# Deep sub-arcsecond wide-field imaging of the Lockman Hole field at 144 MHz

F. Sweijen, R. J. van Weeren, H. J. A. Röttgering, L. K. Morabito, N. Jackson, A. R. Offringa, S. van der Tol, B. Veenboer, J. B. R. Oonk, P. N. Best, et al.

► To cite this version:

F. Sweijen, R. J. van Weeren, H. J. A. Röttgering, L. K. Morabito, N. Jackson, et al. Deep sub-arcsecond wide-field imaging of the Lockman Hole field at 144 MHz. Nature Astronomy, 2022, 6, pp.350-356. 10.1038/s41550-021-01573-z. insu-03719755

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https://insu.hal.science/insu-03719755v1

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Deep sub-arcsecond widefield imaging of the Lockman Hole field
at 144 MHz

F. Sweijen¹*, R. J. van Weeren¹, H. J. A. Röttgering¹, L. K. Morabito², N.Jackson³, A.
R. Offringa⁴, S. van der Tol⁴, B. Veenboer⁴, J. B. R. Oonk¹,⁴,⁵, P. Best⁶, M. Bondi⁷,
T. W. Shimwell¹,⁴, C. Tasse⁸,⁹,¹⁰, A. P. Thomson¹¹

★ e-mail: sweijen@strw.leidenuniv.nl

¹Sterrewacht Leiden, University of Leiden, 2300 RA, Leiden, The Netherlands

²Centre for Extragalactic Astronomy, Department of Physics, Durham University, DH1 3LE, UK

³Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, University of Manchester, Manchester M13
9PL, UK

⁴ASTRON, Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk 4, Dwingeloo NL-7991 PD, The
Netherlands

⁵SURF, P.O. Box 94613, 1090 GP Amsterdam, The Netherlands

⁶Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, UK

⁷INAF - Istituto di Radioastronomia, Via P. Gobetti 101, 40129, Bologna, Italy

⁸GEPI, Observatoire de Paris, CNRS, Université Paris Diderot, 5 place Jules Janssen, 92190 Meudon, France

⁹Centre for Radio Astronomy Techniques and Technologies, Department of Physics and Electronics, Rhodes University,
Graham-stown 6140, South Africa

¹⁰USN, Observatoire de Paris, CNRS, PSL, UO, Nançay, France

¹¹Jodrell Bank Centre for Astrophysics, The University of Manchester, SK11 9DL, UK

February 4, 2022

High quality low-frequency radio surveys have the promise of advancing our understanding of many important topics in astrophysics, including the life cycle of active galactic nuclei (AGN), particle acceleration processes in jets, the history of star formation, and exoplanet magnetospheres. Currently leading low-frequency surveys reach an angular resolution of a few arcseconds. However, this resolution is not yet sufficient to study the more compact and distant sources in detail. Sub-arcsecond resolution is therefore the next milestone in advancing these fields. The biggest challenge at low radio frequencies is the ionosphere. If not adequately corrected for, ionospheric seeing blurs the images to arcsecond or even arc

cminute scales. Additionally, the required image size to map the degree-scale field of view of low-frequency radio telescopes at this resolution is far greater than what typical soft- and hardware is currently capable of handling. Here we present for the first time widefield sub-arcsecond imaging at low radio frequencies. We derive ionospheric corrections in a few dozen individual directions. Using a recently developed imaging algorithm [19, 26, 32, 31] we then efficiently apply these corrections during imaging. This algorithm also reduces the computational cost of imaging, allowing us to efficiently map a large area of the sky. We demonstrate our method by applying it to an eight hour observation of the *International*
Low Frequency Array (LOFAR) Telescope (ILT) [28]. Doing so we have made a sensitive 7.4 deg² 144 MHz map at a resolution of 0.3". The estimated 250,000 core hours used to produce this image, fit comfortably in the budget of available computing facilities. This breakthrough will allow us to map the entire northern low-frequency sky at sub-arcsecond resolution.

Endeavours in expanding sub-arcsecond resolution observations to both go deeper and wider have had successes at both 300 MHz [12] and GHz [6, 17, 16, 3, 13, 20, 18] frequencies using Very Long Baseline Interferometry (VLBI). However, producing large contiguous images of significant sky areas (i.e. many degrees) at sub-arcsecond resolution, has not been done at low frequencies and is still tedious and time consuming at higher frequencies due to the limited field of view. This was further complicated due to a combination of a lack of known compact sources (limiting available calibrators and blind surveying), the long observing times required (limiting survey speed) or simply due to computational limits because of the amount of resources required to produce such images. Recent results, however, show that the vast majority of 150 MHz detected sources are still unresolved or barely resolved at the now routinely obtained angular resolution of 6" [23]. The typical size of the µJy population is of the order of a few kpc, which translates to sub-arcsecond angular sizes on the sky[1]. Sub-arcsecond resolution surveys will be able to start resolving these physical scales and are therefore an important and valuable resource.

