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The LOFAR EoR Data Model: (I) Effects of Noise and Instrumental Corruptions on the 21-cm Reionization Signal-Extraction Strategy PDF

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Mon.Not.R.Astron.Soc.000,1–20(2008) Printed21January2009 (MNLATEXstylefilev2.2) The LOFAR EoR Data Model: (I) Effects of Noise and Instrumental Corruptions on the 21-cm Reionization Signal-Extraction Strategy P. Labropoulos1∗, L. V. E. Koopmans1, V. Jelic´1, S. Yatawatta1, R. M. Thomas1, 9 G. Bernardi1, M. Brentjens2, A.G. de Bruyn1,2, B. Ciardi3, G. Harker1, A.R. Offringa1, 0 V. N. Pandey1, J. Schaye4, S. Zaroubi1 0 2 1KapteynAstronomicalInstitute,UniversityofGroningen,P.O.Box800,9700AVGroningen,theNetherlands n 2ASTRON,Postbus2,7990AADwingeloo,theNetherlands a 3Max-PlanckInstituteforAstrophysics,Karl-Schwarzschild-Straße1,85748Garching,Germany J 4LeidenObservatory,LeidenUniversity,POBox9513,2300RALeiden,theNetherlands 1 2 21January2009 ] M I . ABSTRACT h p Anumberofexperimentsaresettomeasurethe21-cmsignalofneutralhydrogenfrom - o theEpochofReionization(EoR).Thecommondenominatoroftheseexperimentsarethelarge r datasetsproduced,contaminatedbyvariousinstrumentaleffects,ionosphericdistortions,RFI t s andstrongGalacticandextragalacticforegrounds.Inthispaper,thefirstinaseries,wepresent a theDataModelthatwillbethebasisofthesignalanalysisfortheLOFAR(LowFrequency [ Array)EoRKeyScienceProject(LOFAREoRKSP).Usingthisdatamodelwesimulatereal- 1 isticvisibilitydatasetsoverawidefrequencyband,takingproperlyintoaccountallcurrently v knowninstrumentalcorruptions(e.g.direction-dependentgains,complexgains,polarization 9 effects,noise,etc).Wethenapplyprimarycalibrationerrorstothedatainastatisticalsense, 5 assuming that the calibration errors are random Gaussian variates at a level consistent with 3 ourcurrentknowledgebasedonobservationswiththeLOFARCoreStation1.Ouraimisto 3 demonstratehowthesystematicsofaninterferometricmeasurementaffectthequalityofthe . 1 calibrateddata,howerrorscorrelateandpropagate,andinthelongrunhowthiscanleadto 0 new calibration strategies. We present results of these simulations and the inversion process 9 and extraction procedure. We also discuss some general properties of the coherency matrix 0 and Jones formalism that might prove useful in solving the calibration problem of aperture : synthesis arrays. We conclude that even in the presence of realistic noise and instrumental v i errors, the statistical signature of the EoR signal can be detected by LOFAR with relatively X smallerrors.Adetailedstudyofthestatisticalpropertiesofourdatamodelandmorecomplex r instrumentalmodelswillbeconsideredinthefuture. a Keywords: telescopes-techniques:interferometric-techniques:polarimetric-cosmology: observations-methods:statistical-methods:dataanalysis 1 INTRODUCTION galacticmedium(IGM),carvingout‘bubbles’intheotherwiseneu- tralhydrogen-filledUniverse.Thesebubblesgrewrapidly,bothin Recent years have seen a marked increase in the study, both the- size and number, and caused a phase transition in the hydrogen- oreticalandobservational,oftheepochinthehistoryofourUni- ionized fraction of our Universe at redshifts 6<z<20 (Sunyaev verseafterthecosmologicalrecombinationera:fromthesocalled &Zeldovich1975).AlthoughtheEoRspannedarelativelysmall ‘Dark Ages’ to the Epoch of Reionization (EoR) (Hogan & Rees fraction, in time, of the Universe’s age, its impact on subsequent 1979;Scott&Rees1990;Madau,Meiksin,&Rees1997).Acold structureformation(atleastbaryonic)iscrucial.Hence,studying anddarkUniverse,aftertherecombinationera,wasilluminatedby the EoR directly influences our