Large-scale 21-cm Cosmology with LOFAR and AARTFAAC Gehlot, Bharat Kumar
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Modelling Degree-Scale Galactic Radio Emission at 122 MHz around the North Celestial Pole with LOFAR-AARTFAAC
B. K. Gehlot, L. V. E. Koopmans, M. Kuiack, A. R. Offringa, S. Yatawatta, F. G. Mertens, V. N. Pandey, and R. A. M. J.
Wijers
In preparation.
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Abstract
Removal of bright diffuse Galactic thermal and non-thermal radio emission is a key challenge in experiments that aim to detect the 21-cm signal of neu- tral hydrogen from the Cosmic Dawn (CD) and Epoch of Reionization (EoR).
Furthermore, it is also crucial to either avoid or include a model of this dif- fuse emission, in direction and direction-independent gain calibration in order to avoid biases due to sky-model incompleteness. In this chapter, we examine very large scale diffuse Galactic emission at 122 MHz, around the North Celes- tial Pole, using the Amsterdam-ASTRON Radio Transient Facility and Anal- ysis Centre (AARTFAAC)- High Band Antenna (HBA) system in A12 mode.
The system cross-correlates the inner 576 HBA-tiles, yielding a ∼25 degree field of view. The resulting wide-field images are the first-ever images pro- duced with the AARTFAAC-HBA system. We demonstrate two methods to model this diffuse emission: multiscale CLEAN and shapelet decomposition.
We find that the multiscale CLEAN is suitable to model the smaller scale struc- tures (on scales smaller than a few degrees) embedded in the diffuse structure, whereas the shapelet decomposition method better models the larger scales which are of the order of a few degrees or more. The angular power spectrum of the emission in the field is dominated by the point sources on scales with ℓ≳ 200, and the diffuse emission dominates on scales with ℓ ≲ 200. The dif- fuse emission has a brightness temperate variance of ∆2ℓ=180= (13.8± 0.6 K)2 at 122 MHz on angular scales of 1 degree, and is consistent with a power-law following Cℓ∝ ℓ−2.0.
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5.1: Introduction 127
5.1 Introduction
Observations of the redshifted 21-cm hyperfine transition line of neutral hy- drogen (HI hereafter) during the Cosmic Dawn (CD; 12≲ z ≲ 30) and Epoch of Reionization (EoR; 6 ≲ z ≲ 12) are the most promising probes of the HI distribution in the Inter-Galactic Medium (IGM) and hence the evolution of large-scale structure in the nascent Universe (Madau, Meiksin & Rees 1997;
Shaver et al. 1999; Furlanetto, Oh & Briggs 2006). Several current and next- generation interferometers, such as the LOw Frequency ARray1(LOFAR; van Haarlem et al. 2013), the Giant Meterwave Radio Telescope2(GMRT; Paciga et al. 2011), the Murchison Widefield Array3(MWA; Tingay et al. 2013; Bow- man et al. 2013), the Precision Array for Probing the Epoch of Reioniza- tion4(PAPER; Parsons et al. 2010), the Hydrogen Epoch of Reionization Ar- ray5 (HERA; DeBoer et al. 2017), NENUFAR6(New Extension in Nançay Upgrading loFAR; Zarka et al. 2012), and the Square Kilometre Array7(SKA;
Mellema et al. 2013; Koopmans et al. 2015) are currently seeking a statisti- cal detection of brightness temperature fluctuations in the cosmological 21-cm signal from the CD and EoR.
The expected 21-cm signal from high redshifts (z = 6− 30) is extremely faint (Furlanetto, Oh & Briggs 2006; Pritchard & Loeb 2012) and is contaminated by bright astrophysical foregrounds, e.g. Galactic diffuse emission compris- ing of synchrotron and free-free emission, and extra-galactic compact sources which comprise of radio-galaxies, supernova remnants and other sources (Di Matteo et al. 2002; Zaldarriaga, Furlanetto & Hernquist 2004; Bernardi et al.
2009, 2010; Ghosh et al. 2012). These foregrounds are several orders of magni- tude brighter than the expected signal. Subtraction of these bright foregrounds poses a significant challenge in all 21-cm signal experiments. It is crucial to remove these bright foregrounds with high accuracy in order to obtain an ac- curate and precise estimate of the power spectrum of the 21-cm signal (Jelić et al. 2008; Harker et al. 2009; Chapman et al. 2013; Bonaldi & Brown 2015;
Mertens, Ghosh & Koopmans 2018). Experiments such as PAPER, MWA and HERA follow a so-called “foreground-avoidance” technique for the separation of the 21-cm signal from the foregrounds. This technique takes advantage of the fact that the smooth foregrounds tend to reside within a “wedge-shaped”
region of the two-dimensional power spectrum, whereas the “EoR-window”,
1 http://www.lofar.org/
2 http://gmrt.ncra.tifr.res.in/
3 http://www.mwatelescope.org/
4 http://eor.berkeley.edu/
5 http://reionization.org/
6 https://nenufar.obs-nancay.fr/
7 http://skatelescope.org/
5
complementary to the “wedge” region of the power spectrum, is dominated by 21-cm signal (Datta, Bowman & Carilli 2010; Morales et al. 2012; Vedan- tham, Udaya Shankar & Subrahmanyan 2012; Trott, Wayth & Tingay 2012;
Parsons et al. 2012; Hazelton, Morales & Sullivan 2013; Dillon et al. 2014).
