The 1.28 GHz MeerKAT DEEP2 Image

MeerKAT is an array of 64 13.5 m diameter dish antennas spread out over roughly 8 km centered near latitude 30°42' S and longitude 21°23' E. Each dish has a 3.8 m offset Gregorian subreflector and a receiver indexer located just below the subreflector to ensure a completely unblocked aperture. A conical skirt extends below the subreflector to deflect radiation from the surrounding ground away from the receiver. The unblocked aperture is essential for deep continuum imaging at L band because it improves dynamic range by (1) lowering the sensitivity of primary-beam sidelobes to strong sources and radio frequency interference (RFI) outside the main beam and (2) reducing the system noise temperature by limiting pickup of ground radiation. The observations described in this paper were all carried out with the dual linear polarization (horizontal and vertical) L-band (856–1712 MHz) receivers (Lehmensiek & Theron 2012, 2014). Each antenna has a measured system noise temperature T_sys ≈ 20 K and a remarkably low system equivalent flux density SEFD ≈ 430 Jy on cold sky.

The MeerKAT antennas are named "m000" through "m063," the order of which roughly follows their distances from the array center. The inner 48 antennas are located within a 1 km diameter central "core" region, and the remaining 16 are spread beyond this central area out to a radius of nearly 4 km. The distribution of baseline lengths between the 2016 antenna pairs (Figure 1) is peaked at lengths shorter than 1000 m, and roughly half of all MeerKAT baselines are between antennas in the core. MeerKAT and large proposed arrays such as the SKA and ngVLA have fixed antennas in centrally concentrated multiscale configurations, which are compromises designed to satisfy conflicting demands for high surface brightness sensitivity and high angular resolution. MeerKAT provides excellent L-band surface brightness sensitivity on angular scales ≳1' at the cost of a very broad naturally weighted synthesized beam (Figure 2). To achieve θ ≈ 76 FWHM resolution in our DEEP2 image, we observed only when most of the outer antennas were working, and we gave up some sensitivity by heavily downweighting the (u, v) data from intracore baselines.

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Visibilities are transported to the archive located in the Centre for High Performance Computing 600 km away in Cape Town, where they are converted to formats used by major radio astronomy imaging and analysis packages (e.g., Measurement Set or AIPS UV). Additional information about MeerKAT and its specifications can be found in Jonas & MeerKAT Team (2016) and Camilo et al. (2018).

To verify the accuracy of our early MeerKAT data, we made a series of snapshot images offset by 10', 20', 30', 40', and 50' to the north, south, east, and west of the calibration source PKS B1934−638. At the θ ≈ 8'' resolution of MeerKAT, PKS B1934−638 is a point source. It is also strong (S ≈ 15.1 Jy) at 1284 MHz and has a well-established radio spectrum (Reynolds 1994). The usefulness of such offset snapshots is described in Condon et al. (1998): position errors can reveal incorrect (u, v) coordinates or frequency labeling in the data, and the variation in amplitude with position in the primary beam can be used to measure the primary-beam attenuation patterns and pointing errors of the MeerKAT dishes.

A 14-minute observation cycling through the offset pointings was made with a start time chosen to ensure that the scans were as close to the transit of PKS B1934−638 as possible (azimuth ≈ 180°, elevation ≈ 57°). Images made from individual pointings have rms noise ${\sigma }_{n}\sim 1\,\mathrm{mJy}\,{\mathrm{beam}}^{-1}$. The flux density and position of the source were measured in each image by fitting an elliptical Gaussian using the AIPS task JMFIT. At all offsets the source is never attenuated to S_a < 2.3 Jy beam⁻¹ by the MeerKAT primary beam, so the source signal-to-noise ratio is always (S/N) ≳ 2300:1. The noise component of fitting errors should be <0006 in position and <0.04% in flux density assuming a point source and a circular 8'' beam (Condon 1997), so errors in the measured positions and flux densities are dominated by calibration errors.

2.2.1. Positions

Geometric errors can be introduced into source positions on the image plane by incorrect calculations of the (u, v) coordinates associated with measured visibilities. The (u, v) coordinates used during imaging were calculated at a reference frequency of ν₀ = 886 MHz. If ν₀ differs from the actual array reference frequency ν_c, this will rescale the image radial offsets of source positions from the phase center ρ_m from their true offsets ρ (Condon et al. 1998). Furthermore, if the time stamps used to calculate the (u, v) coordinates differ from the true time stamps of observation by Δt s, an image near the celestial pole will be rotated about its phase center by an angle θ ≈ 2π Δt/T_d, where T_d ≈ 86,164 s is one sidereal day.

Figure 3 compares the measured offsets (circles and red points) of the 21 pointings with their commanded offsets (plus signs). The red points indicate the measured offsets in our initial data set before any corrections; they clearly show both a rotation and a radial scaling. The rotation revealed a 2 s shift in the time stamps of the raw correlated visibilities. The radial scaling was caused by the frequency labeling being offset by 0.5 channels during conversion of the data to AIPS UV. Correcting these errors moved the red points to the black circles in Figure 3. To ensure that all MeerKAT data do not require these time and frequency corrections, the errors have now been fixed in the MeerKAT correlator and in the software.

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The mean ratio of the corrected measured (circles) to the commanded (plus signs) radial offsets is consistent with unity:

$$\begin{eqnarray}&&\langle {\rho }_{{\rm{m}}}/\rho \rangle ={\nu }_{c}/{\nu }_{o}=1-(4.0\pm 4.9)\times {10}^{-5}\,.\end{eqnarray} \tag{ 1 }$$

Likewise, the circles in Figure 3 are consistent with no rotation about the center.

The upper right inset of Figure 3 shows the distribution of position errors in R.A. and decl. Their standard errors and means are

$$\begin{eqnarray}\begin{array}{rcl}{\sigma }_{{\rm{\Delta }}\alpha } & = & 0\buildrel{\prime\prime}\over{.} 11,\qquad \langle {\rm{\Delta }}\alpha \rangle =0\buildrel{\prime\prime}\over{.} 001,\\ {\sigma }_{{\rm{\Delta }}\delta } & = & 0\buildrel{\prime\prime}\over{.} 06,\qquad \langle {\rm{\Delta }}\delta \rangle =0\buildrel{\prime\prime}\over{.} 02.\end{array}\end{eqnarray} \tag{ 2 }$$

Thus, we expect that individual strong sources within ∼50' of the phase centers of L-band MeerKAT images will have rms position errors of ∼01 in each coordinate, and the image frames will have astrometric uncertainties of ∼001.