understanding of issues in con- sourcesofradiation,beitstars,quasarsordarkmatterannihilation. temporary astrophysical research such as metal-poor stars, early These ‘first objects’ ionized and heated their surrounding inter- galaxyformation,quasarsandcosmology(Nusser2005;Zaroubi& Silk2005;Kuhlen&Madau2005;Thomas&Zaroubi2008;Field 1958,1959;Scott&Rees1990;Kumar,Subramanian,&Padman- ∗ E-mail:[email protected] (cid:13)c 2008RAS 2 Labropoulosetal. abhan1995;Madau,Meiksin,&Rees1997).Foradetailedreview becausestronglyradiatingsourcescreatebubblesofionizedgasin- oftheEoRandourcurrenteffortstodetectit,wereferthereader sidetheneutralIGM,oneshouldobservefluctuationsinthe21-cm toFurlanetto,Oh,&Briggs(2006)andthereferencestherein. emissionduetoreionizationthatdeviatefromthoseofneutralgas Given the recent progress in developing a concrete theoreti- tracingthedark-matterdistribution,evendeepintothehighlylinear cal framework, and simulations based thereon, the EoR from an regime. observational point of view is still very poorly constrained. De- Developments in radio-wave sensor technologies in recent spite a wealth of observational cosmological data made available yearshaveenabledustoconceiveofanddesignextremelylarge, during the past years (e.g. Spergel et al. 2007; Page et al. 2007; highsensitivityandhighresolutionradiointerferometers,adevel- Beckeretal.2001;Fanetal.2001;Pentericcietal.2002;White opment which is essential to conduct a successful 21-cm experi- et al. 2003; Fan et al. 2006), data directly probing the EoR have menttoimagetheEoR.Aseriesofradiotelescopesarebeingbuilt eluded scientists and the ones that constrain the EoR are indi- similartoLOFAR,suchasMWA2,PAPER3,21CMA4andfurther rect and very model-dependent (Barkana & Loeb 2001; Loeb & inthefuturetheSKA5,allwithoneoftheirprimarygoalsbeingthe Barkana 2001; Ciardi, Ferrara, & White 2003; Ciardi, Stoehr, & detectionoftheredshifted21-cmsignalfromtheEoR.TheGMRT6 White2003;Bromm&Larson2004;Ilievetal.2007;Zaroubiet hasaprogrammealreadyunderwaytodetecttheEoRoratleastto al.2007;Thomas&Zaroubi2008).Currentlytherearetwomain constraintheforegroundsthatmayhampertheexperiment(Penet observationalconstraintsontheEoR:first,thesuddenjumpinthe al.2008). Lyman-αopticaldepthintheGunn–Petersontroughs(Gunn&Pe- Calculationspredictthecosmological21-cmsignalfromthe terson 1965), observed in the quasar spectra of the Sloan Digital EoR to be extremely faint. Apart from the intrinsic low strength SkySurvey(SDSS)(Beckeretal.2001;Fanetal.2001;Pentericci of the 21-cm signal, the experiment is plagued by a myriad of et al. 2002; White et al.q 2003; Fan et al. 2006) which provides signal contaminants like man-made and natural (e.g. lightnings) alimitonwhenreionizationwascompleted.Currentconsensusis interference, ionospheric distortions, Galactic free-free and syn- thatreionizationendedaroundaredshiftofsix.Second,thefive- chrotron radiation, clusters and radio galaxies along the path of yearWMAPdataonthetemperatureandpolarizationanisotropies the signal. Thus long integration times, exquisite calibration and ofthecosmicmicrowavebackground(CMB)(Spergeletal.2007; well-designed RFI mitigation techniques are needed in order to Pageetal.2007)whichgivesanintegralconstraintontheThomson ensure the detection of the underlying signal. It is also impera- opticaldepthforscatteringexperiencedbytheCMBphotonssince tivetoproperlymodelalltheseeffectsbeforehand,inordertode- the EoR. A maximum likelihood analysis performed by Spergel velopsophisticatedschemesthatwillbeneededtocleanthedata etal.