The foreground-avoidance technique has several limitations, such as access to only a limited number of 21-cm signal wave modes for power spectrum estima- tion and the potential contamination of the “EoR-window” by signal leakage of the foregrounds, caused by effects which are non-smooth in frequency e.g.
polarization leakage (Asad et al. 2015, 2016, 2018; Nunhokee et al. 2017), systematic biases (Patil et al. 2016; Ewall-Wice et al. 2016), cable reflections (Beardsley et al. 2016) and multi-path propagation effects (Neben et al. 2015).
On the other hand, the LOFAR-EoR project and also the planned SKA fol- low a foreground-removal approach in their analysis strategy (Patil et al. 2017;
Gehlot et al. 2018b). To achieve this, various image deconvolution e.g. CLEAN (Högbom 1974; Clark 1980; Cornwell 2008; Offringa & Smirnov 2017) and source extraction algorithms e.g. Duchamp, PyBDSF (Whiting 2012; Mohan
& Rafferty 2015) allow us to model the extra-galactic compact sources as delta functions and Gaussians. These sources can be removed from the ob- served signal using Direction Dependent (DD) calibration (see e.g. Yatawatta et al. 2013; Patil et al. 2017; Gehlot et al. 2018b). However, on short inter- ferometer baselines, the large-scale diffuse emission, extended source and the sources below confusion noise which dominate the observed signal are much more difficult to model with these traditional methods that have largely been developed for longer-baseline high brightness-temperature data. Several para- metric (Jelić et al. 2008; Bonaldi & Brown 2015) and non-parametric (Harker et al. 2009; Chapman et al. 2013) blind foreground removal techniques exploit the frequency smoothness of the foregrounds to remove them from the ob- served signal. The Gaussian Process Regression (GPR) technique (Mertens, Ghosh & Koopmans 2018) uses Gaussian processes to very efficiently model the various foreground components and is capable of removing the smooth intrinsic foregrounds, as well as instrumental mode-mixing component caused by the chromatic response of the instrument and additional side-lobe noise due to calibration errors, from the data.
Besides the intrinsic interest of studying this diffuse largely Galactic emission, it is important to include the diffuse emission component in the instrumen- tal gain calibration step as well. Patil et al. (2016), Barry et al. (2016) and Ewall-Wice et al. (2017) showed that using an incomplete sky-model in cali- bration step leads to significant suppression of the diffuse emission as well as the 21-cm signal of interest on shorter baselines that are dominated by dif- fuse emission (Mouri Sardarabadi & Koopmans 2018). Since shorter baselines provide most of the sensitivity in the 21-cm signal power spectrum, it is im-
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5.2: Observations and preprocessing 129
portant to calibrate these baselines accurately to avoid any suppression of the signal. One option to mitigate this effect is to remove these short baselines from calibration and include only longer baselines where the diffuse emission is resolved out. This approach was utilized in Patil et al. (2017). However, exclusion of short baselines from calibration leads to a so-called “excess-noise”
on the excluded baselines in calibration (Patil et al. 2016; Barry et al. 2016;
Ewall-Wice et al. 2017; Gehlot et al. 2018a,b), impacting the power spectrum estimation. Therefore, it is also crucial to include diffuse emission in the cal- ibration step to avoid suppression and “excess-noise”, which is a non-trivial task (Mouri Sardarabadi & Koopmans 2018).
In this chapter, we explore various methods to model the diffuse emission in an extended region around the North Celestial Pole (NCP) using High Band Antenna (HBA) observations of the LOFAR Amsterdam-ASTRON Ra- dio Transients Facility And Analysis Centre (AARTFAAC) wide-field imager (Prasad et al. 2016) at 122 MHz. These models can be utilised in the cali- bration process of the LOFAR-EoR project in order to further improve the current calibration sky-model. They can also be used to remove these fore- grounds from the data, before direction-dependent gain calibration, in order to reduce signal suppression and gain bias.
The chapter is organised as follows: Section 5.2 briefly describes the observa- tional setup and preprocessing steps. Section 5.3 describes the steps involved in the calibration and imaging scheme. The two diffuse foreground modelling methods and their comparison are presented in Section 5.4. Section 5.6 sum- marises the main results of the chapter and future work.