Currently, an ideal instrument for this is the ILT. It is most sensitive at frequencies between 120 and 168 MHz. Stations throughout the European continent form baselines as short as 68 m and up to approximately 2000 km. This results in a theoretical angular resolution of approximately 0.3" at 150 MHz<fnref n="1" />. The size of the international stations set the field of view at 2.6° full width at half maximum at 120 MHz. Over the past decade innovative algorithms have been developed to overcome the ionospheric blurring sufficiently to allow high quality arcsecond 6" resolution imaging and novel computer science approaches have been designed

to efficiently process data on large-scale parallel-compute infrastructure [5]. Furthermore, recently developed pipelines have demonstrated our ability to produce low-frequency sub-arcsecond resolution images of individual sources that span regions of a few arcminutes [15]. Following this achievement, two major outstanding questions become: (i) can calibration of ~10³ km baselines be reliably extended to many sources, correcting for ionospheric effects at sub-arcsecond level across the full field of view and (ii) can we subsequently image the entire field of view at full resolution?

Here we present our main result: the first deep widefield sub-arcsecond image at low radio frequencies. This image is shown in Fig. 1. It covers 7.4 deg² of the Lockman Hole field, centred at α =10h45m00s, δ =+58d05m00s, where there is a wealth of available ancillary data available (e.g. deep optical, near and far-infrared data). The ionospheric conditions during this observations were typical, based on the 6" image quality.

We demonstrate the feasibility of calibrating and imaging of the ILT's full international station field of view at sub-arcsecond resolution, providing a general strategy for widefield high-resolution imaging at low radio frequencies for dense interferometers such as the ILT.

The procedure we have developed to calibrate and image the full ILT field of view begins with a direction independent calibration on a known calibrator source (ILTJ104940+583529) from the Long Baseline Calibrator Survey (LBCS) [9, 10]. This direction-independent calibration is not sufficient to fully calibrate the FoV, however, and direction dependent effects (DDEs) remain. These mainly come from the ionosphere, which varies strongly over time and across the sky. Successfully correcting for these DDEs hinges on two conditions: the direction independent calibration needs to be approximately valid over the area that is to be calibrated and there need to be enough sources with a sufficiently high signal-to-noise ratio (SNR) in terms of compact structure to allow self-calibration to succeed. The former depends on the isoplanatic patch size while the latter depends on the amount of compact emission in sources.

Judging the amount of compact emission at sub-arcsecond scales in sources is difficult at a resolution of 6". Therefore, to facilitate the selection of DDE calibrator candidates, an intermediate reso-

<footnote n="1">¹The exact resolution varies depending on the visibility weighting scheme.</footnote>
lution image with an angular resolution of approximately 1" was created. This gives a rough indication of the validity of the direction independent calibration across the field and of which sources are likely to retain compact emission at sub-arcsecond resolution. We selected sources that had a peak intensity of 25 mJy or higher in the 1" image, resulting in a sample of 46 potential calibrators. The data were then phasedifted to these sources, creating individual datasets, and self-calibrated.

In the direction-dependent self-calibration process we leverage the fact that the dominant remaining effects are the ionosphere and slowly varying effects, such as errors in the primary beam model. A starting model was automatically generated for each source by imaging it before the first iteration started. Subsequent self-calibration is an iterative process where in each iteration calibration solutions are obtained that provide an updated model, which can in turn provide better calibration solutions. Dominated by the ionosphere, we can constrain the “phase-only” self-calibration iterations to calibrate for differential Total Electron Content (dTEC), reducing the degrees of freedom by introducing the known frequency dependence of the ionospheric delay as the functional constraint given by Equation 1.

$$ \phi = -8.44797245 \times 10^9 \times \frac{\mathrm{dTEC}}{\nu}, \quad (1) $$

Here $\phi$ is the phase in radian and $\nu$ is the frequency in Hz. For every source, now only a single dTEC value is determined per time interval, using the entire bandwidth, instead of frequency dependent phases. This reduces the degrees of freedom 48 times compared to, for example, solving for scalar phases at a frequency interval of at 1 MHz (a typical solution interval for 6" calibration), significantly increasing the effective SNR. Once the ionosphere was calibrated for, long-timescale amplitude corrections were determined to solve remaining amplitude effects. For the ILT an imperfect primary beam model is one of the dominant causes of residual amplitude errors.

