LOW-FREQUENCY IMAGING OF FIELDS AT HIGH GALACTIC LATITUDE WITH THE MURCHISON WIDEFIELD ARRAY 32 ELEMENT PROTOTYPE

The MWA antenna tile architecture poses several nontraditional calibration issues due to both the nature of the primary beam and the wide field of view. The primary beam is formed by the summation of beamformer-delayed zenith-centered dipole responses. The beamformer delays are periodically changed to track a field across the sky. Although this moves the center of the primary beam as intended, the overall shape of the beam changes as well. For a given set of beamformer delays, the beam is fixed relative to the tile, and therefore moves relative to the sky as the Earth rotates. As a further complication, the time and direction dependence of the primary beam response is different for the two polarizations of the crossed dipoles in the array. This leads to apparent polarization in inherently unpolarized sources due to the different responses in the two orthogonal dipole polarizations unless the appropriate corrections are applied in the analysis procedure.

Methods for measuring and calibrating the primary beam have been developed for the full MWA system by Mitchell et al. (2008), who plan to use a real-time system (RTS) to calibrate and image MWA data. Their method performs a calibration of the instrument in real time, solving for direction- and frequency-dependent gains for each antenna based on the simultaneous measurement of multiple known bright sources across the field of view. This method was developed for use in a 512-tile array, where the instantaneous sensitivity and uv-plane coverage enable the measurement of several hundred sources in each 8 s iteration (Mitchell et al. 2008). Ord et al. (2010) have successfully demonstrated the use of a modified version of the RTS in order to calibrate and image data from the MWA-32T. However, the reduced sensitivity of the MWA-32T array makes this full calibration challenging. For the MWA-32T system, the data rate is sufficiently low that real-time calibration and imaging are not necessary, and the raw visibility data can be captured and stored. We have chosen to pursue an alternative data reduction pipeline based on more traditional calibration and imaging software, which allows us to use the full visibility data set in order to perform a detailed investigation of the calibration and imaging performance of the MWA-32T instrument.

Without the ability to directly measure the primary beam for each tile, we instead assume a model and use it to account for the instrumental-gain direction dependences. Knowledge of the primary beams is also needed for an optimized weighting of the maps when combining them to obtain deeper maps (see Section 6.4). It is likely that precise characterization of individual tile beams will be necessary to achieve dynamic range sufficient for accurate foreground subtraction and EoR detection, but this is not attempted in the work described herein.

For present purposes, we assume that the polarized primary beam patterns are identical across all tiles, and can be modeled by simply summing together the direction-dependent complex gains of the individual dipoles in the tile, i.e., mutual coupling between elements and tile-to-tile differences are ignored. We model the complex beam patterns of an isolated individual dipole for both the north–south (Y) and east–west (X) electric field polarizations using the WIPL-D Pro²² electromagnetic modeling software package. A tile beam pattern is then computed by summing the 16 dipole responses with the dipoles assumed to be at their nominal locations in a tile and with the individual responses modified by the nominal amplitudes and phases introduced by the beamformer for the given delay settings. Figure 1 shows cuts through power patterns (square modulus of the complex beam) at zenith and at a pointing direction 28° east of zenith for both the X and Y polarizations. Model beam patterns were calculated at frequency intervals of 2 MHz, since they vary significantly across the MWA frequency band.

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We assume that this model fully describes the direction dependence of each tile. We do, however, allow for a different overall, i.e., direction-independent, complex gain for each tile. We follow the Jones matrix formalism as presented by Hamaker et al. (1996). The instrumental model then takes the following form for a single tile at a single frequency:

$$\begin{equation} {\mathbf v}_{\rm A}={\mathbf G}_{\rm A} {\mathbf B} {\mathbf e}_{\rm A}, \end{equation} \tag{ 1 }$$

where

$$\begin{equation} {\mathbf e}_{\rm A} = \left(\begin{array}{@{} c @{}}e_{\rm x} \\ e_{\rm y}\end{array}\right) \end{equation} \tag{ 2 }$$

is the incident electric field at tile A, decomposed into linear E–W and N–S polarizations,

$$\begin{equation} {\mathbf v}_{\rm A} = \left(\begin{array}{@{} c @{}}v_{\rm x} \\ v_{\rm y}\end{array}\right) \end{equation} \tag{ 3 }$$

is the vector of measured antenna voltages,

$$\begin{equation} {\mathbf G}_{\rm A} = \left(\begin{array}{@{} cc @{}}g_{\rm A,x} & 0 \\ 0 & g_{\rm A,y}\end{array}\right) \end{equation} \tag{ 4 }$$

is the matrix of direction-independent complex gains for an antenna, and

$$\begin{equation} {\mathbf B} = \left(\begin{array}{@{} cc @{}}b_{\rm x}(\theta ,\phi) & 0 \\ 0 & b_{\rm y}(\theta ,\phi)\end{array}\right) \end{equation} \tag{ 5 }$$

is the matrix of direction- and frequency-dependent but tile-independent gains due to the primary beam shape (we represent spatial coordinates with θ and ϕ). We neglect the feed-error "D" matrix of Hamaker et al. (1996); in other words we assume that the sensitivity of the X-polarization response of a dipole to Y-polarization radiation (the cross-polarization) is zero, and vice versa. This is likely to be a good approximation since ideal dipoles have zero cross-polarization by definition. In reality, various effects, such as the finite thickness of our dipole elements, interactions between structures in neighboring dipoles, or projection effects may produce a nonzero cross-polarization response. In this paper we restrict our imaging and analysis to these two senses of linear polarization and their combination as total intensity. Errors caused by neglecting cross-polarization effects are second order in the small off-diagonal elements of the D matrix.

The strategy we adopt for the data reduction is first to analyze short snapshots wherein the settings in the analog beamformer were static so that the primary beam pattern can be taken as constant over the duration of each snapshot, and any gain changes due to the sidereal motion of the sky relative to the beam can be neglected. In this regime, the direction-dependent gain can be factored out of the response and corrected in the image plane in the resulting map. Under this approximation, we are able to use standard tools for radio astronomical data analysis for much of the processing. Finally, the frequency dependences of the overall antenna complex gains are in principle determined by modeling the summed spectra of the bright sources in the field used in the calibration.