2.2.2. The MeerKAT Primary Beam

The attenuated flux densities of PKS B1934−638, observed at various pointing offsets ρ, divided by its flux density at the pointing center, measure the primary-beam attenuation pattern ${a}_{{\rm{b}}}(\rho )\equiv S(\rho )/S(\rho =0)$ of the MeerKAT antennas. The attenuation pattern derived from these data and also from a horizontal slice through holographic measurements of the MeerKAT Stokes I beam at 1.5 GHz (M. de Villiers 2019, in preparation) is well matched by the attenuation pattern resulting from cosine-tapered field (or cosine-squared power) illumination (Condon & Ransom 2016):

$$\begin{eqnarray}&&{a}_{{\rm{b}}}(\rho /{\theta }_{{\rm{b}}})={\left[\displaystyle \frac{\cos (1.189\pi \rho /{\theta }_{{\rm{b}}})}{1-4{\left(1.189\rho /{\theta }_{{\rm{b}}}\right)}^{2}}\right]}^{2}\,.\end{eqnarray} \tag{ 3 }$$

For the purpose of comparing the observed attenuation pattern at 1.5 GHz with Equation (3), we set the FWHM of the horizontal slice through the primary power pattern to

$$\begin{eqnarray}&&{\theta }_{{\rm{b}}}=57\,\buildrel{\,\prime}\over{.} 5{\left(\displaystyle \frac{\nu }{1.5\mathrm{GHz}}\right)}^{-1}\,.\end{eqnarray} \tag{ 4 }$$

Figure 4 shows that the attenuation pattern of the cosine illumination taper (Equation (3)) is a good match out to ρ = 2 5. Equation (3) also has the practical advantage of expressing a_b(ρ/θ_b) as an elementary function, so we used it to fit our PKS B1934−638 data and in all subsequent analyses requiring narrowband beam patterns at frequency ν.

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Figure 5 shows fits of Equation (3) to the flux densities measured from our PKS B1934−638 observations. During fitting, we inserted the mean pointing offset Δ as a free parameter, replacing (ρ/θ_b) in Equation (3) by [(ρ − Δ)/ θ_b]. The horizontal and vertical fits of the Equation (3) attenuation power pattern are an excellent match to the data for all ${\mathrm{log}}_{10}({a}_{{\rm{b}}})\gt -0.70$.

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The primary-beam pattern is slightly elliptical; its vertical θ_b is larger than its horizontal θ_b. Simulations of the MeerKAT antenna optics predict that asymmetries in the horizontal and vertical linearly polarized L-band feeds result in an ellipticity in the Stokes I primary-beam pattern. Our measurement of the ellipticity as a function of frequency (Table 1) agrees with these simulations to within 1%. We conclude that feed asymmetry is the reason for the ellipticity in the measured Stokes I MeerKAT L-band primary beam.

Table 1.  Frequency Dependence of the MeerKAT Primary Beamwidths θ_b and Typical Pointing Errors Δ

		Vertical		Horizontal
Subband	ν	θb	Δ	θb	Δ
Number	(MHz)	(arcmin)	(arcsec)	(arcmin)	(arcsec)
1	908.04	100.1	−17.1	96.2	−21.7
2	952.34	94.7	−5.1	91.1	−24.1
3	996.65	90.5	−2.4	87.1	−23.4
4	1043.46	86.1	6.2	82.8	−25.4
5	1092.78	81.7	14.0	78.6	−27.8
6	1144.61	78.2	17.4	75.2	−27.0
7	1198.94	73.4	21.8	70.5	−32.2
8	1255.79	70.0	21.1	67.3	−29.5
9	1317.23	65.7	19.7	63.2	−25.9
10	1381.18	63.1	12.1	60.6	−32.6
11	1448.05	60.6	−29.7	58.3	−68.3
12	1519.94	58.9	−30.2	56.8	−63.6
13	1593.92	56.2	−19.4	54.3	−39.4
14	1656.20	55.4	−50.4	53.6	−43.2

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We used wideband images of PKS B1934−638 covering the frequency range 886–1682 MHz centered on ν = 1.28 GHz to fit the primary beam in Figure 5. We have further split the band into 14 narrow subbands and repeated the beam fitting described above for each. Table 1 summarizes the results. Note that the narrow subbands defined in this table are the same as the subbands in Table 5 used for the wideband imaging of the DEEP2 field.

Figure 6 shows that the product θ_bν varies by <±3% across the band for both the vertical and horizontal cuts through the beam. The best L-band approximations with fixed θ_bν are

$$\begin{eqnarray}\begin{array}{rcl}{\theta }_{{\rm{b}}} & = & 89\,\buildrel{\,\prime}\over{.} 5{\left(\displaystyle \frac{\nu }{\mathrm{GHz}}\right)}^{-1}\qquad (\mathrm{Vertical})\\ {\theta }_{{\rm{b}}} & = & 86\,\buildrel{\,\prime}\over{.} 2{\left(\displaystyle \frac{\nu }{\mathrm{GHz}}\right)}^{-1}\qquad (\mathrm{Horizontal}).\end{array}\end{eqnarray} \tag{ 5 }$$

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The DEEP2 field centered on J2000 α = 04^h 13^m 264, δ = −80° 00' 00'' was observed at L band in 12 separate sessions between 2018 April 27 and 2019 January 20 for a total of 155.2 hr (Table 3). We always required that at least 58 of the 64 antennas and at least seven of the nine outer-ring antennas (those providing the longest baselines) be available. This ensured sufficient long- and intermediate-baseline coverage to produce a fairly clean "dirty beam" point-spread function (PSF) with θ ≲ 8'' FWHM resolution. We preferentially observed during the night, though sessions with longer duration and other scheduling constraints meant that ∼30% of the observations occurred in the daytime. The −80° decl. of the DEEP2 field ensures that it is never <55° from the Sun.

Scans on the DEEP2 target lasted 15 minutes and were interleaved with scans on the $S(1284\,\mathrm{MHz})\approx 6.1$ Jy phase and secondary gain calibrator PKS J0252−7104 located ≈10° from the DEEP2 field center. Scans on PKS J0252−7104 were 2 minutes long before July 25, after which we shortened them to 1 minute because we found that we were getting sufficient S/Ns on the calibrator gain solutions. The primary flux density and bandpass calibrator PKS B1934−638 was observed for 10 minutes at the start of each observation and then subsequently every 3 hr until it set. Its assumed flux densities from Reynolds (1994) are listed as a function of frequency in Table 4.

Table 3 summarizes our observations of the DEEP2 field. From the total 155.2 hr, calibration/slewing overheads took 17% and left 128.8 hr on the DEEP2 target. Our observations had an integration period of 8 s except for the initial April 27 observation that we averaged from 4 to 8 s prior to any calibration. Observations were all carried out in the 4096 × 208.984 kHz channel L-band continuum mode. The raw data volume was typically ≈2 TB for a 12 hr observation. Each data set was calibrated separately prior to imaging.