(2007)estimatesthepeakofreionizationtohaveoccuredat cubesfromthesecontaminants(Shaveretal.1999;DiMatteo,Cia- 11.3whenthecosmicagewas365Myr.Thus,weseethatcurrent rdi, & Miniati 2004; Di Matteo et al. 2002; Oh & Mack 2003; astronomicaldataisonlyabletoprovideuswithcrudeboundaries Cooray & Furlanetto 2004; Zaldarriaga, Furlanetto, & Hernquist withinwhichreionizationoccured.Inordertoproperlycharacterize 2004;Gleser,Nusser,&Benson2007).Duetothelowsignal-to- theonset,evolutionandcompletionoftheEoRandderiveresults noise ratio per resolution element (of the order of 0.2 or for LO- onitsimpactonsubsequentevolutionofstructuresintheUniverse, FARandandevenlessfore.g.MWA),theinitialaimofallcurrent weneedmoredirectmeasurementsfromtheEoR. experimentsistoobtainastatisticaldetectionofthesignal.Bysta- Observations of the hydrogen 21-cm hyperfine “spin-flip” tistical, we mean a global change in a property of the signal, for transition, using radio interferometry, provide just such a direct examplethevariance,asafunctionoffrequencyandangularscale. probeofthedarkagesandtheEoRoverawidespatialandredshift Notethatthistaskinvolvesdistinguishingthesestatisticalproper- range.Itisworthmentioningthatthe“spatialrange”hereimplies tiesfromthoseofthecalibrationresidualsandthethermalnoise. thetwodimensionsonthesky,whichisafunctionofthebaseline In addition to the above-mentioned astrophysical and terres- lengthsofaninterferometer,andthethirddimensionalongthered- trialsourcesofcontamination,onealsohastofaceissuesarising shift direction, which depends on the frequency resolution of the instandardsynthesisimaging.Forthatitiscrucialtodescribeall observation.The21-cmemissionlinefromtheEoRisredshifted physicaleffectsonthesignalthatdeterminethevaluesofthemea- by1+z,becauseoftheexpansionoftheUniverse,towavelengths suredvisibilities.Thestudyofpolarizedradiationfallswithinthe in the meter waveband. For example, at a redshift of z = 9 the regime of optics: Hamaker & Bregman (1996) provided such a 21-cmlineisredshiftedto2.1metres,whichcorrespondstoafre- unifying model for the Jones and Mueller calculi in optics (Born quencyofabout140 MHz.Computersimulationssuggestthatwe & Wolf 1999) and the techniques of radio interferometry based may expect a complex, evolving patch-work of neutral (H ) and on multiplying correlators. Because low frequency phased-array I ionizedhydrogen(H )regions.Ifwemanagetosuccessfullyim- dipole antennas are inherently polarized, one has to consider po- II agetheUniverseatthesehighredshifts(6 < z < 12)weexpect larimetry from the beginning. The Measurement Equation (ME) tofindH regions,createdbyionizingradiationfromfirstobjects, of Hamaker & Bregman (1996) is therefore a natural way to de- II to appearas “holes”in anotherwise neutralhydrogen-filled Uni- scribeLOFAR.Theirtreatment(Hamaker&Bregman1996;Sault, verse; the so-called Swiss-cheese model. Current constraints and Hamaker,&Bregman1996;Hamaker2000c,2006)formstheba- simulationsconvergeonreionizationhappeningforalargepartin sisofourdatamodeldescriptionandwewillpresentitinawider theredshiftrangez ≈11.4(∼115 MHz)toz ≈6(∼203 MHz), setting,givingtheconnectiontophysics.TheHamaker–Bregman– whichistherangeprobedbyLOFAR1,aradiointerferometercur- Sault measurement equation acts on the astronomical signals as rentlybeingbuiltnearthevillageofExloointheNetherlands.The 21-cmradiationcannotonlytracethematterpowerspectruminthe periodafterrecombination,butalsocanconstrainreionizationsce- 2 MurchisonWidefieldArray:http://www.haystack.mit.edu/ast/arrays/mwa 3 Precision Array to Probe Epoch of Reionization: narios(Thomas&Zaroubi2008;Barkana&Loeb2001).Notethat http://astro.berkeley.edu/dbacker/eor/ 4 21-cmArray:http://web.phys.cmu.edu/past/ 5 SquareKilometerArray:http://www.skatelescope.org 