Manual inspection of the 46 DDE calibrators concluded that 44 converged after the dTEC iterations and 41 allowed for subsequent amplitude corrections. The remaining 2 sources did not have sufficient compact flux to perform self-calibration on and were thus discarded.

Figure 2 shows the progression of self-calibration on three sources of varying complexity and distance from the initial LBCS calibrator. The fact that self-calibration converged for 44 directions across the entire 7.4 deg² area confirms that the density of compact sources is high enough to correct direction-dependent effects across the field of view. Furthermore, the fact that a sufficiently accurate skymodel for initiating self-calibration could be created from a direction-independently calibrated dataset indicates that for similar ionospheric conditions a single correction is likely to be approximately valid over the entire field of view. Finally, the solutions were interpolated using radial basis function interpolation into a 2D map of corrections. This provides spatially smooth “screens”, providing both dTEC and amplitude calibration solutions for every location in the field.

Even with perfect calibration, imaging the entire field of view is challenge of its own due to the sheer amount of pixels required. We chose a pixel scale of 0.11"px⁻¹ in order to Nyquist sample the main lobe of the PSF. Covering the ILT’s field of view at 0.3" angular resolution would then require a total of 6–7 × 10⁹ pixels. To put this in perspective, this is more than the entire *Very Large Array* (VLA) *Faint Images of the Radio Sky at Twenty-cm* (FIRST) survey which covers about 10⁴ square degrees using roughly 4 × 10¹⁰ pixels [33].

At the time of writing it is not feasible to make a single image at the required size. Therefore the field was split in 25 facets that were imaged individually. Each facet image covered a 0.69° × 0.69° area that could be imaged at a manageable 22 700 × 22 700 pixels. For each facet, the 1" resolution image was used to subtract sources outside of the central 0.55° × 0.55° region. This helps suppress artifacts that would arise from undeconvolved emission outside the image boundary. The imaged area extends beyond this subtraction border to account for remaining source structure due to imperfect subtraction. Lastly, the inner 0.55° × 0.55° regions of the facets overlap with each other to avoid gaps in the mosaic. This process resulted in a final mosaic of approximately 83 950 × 83 500 pixels covering 7.4 deg² on the sky.

Imaging was done using WSClean in combination with the newly developed Image Domain Gridder (IDG) [19, 26, 32]. Multi-frequency synthesis was used to account for the large bandwidth and multi-
scale clean was used to account for both extended and compact emission. One of the bottlenecks for imaging is the need to place visibilities measured by a radio telescope on a regular grid in a computationally expensive process called gridding. The IDG algorithm is a novel approach to gridding that is computationally efficient as it circumvents the need to calculate expensive convolution functions during this process. It simultaneously allows for on-the-fly application of primary beam corrections and other direction-dependent corrections at little extra cost. Finally the IDG algorithm can leverage a GPU for the gridding calculations, giving a significant boost in speed compared to gridding on a CPU. Each of the facets took between 5 and 7 days of wall clock time to image. Using both local<fnref n="2" /> and national<fnref n="3" /> compute resources imaging ran in parallel and the effective wall time was significantly reduced.

A total of approximately 250 000 core hours were required for the data reduction process in order to arrive at the final image. This was split between calibration and imaging as 34% and 76%, respectively. The image covers a 7.4 deg<fnref n="2" /> sky area at an angular resolution of 0.3'' × 0.4'' and a central rms noise level of 32 µJy beam⁻¹ is reached. This equates to a 1.4 GHz sensitivity of 7 µJy beam⁻¹, assuming a spectral index of α = −0.7. This is comparable in depth to the deepest 1.4 GHz VLA image of the field [8].

7143 sources are detected at 6" in the covered area using a single 8 hour observation with an rms noise level of 70 µJy beam⁻¹. Of these 6397 (90%) are unresolved and 746 (10%) are resolved. Of those unresolved sources, 2568 (40%) had a high resolution counterpart detected at a SNR > 5. 293 of these sources are resolved (11%) and the other 2275 remain compact (89%).

Our demonstration of wide-field, low-frequency, sub-arcsecond resolution imaging opens up important new scientific parameter space to be explored by the ability to conduct a blind all-sky sub-arcsecond survey. This resolution is a valuable tool to help, for example, disentangle radio-quiet AGN from star-forming galaxies, which exhibit similar unresolved radio properties and allow the study of radio sources nearby and far in new detail. Low-

frequency data is especially important for broad-band spectral modelling and at this resolution the ILT is well matched to other telescopes such as e-MERLIN or the VLA.