The uncalibrated visibilities from our observations are publicly available and were obtained from the SARAO archive at https://archive.sarao.ac.za. Readers interested in obtaining these data can do so by searching the archive for Proposal ID: SCI-20180426-TM-01.

The full MeerKAT L band covers the frequency range 856–1712 MHz with 4096 spectral channels. We trimmed 144 channels each from the lower and upper ends of the full band because the receiver response is too weak at the band edges to be useful. The remaining frequency range of our data is 886–1682 MHz. This band is contaminated by various sources of strong RFI, the broadest of which are difficult to detect automatically. We therefore developed an empirical mask that flags at all times those channels most contaminated by RFI. Figure 7 shows an example of the raw MeerKAT bandpass on two baselines, one short (319 m) and one long (7566 m). The long baselines are typically not as badly affected by RFI as the short ones, so we chose to apply the mask only to baselines shorter than 1000 m. The mask rejects 34% of the trimmed L band, or 17% of the full data set when applied only to the subset of short baselines.

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The unmasked data were searched in time and frequency for deviations and flagged. Our flagging method is similar to the SumThreshold technique described by Offringa et al. (2010). A smooth background was fitted to the data by convolving them with a Gaussian whose widths in frequency and time are larger-than-expected RFI spike widths and are smaller than any variations in the bandpass or changes in amplitude with time. The smoothed background was then subtracted from the data, and outliers in the residuals were detected in increasing averaged widths in both time and frequency.

After the data were masked and initially edited, we used the Obit package (Cotton 2008) for further editing and calibration. Prior to calibration, the raw data were smoothed with a Hanning filter to prevent Gibbs ringing. This filter combines adjacent channels with weights (1/4, 1/2, 1/4) and effectively doubles the channel width to $2\times 208.984\,\mathrm{kHz}=417.968\,\mathrm{kHz}$. It makes neighboring frequency channels degenerate, so we excised every second channel from the smoothed data prior to calibration.

Our calibration consisted of the following steps:

• 1.  
  The variations in group delays were calculated from the observations of PKS B1934−638 and PKS J0252−7104 and were interpolated to all of the data.
• 2.  
  A bandpass calibration was determined to remove residual variations in gain and phase as a function of frequency. This was based on a CLEAN component model within 1° of PKS B1934−638. The average correction from all scans on PKS B1934−638 was applied to the data.
• 3.  
  The observations of PKS J0252−7104 were used to correct the phases and amplitudes on the target for each calibration subband as a function of time. The amplitudes of PKS B1934−638 were derived from the model of Reynolds (1994) in each subband (see Column (3) of Table 4) and used to determine the amplitude spectrum of PKS J0252−7104. Amplitude and phase corrections were then interpolated in time and applied to the entire data set.
• 4.  
  Some residual errors that were not detected during earlier editing were found by searching for gains with amplitudes that are discrepant by more than 20σ and flagging them.
• 5.  
  The fully calibrated data were edited once more. At this stage the few visibilities with extremely discrepant Stokes I amplitudes (>200 Jy) and outliers from a running median in time and frequency were flagged.
• 6.  
  The previous steps were repeated after resetting the calibration while retaining the flags on the calibrated data.

After the editing steps listed above, ∼35% of the data were flagged. Prior to imaging, the calibrated visibilities were spectrally averaged again, to $2\times 417.968\,\mathrm{kHz}=835.936\,\mathrm{kHz}$. The calibrated, flagged, and averaged data volume for the target was ≈200 GB in a typical 12 hr observation.

The DEEP2 data were collected during 12 observing sessions spread over 9 months (Table 3) and were individually phase-referenced to PKS J0252−7104, which is ∼10° from the DEEP2 field center at J2000 α = 04^h 13^m264, δ = −80° 00' 00''. The resulting phases were corrected to the direction and times of the target observations and astrometrically aligned with each other before imaging. The data from one day (2018 July 27) were imaged using two iterations of phase-only self-calibration with a 30 s averaging time. The resulting model of the DEEP2 field was used as the initial model to start the phase self-calibration of all data sets. Starting the self-calibrations of other days from one model ensures that images from all days are astrometrically aligned. We avoided amplitude self-calibration for two reasons: (1) amplitude self-calibration does not work well in a field containing many faint sources and no dominant point source, and (2) the external gain calibration based on observations of PKS B1934−638 worked very well. We measured the flux density of the $S\approx 12\,\mathrm{mJy}$ point source at J2000 α = 04^h 15^m 0821, δ = −79° 59'410 on the 12 daily DEEP2 images (Table 3); its rms variation is only 2% over the full 9-month observation period.

The data from each session were then imaged and deconvolved without further self-calibration to ensure good data quality. As a final editing step, the models derived from these preliminary images were Fourier transformed and subtracted from the calibrated (u, v) data. Residual amplitudes >0.5 Jy in the difference data were flagged in the calibrated session data. Next, the data were time averaged in a baseline-dependent fashion using Obit/UVBlAvg with the constraints of <1% amplitude loss within 15 of the field center and averaging time <30 s. Finally, the averaged data sets were concatenated by Obit/UVAppend into a single data set for imaging.

The concatenated data set was imaged using the Obit task MFImage (Cotton et al. 2018) without further self-calibration. MFImage used small planar facets to cover the wide field of view and made separate images in the 14 imaging subbands (Table 5) having fractional bandwidths Δν/ν < 0.05 small enough to accommodate the frequency dependence of sky brightness and primary-beam attenuation. Dirty/residual images were formed in each subband, but a weighted average image was used to drive the CLEAN process. CLEAN components included the pixel flux densities from each subband, and the subtraction during the major cycles used a spectral index fitted to each component to interpolate between the subband center frequencies listed in Table 5. The joint deconvolution of the subband images requires that the width of the dirty PSF be nearly independent of frequency. This was accomplished by a frequency-dependent (u, v) taper that downweighted the longer baselines at the higher frequencies.

Table 5.  DEEP2 Imaging Subband Frequencies and Weights

Subband	νi	${\sigma }_{{\rm{n}},i}$	wi for	wi for
Number	(MHz)	($\mu \mathrm{Jy}\,{\mathrm{beam}}^{-1}$)	Min ${\sigma }_{n}^{2}$	Max S/N
i = 1	908.04	4.22	0.0225	0.0378
2	952.34	5.04	0.0158	0.0248
3	996.65	3.20	0.0393	0.0580
4	1043.46	2.88	0.0483	0.0669
5	1092.78	2.76	0.0526	0.0683
6	1144.61	2.58	0.0603	0.0733
7	1198.94	4.20	0.0227	0.0259
8	1255.79	3.98	0.0253	0.0271
9	1317.23	1.85	0.1171	0.1170
10	1381.18	1.64	0.1486	0.1389
11	1448.05	1.55	0.1672	0.1463
12	1519.94	1.87	0.1147	0.0938
13	1593.92	2.89	0.0481	0.0368
14	1656.20	1.85	0.1173	0.0850

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The facet images were reprojected during gridding to form a coherent grid of pixels on the plane tangent to the celestial sphere at the pointing center. This allows a joint CLEAN of many facets in a given major cycle.