Standard calibration procedures rely on being able to observe a field containing a strong source with easily modeled structure that substantially dominates the visibilities. For the two fields presented in this work, Hydra A is the strongest source in the field. Lane et al. (2004) present low-frequency images that show that while it is quite extended at the VLA's resolution, most of the flux is contained within a region that is a few arcminutes in radius. Since this extent is smaller than the angular resolution of MWA-32T, we were able to treat Hydra A as a point source in our calibration analysis. One might expect that for a large field-of-view instrument such as MWA, we would also need to include several or even many additional strong sources in the calibration model with known direction-dependent gains. We therefore experimented with calibration models that included several point sources in addition to Hydra A, but we found that the complex tile gain solutions were not significantly changed. We therefore simply used Hydra A as the only calibration source in subsequent analysis. It should be noted that this is a potential source of error.

At the low radio frequencies of MWA, position- and time-dependent variations in the electron density of the ionosphere cause variations in propagation times, which appear in the visibility data as frequency- and time-dependent phase shifts. For the short baselines of the MWA-32T, these variations are, except at times of extreme ionosphere disturbance, refractive in effect, i.e., they simply cause apparent changes in the positions of point sources on the sky. These position shifts may be different in different directions, especially over a wide field such as that of the MWA, and consequently may lead to distortions in the derived images.

Ionospheric effects have been quantified by studies with other low-frequency interferometers. Baldwin et al. (1985), using the Cambridge Low-Frequency Synthesis Telescope at 150 MHz, found that the ionosphere typically caused 5° (rms) phase variations on 1 km baselines on sub-day timescales, which were uncorrelated from day to day. They also remarked that ionospheric irregularities on large spatial scales, most likely related to the day–night cycle and strongly correlated from one day to the next, could induce apparent position shifts of sources of up to 20'. Kassim et al. (2007) found, with the VLA operating at 74 MHz, that during times of moderate ionospheric disturbance relative position variations on short timescales across a 25° field of view were at most 2'. Similarly, Parsons et al. (2010), using observations of bright sources with the PAPER array at 150 MHz, found short-term small (typically less than 1') position offsets that were not correlated from day to day, and long-term large (up to 15') position offsets which were correlated from day to day, and were mainly in the zenith direction.

In our analysis, we average snapshots taken with the center of the field within several hours of the local meridian over a period of eight days. The results in the papers cited above suggest that uncorrelated short-term variations in source directions will be significantly smaller than our beam size of ∼15' and, furthermore, that they will tend to average out when images derived from individual snapshots are combined. We therefore neglect them. Long-term correlated variations in source directions may be comparable to our beam size, and may not average to zero as we combine snapshot images. However, our calibration strategy, described in Section 5.1, will tend to remove any ionospheric offset at the position of Hydra A through the phase terms in the direction-independent gain solutions. It is possible, depending on the behavior of the ionosphere during the present observations, that long-term differential position shifts of several arcminutes might be present in our final images. We investigate this possibility through comparisons of the positions of our extracted sources with source positions listed in published catalogs.

Our neglect of short-term ionospheric effects is justified only because of the small baselines (<350 m) of the MWA-32T array. For the longer baselines (∼3 km) of the full MWA, we believe that these effects will need to be corrected to achieve the dynamic range required for many of the science goals.

We developed a calibration and imaging pipeline based on the NRAO Common Astronomical Software Applications²³ (CASA) package and additional tools that we developed in Python and IDL. The pipeline uses a series of short observations to generate "snapshot" images which are weighted, combined, and jointly deconvolved to produce final integrated maps.

In the first stage of the pipeline, the visibilities were averaged over 4 s intervals and converted from the MWA instrumental format into UVFITS files. The MWA-32T correlator does not perform fringe-stopping (the correlation phase center is always at zenith), so phase rotations were applied to the visibilities to track the desired phase center. As a part of this process, data corrupted by RFI were flagged for later exclusion from the analysis. The data were then imported into CASA. Additional editing was done to flag data affected by known instrumental problems or RFI. Approximately 25% of the data were flagged at this stage, mainly because a problem in the data capture software corrupted 480 of the 2080 correlation products.

Calibration was performed separately for each snapshot with CASA, using Hydra A as the gain and phase reference (as discussed in Section 5.1). Although Hydra A was not at the center of the primary beam during observations of the EoR2 field, it was still strong enough to substantially dominate the visibilities. Model visibilities were calculated using a point-source model for Hydra A, assuming an unpolarized flux of unity. The overall frequency-dependent flux scale of the data was set at a later point, along with a correction for the direction-dependent gain. Time-independent channel-by-channel gain factors were calculated for each tile using the task bandpass. After this was done and the visibilities corrected for the gain factors, time-dependent overall tile gain factors were calculated on a 32 s cadence using the task gaincal. The factors determined in the two tasks give the frequency- and time-dependent g_{A, x} and g_{A, y} terms of Equation (4).

The g_{A, x} and g_{A, y} terms were examined for temporal stability and spectral smoothness; regions where deviations were apparent were flagged. Such deviations were rare, and an important outcome of this analysis is the recognition that the MWA antenna gains are quite stable over frequency and time. In fact, the gains even tended to be stable from one day to the next. However, some complicated variations in gain as a function of frequency were identified. These were associated with damaged cables and connectors that have since been replaced or repaired.

The data from each snapshot were subdivided into 7.68 MHz wide frequency bands, and multifrequency synthesis imaging was performed for each snapshot using the CASA task clean. Images were made with a 3' cell size over a ∼51° × ∼51° patch of sky in order to cover the majority of the main lobe of the primary beam. The "XX" and "YY" polarizations were imaged separately. Conversion to the standard Stokes parameters was not performed at this stage, since, as discussed in Section 5.1, the gains for the two polarizations have different direction dependence. The "w-projection" algorithm (Cornwell et al. 2008) was used to correct for wide field-of-view effects, and to produce an image with an approximately invariant point-spread function (PSF) in each of the snapshot images. The images were deconvolved using the Cotton–Schwab CLEAN algorithm (see Schwab 1984) down to a threshold of 1% of the peak flux in the image.