Over the past years the ILT has been conducting the LOFAR Two-Metre Sky Survey (LoTSS), a sensitive 6'' survey of the northern sky [22]. The data recorded for this survey holds incredible legacy value for a future sub-arcsecond survey. Over 75% of the sky has already been observed for LoTSS, using the same duration, bandwidth and averaging as the data presented here. Weighted by integration time, only 3% of this data contains no international stations. 54% has 12 or more international stations per hour of observing time and 84% has 10 or more international stations per hour of observing time. This means that a significant amount of data is already present in the LTA, ready to be processed. Relatively little extra or repeat observing will thus be necessary for a sub-arcsecond counterpart to LoTSS.

The work presented here demonstrates a general strategy for widefield low-frequency sub-arcsecond imaging. By combining advanced calibration strategies with new state of the art imaging techniques we have demonstrated complete direction-dependent calibration and widefield imaging at sub-arcsecond resolution at frequencies between 120 and 168 MHz. Our final result is a 7 gigapixel image of the ILT’s full international station field of view. This paves the road towards routine, pipelined data processing and a sub-arcsecond successor to the northern-sky LoTSS survey.

Correspondence and requests for materials should be addressed to F. Sweijen (sweijen@strw.leidenuniv.nl).

<footnote n="2">²ALICE; Leiden Observatory</footnote>

<footnote n="3">³Spider; SURFsara</footnote>
<figure><img src="image_2.png" /></figure>

Figure 1: An overview of the field illustrating of both the widefield and high resolution aspects. The central image shows a 6" image of the covered part of field. The bottom two panels zoom in on a small portion of the field at 6" and a particular source at 0.3" in the bottom left and right, respectively. Surrounding the central image are highlights of various other sources detected in the 0.3" resolution map. Moon image credit: NASA.
<figure><img src="image_3.png" /></figure>

Figure 2: Illustration of the direction-dependent self-calibration process for different sources across the field and various degrees of extendedness. From left to right: the initial image with only direction-independent corrections applied, the image after dTEC corrections have been applied and finally the image after amplitude corrections have been applied. A significant improvement is seen after correcting ionospheric errors, while the amplitude corrections provide a more subtle improvement. The top and middle images cover a 15" × 15" area and the bottom image covers 25" × 25". A scale bar is shown for size.
# 1 Methods

## 1.1 Observation

For this work we used an observation of the Lockman Hole field located at $\alpha = 10h47m00s, \delta = 58^\circ05'00''$. This pointing was observed on July 12, 2018 from 11:08:10 to 19:08:10 UTC. In total 8 hours were spent on the target field, which was bookended by two 15 minute calibrator scans of 3C 196 and 3C 295, respectively. The observing setup was the standard LoTSS HBA setup with 48 MHz of bandwidth spanning a frequency range of 120-168 MHz. With the exception of DE609, all 12 other international stations partook in the observation. We will use the term *Dutch LOFAR Telescope* (DLT) to refer to the core and remote stations.

## 1.2 Calibration and imaging

Calibration of the radio data started from the raw data stored in the Long Term Archive (LTA). First the data was processed using the de facto pipelines for calibrating the Dutch stations: PREFACTOR and DDF-PIPELINE[5, 23, 25]. Subsequently a suitable infield *Long Baseline Calibrator Survey* (LBCS) calibrator was selected. From thereon no dedicated pipeline existed. First direction independent calibration of the international stations was carried out. An intermediate resolution image at $\sim 1''$ was used to select sources for direction-dependent calibration, which were subsequently self-calibrated. Finally the field was split up in “facets” and imaged at the full native resolution.

### 1.2.1 Direction independent calibration of the DLT

The PREFACTOR pipeline consists of two workflows for the calibrator scan and target scan respectively. A detailed overview of the calibration strategy is given by [5]. A short overview of these workflows will be given next.

First the calibrator workflow determined the following corrections for both Dutch and international stations: a phase offset to align XX and YY polarisations (based on the assumption that the calibrator is unpolarised), a bandpass translating correlator units to physical units and a clock offset in

nanoseconds to synchronize the clocks of each station to the same reference. These solutions were determined on a 4 s time interval and a 48.82 kHz frequency interval. Finally they were transferred to the target dataset and the target workflow determined XX and YY phase solutions for the Dutch stations by calibrating against a skymodel from the TFIR GMRT Sky Survey (TGSS, [7]) on an 8 s time interval and a 195.28 kHz frequency interval.

