Direct detection of the Epoch of Reionization via the redshifted 21-cm line
will have unprecedented implications on the study of structure formation in the
early Universe. To fulfill this promise current and future 21-cm experiments
will need to detect the weak 21-cm signal over foregrounds several order of
magnitude greater. This requires accurate modeling of the galactic and
extragalactic emission and of its contaminants due to instrument chromaticity,
ionosphere and imperfect calibration. To solve for this complex modeling, we
propose a new method based on Gaussian Process Regression (GPR) which is able
to cleanly separate the cosmological signal from most of the foregrounds
contaminants. We also propose a new imaging method based on a maximum
likelihood framework which solves for the interferometric equation directly on
the sphere. Using this method, chromatic effects causing the so-called "wedge"
are effectively eliminated (i.e. deconvolved) in the cylindrical ($k_{\perp},
k_{\parallel}$) power spectrum.Comment: Subbmited to the Proceedings of the IAUS333, Peering Towards Cosmic
  Dawn, 4 pages, 2 figure

Ghosh, A.

Koopmans, L. V. E.

Mertens, F.

English

arXiv

Direct detection of the Epoch of Reionization via the redshifted 21-cm line will have unprecedented implications on the study of structure formation in the early Universe. To fulfill this promise current and future 21-cm experiments will need to detect the weak 21-cm signal over foregrounds several order of magnitude greater. This requires accurate modeling of the galactic and extragalactic emission and of its contaminants due to instrument chromaticity, ionosphere and imperfect calibration. To solve for this complex modeling, we propose a new method based on Gaussian Process Regression (GPR) which is able to cleanly separate the cosmological signal from most of the foregrounds contaminants. We also propose a new imaging method based on a maximum likelihood framework which solves for the interferometric equation directly on the sphere. Using this method, chromatic effects causing the so-called “wedge” are effectively eliminated (i.e. deconvolved) in the cylindrical (k⊥, k∥) power spectrum.<br/

Koopmans, L.~V.~E.