A position-dependent "noise" map was also computed for each of the 7.68 MHz wide snapshot images by selecting a 64 pixel by 64 pixel window around each pixel in the image and fitting a Gaussian to the central 80% of a histogram of the pixel values. This procedure was employed because of the high point-source density in these maps. Throughout much of the area of these maps, it was impossible to identify a source-free region from which to estimate the background noise fluctuations, and the presence of sources artificially skewed the noise estimates calculated strictly as the rms of the pixel values. We found that this clipped histogram fitting procedure provided a more robust estimate of the rms of the background noise. These noise maps were smoothed on a 1° scale to remove local anomalies introduced by extended or clustered sources. However, despite these procedures, some areas, particularly in especially crowded regions, still had anomalously high-noise estimates.

As discussed above, each snapshot was only ∼5 minutes in duration, and was obtained while the delay line settings in the analog beamformers were fixed. This allowed us to model the primary beam pattern of each tile as fixed relative to the sky for the duration of the snapshot. Our calculated model beams for each polarization formed the frequency-dependent b_x(θ, ϕ) and b_y(θ, ϕ) terms of Equation (5). These terms are time dependent only in that they are different for each snapshot.

Deeper images were obtained by combining snapshot maps from a particular 7.68 MHz wide band according to

$$\begin{equation} I_{\rm dirty}(\theta ,\phi)=\frac{ \sum _i \frac{D_i(\theta ,\phi) B_i(\theta ,\phi)}{\sigma _i^2(\theta ,\phi)}}{\sum _i \frac{B_i^2(\theta ,\phi)}{\sigma _i^2(\theta ,\phi)}}, \end{equation} \tag{ 6 }$$

where I_dirty is the integrated, primary beam corrected, dirty map, the snapshots are distinguished by index i, D_i is a snapshot dirty map, B_i is the primary beam calculated for that snapshot, and σ_i is the fitted rms noise obtained from the noise map for snapshot i. This combination optimizes the signal-to-noise ratio (S/N) of the final image. Beam patterns are calculated independently for each 7.68 MHz channel. The same weighting scheme was used to combine the "clean components" and residual maps of the individual snapshots.

A variant of the Högbom CLEAN algorithm (Högbom 1974) was used to further deconvolve the integrated residual maps, using a position-dependent PSF calculated using the same weighting scheme:

$$\begin{equation} P_{{\rm integrated},j}(\theta ,\phi)=\sum _i \frac{B_i(\theta ,\phi) B_i(\theta _j,\phi _j) P_i(\theta -\theta _j,\phi -\phi _j)}{\sigma _i^2(\theta ,\phi)}, \end{equation} \tag{ 7 }$$

where P_{integrated, j} is the PSF for a source at position (θ_j, ϕ_j), B_i is the primary beam pattern for the ith snapshot with a PSF given by P_i, and the peak of the function is normalized to unity. CLEAN components were selected by choosing the pixel in the residual map with the largest S/N (determined by dividing the residual map by its noise map). The PSF was scaled to a peak of 10% of the flux of the pixel. The images were restored with a Gaussian beam determined by a fit to the weighted average of the individual snapshot PSFs at the field center.

In order to increase the S/N and image fidelity for source detection and characterization, the individual 7.68 MHz maps, after deconvolution and restoration, were averaged together. An approximate flux scale for the maps was first set by scaling the surface brightness of the maximum pixel at the location of Hydra A to a value of 296 × (ν/150 MHz)^−0.91 Jy beam⁻¹ (this model was derived from fitting a power law to other low-frequency measurements). For each field, 30.72 MHz bandwidth maps centered at 123.52 MHz, 154.24 MHz, and 184.96 MHz were each made from four 7.68 MHz maps. Before averaging, the 7.68 MHz map fluxes were scaled to the averaged map frequency using an assumed spectral index of α = −0.8 (where S∝ν^α). The averages were computed in a weighted sense using the integrated primary beam weights from each map. A full-band (92.16 MHz bandwidth) weighted average map was made from the three 30.72 MHz bandwidth maps after scaling them to a common reference frequency of 154.24 MHz, again using a spectral index of α = −0.8. The portion of each field within 25° of the field center was used for the subsequent analysis.

Sources were identified in each full-band, i.e., 92.16 MHz wide, averaged map using an automated source extraction pipeline. The first step of this pipeline was the calculation of a position-dependent noise map using the method described in Section 6.4. The full-band map was then divided by the noise map to produce an S/N map. An iterative process then was initiated by identifying contiguous regions of pixels above a certain S/N detection threshold and, for each such region, defining a fitting region that extended beyond the set of connected pixels by several synthesized beamwidths. A two-dimensional Gaussian fit was performed on the corresponding region in the full-band map. The parameters determined in each fit included the background level, peak position, peak amplitude, major-axis width, minor-axis width, and position angle. If the fit converged to a Gaussian centered within the fitting region, then the source was subtracted from the map, and the extracted source parameters were recorded. After fitting all regions identified for a certain S/N level, the detection threshold was reduced and the process was repeated. Regions above the detection threshold for which a fit failed to converge are refit in subsequent iterations at lower detection thresholds (where the fitting regions are typically larger in size). For completeness, sources were extracted down to a detection S/N threshold of 3. It should be noted that in this fitting procedure, each region that is fit by a single two-dimensional Gaussian is taken to correspond to a separate source. Sources that are too close together to be resolved into separate components will be fit with a single component and erroneously taken to be a single source that is a "blend" of the two components.

Sources that were identified in the full-band average map were then extracted from each of the 30.72 MHz bandwidth maps. The sources were sorted by their detection signal-to-noise level and, for each of the three maps, were fitted in descending order. For each source, the position and shape (axial ratio and position angle) parameters of the Gaussian fitting function were held fixed to the values determined in the full-band map extraction, while the peak value, background level, and a scaling factor for the widths of both the major and minor axes of the Gaussian were allowed to vary. This procedure was performed for all sources. When the fit successfully converged, the best-fit model was subtracted from the sub-band map. A total of 908 sources were extracted in the Hydra A field and 1100 sources were extracted from the EoR2 field.

We compared the positions and relative fluxes of the sources identified in the full-band maps with the positions and fluxes of possible counterparts in other catalogs. These comparisons formed the basis for astrometric corrections, and for the determination of the overall MWA flux scale. Counterparts to MWA-32T sources were identified at 408 MHz by locating sources in the MRC which were within 15' of an MWA source. Although we expect our astrometric accuracy to be much better than 15', this value was chosen to be comparable to the size of the MWA-32T synthesized beam major-axis FWHM response, viz., 13' for the Hydra A field and 14' for the EoR2 field, and to allow for some degree of systematic error in the MWA source positions. To avoid possible blending issues, we only considered an identification to be secure when there was precisely one source in the MRC within 30' of the (pre-adjustment) MWA source position. A total of 419 sources were matched uniquely to the MRC in the Hydra A field and 520 sources were matched in the EoR2 field.