### 1.2.2 Direction dependent calibration of the DLT

Next the DDF-PIPELINE was run for direction-dependent calibration of the Dutch stations. See [23, 24, 25] for a detailed description of this pipeline. First the direction-independent calibration was refined by means of self-calibration of the entire Dutch station field of view. Following this, both phase and amplitude calibration solutions were determined in 45 directions across the Dutch field of view, following a facet-based approach. This provided a high-quality $6''$ resolution image of the target field covering the field of view of the Dutch stations.

### 1.2.3 Direction independent calibration of the ILT

To start calibration of the international stations, we followed the strategy as the LOFAR-VLBI pipeline [15]. Calibration solutions from PREFACTOR and DDF-PIPELINE were applied, data was combined into 24 manageable 2 MHz chunks of 32 GB and finally a suitable LBCS calibrator was selected within the international station field of view [10]. This calibrator source was then split off from the main dataset by phase shifting the visibilities towards this source. After phase shifting, the core stations were all combined into a single “super station”, ST001. The dataset was also averaged down to a 16 s time interval and a 195.28 kHz frequency interval. This heavy averaging causes a $\sim 50\%$ intensity loss due to smearing at a distance of $\sim 4'$ from the phase centre, which considerably reduces the influence of other sources. ST001 has a field of view of the order of $\sim 4'$ as well, further significantly suppressing the effects of other sources on every baseline with this station. Self-calibration was performed using the routine described in [29].
This uses DPPP ([27]) to determine calibration solutions and WSClean ([19]) for imaging. First 9 “fast” iterations to solve for rapid phase variations were performed. The solution interval was 16 s (the resolution of the data) and the solution type was **scalarphase**, i.e. a phase as function of time and frequency, but independent of polarisation. Then 7 more iterations were performed where in addition to the fast solves, a “slow” solve using DPPP’s **complexgain** mode was done. This determined time, frequency and polarisation dependent amplitude and phase solutions on a 15 min timescale and a 195.28 kHz frequency interval, for the XX and YY polarisations. The starting model for the first phase-only iteration was a point source, due to the lack of more information, with the flux density taken from the low-resolution 6" image. While such a starting model is likely naive possibly incorrect, after each iteration the model was updated and the self-calibration cycle converged to believable double lobed source structure. These solutions were then applied to the target field dataset as initial direction-independent calibration for the international stations and refined direction-independent calibration solutions for the remote and core stations. The core stations were corrected with the solutions for ST001.

### 1.2.4 Intermediate resolution imaging and faceting

Due to its sheer size, the full field of view could not be covered with a single image at the ILT’s native resolution. Therefore the field was split in 25 facets that were imaged separately. To help suppress interference from undeconvolved sources outside of each facet and to determine suitable candidate sources for direction-dependent calibration, an intermediate resolution image at approximately 1" was made by tapering the data with a Gaussian taper. The resulting image was 25000 × 25000 pixels in size and covered the full field of view.

In preparation for high-resolution imaging, 25 datasets ("facets") were then split out with their phase centres spread over the field of view in a 5 × 5 grid, with each facet spanning 0.55 × 0.55 deg² on the sky, separated by 0.5 deg. For each facet the intermediate resolution map was subtracted outside this 0.55 × 0.55 deg² area. The 0.05 deg on each side ensured some overlap between the facets to not

have gaps in the sky coverage. The data were then phase shifted to each of these phase centres and averaged to 4 s and 48.82 kHz time and frequency resolution. These were the final 25 datasets that would be imaged at high resolution. The data for each facet were about 250 GB in total size.

### 1.2.5 Direction dependent calibration of the ILT

Sources that remain bright and compact at 1" resolution have a better chance of remaining compact down to 0.3" compared to a selection at 6". From the intermediate resolution image sources with a peak intensity above 25 mJy/beam were selected as candidate DDE calibrators. These were then split off into separate datasets similarly to the LBCS calibrator, but averaged more in time to a 1 min interval. For the brightest sources this balanced well the signal to noise ratio while still retaining the time resolution to correct most of the residual ionospheric distortions properly. The calibration solutions derived on the LBCS calibrator were preapplied and a similar self-calibration routine as for the LBCS calibrator was subsequently used to self calibrate these sources, outlined below.

First an initial image was made from which a starting model was derived. As the majority of phase-related errors had been corrected on the in-field calibrator, it was assumed that the remaining direction dependent effects were dominated by ionospheric differences. Therefore in the fast iterations DPPP’s tec mode was used to determine dTEC values rather than phases. In the slow iterations, DPPP’s complexgain mode was used to solve for remaining amplitude errors due to the slowly varying station beams. Solution intervals for both solves were calculated automatically based on the intensity detected on 800 km baselines [29].

