effect.