An astrometric correction was then calculated by allowing for a linear transformation of the MWA source coordinates in order to minimize the positional differences between corresponding MWA and MRC sources. The transformation permits offsets in both right ascension and declination as well as rotation and shear with respect to the field center. Optimal transformation parameters were determined by performing weighted least-squares fits. The results from initial fits indicated that there were errors in the (Earth-referenced) coordinates of the MWA tiles and in the conversion of coordinates in the maps from the epoch of observation frame to the J2000 frame. These errors were then corrected. The final fits were found to be consistent solely with offsets of the MWA source coordinates, with no shear or rotational effects. We therefore applied offsets of Δα = −06 and Δδ = 16 to the positions of the sources in the Hydra A field, and of Δα = 22 and Δδ = 19 to those in the EoR2 field. We believe that the coordinate offsets are likely due to a combination of the effects of structure in Hydra A and to ionospheric refraction. Figure 2 shows the post-offset-correction differences in position between corresponding MWA and MRC sources in the EoR2 field. Histograms of residual positional differences from both the Hydra A and EoR2 fields are plotted in Figure 3. For this figure the difference for each source is normalized by the expected error based on the S/N for the intensity and the 32T synthesized beamwidth (with the assumption of a circular source; see Condon 1997). The residual position differences are generally consistent with those expected, even though there are a small number of large position differences.

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A final flux scale was set for each map using the MWA sources which had counterparts in both the MRC and Culgoora source lists. Only sources in the MWA catalog above a detection S/N threshold of 5 were used. Hydra A was excluded for reasons discussed in Section 7.1. A prediction for each source was obtained by fitting a power-law spectrum to the 408 MHz MRC flux and the 80 MHz and 160 MHz Culgoora fluxes. Under these criteria, measurements in all three bands were found for a total of 64 uniquely matching sources in the Hydra A field and 81 uniquely matching sources in the EoR2 field. Using these flux predictions, a flux scale correction was calculated of the form

$$\begin{equation} S_{\rm calibrated}=C_{\nu } \times S_{\rm uncalibrated}. \end{equation} \tag{ 8 }$$

These calibration terms were calculated independently for each of the three sub-band maps as well as for each averaged map for each field. For the Hydra A field, the calculation yielded C_ν = 1.26 for the full-band average map, 1.17 for the 123.52 MHz map, 1.20 for the 154.24 MHz map, and 1.18 for the 184.96 MHz map, and for the EoR2 field, the calculation yielded C_ν = 1.24 for the full-band average map, 1.19 for the 123.52 MHz map, 1.23 for the 154.24 MHz map, and 1.17 for the 184.96 MHz map. The residuals after applying these flux scale corrections were analyzed to determine if additional biases were present as a function of position in the image (biases would potentially be seen, e.g., if the assumed primary beam model was incorrect). An example of these plots is shown in Figure 4, which shows no evidence for a radially increasing flux bias.

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The magnitude of the post-correction residual differences between the MWA measurements and the predicted fluxes for calibration sources are still larger on average than expected under the assumption that the uncertainty in each MWA-32T flux measurement is due to the rms map noise at the source location, and that the uncertainty in the predicted flux of each source is propagated for the power-law fitting procedure. This excess in the differences could be due to errors in the spectral model for the calibration sources, temporal variability of the sources, or to as yet unidentified errors. Since this is the first work based on MWA-32T data to report the fluxes of a large number of sources, we make the conservative assumption that these excessively large residuals are due solely to errors in the MWA-32T flux measurements. We assume that the flux uncertainties follow a Gaussian distribution which includes the effects of the rms map noise added in quadrature with an additional component proportional to the measured flux of the source:

$$\begin{equation} \sigma _{\rm MWA}^2 = \beta ^2 S_{\rm MWA}^2 + \sigma _{\rm Map}^2, \end{equation} \tag{ 9 }$$

where σ_MWA is the 1σ flux uncertainty for a particular source, β is the fractional flux uncertainty, S_MWA is the measured source flux, and σ_Map is the rms map noise at the position of the source. We evaluated the standard deviation, σ_D, of the fractional flux difference, D, where D is calculated as

$$\begin{equation} D=\frac{S_{\rm MWA} - S_{\rm Predicted}}{S_{\rm Predicted}}. \end{equation} \tag{ 10 }$$

We then solved for the fractional uncertainty in the MWA measurements which would be needed to account for the magnitude of the measured value of σ_D:

$$\begin{equation} \beta ^2=\sigma _D^2 \left(\frac{S_{\rm Predicted}}{S_{\rm MWA}}\right)^2 - \left(\frac{\sigma _{\rm Predicted}}{S_{\rm Predicted}}\right)^2 - \left(\frac{\sigma _{\rm Map}}{S_{\rm MWA}}\right)^2. \end{equation} \tag{ 11 }$$

We calculated the average value of β separately for each field. For the higher frequency maps, we found that the values of β were much larger far from the field center where the primary beam approaches the first null; for these maps the sources were separated into inner and outer region sets using a cutoff of 18°, and β was calculated separately for each region. Using these results, we assign fractional flux uncertainties of 30% for all sources in the full-band average maps, 35% for all sources in the 123.52 MHz maps, 35% for sources in the inner region of the 154.24 MHz maps, 60% for sources in the outer region of the 154.24 MHz maps, 35% for sources in the inner region of the 184.96 MHz maps, and 80% for sources in the outer region of the 184.96 MHz maps. These fractional uncertainty values are applied to all sources in the catalog by adding them in quadrature to the map rms values as described in Equation (9).

The full-band average maps of the Hydra A and EoR2 fields are displayed in Figures 5 and 6. These images overlap partially. Together, they cover ∼2700 deg². The synthesized beam for the Hydra A field has a major-axis width of 19' in the 124.52 MHz map, 14' in the 154.24 MHz map, 12' in the 184.96 MHz map, and 13' in the full-band average map. For the EoR2 field, the major-axis beam widths are 18' in the 124.52 MHz map, 16' in the 154.24 MHz map, 13' in the 184.96 MHz map, and 14' in the full-band average map.