5.2 Observations and preprocessing
We use the AARTFAAC-HBA wide-field imager to observe an extended re- gion of ∼20-degree radius around the NCP8 in the frequency range of 114− 126 MHz, which is the primary observation window of the LOFAR-EoR KSP (Bernardi et al. 2010; Yatawatta et al. 2013; Patil et al. 2017). The obser- vational setup and the preprocessing steps such as applying phase-tracking, flagging and averaging are discussed briefly in the following subsections.
5.2.1 The AARTFAAC wide-field Imager
The Amsterdam-ASTRON Radio Transient Facility and Analysis Centre (AART- FAAC) is a LOFAR based all-sky radio transient monitor (Prasad & Wijnholds 2012; Kuiack et al. 2018). It piggybacks on ongoing LOFAR observations and
8 AARTFAAC usually observes in drift-scan mode. The phase tracking can be applied during data preprocessing though.
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Table 5.1 – Observational details of the data.
Parameter value
Telescope LOFAR-AARTFAAC
Antenna configuration A12
Number of receivers 576 (HBA tiles)
Observation start times (UTC) December 15, 2017; 18:30:00 December 23, 2017; 18:30:00 December 25, 2017; 18:30:00 December 28, 2017; 18:30:00
Number of pointings 1
Phase centre (α, δ; J2000): 00h00m00s, +90◦00′00′′
Duration of observation 11 hours
Minimum frequency 114.06 MHz
Maximum frequency 126.37 MHz
Target bandwidth 12.4 MHz
Primary Beam FWHM 28◦ at 120 MHz
Field of View 640 deg2 at 120 MHz
Polarization Linear X-Y
Time, frequency resolution:
Raw Data 2 s, 195.3 kHz
After flagging 12 s, 195.3 kHz
taps the digital signal streams of individual antenna elements from six or twelve core stations, depending on user requirements. AARTFAAC operates in two modes viz. A6 where the six innermost stations (also called the “supert- erp”) of the LOFAR core are used and A12 where the twelve central stations of the LOFAR core are used. The A6 mode consists of 288 dual-polarization receivers (e.g. Low Band Antenna (LBA) dipoles or HBA tiles) within a 300 m diameter circle, and the A12 mode consists of 576 such receivers spread across 1.2 km diameter. The inner region of the uv-plane is densely sampled, and the array is co-planar at the centimetre level within 0− 300 m, which is optimal for wide-field imaging. In addition to this, the baselines up to 1.2 km support intermediate resolution imaging which helps to improve calibra- tion and better capture compact structure in the sky. Each of the twelve LOFAR core stations consist of 96 LBA dipoles 9 (only 48 out of 96 dipoles can be used at a time) and 48 HBA tiles10. The digitized signals from these
9 LBA dipoles are dual-polarization (X-Y) dipoles optimized to operate between 30-80 MHz
10 HBA tiles consist of 16 dual-polarization dipoles arranged in a 4× 4 grid, which are analogue beam-formed to produce a single tile beam. HBA tiles are optimized to operate between 110-240 MHz
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5.2: Observations and preprocessing 131
antenna elements are tapped and transported to the AARTFAAC correlator prior to beam-forming. Due to network limitations, only 16 sub-bands can be correlated in the 16-bit mode, being limited by the current network capacity.
Each sub-band is 195.3 kHz wide and consists of up to 64 channels provid- ing a maximum frequency resolution of 3 kHz, with an instantaneous system bandwidth of 3.1 MHz. The correlator subsystem located at CIT in Gronin- gen (the Netherlands) is a GPU based correlator, which produces correlations (XX, XY, YX, YY) for all receiver pairs for every frequency channel with 1-second integration. The correlator has 1152 input streams with 576 signal streams per polarization. The output correlations can either be dumped as raw correlations on the AARTFAAC storage/compute cluster or can be routed to the AARTFAAC real-time calibration and imaging pipeline for transient de- tection. AARTFAAC can only observe in a drift-scan mode. However, phase tracking can be applied to the raw data during or after preprocessing. Readers may refer to Prasad et al. (2016) for detailed information about AARTFAAC system design and capabilities, and van Haarlem et al. (2013) for observing capabilities of LOFAR.
5.2.2 Observational setup
We use the AARTFAAC-HBA system in A12 mode (A12-HBA hereafter) to observe the field around the NCP. For our observations, we target the 114- 126 MHz frequency range which corresponds to the redshift range of z = 10.2− 11.4. Due to the currently limited bandwidth of 3.1 MHz, we adopt an observation strategy where we combine four different observations recorded within a span of a few weeks to cover the 12 MHz band fully. For each observa- tion, the sub-bands are sparsely spread over the 12 MHz band in a frequency- comb configuration but covering interlacing and non-overlapping sets of fre- quency channels for each observation. Figure 5.1 shows a schematic of the frequency-comb configuration that we adopted. Each observation is about 11 hours long and has a time and frequency resolution of 2 s and 195.3 kHz (one channel per sub-band). Although higher time and frequency resolution are preferred, the limited storage capacity of the AARTFAAC storage/compute cluster restricts the resolution to lower values for longer duration observations.