The tec iterations converged for 44 sources and the complexgain iterations for 41 sources. So-called “jumps”, manifesting as solutions that are offset by a multiple of a specific “jump” value, were present in the tec solutions. These jumps are a consequence of local $\chi^2$ minima in the TEC-fitting [30]. This jump value was subtracted from the offending solution blocks iteratively until no jumps remained. The dTEC values were then spatially interpolated to a smooth screen over RA and DEC. A simple approach to convert discrete directions into
screens would be a Voronoi tessellation based on the
calibrator locations. This would, however, intro-
duce sharp transitions, which would negatively im-
pact IDG imaging later on. Therefore the solutions
were interpolated with a multiquadric radial basis
function instead, using SciPy's `interpolate.Rbf`
function. Interpolation was constrained such that
at the location of a calibrator source the screen so-
lutions matched with the original solutions. Sim-
ilarly the `complexgain` solutions were also inter-
polated to a screen. These two screens were used
to apply direction-dependent calibration solutions
during the final imaging.

1.2.6 Final imaging

Final high-resolution imaging of the data was done
using WSClean in combination with IDG in CPU
mode. Since the IDG algorithm can leverage a
GPU for the gridding calculations, this amount can
likely be significantly reduced [31]. A 22500 × 22500
image of a similar dataset finished four times faster
on a 22-core Intel (R) Xeon(R) Gold 6226 system
with an Nvidia Tesla V100 compared to an identi-
cal system without one. Each of the facets was im-
aged separately with robust −1 weighting, a 0.11"
pixel size and a 22700 × 22700 pixel image size.
Multiscale multi-frequency synthesis clean was used
together with spectral fitting using a second-order
polynomial (through -fit-spectral-pol 3). Dif-
ferential primary beam, dTEC and gain corrections
were all applied on the fly during imaging. The
beam corrections were applied every 10 min, dTEC
solutions were applied every minute and gain solu-
tions every hour. This could be done efficiently
through the use of IDG. The data was imaged us-
ing hardware on the Spider platform at surfSARA
and the ALICE cluster in Leiden. Imaging took
5 – 7 days wall clock time per facet on average
using 24 cores on a node with two 12-core Xeon
Gold 6126 running at 2.6GHz and 384 GB of RAM.
Effective wall clock time was significantly reduced
however due to the availability of many compute
nodes, allowing imaging to run in parallel where
each compute node imaged a facet independently.
From start to finish the entire data reduction proce-
dure consumed approximately 250,000 core hours.

1.3 Source detection and catalogues

After imaging source detection was done using
PyBDSF[14]. Because source detection was done
on each facet separately, there will be duplicate
detections. Therefore the catalogues were first
merged and then cleaned of duplicate detections.
The compact sources and extended sources were
treated separately for this. First we processed the
compact sources, starting with a cross-match to the
6" resolution catalogue of the LOFAR Deep Fields
survey with key optical host properties [11, 25].
Next duplicate detections were removed using an
internal cross-match within a radius of 0.15" of the
fitted high-resolution position (i.e. approximately
within the restoring beam of the high resolution
map). The match closest to its facet centre was
kept, while the others were discarded. Next a size
constraint was introduced. Sources with fitted ma-
jor or minor axes that were larger than 6" were
discarded as poor fits as by definition the unre-
solved sources cannot be larger than the low res-
olution restoring beam. Next, a signal-to-noise ra-
tio cut was made, constraining the peak intensity
to be ≥ 5σ<sub>rms</sub>, where σ<sub>rms</sub> is the rms noise around
the source as measured by PyBDSF. Finally, the
compact and extended source catalogues were com-
bined into a single catalogue and a final duplicate
removal was done. The final catalogue contains
2483 sources.

1.4 Astrometric corrections

As phases hold information about source positions,
the self-calibration process may have shifted the as-
trometry away from a well defined reference. Dur-
ing calibration of the LBCS calibrator we noted
self-calibration had “recentered” the image near
one of the lobes, due to the use of a point-source
starting model, introducing a noticeable offset that
needs correcting. Therefore the final image needs
to be realigned to a reference. As our reference
we used the final calibrated and aligned radio cat-
alogue of Lockman Hole from the LOFAR Deep
Fields survey.