1 LowFrequencyArray:http://www.lofar.org 6 GiantMeterwaveRadioTelescope:http://www.gmrt.ncra.tifr.res.in (cid:13)c 2008RAS,MNRAS000,1–20 TheLOFARDataModel 3 1.5 1.5 1 1 X X X X X X X X X X X X 0.5 0.5 X X X X y [m]k 0 v []!k 0 ~5 [m] !0.5 !0.5 0.2 !1 0 !1 !0.2 31 [m] !0.2 0 0.2 !1!.51.5 !1 !0.5 0 0.5 1 1.5 !1!.51.5 !1 !0.5 0 0.5 1 1.5 x [km] u [k!] Figure1.Left:TheLOFARcompact-corelayoutsituatednearExloointheNetherlands.Theopencircle-sizecorrespondstotheHighBandAntenna(HBA) stationsizeofapproximately31metresindiameter.The6innerstations(“six–pack”)areshowninthetheinsetfigure.Middle:Thesnapshotuv-coverageof theLOFARcompactcoreatthezenithandatafrequency150 MHZ.Right:ThelayoutofaLOFARHBAstation.Eachsquarerepresentsatilethatconsists offourbyfour,orthogonalbowtiedipoles,asshownintheblown–upinset.Eachdipolepaircanberotatedindividuallyduringtheinstallationofthetile. a“black-box”:theinterferometerconvertstheinputsignalStokes 2 DESCRIPTIONOFTHELOW-FREQUENCYARRAY vectorstothefinaloutputatthecorrelator.Thisisdoneviaase- TheimmediatesciencegoalsoftheLOFAREoR-KSP,thatdrive quenceoflineartransformationsandthusenablesustosystemat- someoftheconsiderationsaboutthedesignofLOFAR(Bregman ically model the series of effects that modify the signal while it 2002),are:(1)extractthe21-cmneutralhydrogensignalaveraged propagatesthroughtheionosphereandthereceivers. alonglinesofsight,i.e.the‘globalsignal’(e.g.Shaveretal.1999; Jelicetal.2008),(2)determinethespatial-frequencypowerspec- trum of the brightness temperature fluctuations on angular scales Calibration(Morales&Matejek2008)oftheobservedvisibil- of about 1 arcminute to 1 degree and frequency scales between itydatasetisgenerallyaimedatdetermininginstrumentalparame- 0.1and10MHzintheredshiftrangeof∼6-11and(3)searchfor tersoftheantennasatalevelsufficienttodetectsignalsatseveral Stro¨mgrenionizationbubblesaroundbrightsourcesandthe21-cm timesthenoiselevel.Whilethisisthetraditionalapproach,amuch absorption-line forest (de Bruyn et. al 2007). In order to achieve more thorough understanding of the instrumental response is re- thesesciencegoals,LOFARrequiresagooduv-coverage,agood quiredincurrentlydesignedexperimentssuchastheLOFAREoR frequencycoverageandalargecollectingarea. Key-ScienceProject,whereunprecedentedhighdynamicrangeand Below, we give a short summary of the aspects of LOFAR sensitivityhavetobeachieved,andthesignalisfarbelowthenoise whicharerelevanttotheEoRexperimentandourdatamodel.For level.Forexample,theCLEANalgorithmandvariationsonithave more details we refer to the project paper by de Bruyn et al. (in been used extensively, as an integral part of the SELFCAL pro- preparation). cess(Pearson&Readhead1984).Whilecomputationallyefficient, itdoesnotprovideastatisticallyoptimalsolution(Schwarz1978; Starck,Pantin,&Murtagh2002).Inthisworkweshalltherefore 2.1 Stationconfigurationanduv-coverage consideramaximum-likelihoodsolutiontothemeasurementequa- tioninversionproblem(Boonstra2005;Leshem,vanderVeen,& Initscurrentlayout,theLOFARtelescope(deBruynet.al2007; Boonstra 2000; Leshem & van der Veen 2000; Wijnholds, Breg- Falcke et al. 2007) will consist of up to 48 stations of which ap- man, & Boonstra 2004), which takes into account realistic zero- proximately24willbelocatedinthecoreregion(Figure1),near meancalibrationresidualsandnoise.Wearealsoexaminingalter- the village of Exloo in the Netherlands. The core marks an area nativesolutions(Yatawattaetal.2008),however. of1.7by2.3kilometres.EachHighBandAntennastation(HBA station;110–240MHz;seenextsection)inthecoreisfurthersplit