The MeerKAT antenna array is centrally concentrated, so a relatively strong ROBUST weighting (ROBUST = −1.3 in AIPS/Obit usage) was used to downweight the shortest baselines to give a θ = 76 FWHM synthesized beamwidth. The DEEP2 field was completely imaged out to a radius of 15, and facets out to 2° were added as needed to cover outlying strong sources selected from the SUMSS (Mauch et al. 2003) catalog. The source density in our DEEP2 image is so high that no CLEAN windowing was used.

CLEAN used a loop gain of 0.15 and found 250,000 point components stronger than 7 μJy for a total CLEAN flux density of S = 1.301 Jy. Prior to restoring the CLEAN components with a circular Gaussian of FWHM θ = 76, the residuals in each subband and facet were convolved with a Gaussian with widths calculated to give a dirty PSF having approximately the same FWHM as the restoring beam. First, the CLEAN components appearing in each facet of each subband image were restored, and then the subband facets were collected into a single subband plane. The output of this imaging process is an image cube containing the 14 CLEANed and restored subband images.

We took weighted averages of the subband images, all of which have well-defined center frequencies ν_i and small fractional bandwidths Δν/ν_i ≲ 0.05, to produce single-plane wideband images using two different weighting schemes. If the rms noise in each subband image is σ_i, then subband weights ${w}_{i}\propto {\sigma }_{i}^{-2}$ minimize the wideband image noise variance

$$\begin{eqnarray}&&{\sigma }_{{\rm{n}}}^{2}=\displaystyle \sum _{i=1}^{14}{w}_{i}{\sigma }_{i}^{2}\,/\displaystyle \sum _{i=1}^{14}{w}_{i}\,.\end{eqnarray} \tag{ 14 }$$

More generally, subband weights ${w}_{i}\propto {({\nu }_{i}^{\alpha }/{\sigma }_{i})}^{2}$ maximize the wideband image S/N for sources with spectral index $\alpha \equiv +d\mathrm{ln}(S)/\,d\mathrm{ln}(\nu )$:

$$\begin{eqnarray}&&{\left({\rm{S}}/{\rm{N}}\right)}^{2}\propto \displaystyle \sum _{i=1}^{14}{w}_{i}{\left({\nu }_{i}^{\alpha }/{\sigma }_{i}\right)}^{2}\,/\displaystyle \sum _{i=1}^{14}{w}_{i}\,.\end{eqnarray} \tag{ 15 }$$

Minimizing σ_n (Equation (14)) is equivalent to choosing α = 0 in Equation (15). The median spectral index of faint sources selected at frequencies ν ∼ 1.4 GHz is $\langle \alpha \rangle \approx -0.7$ (Condon 1984), so this is the best choice of α for maximizing the S/N. The first three columns of Table 5 list the subband numbers, center frequencies ν_i (MHz), and rms noise values σ_i ($\mu \mathrm{Jy}\,{\mathrm{beam}}^{-1}$). Column (4) tabulates the weights w_i that minimize ${\sigma }_{n}^{2}$, and Column (5) shows the different weights w_i that maximize the S/N in the wideband DEEP2 image. Both sets of weights have been normalized to make ${\sum }_{i=1}^{14}{w}_{i}=1$ for convenience.

If a source near the pointing center has flux density S ∝ ν^α, its flux density in the weighted wideband image will be

$$\begin{eqnarray}&&S\propto \displaystyle \sum _{i=1}^{14}{w}_{i}{\nu }_{i}^{\alpha }/\displaystyle \sum _{i=1}^{14}{w}_{i}\,.\end{eqnarray} \tag{ 16 }$$

We define the "effective" frequency ν_e of a weighted wideband image as the frequency at which the image flux density equals the source flux density ${\nu }_{{\rm{e}}}^{\alpha }$:

$$\begin{eqnarray}&&{\left.{\nu }_{{\rm{e}}}=\left(\displaystyle \sum _{i=1}^{14}{w}_{i}{\nu }_{i}^{\alpha }\right./\displaystyle \sum _{i=1}^{14}{w}_{i}\right)}^{1/\alpha }\,.\end{eqnarray} \tag{ 17 }$$

Thus, different weighting schemes yield slightly different effective frequencies for the wideband images. The effective frequencies of our DEEP2 images are ν_e ≈ 1329 MHz for minimum ${\sigma }_{n}^{2}$ and ν_e ≈ 1278 MHz for maximum S/N weighting.

The rms noise in the S/N-weighted DEEP2 image was estimated in five widely separated and apparently source-free regions covering 14,373 CLEAN beam solid angles at offsets ρ ∼ 16 from the pointing center, where the primary attenuation is a_b < 0.01. The distribution of their peak flux densities S_p is nearly Gaussian (Figure 8) with rms ${\sigma }_{n}=0.55\pm 0.01\,\mu \mathrm{Jy}\,{\mathrm{beam}}^{-1}$. The noise in a synthesis image not corrected for primary-beam attenuation should be uniform across the whole image.

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In each narrow subband the primary-beam attenuation pattern ${a}_{{\rm{b}},i}(\rho )$ is well defined and was measured accurately in the horizontal and vertical planes (Section 2.2.2) of the alt-az mounted MeerKAT dishes. The primary beam is slightly elliptical and rotates with parallactic angle on the sky. For the long DEEP2 tracks we approximated the ellipse by a circle whose diameter is the geometric mean of the horizontal and vertical diameters. The primary beamwidth is inversely proportional to frequency, so the effective primary pattern a_b(ρ) of the weighted wideband image must be calculated from

$$\begin{eqnarray}&&{a}_{{\rm{b}}}(\rho )=\displaystyle \sum _{i=1}^{14}{w}_{i}{a}_{{\rm{b}},i}(\rho ){\nu }_{i}^{\alpha }/\displaystyle \sum _{i=1}^{14}{w}_{i}\,.\end{eqnarray} \tag{ 18 }$$

The circularized attenuation pattern for the wideband DEEP2 image weighted for maximum S/N is shown in Figure 9. Its FWHM is θ_b ≈ 68'. Numerically it can be approximated within 0.1% for all a_b > 0.25 by the polynomial

$$\begin{eqnarray}&&\begin{array}{l}{a}_{{\rm{b}}}(\rho )\approx 1.0-0.3514x/{10}^{3}+0.5600{x}^{2}/{10}^{7}\\ \quad -\,0.0474{x}^{3}/{10}^{10}+0.00078{x}^{4}/{10}^{13}+0.00019{x}^{5}/{10}^{16},\end{array}\end{eqnarray} \tag{ 19 }$$

where $x\equiv {[\rho (\mathrm{arcmin}){\nu }_{{\rm{e}}}(\mathrm{GHz})]}^{2}$ and ν_e ≈ 1. 278 GHz. This primary-beam attenuation can be used by the AIPS task PBCOR via the parameter PBPARM = 0.250, 1.0, −0.3514, 0.5600, −0.0474, 0.00078, and 0.00019. The wideband attenuation pattern is close to the narrowband attenuation pattern at ν = ν_e ≈ 1278 MHz shown by the dotted curve in Figure 9, but it has slightly broader wings.