Offsets to align the high resolution image to
the low resolution image were determined for each
facet separately. First, point sources with SNR :=
$I_{\text{peak}}/\sigma_{\text{rms}}^{\text{local}} > 5$ were selected. Secondly an isola-
tion constraint for no other sources to be present
within 30'' was applied. Astrometric offsets were
then determined by the median value of the differ-
ence with the 6'' position for right ascension and
declination, respectively. Supplementary Figure 1
shows the offsets for each facet. There is a roughly
systematic offset due to the aforementioned recen-
tring behaviour during calibration of the LBCS
source. The remaining differences are due to ad-
ditional offsets in those facets, but are small and of
the order of one resolution element or less.

A single offset using only SNR > 25 sources was
also determined as a check. The median values ob-
tained from this agree with the median of all indi-
vidual facets combined. Therefore the facet-based
offsets are adopted to allow for inter-facet varia-
tions.

1.5 Smearing

A variety of effects can degrade the effective reso-
lution in the image, referred to as the “smearing”
of sources. This results in a reduction of peak in-
tensity and artificial broadening of sources. These
effects include time and bandwidth smearing, iono-
spheric disturbances and interstellar or interplan-
etary scattering (ISS/IPS). Time and bandwidth
smearing is an “instrumental” effect that occurs
when averaging the data. It artificially broadens
sources in the image radially (bandwidth smear-
ing) and azimuthally (time smearing). Supplemen-
tary Figure 2 illustrates the smearing losses on
the longest baseline[2]. Ionospheric disturbances
are mostly corrected for in both the direction-
independent and direction-dependent calibration
process, but are difficult to perfectly remove as they
are limited by the density of DD calibrators, avail-
able SNR, solution intervals and source complexity.
The result is similar to atmospheric seeing. Finally
interstellar and interplanetary scattering scramble
astronomical radio signals before they even reach
the ionosphere and hence imprint an inherent, un-
correctable smearing on the data[21]. While the
former can technically be resolved with better cal-
ibration, the latter sets a fundamental limit to the
effective resolution that can be achieved. The com-
bination of ionospheric seeing and interstellar and
-planetary scattering manifests itself as the broad-
ening of sources near the phase centre, where time
and bandwidth smearing is non-existent.

If smearing is not accounted for, peak intensity

and intrinsic size measurements will be off due to
a wrong assumed PSF. Measuring the most com-
pact sources in the centre of the field, where time
and bandwidth smearing is negligible, will indicate
whether broadening from ISS/IPS is an issue for
this observation. The estimated PSF from WS-
Clean is 0.38" × 0.30", while the smallest measured
source in the central facet is 0.39" × 0.31". The ex-
pected broadening from ISS/IPS is ~ 50 masec[4].
We thus conclude that we achieve the theoretical
angular resolution with possible minor broadening
from ISS/IPS.

Broadening from time and bandwidth smearing
will vary across the field. The net effect is a combi-
nation of the local PSF and smearing. We give the
theoretical reduction in peak intensity as a mea-
sure of how severe the smearing is in Supplemen-
tary Figure 2. For accurate deconvolved source
sizes the smearing would need to be taken into ac-
count. A significant reduction in time smearing can
be achieved by processing the data at its archived
1 s time resolution.

1.6 Flux scale corrections

A correction to the flux scale was necessary, as the
reference bandpasses that were used were not tied
to any particular flux density scale. For a reliable
flux scale correction sources with the same flux den-
sity at low and high resolution were needed. A se-
lection was made based on high-significance sources
in the high-resolution image. Compact single-
gaussian sources with SNR > 25 were selected and
the median ratio S<sub>ILT</sub>/S<sub>DLT</sub> was determined. The
selection of bright high-SNR and compact sources
ensures no flux density is lost. Smearing effects are
mitigated by using the flux density, which should
be conserved, instead of peak intensity. The correc-
tion factor was determined to be 1.21 ± 0.19 from
the median flux density ratio. After scaling, the
mean ratio is S<sub>ILT</sub>/S<sub>DLT</sub> = 0.99<sup>+0.26</sup><sub>-0.34</sub> and the me-
dian ratio is S<sub>ILT</sub>/S<sub>DLT</sub> = 1.00<sup>+0.14</sup><sub>-0.18</sub> (0.82 – 1.14).
Both the uncertainty and the values in parentheses
reflect the 16<sup>th</sup> and 84<sup>th</sup> percentiles, respectively.
The distribution of the ratios over all facets after
scaling can be seen in Fig. 1. A small number of
sources is seen to have a ratio significantly smaller
than unity. These are most likely sources that have
extended emission below the detection threshold.
This is a small number of outlying sources however,
and has not affected the scaling in a significant way.