The raw data is converted to standard Measurement Set (MS) format using aartfaac2ms11 which also applies offline phase tracking. We phase the raw visibilities to the NCP before preprocessing. The observational details of the data are summarised in table 5.1.
11 A custom software package designed by André Offringa; https://github.com/
aroffringa/aartfaac2ms
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Figure 5.1 – Schematic of the frequency comb configuration utilised for the observa- tions. Each colour represents a different observation. The vertical blocks correspond to sub-bands spanning the bandwidth.
5.2.3 Data preprocessing
The preprocessing steps include RFI-excision and averaging raw visibilities.
We use the AOFlagger software (Offringa et al. 2010; Offringa, van de Gronde
& Roerdink 2012) to flag RFI corrupted data. The RFI-excision is performed on highest available resolution (2 seconds, 195 kHz) to minimize information loss. We also flagged all the visibilities corresponding to non-working tiles in the datasets. The remaining data were averaged to a lower time resolution of 12 seconds. The data volume of an 11-hour observation after averaging is∼ 1.7 TeraBytes. Unfortunately, we observed that 5 out of 6 stations outside the
“superterp” strangely had much lower visibility amplitudes in all observations.
Furthermore, the visibilities corresponding to these stations showed extremely erratic behaviour even after the calibration step. Therefore, we also flagged the tiles corresponding to these five stations before proceeding with the calibration step.
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5.3: Calibration and imaging scheme 133
5.3 Calibration and imaging scheme
The visibilities that are measured by a radio interferometer are corrupted by errors due to instrumental imperfections and environmental effects. These corruption effects are classified broadly in two categories, viz. Direction Inde- pendent (DI) effects such as complex receiver gains, global band-pass and a global ionospheric phase, and Direction Dependent (DD) effects which change with the direction of the incoming electromagnetic signals due to the antenna voltage patterns, ionospheric phase fluctuations and Faraday rotation. Cal- ibration refers to the estimation of these gain errors in order to obtain a reliable estimate of the true visibilities. We perform gain calibration, in a self-calibration manner, using the following steps:
1. We first remove the bright sources Cas A and Cyg A, which dominate the visibilites on short baselines and superpose significant PSF side-lobes over the field. We subtract these sources with DD-calibration using SAGECal-CO 12 (Yatawatta 2015, 2016; Yatawatta, Diblen & Spreeuw 2017; Yatawatta 2018), which regulates frequency smoothness as a con- straint in the gain calibration in order to improve calibration quality.
We use the Cas A and Cyg A shapelet model adopted from the LOFAR- EoR calibration sky-model to subtract these sources using their inferred directional gains. We use a solution interval of 2 minutes and 5 ADMM iterations with a regularization factor ρ = 5 (Yatawatta 2016). We re- move the baselines|u| < 10λ from the calibration to avoid the large-scale diffuse emission biasing the calibration.
2. We perform DI-calibration on the resulting visibilities after step 1. The calibration model consists of six sources, viz. 3C061.1 (38 Jy at 151.5 MHz; Baldwin et al. 1985), 3C220.3 (38 Jy at 150 MHz; Cohen et al.
2007), LQAC 244+085 001 (6.2 Jy at 151.5 MHz; Baldwin et al. 1985), NVSS J011045+873822 (5.1 Jy at 151.5 MHz; Baldwin et al. 1985), NVSS J190350+853648 (4.9 Jy at 151.5 MHz; Baldwin et al. 1985) and NVSS J062205+871948 (4.9 Jy at 151.5 MHz; Baldwin et al. 1985). All sources, except for 3C061.1, are represented by delta functions, and their corresponding source spectra are represented by a power law with an as- sumed, but typical, spectral index of −0.8. The radio galaxy 3C061.1 has a complicated morphology, and its model is adapted from the intrin- sic sky-model used in the LOFAR-EoR calibration pipeline (Patil et al.
2017). The 3C061.1 model uses delta functions and shapelets to repre- sent the source, and a third order log-polynomial represents the spec- trum. We use a calibration solution interval of 2 minutes to maintain
12 http://sagecal.sourceforge.net/
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Figure 5.2 – The left panel shows the Stokes I dirty image of a ∼20 degree radius field around the NCP using a single sub-band (122 MHz) of A12-HBA data and the baselines in the range u = 0−500λ). The right panel shows the Stokes I image for the same field with same imaging parameters and baseline range, but using LOFAR-HBA station beam-formed data. The dotted circles represent different declinations, with a spacing of 5◦ between them.
a significant Signal-to-Noise ratio per solution and carry out 5 ADMM iterations with a regularization parameter of ρ = 5 (Yatawatta 2016).
