Sensitivity calculators

 

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: These calculators supersede the ipython notebooks released in a previous Call for Proposals and should be used for the preparation of any observation time request for MeerKAT. L- and UHF bands are currently covered.

Contents



It is strongly recommended that the Chrome browser be used to access all SARAO apps. We cannot guarantee performance on any other browser.

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.

Integration time

Thermal noise at robust=-0.5 (uJy/beam)

excluding persistent RFI channels

Integration time

Thermal noise at robust=-0.5 (uJy/beam)

excluding persistent RFI channels

 

L-band

UHF band

12 minutes

20.4

26.6

1 hour

9.1

11.9

8 hours

3.2

4.2

 

The calculators give the recommended time on target only. Calibration and slewing overheads need to be added to your time request. For longer (> 5 hours) single target observations following the standard calibration scheme an overhead of 25% can be assumed. Shorter observations, or slews to multiple targets, may incur higher overheads. Please refer to our page on average overheads for further details.

Your observation can be simulated in the Observation Planning Tool (OPT).

 


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

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. Note that the tapering field can be left blank (see empty option in the dropdown menu) and only Briggs weighting will be applied.

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

  • 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:

 

L-band

UHF band

Total bandwidth

856 MHz

544 MHz

RFI loss

45% (mainly GNSS / GSM downlink)

10% (mainly GSM downlink)

Rolloff low

44 MHz

36 MHz

Rolloff high

42 MHz

73 MHz

Effective available bandwidth

385 MHz

380 MHz

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).

Only fitted beams where the PSF main lobe has a well-defined Gaussian profile1 are shown in the calculator i.e. there are certain combinations of robustness and tapering that will not yield a viable synthesised beam.

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

MeerKAT PSF (“synthesized/dirty beam”) at various untapered Briggs robust settings (DDFacet/WSClean/CASA scale). Profile cuts along the x=0 and y=0 are indicated to the right and bottom respectively of each of the heatmaps showing the PSF structure.

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.

1We assume that the vast majority of our users will use variants of the CLEAN algorithm (e.g. the MFS CLEAN implemented in DDFacet / WSClean or CASA). Since CLEAN fits the main lobe of the PSF for a restoring beam it is necessary to sufficiently taper off the long spacings when imaging to ensure that the main lobe is sufficiently Gaussian-like for the restoring step to be successful. Failing to do so may result in flux scale errors on slightly extended emission.

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 (βmin) of beam solid angles (Ωb) per reliably detectable source is roughly βmin = 25 for a power-law differential source count n(S) ∝ S 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 < S1.4GHz < 1 Jy.

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

We calculate the flux density Smin at which

for a circular Gaussian restoring beam FWHM θ with solid angle Ωb =πθ2/(4 ln(2)) and compute σc = Smin/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.

 

Note that for short observations (ie. not dominated by confusion noise) the quoted beam sizes in our calculator may be different from those measured.

Reaching predicted sensitivity may not be  possible under the following conditions:

  • Snapshot imaging or imaging at equatorial or higher declinations. In this case, the PSF has substantial sidelobe structure that, when coupled with residual calibration error, may adversely affect deconvolution of your target fields. We recommend that you perform simulations to verify that you will be able to reach the required dynamic range for your target field.

  • The noise estimates we show in the calculator are valid for a measurement of noise in the sidelobes of the primary beam, where the AGN population is sufficiently modulated by the far-field response of the antennas to “bury” the calibration errors underneath the system noise. In a moderate field (with sources in the 1-30mJy apparent level) this typically increases the noise by around 2-3x compared to noise in the sidelobes. Having brighter sources within the FWHM may raise this further. This will need to be addressed by applying direction-dependent calibration.

  • In the presence of strong off-axis sources, pointing error will usually dominate calibration errors surrounding the source. Depending on how distant the source is from the phase centre, the observer may experience sidelobe interference at the position of the science target. Users are advised to check NVSS, SUMSS or RACS for the presence of strong sources in the field of view.

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

Screenshot of the spectral line calculator for a narrowband observation of a target at -34° declination and a synthesized beamwidth of 20″.

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.

Screen shot of the spectral line calculator with drop-down menu for channel widths depending on mode. Note that you can also enter your own value depending on how many channels you intend to bin.

 

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 band.

Mode

Channels

L-band channel width

UHF-band channel width

Mode

Channels

L-band channel width

UHF-band channel width

 

 

(kHz)

velocity width at 1420 MHz

(km/s)

(kHz)

velocity width at 816 MHz

(km/s)

Wideband coarse

4096 (4K)

208.984 kHz

44.12

132.812 kHz

36.41

Wideband fine

32768 (32K)

26.123 kHz

5.52

16.602 kHz

4.55

Narrowband (107 MHz bandwidth)

32768 (32K)

3.3 kHz

0.70

-

-

Narrowband (54 MHz bandwidth)

32768 (32K)

1.633 kHz

0.35

-

-

 

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.

  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.

Due to performance issues on the server in processing a large number of data points, we have set a minimum separation of 0.4 deg (which is somewhat smaller than the standard recommendation for the top of L-band). Please contact us if you have reason to use a smaller separation.

Annotated screenshot of the landing page of the calculator. Note that there are two tabs on this calculator: ‘Targets’ and ‘Sensitivity’.

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.

Steps to follow to obtain a recommended separation for your mosaic pointings.

 

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

1 2 3 4 13:44:00,-28:00:00 13:30:00,-28:00:00 13:30:00,-32:00:00 13:44:00,-32:00:00

The result is as follows:

Screenshot of mosaic calculator with a rectangular area set.

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

 

Screenshot of completed workflow on ‘Targets’ tab of mosaic calculator

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.

Screenshot with previous calculation with a rotation of 30°.

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.

Screenshot of opening view of sensitivity tab.
Calculation result for a 7-point mosaic optimised at a frequency of 1280 MHz.
Calculation result for a higher frequency, using pointings optimised for the centre of L-band. The smaller primary beam is evident, resulting in attenuated sensitivity at the edges of the mosaic.

 

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.

*A minimum of two samples spaced by half of the bandwidth centred on the observation frequency, or spaced by 1/20th of the observing frequency if that is the smaller quantity.