The uncertainty on measured flux densities is compounded by two main effects: an uncertainty in the general flux density scale ($\sigma_{cal}$) and an uncertainty in our scaling factor ($\sigma_{scale}$). For the former we assume a conservative error of 20%[23]. This was added in quadrature with the other uncertainty to arrive at the total uncertainty on the flux density of $\sigma_{total} = \sqrt{\sigma_{cal}^2 + \sigma_{scale}^2} = 30\%$.

## Acknowledgements

We thank the (anonymous) reviewers for their comments. FS would like to thank SURFsara for beta access to the Spider platform. This paper is based (in part) on data obtained with the International LOFAR Telescope (ILT) under project code LT10_012. LOFAR (van Haarlem et al. 2013) is the Low Frequency Array designed and constructed by ASTRON. It has observing, data processing, and data storage facilities in several countries, that are owned by various parties (each with their own funding sources), and that are collectively operated by the ILT foundation under a joint scientific policy. The ILT resources have benefitted from the following recent major funding sources: CNRS-INSU, Observatoire de Paris and Université d'Orléans, France; BMBF, MIWF-NRW, MPG, Germany; Science Foundation Ireland (SFI), Department of Business, Enterprise and Innovation (DBEI), Ireland; NWO, The Netherlands; The Science and Technology Facilities Council, UK[7]. RJvW acknowledges support from the ERC Starting Grant ClusterWeb 804208. This work was supported by the Medical Research Council [grant MR/T042842/1]. This work made use of the Dutch national e-infrastructure with the support of the SURF Cooperative using grant no. EINF-251. PNB is grateful for support from the UK STFC via grants ST/R000972/1 and ST/V000594/1. This work was performed using the compute resources from the Academic Leiden Interdisciplinary Cluster Environment (ALICE) provided by Leiden University. This work also made use of SciPy, NumPy, Matplotlib and TOPCAT.

## Author contribution statements

FS led the paper, reduced the data and produced the images. RJvW developed the self-calibration routine and helped expanding it to international baselines. HJAR helped scope and write the paper. LKM and NJ conducted the LBCS survey and developed the LOFAR-VLBI pipeline which served as the foundation for this work. ARO, SvdT and BV maintain WSClean, IDG and were of great help in fixing problems and adding features to the software. JBRO helped secure resources on SURFsara and provide support for our data reduction on their Spider platform. PNB led the proposal that obtained the data and contributed to editing of the paper. MB is a member of the long-baseline working group and contributed to editing of the paper. TWS contributed towards some of the various data processing pipelines used in this work and produced the deep Lockman Hole 6" image. CT developed the DDFacet software used and produced the deep Lockman Hole 6" image. APT is a member of the long-baseline working group and helped determining the flux scale.

## Competing interest statement

The authors declare no competing interests.

## Data Availability

A source catalogue and the full-resolution images are accessible through the LOFAR Surveys Key Science Project webpage: https://www.lofar-surveys.org/hdfields.html

## Code Availability

The various software and pipelines used in this work are publicly available at https://github.com/lofar-astron/prefactor (PREFACTOR), https://github.com/cyriltasse/DDFacet (DDFacET), https://github.com/lmorabit/lofar-vlbi (LOFAR-VLBI), https://github.com/mhardcastle/ddf-pipeline (DDF-PIPELINE) and https://gitlab.com/astron-idg/idg (IDG). Code from other parts not using these pipelines is
not directly available, because it is not part of a
complete pipeline, but is available upon reasonable
request to the authors.
<figure><img src="image_4.png" /></figure>

Supplementary Figure 1: Astrometric offsets per facet and flux density distributions after setting the absolute flux scale. **Left:** Offsets in arcsecond to align the high-resolution catalogue with the 6'' catalogue. Each square is one facet. The black arrows are to scale with respect to each other and indicate the magnitude and direction of the applied offset. **Top right:** The distribution of flux density ratios of compact SNR > 5 sources, after scaling. The median, 16th and 84th percentiles are indicated by the dotted and dashed gray lines, respectively. The black line indicates unity. **Bottom right:** Flux densities measured with the international array versus those measured with the Dutch array. The black line indicates unity.
<figure><img src="image_5.png" /><figcaption>Supplementary Figure 2: Time and bandwidth smearing losses on the longest baseline (1986 km, $\theta = 0.2''$ at 144 MHz), for the central frequency, as a function of distance to the phase centre. Losses are expressed as the remaining peak intensity $I$ as a fraction of the original peak intensity $I_0$. The dotted line indicates the reduction in peak intensity from bandwidth smearing, the dashed line the reduction from time smearing and the solid line the total reduction from both effects. The top panel shows smearing at the averaging of this dataset, while the bottom panel shows the smearing at the archived averaging parameters.</figcaption></figure>
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