We also remove the |u| < 10λ baselines. We do not use a beam model during calibration, and the flux scale of the visibilities post-calibration is on an apparent scale (i.e. uncorrected for the average beam). Currently, the primary beam model for AARTFAAC-HBA tile does not exist, and efforts are ongoing to utilize the present LOFAR-HBA tile beam model for AARTFAAC-HBA calibration. Therefore, to avoid notable differen- tial primary beam effects, all six sources chosen for the gain calibration reside within a∼ 7◦ radius around the NCP.
3. We deconvolve (i.e. Clean) and image the visibilities using WSClean (Of- fringa et al. 2014; Offringa & Smirnov 2017), using a cleaning threshold of 0.7σ, with a ‘Briggs –0.1’ weighting-scheme and baselines |u| > 50λ in order to avoid any residual diffuse emission affecting the Cleaning process.
4. The steps 2 and 3 were repeated in a ”self-cal” loop using the improved calibration model (updated with the clean components) obtained after each iteration. The final Clean model contains∼ 5000 components (delta functions and Gaussians) to represent the compact sources in the field.
5. We finally image the DI-calibrated visibilities obtained after step 4 with
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5.4: Modelling the Diffuse Galactic Emission 135 WSClean using a ‘Briggs +0.5’ weighting scheme in all subsequent anal- yses.
Note that we only use a single sub-band at 122.06 MHz throughout the analy- sis. We are working on expanding the analysis to the full frequency range and include Stokes Q and U imaging, but currently, we leave that analysis for the future. Figure 5.2 shows single sub-band (at 122.06 MHz) Stokes I dirty images of the NCP field produced using A12-HBA (left panel) as well as LOFAR-HBA data (right panel). The A12-HBA system has a∼ 25 times larger field of view compared to the LOFAR-HBA system. Galactic large-scale diffuse emission is visible in the A12-HBA image. Although the diffuse emission is also partly present in LOFAR-HBA, it is not visible because LOFAR-HBA has a much lower short-baseline density and a total absence of baselines with |u| < 70 m compared to A12-HBA.
5.4 Modelling the Diffuse Galactic Emission
In this section, we present two methods to model the compact and diffuse structure around the NCP as seen in figure 5.2. In the first method, we use multiscale CLEAN deconvolution with WSClean (Offringa & Smirnov 2017) to model the diffuse structure with delta functions and Gaussians. In the second method, we use orthonormal basis functions (shapelets) (Yatawatta 2011) to model the diffuse structure. We use a single sub-band at 122.06 MHz for the analysis throughout the chapter.
5.4.1 Removing compact sources
Before proceeding with describing the diffuse foreground modelling, the com- pact sources need to be removed from the map. We use the CLEAN compo- nent model obtained during the gain calibration (see step 4 in Section 5.3) to subtract 5000 components with (apparent) fluxes greater than 20 mJy from the data. Figure 5.3 presents a single sub-band Stokes I dirty image of the NCP field at 122 MHz before and after subtraction of the CLEAN model. The images are produced using ’Briggs –0.1’ weighting scheme with baselines that have u > 50λ. We see that subtracting the CLEAN model removes most bright sources, leaving only faint residuals of the order of a few hundred mJy rms. The images do not reveal signatures of diffuse emission because of the baseline cut used during the imaging process. We also see that the subtrac- tion of some of the brightest sources leaves holes or ring-like artifacts near the source location. Factors such as imperfect calibration and imperfect source modelling are the likely cause. These artifacts can be further mitigated by using DD-calibration to subtract the bright sources. We plan to employ the
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Figure 5.3 – The left panel shows a single sub-band Stokes I dirty image of the NCP field at 122 MHz, produced using a ’Briggs –0.1’ weighting scheme with baselines that have u > 50λ. The right panel shows the Stokes I residuals after subtraction of the 5000 component CLEAN model from the data. The residuals are consistent with the confusion noise.
DD-calibration for the subtraction of bright sources using several directions in future analyses. For the purpose of this work, i.e. analysing the properties of the diffuse foregrounds, these minor artifacts are not of much importance. We derive an approximate estimate of the confusion noise for AARTFAAC from the classical confusion limit (σc) using the relation from (van Haarlem et al.
2013):
σc= 30 ( θ
1′′
)1.54( ν 74 MHz
)−0.7
[µJy beam−1], (5.1) where θ is the angular resolution (FWHM), and ν is the observation frequency.
A12-HBA has an angular resolution of∼ 7′ at 122 MHz, yielding a confusion limit of σc ∼ 230 mJy beam−1. The standard deviation of the residuals, af- ter cleaning, for the inner 8◦ region of the field is ∼ 200 mJy. These two values are consistent with each other assuming that the primary beam cor- rection is small for this inner 8◦ region. In the future, we plan a deeper multi-frequency CLEAN by combining the four frequency-combs along with an improved direction-dependent calibration step which includes an HBA-tile beam model to improve the sky-model further.