intotwo“half-stations”ofhalfthecollectingarea(∼31metredi- AftergivingashortdescriptionoftheLOFARarray,withem- ameter),separatedby∼130metres.Thissplitfurtherimprovesthe phasisonthehighband(HBA)aspectsofthedesigninSection2, uv-coverage,thoughatthecostofquadruplingtheBlueGene/Pcor- wereviewthebasicrelationbetweentheobservedvisibilitiesand relator demands (i.e. there are four times more base-lines). The theskyintensityinsection3.Weemphasizethepolarized,matrix central region of the core consists of six closely-packed stations, formulationofthemeasurementequationandthemathematicalas- “thesix-pack”,toensureimprovedcoverageoftheshortestbase- pectsofcoherencymatrix(Hamaker&Bregman1996).InSection lines necessary to map out the largest scales on the sky, such as 4webrieflydiscusstherelevantMeasurementEquation(ME)pa- the Milky Way. The station-layout yields a snapshot uv-coverage rametersforLOFAR.Usingthismeasurementequationasourdata atthezenithasshowninFigure 1.Theuv-coveragesforatypical model,inSection5,weproduceanumberofsimulationsfordif- synthesistimeof4hrsareshowninFigure 2. ferentinstrumentalparameters.Thisistheforwarduseofthedata A good uv-coverage is crucial for several reasons. First to model.Wealsotrytoinvertthedatamodel,giventheinstrumen- improvesamplingofthepowerspectrumoftheEoRsignal(San- talparametersandtheirerrordistributions,inordertorecoverthe tos,Cooray,&Knox2005;Hobson&Maisinger2002;Bowman, original data. Our goal is to test the calibration and inversion re- Morales, & Hewitt 2008, 2006, 2005; Morales, Bowman, & He- quirements using realistically generated data cubes. In Section 6 witt2005).Second,toobtainpreciseLocal(Nijboer,Noordam,& wediscusstheresultsandgiveourconclusions. Yatawatta2006)andGlobal(Smirnov&Noordam2004)Skymod- (cid:13)c 2008RAS,MNRAS000,1–20 4 Labropoulosetal. Figure2.Thefirstcolumnoffiguresshowstheuv-coverageoftheLOFARcore,thatwillbeusedfortheEORexperiment,fordiffrentvaluesofthedeclination δ.Thesizeofeachpointcorrespondstothestationdiameter.Theuvcoverageiscalculatedfor4hoursofsynthesiswith10saveragingat150 MHz.The secondcolumnshowsthecorrespondinguvpointdensity.Theuvplaneisgriddedwithacellsizeof8.5wavelengthssquared.Inthelastcolumnahorizontal cutofthe“dirty”beamisshown. els (LSM/GSM; i.e. catalogues of the brightest, mostly compact, pectedinterferencelevelsoperatingateither160or200MHzinthe sourcesinandoutsideofthebeam,i.e.localversusglobal).Afur- first,secondorthirdNyquistzone(i.e.,0–100,100–200,or200– thercomplicationistheextractionoftheGalacticandextragalactic 300MHzbandrespectivelyfor200MHzsampling).Thedatafrom foregrounds.Thisisavitalstepintherecoveryofthesignaland thereceptorsarefilteredin512,195kHzsub-bands(156kHzsub- requires good sampling of the uv-plane at all frequencies (Bow- bandsfor160MHzsampling)ofwhichatotalof32MHzband- man,Morales,&Hewitt2008,2006;Morales,Bowman,&Hewitt width(164channels)canbeusedatonetime.Sub-bandsfromall 2005). antennasarecombinedatthestationlevelinadigitalbeamformer allowingmultiple(4–6)independentlysteerablebeams,whichare senttothecentralprocessorviaaglass-fibrelinkthathandles0.7 2.2 TheHigh-BandAntennas Tbit/sdata.Thebeamsfromallstationsarefurtherfilteredinto1 kHzchannels,cross-correlatedandintegrated.Theintegratedvisi- LOFAR will have two sets of dipoles, the Low Band Antennas bilitiesarethencalibratedon1secondintervals,tocorrectforthe (LBA)andtheHighBandAntennas(HBA).FortheEoRexperi- effectsoftheionosphere,andsubsequentlyimagesareproduced. mentwearemostlyinterestedintheHBAdipoleswhichcoverthe Channelswithdisturbingradiofrequencyinterference(RFI)areex- 110to220MHzfrequencyrange.Eachdipoleisacrosseddipole cised(Leshem,vanderVeen,&Boonstra2000;Veen,Leshem,& which enables X and Y polarization