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The wideband DEEP2 images are so sensitive (${\sigma }_{{\rm{n}}}\approx 0.55\,\mu \mathrm{Jy}\,{\mathrm{beam}}^{-1}$) that they are densely covered by sources with flux densities S ≫ σ_n (Figures 10 and 11). The total flux density in a dirty interferometer image is always zero because there are no zero-spacing (u, v) data, only sinusoidal fringes with zero means. The dirty image of a single strong point source contains a negative "bowl" whose angular size is inversely proportional to the smallest (u² + v²)^1/2 sampled. For our DEEP2 data, the bowl surrounding each source is much wider than the source spacing but much narrower than the primary attenuation pattern a_b(ρ) (Figure 9). Faint sources have a fairly uniform random distribution on the sky, and their attenuated brightness distribution in the DEEP2 images is multiplied by the primary attenuation pattern. Consequently, each dirty DEEP2 image has a negative bowl whose size and shape closely match the primary beam and whose depth exceeds σ_n. Partial CLEANing reduces the depth of the bowl but does not completely eliminate it.

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We estimated the central depth of the bowl of our S/N-optimized DEEP2 image using the mode of the brightness distribution in the 6' × 6' square at the center of the wideband CLEAN image, where the primary attenuation is confined to the narrow range 0.98 < a_b < 1.00 (Figure 9); it is $-1.4\pm 0.1\,\mu \mathrm{Jy}\,{\mathrm{beam}}^{-1}$. To fill in the bowl and flatten the image baseline level, we added the weighted wideband primary attenuation pattern a_b(ρ) (Equation (19)) multiplied by 1.4 × 10⁻⁶ to the wideband image, whose brightness units are Jy beam⁻¹.

The central square cutout $4640\,\mathrm{pixels}\times 1\buildrel{\prime\prime}\over{.} 25\,{\mathrm{pixel}}^{-1}=5800^{\prime\prime}$ on a side from the S/N-weighted 1.28 GHz wideband DEEP2 image with the bowl removed, but not corrected for primary-beam attenuation, is available in FITS format at https://archive.sarao.ac.za.

The differential number n(S) of sources per unit flux density per steradian is a statistical quantity that can be calculated directly from the distribution of image brightnesses expressed in units of flux density per beam solid angle. The term "confusion" describes the brightness fluctuations caused by radio sources, and an image is said to be confusion limited if the confusion fluctuations are larger than the rms noise fluctuations σ_n. The normalized probability distribution of confusion brightness is traditionally called the P(D) distribution, where D originally stood for the pen deflection on a chart recording but now refers to image peak flux density S_p, that is, flux density per beam solid angle.

The noiseless P(D) distribution for point sources can be derived analytically only for power-law differential source counts n(S) = kS^−γ, where k is the overall source density parameter and γ is the power-law exponent (Condon 1974). Power-law distributions are scale-free, so the shape of a power-law P(D) distribution depends only on γ. The amplitudes and widths of power-law P(D) distributions obey the scaling relation

$$\begin{eqnarray}&&P\left[{\left(k{{\rm{\Omega }}}_{{\rm{e}}}\right)}^{1/\left(\gamma -1\right)}D\right]={\left(k{{\rm{\Omega }}}_{{\rm{e}}}\right)}^{-1/\left(\gamma -1\right)}P(D)\,,\end{eqnarray} \tag{ 20 }$$

where

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{e}}}\equiv \int {[a(\rho ,\phi )]}^{\gamma -1}\,d{\rm{\Omega }}\end{eqnarray} \tag{ 21 }$$

is the "effective" solid angle Ω_e of a beam whose polar attenuation pattern is a(ρ, ϕ). For power-law counts, the P(D) distribution depends on Ω_e but not on the form of the beam attenuation pattern. The effective solid angle of an elliptical Gaussian beam with FWHM axes θ₁ and θ₂ is

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{e}}}=\left(\displaystyle \frac{\pi {\theta }_{1}{\theta }_{2}}{4\mathrm{ln}2}\right)\left(\displaystyle \frac{1}{\gamma -1}\right)=\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{b}}}}{\gamma -1}\,,\end{eqnarray} \tag{ 22 }$$

where Ω_b is the Gaussian beam solid angle

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{{\rm{b}}}\equiv \int a(\rho ,\phi )\,d{\rm{\Omega }}=\displaystyle \frac{\pi {\theta }_{1}{\theta }_{2}}{4\mathrm{ln}2}\,.\end{eqnarray} \tag{ 23 }$$

A convenient normalization for displaying analytic P(D) distributions is $k{{\rm{\Omega }}}_{{\rm{e}}}={\eta }_{1}^{-1}$, where

$$\begin{eqnarray}&&{\eta }_{1}^{-1}=\displaystyle \frac{2{\rm{\Gamma }}(\gamma /2){\rm{\Gamma }}[(\gamma +1)/2]\sin [\pi (\gamma -1)/2]}{{\pi }^{\gamma +1/2}}\end{eqnarray} \tag{ 24 }$$

and Γ(x) is the factorial function (Condon 1974).

The shape of the P(D) distribution varies significantly with the count slope γ. Four examples of P(D) distributions with $k{{\rm{\Omega }}}_{{\rm{e}}}={\eta }_{1}^{-1}$ are shown in Figure 12. The super-Euclidean slope γ = 2.8 is close to the actual slope observed above S ∼ 1 Jy at 1.4 GHz, so the γ = 2.8 curve in the top panel of Figure 12 represents the troublesome confusion that affected early radio surveys such as 2C (Shakeshaft et al. 1955). In the limit $\gamma \to 3-$, the P(D) distribution is Gaussian and would appear as a parabola in the semilogarithmic Figure 12. The γ = 2.8 curve is dominated by its nearly parabolic core and has only a weak tail extending to the right. The sky brightness contributed by point sources diverges for all γ ≥ 2 (Olbers's paradox), so the calculated values of D are relative to the mean deflection $\langle D\rangle$. The γ = 2.1 curve shows how the P(D) peak shifts to the left and the tail becomes more prominent as $\gamma \to 2+$.