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The bright radio galaxy Hydra A is the dominant source in these maps. The flux of the source is measured to be 710 ± 210 Jy in the full-band average map of the Hydra A field, and 550 ± 170 Jy in the full-band average map of the EoR2 field. These measurements of Hydra A are significantly brighter than expected based on previous measurements: the Culgoora 160 MHz measurements give a flux of 243 Jy, and a prediction based off of the Culgoora and MRC measurements (as described in Section 6.7) gives a flux of 284 Jy. Using the VLA, Kassim et al. (2007) find an integrated flux density of 644.2 Jy for Hydra A at 74 MHz, which, when scaled to 154.24 MHz using a spectral index of α ∼ −0.8, is consistent with the prediction from Culgoora and MRC measurements and inconsistent with the MWA results. Although Hydra A is slightly extended in the MWA-32T maps, the structures seen in the previous low-frequency maps of Hydra A presented by Lane et al. (2004) are below the scale of the MWA synthesized beam. We note that the MWA-32T array has a significantly more compact uv distribution than the VLA, Molonglo, or Culgoora telescopes. A flux from Hydra A above what is expected from the Culgoora measurements is also noted in measurements with the PAPER array (D. C. Jacobs 2011, private communication), which has a similar baseline distribution to that of the MWA-32T. Consequently we did not include Hydra A in the final flux scale calibration procedure described in Section 6.7. Resolving this flux discrepancy remains a major outstanding issue.

The behavior of the fitted rms noise in these images is illustrated in Figure 7, which shows annular averages as a function of distance from the field center. We can estimate a lower limit to the rms noise in each map by calculating the classical source confusion limit. Note that this differs from the sidelobe confusion limit; see, e.g., Condon (1974) for a rigorous discussion of classical source confusion in radio telescopes. Di Matteo et al. (2002) present a model of the radio source counts derived from the 6C (Hales et al. 1988) catalog at 150 MHz. Their expression takes the form of a broken power law:²⁴

$$\begin{equation} \frac{dn}{dS} \!=\! \left\lbrace \begin{array}{@{} ll @{}} \! 4000 \left(\frac{S}{1\,{\rm Jy}}\right)^{-2.51}\,{\rm sources}\,{\rm Jy}^{-1}\,{\rm sr}^{-1}, & S > S_0\\ \ms \! 4000 \left(\frac{S_0}{1\,{\rm Jy}}\right)^{-0.76} \left(\frac{S}{1\,{\rm Jy}}\right)^{-1.75}\,{\rm sources}\,{\rm Jy}^{-1}\,{\rm sr}^{-1}, & S < S_0, \end{array}\right. \end{equation} \tag{ 12 }$$

where S₀ = 0.88 Jy. Integration of this expression gives the source density in each beam above a minimum flux value, S_min:

$$\begin{equation} \rho _S=\frac{\pi \theta ^2}{4 \ln {2}} \int _{S_{\rm min}}^{\infty }\frac{dn}{dS} dS\,{\rm sources}\,{\rm beam}^{-1}, \end{equation} \tag{ 13 }$$

where ρ_S is the source density in units of sources per beam and θ is the FWHM synthesized beam size. The source confusion limit then corresponds to the flux, S_min, for which the source density approaches one source per synthesized beam area (typically maps are considered to be source confusion limited when they have a source density of greater than 1 source per 10 synthesized beams). The average rms noise (Figure 7) reaches a minimum value of ∼160 mJy for the full-band average map of the EoR2 field. Using the corresponding flux threshold for an unresolved source of S_min = 160 mJy in combination with the EoR2 full-band map synthesized beam area gives ρ_S ∼ 0.30 sources per synthesized beam, or, in other words, one source of at least 160 mJy in roughly every three synthesized beams. Estimates for the other average and sub-band maps give expected source densities of one source per every three to six synthesized beams. Thus, source confusion is likely the limiting source of noise in the central region of these maps. This explains the relatively flat nature of the central noise floor seen in Figure 7. Near the edges of the images, not far from the first null of the primary beam, we expect the noise to be dominated by receiver noise, and to scale with the inverse of the primary beam power pattern. This is also seen in Figure 7 (the frequency dependence of the noise curves illustrates the frequency dependence of the primary beam).

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A catalog was constructed from the 2008 detections at a detection S/N threshold ⩾3 of potential sources in the EoR2 and Hydra A fields. In the cases where there were detections at corresponding celestial positions in the two fields, the measurement where the source was closer to the observation field center was retained, resulting in a list of 1526 unique source detections. The quality of this source list is assessed in Sections 7.2 and 7.3 as a function of the detection S/N level. The 655 sources detected at an S/N level ⩾5 in the detection images are reported in Table 2. The 871 sources with $3\le {\rm {\rm S/N}} <5$ are considered to be less reliable detections. A list of these candidate detections can be obtained by contacting the authors.