observations. Inside the sta- Boonstra2004b;Wijnholds,Bregman,&Boonstra2004;Fridman tionthedipolesarearrangedintilesof4by4dipoles,with24tiles & Baan 2001). For the correlation we use three racks of an IBM per station inside the core (Figure 1). Radio waves are sampled BlueGene/PmachineinGroningenwithatotalof12288process- witha16-bitanalog-to-digitalconvertertobeabletocopewithex- (cid:13)c 2008RAS,MNRAS000,1–20 TheLOFARDataModel 5 ingcores.LOFARisanewconceptinarraydesign,abroad-band (addingupto32MHzbandwidth)andforasetofshort-timein- aperturearraywithdigitalbeamforming.ThismakesLOFARes- tegrations(addingupto>300hoursofintegration).Theconnec- sentially qualify as a pathfinder for the Square Kilometre Array tion between the correlation matrices R (t ) and the visibilities f k (Falckeetal.2007). V (u,v)isthateachentryR (t )ofR (t )isasampleofthe f ij k f k visibilityfunctionforaspecificcoordinate(u,v)correspondingto thebaselinevectorr =r −r betweentelescopesiandjattime ij i j 3 MATHEMATICALFRAMEWORKOFTHELOFAR tk(Boonstra2005): DATAMODEL V (u −u ,v −v )≡R (t ). f ik jk ik jk ij k Themostimportantpartofanyphysicalmeasurementistofinda Thenoiselessscalarmeasurementequation(notaccountingforin- correspondencebetweenthephysicalquantitiesandthemeasured strumentalandotherdistortingeffects)foroneshort-timeintegra- quantities. In radio interferometry this is achieved through the so tionandnarrow-bandfrequencychannel,assumingtheskycanbe calledmeasurementequations(Hamaker&Bregman1996;Boon- describedbyasetofpoint-sources,canthenbewritteninterms stra2005).Themeasurementequation(ME)describestherelation- ofthecorrelationmatrices7as(Boonstra2005;Leshem&vander shipbetweenthevisibilities(correlationsbetweentheelectricfields Veen2000) fromdifferentantennas)andthebrightnessdistributionofthesky. Wewillbeginbydiscussingbrieflythedifferenttypesofmeasure- Rk,f =Ak,fBfA†k,f (3) mentequationsandtheimplicationsforLOFAR(Smirnov&No- where ordam2006;Noordam2004,2000).Thedatamodelpresentedin thispapercanbeeasilyappliedtoothertelescopesthatoperateat Ak,f =[ak,f(s1),...,ak,f(sd)] lowfrequenciessuchastheMWAandeventuallytheSKA. Astronomicalradiosignalsappearasspatiallywide-bandran- 2 e−i(u1kl+v1km) 3 domnoisewithsuperimposedfeatures,suchaspolarization,emis- ak,f(si)=66 ... 77 sionandabsorptionlines.Thephysicalquantitythatunderliesthis 4 5 kindofmeasurementistheelectricfield,butforconvenienceas- e−i(uNtelkl+vNtelkm) (4) tronomers try to recover the intensity in the direction of the unit pointing vector s, If(s) = (cid:104)|Ef(s)|(cid:105)2. The measured correlation 0 Bf(s1) 1 oftheelectricfieldsbetweentwosensorsiandjiscalledthecom- Bf =B@ ... CA plexvisibility.ForEarth-rotationsynthesisweassumethatthetele- B (s ) f d scopeshaveasmallfieldofview(FOV)andthattheytrackaposi- tiononthesky.Toachievethat,aslowlytime-varyingphasedelay andwherekisthetime–orderedvisibilitynumber,sisthesource hastobeintroducedatthereceivertocompensateforthegeometri- position vector on the celestial sphere and f the observing fre- caldelays.Theresultisthatthereferencelocationappearstobeat quencychannelnumber. thezenith,ortheirchosenpointinthesky(phasereferencecenter). Thevectorfunctionsak,f arecalledarrayresponsevectorsin Foraplanararray,thereceiverbaselinescanbeparametrizedas arraysignalprocessingandtheyarefrequencydependent,butalso c timedependentinthiscaseduetotherotationoftheEarth(Leshem ri−rj =λ[u,v,0], λ≡ 2πν. (1) &vanderVeen2000;vanTrees2002).Theydescribetheresponse ofaninterferometertoasourceatdirections=(l,m)(seeFigure wherer arethestationpositionvectors.Thissystemiswavelength i 3). dependent. The (scalar) measurement equation in (u,v) coordi- Theaboveformalismistrivialaslongasthepositionsofthe natesbecomes telescopes are well known. In reality though, the response of the ZZ V (u,v)= P (l,m)I (l,m)e−i(ul+lm)dldm (2) arrayisnotperfect:telescopesarenotomni-directionalantennas, f f f buteachonehasitsownproperties(i.e.complexbeam-shapeand whereP (l,m)isthecomplexprimarybeamorantennaresponse gainetc.).Inthiscasethearrayresponsevectorsmustberedefined f patternandI (l,m)istheskybrightnessdistribution.Thisequa- as f tion(vanCittert–Zerniketheorem)isintheformofa2–DFourier 0 A (s ) 1 2 e−i(u1k·si) 3 1 i tMraonrsafno,rm&,Swwheinchsoins2a0n01a;ppCraorxoizmziat&ionWfooarna20fl0a8t)s.kTyhe(Tvhiosimbiplistoiens, ak,f(si)=B@ ... CA(cid:12)664 ... 775, (5) aresampledforalldifferentsensorpairsiandj,butalsofordif- ANtel(si) e−i(uNtelk·si) ferentsensorlocationsprojectedonthesky,sincetheEarthrotates. where(cid:12)indicatesaHadamard(element-wise)product.Thesource Hence Earth-rotation synthesis traces uv tracks for each baseline structurecanalsovarywithfrequency.Finally,mostofthereceived (Thompson,Moran,&Swenson2001). signalconsistsofadditivenoise.Whenthenoisehaszeromeanand isindependentamongtheantennas(spatiallywhite),then 3.1 TheScalarMeasurementEquation R =A BA† +σ2I k,f k,f k,f f ThemostwidelyusedMEisstillthescalarformulationoftheME. NoisecanbeassumedtobeGaussianinradio-interferometers,like WebeginthediscussionwiththescalarMEandlaterwewillalso LOFAR. We will assume this in the remainder of this paper, and discussthepolarizedversionthereof.Sinceallprocessingisdone mightaddressnon-Gaussian(Thompson,Moran,&Swenson2001) with digital computers this equation must be transformed into a time-varyingnoiseinfuturepublications.Actually,systemnoiseis moreconvenientdiscretizedform.ThemainoutputoftheLOFAR correlator is a set of correlation matrices (Boonstra 2005; Falcke etal.2007),Rf(tk),forasetofnarrow-bandfrequencychannels 7 Thesymbol”†”standsfortheHermitianconjugationoperator (cid:13)c 2008RAS,MNRAS000,1–20 6 Labropoulosetal. Figure3.Theskybrightnessdistributionbeforeandaftergeometricaldelaycompensation,asseenbyaninterferometer.Risthesourcepositionvectorands therelevantunitvectorinthedirectionofR.risthebaselinevector. slightly different at each receiver. It is reasonable to assume that whereδ(t)istherelativephase.Thisvectorincludesallinforma- noiseisspatiallywhite:thenoisecovariancematrixisdiagonal. tion about the temporal evolution of the electric field. When the parametershavenotimedependencethisiscalledtheJonesvector. Moreover, the coherency (or polarization or density) matrix of a 3.2 ThePolarizationMeasurementEquation lightbeamcontainsalltheinformationaboutitspolarizationstate. Theequationsabovedescribetherelationofthevisibilitiestothe ThisHermitian2×2matrixisdefinedas totalintensityofthesource,i.e.theclassicalStokesI.Totakepo- C≡De(t)⊗e†(t)E= larizationintoaccount,severalmodificationsarerequired.Theuse of the polarization information is of great importance for several „ (cid:104)e1(t)e∗1(t)(cid:105) (cid:104)e1(t)e∗2(t)(cid:105) « (7) reasons.First,thisinformationprovidesinsightinthephysicalpro- (cid:104)e2(t)e∗1(t)(cid:105) (cid:104)e2(t)e∗2(t)(cid:105) cessesthatmightexistintheastronomicalobjectofinterest.Sec- Thisisthecoherencymatrixoftheperpendiculardipolesofasin- ond,moderntelescopeslikeLOFARarealsoinherentlypolarized. gleLOFARHBAantenna.