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The bottom panel of Figure 12 shows sample P(D) distributions when γ < 2, and D represents the deflection above the absolute zero of sky brightness contributed by radio sources. The γ = 1.9 curve is similar to the γ = 2.1 curve in form, and its peak D is still significantly offset above zero by contributions from faint sources so numerous that multiple sources are blended together in each beam. This is characteristic of confusion seen at levels $10\,\mu \mathrm{Jy}\lesssim {S}_{1.4\mathrm{GHz}}\lesssim 0.1\,\mathrm{Jy}$.

At the lower value γ ≈ 1.5 observed when ${S}_{1.4\mathrm{GHz}}\ll 10\,\mu \mathrm{Jy}$, the nature of confusion changes again. There are so few sub-μJy sources that the P(D) peak deflection approaches $D\to 0+$, indicating that the extragalactic background has been largely resolved into discrete sources. The long tail of the P(D) distribution so completely dominates the narrow core that the rms confusion σ_c is ill-defined and should not be used to describe the amount of confusion when ${S}_{1.4\mathrm{GHz}}\ll 10\,\mu \mathrm{Jy}$. Confusion in the traditional sense "melts away" at the sub-μJy levels reached by DEEP2. Instead of numerous even fainter sources tending to boost the flux densities of faint sources, faint sources are more likely to suffer obscuration by stronger sources.

The central 1250'' × 1250'' square covering about 2.4 × 10⁴ restoring beam solid angles was extracted from the DEEP2 sky image (Figure 11) and rescaled to 1.4 GHz via the spectral index α = −0.7 typical of faint sources. Its 1.4 GHz P(D) distribution is shown by the red curve in Figure 13. The best least-squares power-law fit with fixed rms noise ${\sigma }_{n}=0.55\pm 0.01\,\mu \mathrm{Jy}\,{\mathrm{beam}}^{-1}$ but free k and γ is n(S) = 1.07 × 10⁵S^−1.52 Jy⁻¹ sr⁻¹ between S = 0.25 and 10 μJy. Letting σ_n be a free parameter and varying σ_n by $2{{\rm{\Delta }}\sigma }_{{\rm{n}}}=\pm 0.02\,\mu {\rm{Jy}}\,{{\rm{beam}}}^{-1}$ has a negligible effect on this fit. The fit and its rms uncertainties (68% confidence region) are bounded by the thick box at the left in Figure 14. The actual source count is not a perfect power law, and the DEEP2 dirty beam is not perfectly Gaussian, so getting more accurate source counts will require numerical simulations based on more realistic non-power-law source counts and non-Gaussian dirty beams (A. M. Matthews et al. 2019, in preparation).

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Smoothly extrapolating our power-law 1.4 GHz count below S₀ ≈ 0.25 μJy is reasonable because fainter sources are evolved from the power-law region below the "knee" in the local RLF (Condon et al. 2019), so simple evolutionary models (Condon 1984) predict nearly power-law sub-μJy source counts. This extrapolation yields an estimate of the Rayleigh–Jeans brightness temperature T_b contributed by all fainter SFGs:

$$\begin{eqnarray}&&{\rm{\Delta }}{T}_{{\rm{b}}}(\lt {S}_{0})=\left[\displaystyle \frac{\mathrm{ln}(10){c}^{2}}{2{k}_{{\rm{B}}}{\nu }^{2}}\right]{\int }_{0}^{{S}_{0}}{S}^{2}n(S)d[\mathrm{log}S]\,,\end{eqnarray} \tag{ 25 }$$

where ${k}_{{\rm{B}}}\approx 1.38\times {10}^{-23}\,{\rm{J}}\,{{\rm{K}}}^{-1}$ is the Boltzmann constant (Condon et al. 2012). For $n(S)=1.07\,\times \,{10}^{5}{S}^{-1.52}\,{\mathrm{Jy}}^{-1}\,{\mathrm{sr}}^{-1}$ at 1.4 GHz, the background contributed by sources fainter than S₀ = 0.25μJy is ΔT_b ≈ 2.5 mK, which is only ≈7% of the T_b ≈ 37 mK total background (Condon et al. 2012) contributed by SFGs and <3% of the background contributed by all extragalactic sources.

An image is said to be confusion limited if the errors caused by confusion exceed the errors caused by Gaussian noise. Both confusion and noise set lower limits to the flux density S of the faintest individual source that can be reliably detected. This section extends earlier calculations of the confusion limit below S ∼ 10 μJy, where γ ≪ 2 and obscuration dominates confusion. This flux density range affects the confusion/obscuration corrections to DEEP2 direct source counts (Section 5.3) and directly impacts the design of future arrays such as the SKA and ngVLA.

A common way to compare confusion and noise is to calculate their variances ${\sigma }_{{\rm{c}}}^{2}$ and ${\sigma }_{{\rm{n}}}^{2}$. This calculation must be done carefully because the confusion variance ${\sigma }_{{\rm{c}}}^{2}={\int }_{0}^{\infty }{D}^{2}P(D)\,{dD}$ formally diverges for all power-law source counts n(S) = k S^−γ and is finite only if the P(D) distribution is truncated above some cutoff deflection D_c. Then (Condon 1974),

$$\begin{eqnarray}&&{\sigma }_{{\rm{c}}}={\left(\displaystyle \frac{k{{\rm{\Omega }}}_{{\rm{e}}}}{3-\gamma }\right)}^{1/2}{D}_{{\rm{c}}}^{(3-\gamma )/2}\,,\,\,1\lt \gamma \lt 3\,.\end{eqnarray} \tag{ 26 }$$

The cutoff should be proportional to σ_c: D_c = q σ_c, where the constant q ∼ 5 is the usual cutoff signal-to-confusion ratio. Then,

$$\begin{eqnarray}&&{\sigma }_{{\rm{c}}}={\left(\displaystyle \frac{k{{\rm{\Omega }}}_{{\rm{e}}}}{3-\gamma }\right)}^{1/(\gamma -1)}{q}^{(3-\gamma )/(\gamma -1)}\,.\end{eqnarray} \tag{ 27 }$$

Note that the rms confusion σ_c still depends on the choice of q. Only as $\gamma \to 3-$ is σ_c nearly independent of q. When γ = 2, σ_c ∝ q. In the sub-μJy regime where γ ∼ 1.5, σ_c ∝ q³ depends so sensitively on q that the very concept of rms confusion is nearly useless.