ARTS repository - University of Groningen

   University of GroningenRobust Foregrounds Removal for 21-cm ExperimentsMertens, F.; Ghosh, A.; Koopmans, L.~V.~E.Published in:Peering towards Cosmic DawnDOI:10.1017/S1743921318000546IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2018Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Mertens, F., Ghosh, A., & Koopmans, L. V. E. (2018). Robust Foregrounds Removal for 21-cmExperiments. In V. Jeli, & T. van der Hulst (Eds.), Peering towards Cosmic Dawn (pp. 284-287). (Peeringtowards Cosmic Dawn; Vol. 333). International Astronomical Union.https://doi.org/10.1017/S1743921318000546CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 11-09-2023Peering towards Cosmic DawnProceedings IAU Symposium No. 333, 2018V. Jelić & T. van der Hulst, eds.c© International Astronomical Union 2018doi:10.1017/S1743921318000546Robust Foregrounds Removalfor 21-cm ExperimentsF. Mertens1, A. Ghosh2,3 and L. V. E. Koopmans11Kapteyn Astronomical Institute, University of Groningen,P. O. Box 800, 9700 AV Groningen, The Netherlands2Department of Physics and Astronomy, University of the Western Cape,Robert Sobukwe Road, Bellville 7535, South Africa3Square Kilometre Array radio telescope (SKA) South Africa,The Park, Park Road, Cape Town 7405, South AfricaAbstract. Direct detection of the Epoch of Reionization via the redshifted 21-cm line willhave unprecedented implications on the study of structure formation in the early Universe. Tofulfill this promise current and future 21-cm experiments will need to detect the weak 21-cmsignal over foregrounds several order of magnitude greater. This requires accurate modeling ofthe galactic and extragalactic emission and of its contaminants due to instrument chromaticity,ionosphere and imperfect calibration. To solve for this complex modeling, we propose a newmethod based on Gaussian Process Regression (GPR) which is able to cleanly separate thecosmological signal from most of the foregrounds contaminants. We also propose a new imagingmethod based on a maximum likelihood framework which solves for the interferometric equationdirectly on the sphere. Using this method, chromatic effects causing the so-called “wedge” areeffectively eliminated (i.e. deconvolved) in the cylindrical (k⊥, k‖) power spectrum.Keywords. methods:data analysis, statistical; techniques:interferometric; cosmology: observa-tions, early universe, large-scale structure of universe1. IntroductionObservations of the redshifted 21-cm signal from neutral hydrogen is a unique probe ofthe early universe and can open the entire redshift window z ∼ 6−30 for astrophysical andcosmological studies, allowing us to directly study the astrophysical processes occurringduring the Epoch of Reionization (EoR) and the Cosmic Dawn (CD). This excitinggoal is challenged by the difficulty of extracting the feeble 21-cm signal buried underastrophysical foregrounds orders of magnitude brighter and contaminated by numerousinstrumental systematics.Several experiments are currently underway aiming at statistically detecting the 21-cmsignal from the Epoch of Reionization (e.g. LOFAR, MWA, PAPER), already achievingincreasingly attractive upper limits on the 21-cm signal power spectra (Patil et al. 2017;Beardsley et al. 2016; Ali et al. 2015), and paving the way for the second generationexperiments such as the SKA and HERA which will be capable of robust power spectracharacterization and for the first time directly image the large scale neutral hydrogenstructures from EoR and CD.To fully exploit the sensitivity of these experiments, accurate removal of the fore-grounds is required. The radiation from our own Galaxy and other extra-galactic sourcesare well-known to vary smoothly in frequency, and this characteristic can be used tomodel and remove them (Jelić et al. 2008; Chapman et al. 2013). However, the interac-tion of the spectrally smooth foregrounds with the Earth’s ionosphere and the observinginstrument create additional “mode-mixing” foregrounds contaminants, which can mimic284https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1743921318000546Downloaded from https://www.cambridge.org/core. Rijksuniversiteit Groningen, on 21 Nov 2018 at 13:03:13, subject to the Cambridge Core terms of use, available atRobust Foregrounds Removal 285Figure 1. Cylindrically averaged power spectra estimated from the simulated visibilities. Thecentral and right panel shows the power spectra of the observed signal (foregrounds, noise and21 cm signal; Stokes I) obtained from the spherical harmonics ML inversion (central) and bygridding the simulated visibilities using WSClean (right). The absence of structure inside the10 Field of View wedge line (red dashed line) in the power spectra estimated using the SpH MLinversion demonstrate that the method effectively compute PSF-deconvolved representation ofthe sky. The power spectra of the noise (Stokes V) is plotted in the right panel.the 21-cm signal. Those are mainly due to the rapid phase and sometime amplitude modi-fications of radio waves caused by small-scale structures in the ionosphere (e.g. Koopmans2010), the inherent chromatic response of the instrument which creates chromatic side-lobe noise (e.g. Vedantham et al. 2012; Thyagarajan et al. 2015; Gehlot et al. 2017),calibration errors and mis-subtraction of sources due to imperfect sky modeling whichalso introduce frequency structure to the otherwise smooth foregrounds (e.g. Patil et al.2016; Ewall-Wice et al. 2017). Mitigating those sources of chromatic noise has proven tobe extremely difficult. The accuracy of the sky model used for calibration is limited bythe instrument confusion noise level and the precision of the beam response model of ourradio receivers. Unavoidably the observed signal will be contaminated by mode-mixing.To reduce the impact of this potential ‘show-stopper’, we have developed a Maximum-Likelihood Power Spectra estimator using the spherical-wave visibility equation (SpHML) and a foreground removal method based on Gaussian Process Regression (GPR).2. Spherical Harmonics Maximum-Likelihood InversionThe SpH ML is a new method to produce Point-Spread-Function (PSF) deconvolvedimages of the sky from radio-interferometric observations. It solves for the interferometricequation using Maximum-Likelihood inversion of the spherical-wave visibility equation,formulated in a full sky setting and including the primary beam and its side-lobes (Carozzi2015; Ghosh et al. 2018):V =∑lmṽm j(kr)Ym (Ωk ). (2.1)The method is tested using simulated LOFAR-HBA full-sky observations which includediffuse astrophysical foregrounds (Jelić et al. 2008), 21-cm signal simulated using 21cm-Fast (Mesinger et al. 2011) and noise level corresponding to 100 nights of 12 hr LOFARintegration time. From the simulated Stokes-I and Stokes-V visibilities, we infer the re-covered spherical harmonics using the SpH ML algorithm and compute the angular powerspectra. Figure 1 presents the cylindrically averaged power spectra. We find the smoothdiffuse foreground in the Stokes I power spectrum mostly dominates at low k‖, wherehttps://www.cambridge.org/core/terms. https://doi.org/10.1017/S1743921318000546Downloaded from https://www.cambridge.org/core. Rijksuniversiteit Groningen, on 21 Nov 2018 at 13:03:13, subject to the Cambridge Core terms of use, available at286 F. Mertens et al.Figure 2. Detection of the EoR signal with the reference simulation. The top panel showsthe spherically averaged power spectra. The central and bottom panels show the cylindricallyaveraged power spectra as a function of k⊥ and k‖ respectively. The simulated observed signal(dark blue) is composed of intrinsic astrophysical foregrounds (dotted dark blue), instrumentalmode-mixing contaminants (dashed light blue), noise (green) and a simulated 21-cm signal(dashed gray). Using our GPR method to model and remove the foregrounds from the simulatedcube, the 21-cm signal (orange) is well recovered with limited bias.most of the foreground power is bound within k‖  0.05hcMpc−1 . The power drops bytwo to three orders of magnitudes in high k‖ regions, where the 21-cm signal plus noisesignal is expected to dominate. In the power spectra obtained from the same simulatedvisibilities but using the more traditional method of gridding the visibilities in uv-space,a wedge like structure is clearly visible (central panel of Figure 1), and is well known tobe due to the frequency dependence of the PSF (Vedantham et al. 2012; Hazelton et al.2013). This demonstrate that by doing a ML fit to non-gridded visibility data sets ateach frequency, we effectively obtain a PSF-deconvolved estimates of the sky sphericalharmonics coefficients.3. Statistical 21-cm Signal SeparationThe source of mode-mixing contaminants are manifold, and we also need to account forthe remaining contaminants related to calibration error and sky-model incompleteness.Ad-hoc modeling is not an option for most of them, and