Table 2. Detected Sources in the Hydra A and EOR 2 Fields

Name	R.A.	Decl.	Savg	S123.52	S154.24	S184.96	Field	rFC	Detection
									S/N Level
J0747−1854	07h47m05s	−18°54'12''	8.8 ± 2.8	10.1 ± 3.7	9.2 ± 6.0	8.9 ± 8.5	HydA	229	8.6
J0747−1919	07h47m27s	−19°19'33''	22.2 ± 6.7	25.2 ± 8.9	23.8 ± 14.5	15.1 ± 13.0	HydA	230	22.4
J0751−1919	07h51m20s	−19°19'30''	7.1 ± 2.2	8.1 ± 3.0	8.5 ± 5.3	4.7 ± 5.1	HydA	221	8.7
J0752−2204	07h52m30s	−22°04'38''	4.4 ± 1.5	4.7 ± 1.8	7.1 ± 4.5	14.4 ± 12.1	HydA	227	5.6
J0752−2627	07h52m30s	−26°27'43''	8.4 ± 2.7	9.8 ± 3.7	3.6 ± 3.1	...	HydA	247	7.0
J0757−1137	07h57m10s	−11°37'10''	3.9 ± 1.3	4.7 ± 1.8	5.1 ± 3.3	2.5 ± 2.5	HydA	198	6.0
J0802−0915	08h02m18s	−09°15'32''	3.4 ± 1.2	5.8 ± 2.2	2.1 ± 1.5	1.9 ± 1.8	HydA	188	5.0
J0802−0958	08h02m34s	−09°58'55''	8.9 ± 2.8	12.4 ± 4.4	6.0 ± 3.7	9.1 ± 7.3	HydA	186	12.6
J0803−0804	08h03m60s	−08°04'48''	4.5 ± 1.4	6.5 ± 2.4	2.4 ± 1.6	6.7 ± 5.4	HydA	187	7.0
J0804−1244	08h04m17s	−12°44'32''	4.6 ± 1.5	4.8 ± 1.8	5.2 ± 3.2	4.5 ± 3.7	HydA	180	9.3
J0804−1726	08h04m42s	−17°26'41''	4.9 ± 1.5	5.8 ± 2.1	4.5 ± 2.8	3.1 ± 2.6	HydA	185	9.1
J0804−1502	08h04m53s	−15°02'54''	2.8 ± 0.9	2.2 ± 1.0	4.6 ± 2.9	1.0 ± 1.1	HydA	180	6.4
J0805−0100	08h05m30s	−01°00'12''	9.4 ± 2.9	10.8 ± 3.9	8.3 ± 5.2	6.4 ± 5.4	HydA	211	10.4
J0805−0739	08h05m40s	−07°39'22''	3.5 ± 1.2	4.4 ± 1.7	3.7 ± 2.4	1.9 ± 1.9	HydA	184	5.5
J0806−2204	08h06m26s	−22°04'43''	3.7 ± 1.2	4.2 ± 1.6	3.8 ± 2.4	3.6 ± 3.1	HydA	198	6.2

Notes. The flux of each source detected in the MWA full-band-averaged maps is presented along with the flux measured in each 30.72 MHz sub-band. Duplicate sources in the region of overlap of the two fields are not listed. Missing data indicate that the automatic source measurement algorithm failed to converge in a flux fit for that source in the sub-band map. The field from which each source measurement comes from is listed, along with the distance of the source from the center of the field (r_FC). We expect systematic errors to be larger for sources far from the field center. The "Detection S/N Level" indicates the signal-to-noise ratio at which the source was detected in the full-band-averaged map. Section 7.2 discusses the reliability of the catalog at different detection S/N levels. This list includes sources identified above a detection S/N threshold of 5. The full source list of all sources above a detection S/N threshold of 3 is available from the authors.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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The reliability of the identified sources was evaluated through comparison with the flux-limited sample from MRC, VLSS, and maps from TGSS. The MRC source list has a well-defined completeness flux limit and covers our entire field; however it gives fluxes at a different frequency (408 MHz) and does not go quite as deep as our survey. VLSS covers a portion of our fields at a lower frequency than MWA (74 MHz), and provides a useful complementary assessment. The TGSS maps are at the same frequency as the MWA observations (150 MHz), but the maps which have been released to date only cover a small fraction of our fields and are based on a significantly different sampling of the visibility function due to the different array baselines.

In order to assess the MWA-32T catalog reliability, we first evaluate the detectability of an MWA source in the external comparison survey. The MWA full-band average flux is extrapolated to the relevant frequency using a spectral index of α = −0.8, and the extrapolated value is then compared to the parameters of the comparison map or catalog to evaluate whether it meets the detection criteria for that survey. If the source is deemed detectable, then we search for a companion source in that catalog or map to see if the source was actually detected by the other survey. Under the assumption that the other surveys are complete, this allows us to assess how many spurious sources are present in the MWA catalog. We define the reliability as

$$\begin{equation} R=\frac{N_{\rm detected} / N_{\rm detectable} -f}{1-f}, \end{equation} \tag{ 14 }$$

where R is the fraction of MWA sources we believe to be reliable, N_detectable is the number of MWA sources which we believe should have been detectable in the comparison survey, N_detected is the number of detectable MWA sources for which we found counterparts in the other survey, and f is the false source coincidence fraction. We determine f by calculating the source density of the comparison survey in the MWA fields, and use our counterpart matching criteria to estimate the probability that a randomly chosen sky location will lead to an association with a source in the comparison survey. This analysis was performed for different MWA source detection thresholds.

For the reliability comparison with the MRC, we used the completeness limit of 1 Jy (Large et al. 1981) to assess the detectability of the extrapolated MWA source fluxes. An MRC counterpart is associated with the MWA source if it is within 10' of the MWA source position. Based off of the counterpart search radius and source densities in the MRC field, we estimate a false coincidence chance of 4%. A fixed flux completeness limit is not given for the VLSS, however Cohen et al. (2007) note that for a typical VLSS rms of 0.1 Jy beam⁻¹, the 50% point-source detection limit is approximately 0.7 Jy. We then assume that the VLSS is complete down to a flux level of 1 Jy, and we again use a 10' source association radius in the reliability calculation. At the present time, the VLSS catalog does not cover the entire combined EoR2 and Hydra A region that we have surveyed. To ensure we are only including sources in the VLSS survey area, we only analyzed MWA sources above a declination of δ = −25°. We estimate a false coincidence chance of 18% for the VLSS matching.

The cumulative and differential catalog reliabilities are listed as functions of the MWA detection level in Tables 3 and 4. We view these reliability estimates as lower limits, particularly at the lower flux levels, because our assessments of comparison survey detectability do not take into account errors in the source flux extrapolation or source time variability. MWA sources which are erroneously calculated as detectable will not, in general, lead to detections of counterparts in the comparison catalog, whereas sources erroneously calculated as undetectable will be omitted from the analysis and, therefore, will not be included in the calculation of the reliability ratio. These catalog comparisons imply a reliability of ≳ 99% for sources detected above a detection S/N of 5.