For imaging with a Gaussian beam, ${{\rm{\Omega }}}_{{\rm{e}}}={{\rm{\Omega }}}_{{\rm{b}}}/(\gamma -1)$ and

$$\begin{eqnarray}&&{\sigma }_{{\rm{c}}}^{\gamma -1}=\displaystyle \frac{k{{\rm{\Omega }}}_{{\rm{b}}}\,{q}^{3-\gamma }}{(\gamma -1)(3-\gamma )}\,.\end{eqnarray} \tag{ 28 }$$

Solving the cumulative source count

$$\begin{eqnarray}&&N(\gt S)=\displaystyle \frac{{{kS}}^{1-\gamma }}{\gamma -1}\end{eqnarray} \tag{ 29 }$$

for k = (γ − 1) N( > S) S^γ−1 and substituting the detection limit S ≈ q σ_c results in

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{{\rm{c}}}}{S}\right)}^{\gamma -1}\approx \left(\displaystyle \frac{{q}^{3-\gamma }}{3-\gamma }\right)N(\gt S){{\rm{\Omega }}}_{{\rm{b}}}=\left(\displaystyle \frac{{q}^{3-\gamma }}{3-\gamma }\right){\beta }^{-1}\,,\end{eqnarray} \tag{ 30 }$$

which can be solved for β_min, the minimum number of beam solid angles per reliably detectable source, in terms of the confusion S/N q:

$$\begin{eqnarray}&&{\beta }_{\min }\approx \displaystyle \frac{{q}^{2}}{3-\gamma }\,.\end{eqnarray} \tag{ 31 }$$

The number of beams per source at the q = 5 confusion limit is shown as a function of γ by the solid curve in Figure 15. In the limit $\gamma \to 3-$, the P(D) distribution is dominated by the very faintest sources and the fluctuations in sky brightness diverge, just as the total sky brightness diverges as $\gamma \to 2-$ (Olbers's paradox). The P(D) distribution becomes nearly Gaussian, the same as the noise distribution, so sources can never be distinguished from noise and $\beta \to \infty$. For sources stronger than S ∼ 1 Jy at 1.4 GHz, a super-Euclidean differential count slope γ ≈ 2.7 implies β_min ≈ 80 beam solid angles per reliable q = 5 source detection. This very severe requirement on β_min was not obvious when the first extragalactic radio surveys were being made, so sources with smaller β were often cataloged as real and later found to be spurious.

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An alternative approach to calculating the confusion β_min uses the probability P_c that a source of flux density S will be confused by a weaker source of flux density between fS and S, where f < 1. For a cumulative source count N( > S) ∝ S^1−γ and a top-hat beam (a(θ, ϕ) = 1 over solid angle Ω_e in the notation of Equation (21)),

$$\begin{eqnarray}&&{P}_{{\rm{c}}}={{\rm{\Omega }}}_{{\rm{e}}}[N(\gt {fS})-N(\gt S)]={{\rm{\Omega }}}_{{\rm{e}}}[N(\gt S)({f}^{1-\gamma }-1)]\,.\end{eqnarray} \tag{ 32 }$$

A Gaussian beam with beam solid angle ${{\rm{\Omega }}}_{{\rm{b}}}={{\rm{\Omega }}}_{{\rm{e}}}(\gamma -1)$ yields exactly the same P(D) distribution, so

$$\begin{eqnarray}&&{P}_{{\rm{c}}}=N(\gt S){{\rm{\Omega }}}_{{\rm{b}}}\left(\displaystyle \frac{{f}^{1-\gamma }-1}{\gamma -1}\right)=\displaystyle \frac{1}{\beta }\left(\displaystyle \frac{{f}^{1-\gamma }-1}{\gamma -1}\right)\end{eqnarray} \tag{ 33 }$$

and

$$\begin{eqnarray}&&{\beta }_{\min }=\displaystyle \frac{1}{{P}_{{\rm{c}}}}\left(\displaystyle \frac{{f}^{1-\gamma }-1}{\gamma -1}\right)\end{eqnarray} \tag{ 34 }$$

is the minimum number of Gaussian beams per source consistent with this confusion requirement. A reasonable choice for f would be f ≈ 0.2, so the confusion is at least S/5. A reasonable choice for the probability of confusion at this level is P_c ≈ 0.159, the probability that Gaussian noise exceeds +σ_n. The dotted curve in Figure 15 shows the resulting β_min as a function of γ.

In the broad flux density range 10 μJy < S < 0.1 Jy covered by most 1.4 GHz surveys, γ ∼ 2 and β_min ∼ 25. Below S ∼ 10 μJy, the differential count slope falls again, to γ ≲ 1.5. While β_min for avoiding confusion by fainter sources continues to fall, the probability of obscuration by stronger sources grows. For a top-hat beam the probability that a source of flux density S will be obscured by a stronger source is P_o = N( > S) Ω_e. For power-law source counts, the P(D) distribution is independent of beam shape and depends only on Ω_e, so for a Gaussian beam with beam solid angle Ω_b = (γ − 1) Ω_e (Equation (22)),

$$\begin{eqnarray}&&{P}_{{\rm{o}}}=N(\gt S)\,{{\rm{\Omega }}}_{{\rm{e}}}=\displaystyle \frac{N(\gt S)\,{{\rm{\Omega }}}_{{\rm{b}}}}{\gamma -1},\end{eqnarray} \tag{ 35 }$$

and the minimum number of beam solid angles per source to minimize obscuration is

$$\begin{eqnarray}&&{\beta }_{\min }\approx {[{P}_{{\rm{o}}}(\gamma -1)]}^{-1}\,.\end{eqnarray} \tag{ 36 }$$

The dashed curve in Figure 15 shows the number β_min of beam solid angles per source corresponding to ${P}_{{\rm{o}}}=0.1$.

Choosing β_min ≈ 25 is a good rule of thumb for detecting individual sources valid in the flux density range below ${S}_{1.4\mathrm{GHz}}\sim 0.1\,\mathrm{Jy}$, where 1.4 ≲ γ ≲ 2. The solid curve in Figure 16 shows the 1.4 GHz flux density S at which β = 25. For example, the 1.4 GHz EMU survey (Norris et al. 2011) has θ = 10'' FWHM resolution, rms confusion $S/{{\rm{\Omega }}}_{{\rm{b}}}\approx 240\,\mathrm{nJy}\,{\mathrm{arcsec}}^{-2}$, and beam solid angle ${{\rm{\Omega }}}_{{\rm{b}}}=\pi {\theta }^{2}/(4\mathrm{ln}2)\approx 113\,{\mathrm{arcsec}}^{2}$. The weakest reliably detectable sources at that resolution have flux densities S ≈ 27 μJy.