the problem boils down tostatistically separating this contaminants from the 21-cm signal.With this aim, we developed a foreground removal method using the technique ofGaussian Process Regression (GPR) (Mertens et al. 2017). In this framework, the dif-ferent components of the observations, including the astrophysical foregrounds, mode-mixing contaminants, and the 21-cm signal, are modeled with Gaussian Process (GP).Formally, a GP is the joint distribution of a collection of normally distributed randomvariables (Rasmussen & Williams 2005). The covariance matrix of this distribution isspecified by a covariance function, which defines the covariance between pairs of obser-vations (e.g. at different frequencies). The covariance function determines the structurethat the GP will be able to model, for example its smoothness. In GPR, we use GP asparameterized priors, and the Bayesian likelihood of the model is estimated by condi-tioning this prior to the observations. Standard optimization or MCMC methods can beused to determine the parameters of the covariance functions.This formulation ensures a relatively unbiased separation of their contribution andhttps://www.cambridge.org/core/terms. https://doi.org/10.1017/S1743921318000546Downloaded from https://www.cambridge.org/core. Rijksuniversiteit Groningen, on 21 Nov 2018 at 13:03:13, subject to the Cambridge Core terms of use, available atRobust Foregrounds Removal 287accurate uncertainty estimation, even in very low signal to noise observations. Whenapplied to simulation datasets, equivalent to LOFAR-HBA 1200 hours of observationsand based on its current assessment of noise and systematic errors, we found that themethod is capable of recovering well the 21-cm signal power spectrum (see Fig. 2).4. Discussions and Future PerspectiveIn this paper we have introduced two new methods that aim at mitigating the fore-grounds contaminants including the mode-mixing.We have shown that using an ML spherical harmonic power spectra estimator, it ispossible to deconvolve the chromatic “wedge” (caused by frequency-dependent side-lobes)in the (k⊥, k‖) power spectrum space.We also introduced a novel signal separation method which uses Gaussian Process Re-gression (GPR) to model the various mixed components of the observed signal, includingthe spectrally smooth sky, mode-mixing, and a 21-cm signal model.The fundamental improvement of GPR resides in its complete statistical descriptionof all components contributing to the observed signal. In the current implementation ofGPR, we use generic covariance models for the 21-cm signal and mode-mixing compo-nents. While this treatment may be sufficient for a detection of the 21-cm signal and itscharacterization with LOFAR, an improved model may be build for future experimentswith e.g. the more sensitive SKA. The mode-mixing model for example can be improvedby integrating the k⊥ dependency of the foreground wedge and folding into the modelthe analytic work describing the effect on the signal of the instrumental chromaticity,calibration errors and sky-model incompleteness. Exploiting the isotropic nature of the21-cm signal and its evolution at different redshift-bins will also ensure a more sensitiveand accurate modeling.ReferencesAli, Z. S., et al. 2015, ApJ , 809, 61Beardsley, A. P., et al. 2016, ApJ, 833, 102Carozzi, T. D., 2015, MNRAS 451, L6Chapman, E., et al. 2013, MNRAS, 429, 165Ewall-Wice, A., Dillon, J. S., Liu, A., & Hewitt, J., 2017, MNRAS, 470, 1849Gehlot, B. K., et al. 2017, preprint, (arXiv: 1709.07727)Ghosh, A., Mertens, F. G., & Koopmans, L. V. E., 2018, MNRAS, 474, 4552Hazelton, B. J., Morales, M. F., & Sullivan, I. S., 2013, ApJ, 770, 156Jelić, V., et al. 2008, MNRAS, 389, 1319Koopmans, L. V. E., 2010, ApJ, 718, 963Mertens, F. G., Ghosh, A., & Koopmans, L. V. E., 2017, preprint, (arXiv: 1711.10834)Mesinger, A., Furlanetto, S., & Cen, R., 2011, MNRAS, 411, 955Patil, A. H., et al. 2016, MNRAS, 463, 4317Patil, A. H., et al. 2017, ApJ, 838, 65Rasmussen, C. E. & Williams, C. K. I., 2005, Gaussian Processes for Machine Learning (AdaptiveComputation and Machine Learning). The MIT PressThyagarajan, N., et al. 2015, ApJ, 804, 14Vedantham, H., Udaya Shankar, N., & Subrahmanyan, R., 2012, ApJ, 752, 137https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1743921318000546Downloaded from https://www.cambridge.org/core. Rijksuniversiteit Groningen, on 21 Nov 2018 at 13:03:13, subject to the Cambridge Core terms of use, available at