For the reliability comparison using TGSS, we used the maps from TGSS Data Release 2 available at the time of our analysis. Although the baseline distribution of GMRT is substantially different from that of MWA-32T, GMRT has a compact central array consisting of 14 antennas within an area of radius 500 m (Swarup et al. 1991) that leads to substantial overlap with MWA-32T regarding the region of the uv plane that was sampled. Twenty-seven TGSS fields overlapped the EoR2 field; none overlapped the Hydra A field. We modeled the effects of source blending by the large MWA beam by convolving the CLEANed and restored TGSS maps to a Gaussian FWHM resolution of 12'. Care was taken in the convolution to preserve the flux density scale. The resulting convolved images have typical rms surface brightness fluctuations of ∼0.4 Jy beam⁻¹ in regions free of sources. Sources in these maps were identified by taking all pixels above 4σ and associating a source with each island of bright pixels. The resulting TGSS source catalog was compared to those sources in the MWA list with flux density greater than the 4σ level in the TGSS field, and thus expected to be detected in the TGSS field. Pairs of sources in the two catalogs coincident within 10' were recorded as sources detected in both surveys. MWA sources without a TGSS counterpart were counted as non-detections. Reliability values are presented in Tables 3 and 4. These values are ratios for each MWA detection S/N bin of the number of TGSS detections to the number of MWA sources expected to be detected in the TGSS field, and are, of course, a function of the threshold chosen in the TGSS maps. As an example of the TGSS comparison we present Figure 8, which plots the positions of MWA and MRC sources on a grayscale image of a convolved TGSS field. It is important to note that these reliability estimates solely test for the presence of a source coincident with the reported position, and do not speak to the fidelity of the fluxes of these sources or whether the MWA sources are due to single objects or blends of multiple fainter objects.

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We used the Culgoora source list to assess the completeness of the MWA catalog presented in this paper. We chose the Culgoora list because it includes observations done at 160 MHz, a frequency not far from the midpoint of the MWA band and because its synthesized beam is similar in size to that of MWA-32T. For each Culgoora source within a field observed by MWA we used the position-dependent rms noise in the MWA full-band-averaged maps to evaluate its detectability in the MWA map. Culgoora sources which should be detectable above a specified S/N level in the MWA images were then checked for a matching source within 10' in the MWA catalog. Since we used the Culgoora source list to assess the completeness, the results are only valid down to a level comparable to the lowest Culgoora fluxes of ∼1.2 Jy. The completeness ratio was calculated similarly to the reliability described in Section 7.2:

$$\begin{equation} C=\frac{N_{\rm detected} / N_{\rm detectable} -f}{1-f}, \end{equation} \tag{ 15 }$$

where C is the completeness percentage, N_detectable is the number of Culgoora sources which we believe should have been detectable in the MWA source list, N_detected is the number of detectable Culgoora sources for which we found counterparts in the MWA list, and f is the false source coincidence fraction calculated from the MWA catalog source density using the 10' source matching criteria. The results are presented in Table 5.

The completeness was analyzed separately for sources within inner and outer regions separated by a circle of radius r = 18° around the field center. All Culgoora sources within the inner region which did not have a corresponding detection in the MWA source list were inspected and found to coincide with a local maximum in the map, implying that the completeness is limited by the robustness of the source extraction algorithm and the flux calibration rather than the intrinsic map quality. It is important to note that because the MWA maps have a sensitivity that varies strongly across the field, this completeness value does not specify a flux limit to the catalog, but rather assesses the efficacy of the source extraction. As with the above reliability estimate, variability and incompleteness act to make this a lower limit on the true completeness.

Radio source counts provide another useful diagnostic test to assess the quality of the catalog and consistency with previous works. As discussed in Section 7.1, Di Matteo et al. (2002) fit the 151 MHz 6C survey results of Hales et al. (1988) to obtain a power-law model for radio source counts. The fit is valid up to ∼10 Jy, but the actual 6C counts fall somewhat below the fit at the high end of the range. Using the MWA-32T catalog generated from the EoR2 field, we calculate the differential source counts, using the noise map of the field to correct for the effects of sensitivity variations on the effective survey area for different flux values (i.e., bright sources can be detected over a larger area than faint sources). No Eddington bias correction is applied (to correct for the artificial enhancement of faint sources due to noise in the map) and the error bars are calculated from the square root of the number of counts in each bin. These source counts are shown in Figure 9, along with the expected source counts from integrating the Di Matteo et al. (2002) model. We note that more sophisticated models of 150 MHz source counts models have been described by, e.g., Wilman et al. (2008) and Jackson (2005). These models have different behavior at the sub-Jansky flux levels that have been probed by high resolution, deep, narrower field-of-view studies such as those described in Intema et al. (2011) and Ishwara-Chandra et al. (2010). We compare our results with the Di Matteo et al. (2002) source counts model because it is commonly used as a basis for studies of EoR foregrounds and sensitivities.

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The results from the catalog presented in this work agree with the Di Matteo et al. (2002) model above flux levels of ∼2 Jy. Below this level, the MWA-32T source counts diverge from the model, likely because of the incompleteness of the MWA source extraction for low flux sources. A power-law fit to the EoR2 field source counts above 2 Jy yields dn/dS = (3500 ± 500)(S/1 Jy)^{−2.59 ± 0.09} sources Jy⁻¹ sr⁻¹ for sources with a detection S/N greater than 5. Fitting for a power law to the full list of sources in the field down to a detection S/N of 3 yields dn/dS = (5700 ± 700)(S/1 Jy)^{−2.76 ± 0.08} sources Jy⁻¹ sr⁻¹.

As an additional test for systematic effects, we have constructed the angular two-point correlation function, w(θ), of the sources in our catalog. This correlation function can show systematic effects that manifest themselves on characteristic angular scales in the catalog—see, e.g., Blake & Wall (2002). We measure w(θ) using the estimator defined by Hamilton (1993):

$$\begin{equation} w(\theta) = \frac{DD(\theta) RR(\theta)}{DR(\theta)} - 1, \end{equation} \tag{ 16 }$$

where DD(θ) is the measured angular autocorrelation function from the MWA source catalog, RR(θ) is the autocorrelation function calculated using a simulated "mock" catalog, and DR(θ) is the cross-correlation between the MWA and the mock catalog. We generate an ensemble of 100 mock catalogs and evaluate the correlation function with each one separately in order to produce a set of normally distributed estimates of w(θ). Each mock catalog is produced using an approach developed to simulate point sources at cosmic microwave background and far-IR frequencies (Argüeso et al. 2003; González-Nuevo et al. 2005), but tailored specifically for the MWA experiment (A. de Oliveira-Costa et al., in preparation), i.e., we drew sources from the observed MWA-32T source counts distribution described above in accordance with the expected low-frequency source clustering statistics (de Oliveira-Costa & Capodilupo 2010; de Oliveira-Costa & Lazio 2010). On the angular scales probed by this survey, no observable clustering is expected. By constructing the mock catalogs in this manner, and correlating them with the observed distribution, the resulting estimate of w(θ) identifies any unexpected correlation which may be due to systematic errors in our survey or in our catalog construction procedure. Figure 10 shows our measurement of w(θ) above a flux limit of S ≈ 3 Jy (black squares). Distances between the observed and/or simulated sources are measured in bins of ∼1°, which is substantially above the MWA resolution. The mean value of w(θ) is shown, along with uncertainties derived from calculating the covariance between w(θ) bins in the mock catalogs. As expected, w(θ) is consistent with zero, implying that there is no excess correlation in our catalog.