Figure 5.4 shows the Stokes I image at 122 MHz of the residual visibilities after 5000 CLEAN model components have been subtracted. These short baselines are dominated by large-scale diffuse emission. We observe several negative ’holes’ at the locations of brightest sources, which could be due to
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5.4: Modelling the Diffuse Galactic Emission 137
Figure 5.4 – The Stokes I dirty image at 122 MHz of the residual visibilities after the subtraction of 5000 CLEAN model components. The image was produced using a
’Briggs +0.5’ weighting scheme (which gives more weight to short baselines to better visualize the diffuse emission), a baseline range of u = 0−120λ, and a Gaussian taper of 30′.
imperfect gain calibration and the imperfect CLEAN model corresponding to those sources. We now model this residual diffuse emission using the two methods mentioned above.
5.4.2 Modeling with multiscale CLEAN
We use multiscale deconvolution algorithm implemented in WSClean (Offringa
& Smirnov 2017) to model the diffuse structure observed in figure 5.4. We perform the deconvolution using Isotropic Undecimated Wavelet Transform (IUWT; Starck, Fadili & Murtagh 2007) with following settings: ‘Briggs +0.5’
weighting scheme, auto-mask of 2σ, cleaning threshold of 0.5σ, and a multi- scale scale bias of 0.7 (which determines the balance between cleaning of large and small scales). The ’Briggs’ weighting scheme with threshold greater than zero gives more weight to short baselines and better visualizes the diffuse emis- sion. Figure 5.5 (top row) shows the Stokes I image of the field before and after subtracting the diffuse foreground model obtained from multiscale CLEAN.
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Figure 5.5 – Top row: The left panel shows Stokes I dirty image before subtraction of the diffuse emission. The middle panel shows the diffuse model obtained with multiscale CLEAN. The right panel shows the residual Stokes I image after subtrac- tion of the CLEAN model. These images were produced using ’Briggs +0.5’ and u = 0− 120λ baselines with a Gaussian taper of 30′. Bottom row: The left panel shows Stokes I dirty image before subtraction of the diffuse emission. The middle panel shows the diffuse model obtained with shapelet decomposition. The right panel shows the residual Stokes I image after subtraction of the shapelet model. These images were produced using the same settings as in figure 5.5.
We observe that there is still a considerable amount of diffuse flux left in resid- uals after the subtraction of the diffuse foreground model. Upon inspection of the CLEAN component model, we observe that multiscale CLEAN is more sensitive to smaller scales, it does model the intermediate scales but leaves the fluctuations on the largest scales unmodeled. The traditional deconvolution methods based on CLEAN use delta functions and Gaussians to represent the compact and extended sources. Although multiscale CLEAN is able to model extended emission at different scales using Gaussians, it is sub-optimal for modelling the larger scales of the order of a few degrees.
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5.4: Modelling the Diffuse Galactic Emission 139
5.4.3 Modelling with Shapelets
Yatawatta (2011) showed that the use of orthonormal basis functions e.g.
Cartesian shapelets (Refregier 2003) or Prolate Spheroidal Wave Functions (PWSF) (Slepian & Pollak 1961; Landau & Pollak 1961) in radio interferomet- ric image deconvolution, provides improved image fidelity and larger dynamic range. In the image domain, (l, m), shapelet basis functions can be written as (Yatawatta 2011):
ϕn1,n2(l, m, β) = 1
2n1+n2πβ2n1! n2!Hn1(l/β) Hn2(m/β)
× exp [−(l2+ m2)/2β2], (5.2) where the functions Hn1 and Hn2 are the Hermite polynomials of order n1and n2 being integer values, the value of β is the model scale factor. We demon- strate the use of Cartesian shapelets to model the diffuse structure observed in A12-HBA data using the Shapelet_gui13tool. We use 25×25 basis functions for the shapelet decomposition using L1-norm regularization. We optimize the model scale during the process. Figure 5.5 (bottom row) presents the Stokes I diffuse-emission images before and after shapelet model subtraction, as well as the shapelet model obtained from the decomposition. We note that the shapelet model nicely captures the large-scale diffuse emission within the pri- mary beam. The two negative-flux holes in the model were masked during the decomposition. The residuals within the primary beam appear similar to the residuals after subtraction of the 5000 component CLEAN model, as shown in figure 5.3. These residual are most likely the unmodeled compact sources and sources below the confusion noise. However, the diffuse structure outside the primary beam is rather poorly modelled. This is probably due to the finite support of the basis functions, and it does not capture the structure outside the primary beam very well. A possible approach to mitigate this effect is to increase the number of basis functions for decomposition.
5.4.4 Comparing the two diffuse emission modelling methods We see that although multiscale CLEAN captures compact sources and diffuse structure at intermediate scales, it is unable to model the emission on scales larger than a few degrees. The shapelet decomposition of the diffuse emission, on the other hand, captures the emission on large-scales much better, but it is unable to model the fluctuations on scales smaller than a degree. This is evident from the sharpness of the details in the two models shown in figure 5.5.