Robust Foregrounds Removal for 21-cm Experiments

   University of GroningenRobust Foregrounds Removal for 21-cm ExperimentsMertens, F.; Ghosh, A.; Koopmans, L.~V.~E.Published in:Peering towards Cosmic DawnDOI:10.1017/S1743921318000546IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2018Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Mertens, F., Ghosh, A., & Koopmans, L. V. E. (2018). Robust Foregrounds Removal for 21-cmExperiments. In V. Jeli, & T. van der Hulst (Eds.), Peering towards Cosmic Dawn (pp. 284-287). (Peeringtowards Cosmic Dawn; Vol. 333). International Astronomical Union.https://doi.org/10.1017/S1743921318000546CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 31-10-2023Peering towards Cosmic DawnProceedings IAU Symposium No. 333, 2018V. Jelić & T. van der Hulst, eds.c© International Astronomical Union 2018doi:10.1017/S1743921318000546Robust Foregrounds Removalfor 21-cm ExperimentsF. Mertens1, A. Ghosh2,3 and L. V. E. Koopmans11Kapteyn Astronomical Institute, University of Groningen,P. O. Box 800, 9700 AV Groningen, The Netherlands2Department of Physics and Astronomy, University of the Western Cape,Robert Sobukwe Road, Bellville 7535, South Africa3Square Kilometre Array radio telescope (SKA) South Africa,The Park, Park Road, Cape Town 7405, South AfricaAbstract. Direct detection of the Epoch of Reionization via the redshifted 21-cm line willhave unprecedented implications on the study of structure formation in the early Universe. Tofulfill this promise current and future 21-cm experiments will need to detect the weak 21-cmsignal over foregrounds several order of magnitude greater. This requires accurate modeling ofthe galactic and extragalactic emission and of its contaminants due to instrument chromaticity,ionosphere and imperfect calibration. To solve for this complex modeling, we propose a newmethod based on Gaussian Process Regression (GPR) which is able to cleanly separate thecosmological signal from most of the foregrounds contaminants. We also propose a new imagingmethod based on a maximum likelihood framework which solves for the interferometric equationdirectly on the sphere. Using this method, chromatic effects causing the so-called “wedge” areeffectively eliminated (i.e. deconvolved) in the cylindrical (k⊥, k‖) power spectrum.Keywords. methods:data analysis, statistical; techniques:interferometric; cosmology: observa-tions, early universe, large-scale structure of universe1. IntroductionObservations of the redshifted 21-cm signal from neutral hydrogen is a unique probe ofthe early universe and can open the entire redshift window z ∼ 6−30 for astrophysical andcosmological studies, allowing us to directly study the astrophysical processes occurringduring the Epoch of Reionization (EoR) and the Cosmic Dawn (CD). This excitinggoal is challenged by the difficulty of extracting the feeble 21-cm signal buried underastrophysical foregrounds orders of magnitude brighter and contaminated by numerousinstrumental systematics.Several experiments are currently underway aiming at statistically detecting the 21-cmsignal from the Epoch of Reionization (e.g. LOFAR, MWA, PAPER), already achievingincreasingly attractive upper limits on the 21-cm signal power spectra (Patil et al. 2017;Beardsley et al. 2016; Ali et al. 2015), and paving the way for the second generationexperiments such as the SKA and HERA which will be capable of robust power spectracharacterization and for the first time directly image the large scale neutral hydrogenstructures from EoR and CD.To fully exploit the sensitivity of these experiments, accurate removal of the fore-grounds is required. The radiation from our own Galaxy and other extra-galactic sourcesare well-known to vary smoothly in frequency, and this characteristic can be used tomodel and remove them (Jelić et al. 2008; Chapman et al. 2013). However, the interac-tion of the spectrally smooth foregrounds with the Earth’s ionosphere and the observinginstrument create additional “mode-mixing” foregrounds contaminants, which can mimic284https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1743921318000546Downloaded from https://www.cambridge.org/core. Rijksuniversiteit Groningen, on 21 Nov 2018 at 13:03:13, subject to the Cambridge Core terms of use, available atRobust Foregrounds Removal 285Figure 1. Cylindrically averaged power spectra estimated from the simulated visibilities. Thecentral and right panel shows the power spectra of the observed signal (foregrounds, noise and21 cm