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Comparing the present results from MWA with results from PAPER (Parsons et al. 2010) is a particularly useful exercise, as both arrays are new, broadband, wide field-of-view instruments with similar uv coverage, and both are intended to be used to make EoR power spectrum measurements. We find that 43 of the sources found in the survey (described in Section 2) of Jacobs et al. (2011) are located in our survey region. A search in our MWA-32T catalog reveals unique counterparts within 30' (the PAPER beam size used by Jacobs et al. 2011 for source association) of the PAPER locations for 31 out of these 43 sources, multiple counterparts in 11 cases, and no counterpart in 1 case. A comparison of the fluxes of sources detected in both the MWA-32T and PAPER catalogs is shown in Figure 11.

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The one source with no MWA counterpart is 24° from the center of the MWA Hydra A field, and corresponds to a local maximum in the MWA image; however, it was not detected by the automatic source finding algorithm. For each of the 10 sources with multiple MWA counterparts, an estimate of the blended flux was obtained by summing the flux of all MWA sources within the PAPER beam. Other than Hydra A, all 41 PAPER fluxes are consistent with the MWA blended fluxes (the Hydra A flux reported in the PAPER catalog was corrupted by the filtering used in the PAPER analysis; D. C. Jacobs 2011, private communication). A weighted average of the ratios of the MWA and PAPER source fluxes yields the average ratio 〈S_MWA/S_PAPER〉 = 1.17 ± 0.10. However, we note that the PAPER flux scale was set using measurements of two calibration sources from the Culgoora source list, whereas the MWA-32T flux scale was set using a fit to an ensemble of Culgoora and MRC measurements. Slee (1977) note that the Culgoora flux scale may be depressed by 10%, with additional flux uncertainties of between 13% and 39% for individual source measurements. If these Culgoora flux uncertainties are taken into account as potential errors on the PAPER flux scale, then the significance of the difference between the MWA and PAPER flux scales is decreased.

The standard deviation of the MWA to PAPER flux ratios after correcting for the different flux scales is ∼25%. This is smaller than our estimate of the MWA flux uncertainties based on flux predictions from the MRC and Culgoora measurements. This indicates that the flux comparison with the MRC and Culgoora lists may be affected to a considerable extent by radio source variability or other systematic effects.

USS radio sources form a compelling class of candidate high-redshift radio sources (De Breuck et al. 2000, 2002; Di Matteo et al. 2004; Broderick et al. 2007). Low-frequency radio observations are particularly sensitive to these objects (see, e.g., Pedani 2003). We have conducted an analysis of the sources detected in the MWA fields in an attempt to identify additional USS radio sources. A table of candidates is presented in Table 6. We calculated spectral indices by using the PMN 4.85 GHz survey (Griffith & Wright 1993) together with the MWA full-band average flux measurements. We associated MWA sources with PMN counterparts if their positions were coincident within 5'. To avoid source confusion and blending issues, we excluded any MWA sources with more than one PMN counterpart within a 30' radius. A total of 331 sources were identified for which we could unambiguously identify a counterpart and extract a spectral index. A histogram of the spectral indices is shown in Figure 12. This histogram appears consistent with the low-frequency-selected spectral index distributions obtained by De Breuck et al. (2000), and plotted in their Figure 7 (however, the De Breuck et al. 2000 analysis used slightly different frequencies). Using a low-frequency-selected distribution results in a sample which is significantly more sensitive to the USS sources.

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We choose a spectral index cutoff of α ⩽ −1.2 for USS source candidates, and find three sources which match the criteria. All three sources have counterparts in the MRC, and the source MWA J1032−3421 has a counterpart in both the PAPER and Culgoora source lists (although the other two USS candidates do not). A further 33 sources are identified which have no counterpart in the PMN catalog within 1°. Using the PMN catalog limiting flux of 50 mJy (Griffith & Wright 1993) for these sources, we find that 25 of these 33 sources have an inferred spectral index of α ⩽ −1.2. Of these 25 sources, 11 have unique counterparts in the MRC (one additional source has multiple counterparts), 9 have counterparts in the Culgoora source list, and none have counterparts in the PAPER source list.

A comparison between the USS source list of De Breuck et al. (2000) and the source list identified in this work finds only one source from their list which matches an MWA USS candidate within 15': MWA J1032−3421. They select this source from an analysis of the MRC and PMN samples, and find a spectral index of −1.23 ± 0.04, which is consistent with the MWA-32T measurements. There are an additional 21 sources from the De Breuck et al. (2000) sample which match MWA sources, however only three of these MWA sources matched uniquely with a PMN counterpart: MWA J1133−2717, MWA J0941−1627, and MWA J0937−2243. These three sources were identified as USS sources in the "TN" sample of De Breuck et al. (2000), which used measurements at 365 MHz and 1.4 GHz, but they did not meet the USS criteria in the MWA-PMN comparison. The flatter MWA-PMN spectra may indicate a high-frequency turnover in the source spectrum, similar to that noted for J0008−421 in Jacobs et al. (2011). This interpretation is supported by 74 MHz VLSS measurements of these sources (Cohen et al. 2007), which imply a spectral flattening at low frequencies, although further follow-up will be important to definitively establish this behavior.

It is important to note that these MWA sources are only candidates, and should not be treated as definitive USS sources. Flux calibration and measurement errors as well as blending issues due to the low MWA resolution and time variability may result in a reclassification of these sources upon more detailed investigation. Additionally, any time variability or errors introduced by the different catalog resolutions and fitting algorithms will add to errors in this candidate list. The MWA instrument is under continued development, and the fidelity of these studies will improve as the systematics are better understood. However, this candidate list serves as a good basis for more detailed follow-up and investigation.