From this qualitative comparison of two methods, we infer that the shapelet
13 designed by Sarod Yatawatta. See LOFAR imaging cookbook for more details: https:
//support.astron.nl/LOFARImagingCookbook/
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Figure 5.6 – Angular power spectrum (ℓ(ℓ + 1)Cℓ/2π) of the A12-HBA Stokes I images. The red curve correspond to the data before subtraction of the 5000 compo- nent CLEAN model. The solid black curve corresponds to the residuals after CLEAN model subtraction. The solid blue curve corresponds to the CLEAN model. The dashed blue and black curves represent the best fit profiles fmodel(ℓ) and fresidual(ℓ) of the CLEAN model and the residuals respectively. The errorbars correspond to the 2σ errors due to sample variance.
decomposition is the more optimal method to model the large-scale (order of several degrees) diffuse Galactic emission, whereas a multiscale CLEAN deconvolution captures diffuse emission on smaller scales, but produces sub- optimal results on large-scales. For the future, we plan a hybrid approach to build the sky model by combining the two methods, where the CLEAN model is used to capture small-scale information and shapelet decomposition is used to represent the large-scale diffuse structures. We are currently working on producing a more complete model of diffuse emission consisting of information on the broad spectrum of spatial scales.
5.5 The Angular power spectrum
The angular power spectrum is a commonly used tool to study the spatial properties of the foreground emission. We use the angular power spectrum, Cℓ, to describe the observed diffuse emission in the NCP field in previous
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5.5: The Angular power spectrum 141
Table 5.2 – Best-fit Parameters Fit parameters Model Residuals
A 1744.91± 101.83 174.28 ± 14.97
B – 26.36± 10.13
β 1.89± 0.22 2.84± 0.39
sections. The angular power spectrum Cℓ is defined as (see e.g., Seljak 1997;
Bernardi et al. 2009):
Cℓ= 1 Nℓ
∑
l
I(l) ˜˜ I∗(l), (5.3)
where ˜I(l) is the spatial Fourier transform of an image I (in units of Kelvins), ℓ is the multipole moment and ℓ = 2π|u| = 180/Θ, where Θ is the angular scale in degrees (Half Width Half Maximum). Nℓ is the number of samples in an azimuthal bin of size ∆ℓ = 8 which corresponds to ∆|u| = 1.27λ or 45◦ FoV. We use the Stokes-I images to estimate Cℓ. The images are Fourier transformed along the spatial axes using a FFT. We normalize the Fourier transform of the images with the PSF to remove the effect of the weighting scheme applied during imaging. Note that we do not apply a beam correction during the estimation of Cℓ, nor correct for curvature of the sky.
Figure 5.6 presents the angular power spectrum ∆2ℓ ≡ ℓ(ℓ + 1)Cℓ/2π of the Stokes-I images of the data before and after subtraction of the 5000 compo- nents of the CLEAN model. We observe that the power on smaller scales (large ℓ values) is dominated by compact sources, which are largely removed after subtraction of the CLEAN model. However, the power on ℓ < 60 remains intact after model subtraction. Moreover, the angular power spectrum of the residuals is flat over a wide range of multipole moments, ranging from ℓ = 20 to 300, which corresponds to angular scales of Θ = 0.5−10◦. To further quan- tify the model and residual power spectra, we fit the corresponding angular power spectra respectively with the following functions:
Model : ∆2ℓ = [ℓ(ℓ + 1)Cℓ/2π] (ℓ) = A ( ℓ
ℓ0 )β
,
Residual : ∆2ℓ = [ℓ(ℓ + 1)Cℓ/2π] (ℓ) = A + B ( ℓ
ℓ0
)β
, (5.4)
where A, B, and β are the free parameters. We choose ℓ0 = 180 which correspond to an angular scale Θ = 1◦. The best-fit parameter values of are listed in Table 5.2. The data and the model show power-law behaviour on ℓ≳ 100 and follow the best fit obtained using equation 5.4, which is consistent with the angular power spectrum of unresolved and unclustered sources (∆2ℓ ∝
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Figure 5.7 – The angular power spectra (ℓ(ℓ + 1)Cℓ/2π) of the data (red curve), model (blue curve) and residual (black curve) images as shown in figure 5.5. The left panel corresponds to the multiscale-CLEAN method. The right panel corresponds to the shapelet-decomposition method.