signal; Stokes I) obtained from the spherical harmonics ML inversion (central) and bygridding the simulated visibilities using WSClean (right). The absence of structure inside the10 Field of View wedge line (red dashed line) in the power spectra estimated using the SpH MLinversion demonstrate that the method effectively compute PSF-deconvolved representation ofthe sky. The power spectra of the noise (Stokes V) is plotted in the right panel.the 21-cm signal. Those are mainly due to the rapid phase and sometime amplitude modi-fications of radio waves caused by small-scale structures in the ionosphere (e.g. Koopmans2010), the inherent chromatic response of the instrument which creates chromatic side-lobe noise (e.g. Vedantham et al. 2012; Thyagarajan et al. 2015; Gehlot et al. 2017),calibration errors and mis-subtraction of sources due to imperfect sky modeling whichalso introduce frequency structure to the otherwise smooth foregrounds (e.g. Patil et al.2016; Ewall-Wice et al. 2017). Mitigating those sources of chromatic noise has proven tobe extremely difficult. The accuracy of the sky model used for calibration is limited bythe instrument confusion noise level and the precision of the beam response model of ourradio receivers. Unavoidably the observed signal will be contaminated by mode-mixing.To reduce the impact of this potential ‘show-stopper’, we have developed a Maximum-Likelihood Power Spectra estimator using the spherical-wave visibility equation (SpHML) and a foreground removal method based on Gaussian Process Regression (GPR).2. Spherical Harmonics Maximum-Likelihood InversionThe SpH ML is a new method to produce Point-Spread-Function (PSF) deconvolvedimages of the sky from radio-interferometric observations. It solves for the interferometricequation using Maximum-Likelihood inversion of the spherical-wave visibility equation,formulated in a full sky setting and including the primary beam and its side-lobes (Carozzi2015; Ghosh et al. 2018):V =∑lmṽm j(kr)Ym (Ωk ). (2.1)The method is tested using simulated LOFAR-HBA full-sky observations which includediffuse astrophysical foregrounds (Jelić et al. 2008), 21-cm signal simulated using 21cm-Fast (Mesinger et al. 2011) and noise level corresponding to 100 nights of 12 hr LOFARintegration time. From the simulated Stokes-I and Stokes-V visibilities, we infer the re-covered spherical harmonics using the SpH ML algorithm and compute the angular powerspectra. Figure 1 presents the cylindrically averaged power spectra. We find the smoothdiffuse foreground in the Stokes I power spectrum mostly dominates at low k‖, wherehttps://www.cambridge.org/core/terms. https://doi.org/10.1017/S1743921318000546Downloaded from https://www.cambridge.org/core. Rijksuniversiteit Groningen, on 21 Nov 2018 at 13:03:13, subject to the Cambridge Core terms of use, available at286 F. Mertens et al.Figure 2. Detection of the EoR signal with the reference simulation. The top panel showsthe spherically averaged power spectra. The central and bottom panels show the cylindricallyaveraged power spectra as a function of k⊥ and k‖ respectively. The simulated observed signal(dark blue) is composed of intrinsic astrophysical foregrounds (dotted dark blue), instrumentalmode-mixing contaminants (dashed light blue), noise (green) and a simulated 21-cm signal(dashed gray). Using our GPR method to model and remove the foregrounds from the simulatedcube, the 21-cm signal (orange) is well recovered with limited bias.most of the foreground power is bound within k‖  0.05hcMpc−1 . The power drops bytwo to three orders of magnitudes in high k‖ regions, where the 21-cm signal plus noisesignal is expected to dominate. In the power spectra obtained from the same simulatedvisibilities but using the more traditional method of gridding the visibilities in uv-space,a wedge like structure is clearly visible (central panel of Figure 1), and is well known tobe due to the frequency dependence of the PSF (Vedantham et al. 2012; Hazelton et al.2013). This demonstrate that by doing a ML fit to non-gridded visibility data sets ateach frequency, we effectively obtain a PSF-deconvolved estimates of the sky sphericalharmonics coefficients.3. Statistical 21-cm Signal SeparationThe source of mode-mixing contaminants are manifold, and we also need to account forthe remaining contaminants related to calibration error and sky-model incompleteness.Ad-hoc modeling is not an option for most of them, and the problem boils down tostatistically separating this contaminants