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The sharp decline of the minimum S caused by the low γ ∼ 1.5 at θ ≪ 10'' means that even the most sensitive SKA images will not be confusion limited at beamwidths θ ≳ 1''. This is fortunate because the median angular size of faint SFGs is $\langle \phi \rangle =0\buildrel{\prime\prime}\over{.} 3\pm 0\buildrel{\prime\prime}\over{.} 1$ with an rms scatter σ_ϕ ≲ 03 (Cotton et al. 2018), and sources with ϕ ≳ θ are more difficult to detect because their peak flux densities are reduced by factors ≳2. Although the Loi et al. (2019) 1.4 GHz simulation of the radio sky for the SKA and its precursors also predicts γ ≲ 1.5 below S = 10 μJy, their estimated rms confusion ${\sigma }_{\mathrm{mJy}/\mathrm{bm}}={(0.237\pm 0.001)({\nu }_{\mathrm{GHz}})}^{-0.8}{(\theta )}^{2.149\pm 0.001}$ (the dotted line in Figure 16 shows their $5{\sigma }_{\mathrm{mJy}/\mathrm{bm}}$) does not take the lower γ into account and overpredicts the rms confusion at nJy flux densities.

Statistical counts using the P(D) distribution can usefully reach much lower β values and hence much lower flux densities. For point sources randomly placed on the sky, the Poisson probability that any beam solid angle will contain no sources stronger than S is ${P}_{{\rm{P}}}=\exp (-1/\beta )$. Half of all beam solid angles (P_P = 0. 5) must satisfy $\beta =1/\mathrm{ln}(2)\approx 1.44$, and the flux density of the strongest source in them, $\langle S\rangle$, is the solution of

$$\begin{eqnarray}&&N(\gt \langle S\rangle )=\displaystyle \frac{\mathrm{ln}2}{{{\rm{\Omega }}}_{{\rm{b}}}}=\displaystyle \frac{1}{\pi }{\left(\displaystyle \frac{2\mathrm{ln}2}{\theta }\right)}^{2}\,.\end{eqnarray} \tag{ 37 }$$

For the θ = 76 FWHM DEEP2 beam and Condon (1984) model 1.4 GHz source count, Figure 17 shows that half of all beam solid angles contain no sources stronger than $\langle S\rangle \approx 0.25\,\mu \mathrm{Jy}$. Although the observed DEEP2 P(D) distribution (Figure 13) is broadened by ${\sigma }_{n}\approx 0.55\,\mu \mathrm{Jy}\,{\mathrm{beam}}^{-1}$ noise, it samples ∼1.2 × 10⁴ independent effective beam areas Ω_e, so it should be able to constrain the 1.4 GHz source count down to the S ≈ 0.25 μJy SKA1 goal for studying the SFHU (Prandoni & Seymour 2015). At 1.4 GHz the ratio of the S ∼ 16 μJy detection limit β_min = 25 for discrete sources to the confusion sensitivity limit $\langle S\rangle \approx 0.25\,\mu \mathrm{Jy}$ is about 64! Another advantage of P(D) counts is that the low minimum β ∼ 1.44 allows relatively large beamwidths, while the larger minimum β ≈ 25 for deep counts of individual sources requires beamwidths small enough that large and difficult corrections for partial source resolution become necessary (Owen 2018).

Figure 16 indicates that the β = 25 limit for reliably detecting individual sources with the DEEP2 θ_1/2 = 76 beam is S ≈ 17 μJy at 1.278 GHz (≈16 μJy at 1.4 GHz). Below that limit, a significant fraction of sources will be confused or obscured. We produced preliminary counts of DEEP2 sources stronger than 10 μJy inside the ${\rm{\Omega }}\approx 1.026\,{\deg }^{2}$ DEEP2 half-power circle. To estimate confusion and obscuration corrections, we simulated a DEEP2 image, counted sources in the simulated image, and compared those counts with the actual counts used in the simulation.

For the simulation, point sources with approximately the correct source count were randomly placed on the image and convolved with the DEEP2 dirty beam. Then, the simulated image was multiplied by the primary attenuation, convolved with $0.55\,\mu \mathrm{Jy}\,{\mathrm{beam}}^{-1}$ rms noise, and CLEANed. A source catalog was extracted from the simulated image, and the source counts from this catalog were compared with the input counts to derive count corrections. These corrections are typically ∼10%, and we estimate that their rms uncertainties are about half of the corrections themselves. The brightness-weighted 1.278 GHz DEEP2 source counts $\mathrm{log}[{S}^{2}n(S)(\mathrm{Jy}\,{\mathrm{sr}}^{-1})]$ in 12 bins of logarithmic width Δ = 0.2 in $\mathrm{log}(S)$ centered on $\langle \mathrm{log}[S(\mathrm{Jy})]\rangle =-4.9,-4.7,...,-2.7$ and their rms errors are listed in Table 6.

Table 6.  DEEP2 1.278 GHz Source Counts

$\langle \mathrm{log}[S(\mathrm{Jy})]\rangle$	nbin	$\mathrm{log}[{S}^{2}n(S)(\mathrm{Jy}\,{\mathrm{sr}}^{-1})]$
−4.9	5572	2.682 ± 0.031
−4.7	4521	2.754 ± 0.031
−4.5	3025	2.762 ± 0.032
−4.3	2008	2.770 ± 0.032
−4.1	1070	2.740 ± 0.033
−3.9	576	2.690 ± 0.036
−3.7	289	2.578 ± 0.040
−3.5	139	2.444 ± 0.047
−3.3	85	2.450 ± 0.055
−3.1	49	2.428 ± 0.066
−2.9	30	2.411 ± 0.080
−2.7	24	2.532 ± 0.087

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We converted the 1.278 GHz counts to 1.4 GHz via a median spectral index −0.7, and these 1.4 GHz counts are plotted as the black points in Figure 14. The top panel of Figure 14 presents the traditional static Euclidean normalization S^5/2n(S), and the bottom panel shows the brightness-weighted normalization S²n(S). The red points are from Prandoni et al. (2018), the blue points are from Hopkins et al. (2003), and the green points are from Smolčić et al. (2017). The DEEP2, Prandoni et al. (2018), and Hopkins et al. (2003) counts agree within the errors. The counts from Smolčić et al. (2017) are somewhat lower in the range $-4\lesssim \mathrm{log}[S(\mathrm{Jy})]\lesssim -3$, perhaps because some extended sources were partially resolved by their 075 beam. Figure 14 also shows that the DEEP2 direct count is slightly higher than the Mitchell & Condon (1985) 68% confidence P(D) box, possibly because DEEP2 is significantly more sensitive (rms noise ${\sigma }_{n}=0.55\,\mu \mathrm{Jy}\,{\mathrm{beam}}^{-1}\approx 7.2\,\mathrm{mK}$) to low-brightness SFGs.