ℓ2) as the best fit power-law index obtained for the CLEAN model is β = 1.89± 0.22. This suggests that the power on baselines u ≳ 16λ or angular scales of Θ ≲ 1.8◦ is mostly dominated by compact sources. The residual power spectrum (∆2ℓ) is flat, suggesting that Cℓ ∝ ℓ−2. This is consistent with the angular power spectrum of the Galactic diffuse emission, Cℓ ∝ ℓ−2.2 observed by Bernardi et al. (2009). However, the power on ℓ > 200 increases which is possibly dominated by the noise. As a reference point, the diffuse emission has a brightness temperate variance of ∆2ℓ=180 = (13.8± 0.6)2K2 at 122 MHz on angular scales of 1 degree, and is consistent with a power-law following Cℓ∝ ℓ−2.0.
5.5.1 Power spectra from diffuse foreground modelling methods The angular power spectrum is also a useful tool to compare the two fore- ground modelling methods, i.e., multiscale CLEAN and shapelet decomposi- tion, we demonstrated in Section 5.4. We use the images shown in Figure 5.5 for the two methods to determine the corresponding angular power spectra ∆2ℓ. We use the same methodology and settings as mentioned above to determine Cℓ. Figure 5.7 shows the angular power spectra ∆2ℓ for the two methods. We observe that the multiscale CLEAN method removes power mostly on ℓ > 30 modes, which corresponds to angular scales of Θ≲ 6◦. The residual spectrum is still flat in shape suggesting that there is still a significant amount of un- modeled diffuse structure present in the residuals, which we also observed in section 5.4.2. On the contrary, the shapelet decomposition removes significant power on ℓ < 100 modes which correspond to Θ≳ 2◦. However, it is unable to
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5.6: Summary and Future work 143
remove power on ℓ > 100 modes. The residuals on these modes behave simi- larly to the angular power spectrum of unresolved sources, which is possibly due to unmodeled sources or sources below the confusion level. This effect is evident from the similarity of the residuals after shapelet decomposition with the residuals shown in figure 5.3.
5.6 Summary and Future work
In this chapter, we presented the first ever wide-field images obtained with the LOFAR AARTFAAC-HBA system (in A12 mode). In particular, we find strong degree-scale diffuse Galactic Stokes I radio emission at 122 MHz within a ∼20-degree radius field around the North Celestial Pole, which is one of the two primary windows of the LOFAR EoR project (Yatawatta 2013; Patil et al. 2017). We have compared two different methods for the modelling of this diffuse emission viz. multiscale CLEAN deconvolution and shapelet decomposition. We use angular power spectrum to quantify the behaviour of different foreground components viz. point sources and diffuse emission. The main results of this chapter are summarised as follows:
• Stokes I radio emission, as seen by the LOFAR AARTFAAC-HBA sys- tem around the NCP at 122 MHz on baselines that are shorter than 120 wavelengths, is fully dominated by large-scale diffuse emission. The an- gular power spectrum of the emission in the field is dominated by the point sources on scales smaller than a degree (ℓ ≳ 200). The residuals after subtraction of point sources are dominated by diffuse emission on scales larger than a degree (ℓ≲ 200). This diffuse emission can have a considerable impact on the calibration of any radio-interferometric in- strument (e.g., LOFAR, MWA, PAPER, HERA, LEDA and SKA) if this emission is not part of sky model during instrumental gain calibration (or filtered out before calibration).
• The diffuse emission has a brightness temperature variance of ∆2ℓ=180= (13.8± 0.6)2K2 at 122 MHz on angular scales of 1 degree, and is consis- tent with a power-law following ∆2ℓ ∝ ℓ−2.0.
• We show that multiscale CLEAN can model the small and intermediate scales, but leaves the large-scales of the order of a few degrees largely unmodeled. Other the other hand, a shapelet decomposition nicely cap- tures the large-scale diffuse emission but it is incapable of modelling the emission on scales smaller than several degrees. We find that a hybrid method is needed to capture both scales properly.
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5.6.1 Future Work
The analysis in this Chapter is admittedly based only on data from a sin- gle sub-band at 122 MHz. Despite this simplification, for the first time using AARTFAAC-HBA as a wide-field imaging instrument, we have been able to convincingly show in this pilot analysis that large-scale diffuse emission dom- inates the Galactic (mostly synchrotron) emission on degree scales in Stokes I.
To further improve the imaging and modelling of this emission, we are cur- rently working on including an HBA-tile beam model in the gain calibration.
The inclusion of this beam model will allow us to (i) set an absolute flux scale, (ii) improve the calibration quality, (iii) perform deeper deconvolutions dur- ing imaging and (iv) improve the modelling of compact sources and diffuse emission. We are also planning to expand this analysis to the full frequency range in order to include the spectral information over the 114-126 MHz band, as well as analyse diffuse polarized emission in Stokes Q and U . Wide-field polarization studies will enable us to understand the behaviour and morphol- ogy of polarized foregrounds on very large scales and how it instrumentally leaks to Stokes I (Asad et al. 2015, 2016, 2018; Nunhokee et al. 2017). Finally, we plan to include the diffuse emission model in the sky-model as part of the LOFAR-EoR calibration pipeline.