from the 21-cm signal.With this aim, we developed a foreground removal method using the technique ofGaussian Process Regression (GPR) (Mertens et al. 2017). In this framework, the dif-ferent components of the observations, including the astrophysical foregrounds, mode-mixing contaminants, and the 21-cm signal, are modeled with Gaussian Process (GP).Formally, a GP is the joint distribution of a collection of normally distributed randomvariables (Rasmussen & Williams 2005). The covariance matrix of this distribution isspecified by a covariance function, which defines the covariance between pairs of obser-vations (e.g. at different frequencies). The covariance function determines the structurethat the GP will be able to model, for example its smoothness. In GPR, we use GP asparameterized priors, and the Bayesian likelihood of the model is estimated by condi-tioning this prior to the observations. Standard optimization or MCMC methods can beused to determine the parameters of the covariance functions.This formulation ensures a relatively unbiased separation of their contribution andhttps://www.cambridge.org/core/terms. https://doi.org/10.1017/S1743921318000546Downloaded from https://www.cambridge.org/core. Rijksuniversiteit Groningen, on 21 Nov 2018 at 13:03:13, subject to the Cambridge Core terms of use, available atRobust Foregrounds Removal 287accurate uncertainty estimation, even in very low signal to noise observations. Whenapplied to simulation datasets, equivalent to LOFAR-HBA 1200 hours of observationsand based on its current assessment of noise and systematic errors, we found that themethod is capable of recovering well the 21-cm signal power spectrum (see Fig. 2).4. Discussions and Future PerspectiveIn this paper we have introduced two new methods that aim at mitigating the fore-grounds contaminants including the mode-mixing.We have shown that using an ML spherical harmonic power spectra estimator, it ispossible to deconvolve the chromatic “wedge” (caused by frequency-dependent side-lobes)in the (k⊥, k‖) power spectrum space.We also introduced a novel signal separation method which uses Gaussian Process Re-gression (GPR) to model the various mixed components of the observed signal, includingthe spectrally smooth sky, mode-mixing, and a 21-cm signal model.The fundamental improvement of GPR resides in its complete statistical descriptionof all components contributing to the observed signal. In the current implementation ofGPR, we use generic covariance models for the 21-cm signal and mode-mixing compo-nents. While this treatment may be sufficient for a detection of the 21-cm signal and itscharacterization with LOFAR, an improved model may be build for future experimentswith e.g. the more sensitive SKA. The mode-mixing model for example can be improvedby integrating the k⊥ dependency of the foreground wedge and folding into the modelthe analytic work describing the effect on the signal of the instrumental chromaticity,calibration errors and sky-model incompleteness. Exploiting the isotropic nature of the21-cm signal and its evolution at different redshift-bins will also ensure a more sensitiveand accurate modeling.ReferencesAli, Z. S., et al. 2015, ApJ , 809, 61Beardsley, A. P., et al. 2016, ApJ, 833, 102Carozzi, T. D., 2015, MNRAS 451, L6Chapman, E., et al. 2013, MNRAS, 429, 165Ewall-Wice, A., Dillon, J. S., Liu, A., & Hewitt, J., 2017, MNRAS, 470, 1849Gehlot, B. K., et al. 2017, preprint, (arXiv: 1709.07727)Ghosh, A., Mertens, F. G., & Koopmans, L. V. E., 2018, MNRAS, 474, 4552Hazelton, B. J., Morales, M. F., & Sullivan, I. S., 2013, ApJ, 770, 156Jelić, V., et al. 2008, MNRAS, 389, 1319Koopmans, L. V. E., 2010, ApJ, 718, 963Mertens, F. G., Ghosh, A., & Koopmans, L. V. E., 2017, preprint, (arXiv: 1711.10834)Mesinger, A., Furlanetto, S., & Cen, R., 2011, MNRAS, 411, 955Patil, A. H., et al. 2016, MNRAS, 463, 4317Patil, A. H., et al. 2017, ApJ, 838, 65Rasmussen, C. E. & Williams, C. K. I., 2005, Gaussian Processes for Machine Learning (AdaptiveComputation and Machine Learning). The MIT PressThyagarajan, N., et al. 2015, ApJ, 804, 14Vedantham, H., Udaya Shankar, N., & Subrahmanyan, R., 2012, ApJ, 752, 137https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1743921318000546Downloaded from https://www.cambridge.org/core. Rijksuniversiteit Groningen, on 21 Nov 2018 at 13:03:13, subject to the Cambridge Core terms of use, available at

Robust Foregrounds Removal for 21-cm Experiments

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