⊗standsfortheKroneckerproductand Third,usingmoreinformationenhancestheresultofthedatare- thebracketsindicateaveragingovertime(Boonstra2005).Theco- ductionprocess.Thestudyofpolarizedlightisthereforebecoming herencymatrixisacorrelationmatrixwhoseelementsarethesec- anincreasinglyimportantissueinastrophysics(Tinbergen1996). ondmomentsofthesignal.Usingtheergodichypothesisthebrack- Manymatricialmodelshavebeendevelopedtostudythepo- etscanbeconsideredasensembleaveraging.Duetoitsstatistical larization properties of light. A proper description of the polar- nature its eigenvalues ought to be non-negative. The normalized ization properties of light relies on the concept of the coherency versionofthismatrixCˆ = C containsinformationaboutthe matrix (Born & Wolf 1999; Hamaker & Bregman 1996). This tr(C) populationandcoherenciesofthepolarizationstates(Fano1957). mathematical formalism holds for every band of the electromag- This object is the equivalent of the single brightness point in the netic spectrum. A usual assumption is monochromatic light, but scalarversionofthetheory. polychromatic light behaves as monochromatic for time intervals longerthanthenaturalperiodandshorterthanthecoherencetime (Gil2007;Barakat1963).Ourmathematicalmodelwillbebased 3.2.2 TheStokesParameters on those introduced in radio-astronomy by Hamaker & Bregman Abovewediscussedthestatisticalinterpretationofthecoherency (1996). matrix.Nowwecanalsointroduceageometricaldescription:the measurable quantities Stokes I, Q, U and V arise as the coeffi- 3.2.1 TheElectric-FieldVectorandCoherencyMatrix cientsoftheprojectionofthecoherencymatrixontoasetofHermi- tiantrace-orthogonalmatrices,thegeneratorsoftheunitarySU(2) Theeffectsoflinearpassivemediaonthepropagatedphotonscan groupplustheidentitymatrix(seeAppendixA1).Parameterswith berepresentedbylineartransformationsoftheelectricfieldvari- direct physical meaning can be derived from the corresponding ables.Thenatureofthoseeffects,thespectralprofileofthelight measurablequantities.TheStokesparametersareusuallyarranged andthechromaticandpolarizingpropertiesofthemediumthrough asa4×1vector, which light passes, all affect the degree of mutual coherence. In 0 1 I generalcoherentinteractionscanberepresentedbytheJonescal- culus (Jones 1941, 1942, 1948), while incoherent interactions of s=BB Q CC. polychromatic light require the Mueller calculus (Barakat 1963), @ U A V sincethelossofcoherenceneedsmoreparameterstobedescribed. Thetwocomponentsoftheelectricfield(e.g.thosereceivedattwo Analternativenotation,thatwillbederivedinAppendixAisthe dipoles)canbearrangedasthecomponentsa2×1complexvector: 2×2Stokesmatrix: e(t)=„ Ex(t) « (6) S= 1„ I+Q U −iV «≡C, (8) Ey(t)eiδ(t) 2 U +iV I−Q (cid:13)c 2008RAS,MNRAS000,1–20 TheLOFARDataModel 7 which relates the measured coherency matrix quantities to the canbestackedintoasinglevectorandthecorrelationmatrixcan Stokesparameters(Born&Wolf1999). begeneralizedtothefollowingform „ (cid:104)E (r )E∗(r )(cid:105) ˙E (r )E∗(r )¸ « V = (cid:104)Ex(ri)Ex∗(rj)(cid:105) ˙Ex(ri)Ey∗(rj)¸ y i x j y i y j 3.2.3 TheJonesFormalism „ « V V = xx xy (12) Anadequatemethodtodescribeanon-depolarizingsystemisthe V V xy yy Jonesformalism.Itrepresentstheeffectsonthepolarizationprop- foralinearpolarizationbasis.NotethateachelementofV,V is ertiesofanEMwaveaftertheinteractionwithsuchasystem.For ij notHermitianfori(cid:54)=j,butV =V†,sothatVremainsHermi- passive,puresystems,theelectricfieldcomponentsofthelightin- ij ji tian(Hamaker&Bregman1996).Herewehavedenotedexplicitly teractingwiththemisgivenbythecorrespondingJonesmatrixJ, thecorrelationfromtheXandYorienteddipoles. e(cid:48) =Je. LetAilbethepositiondependentpolarizationmultiplication matrix. This is the array response matrix (as it is termed in the Asboththeinitialandfinalfieldscanfluctuate,itisusefultode- languageofcommunicationtheory)orintheHamakerformalism scribethepropertiesofpartiallypolarizedlightwiththecoherency theJonesmatrix.Thearrayresponsevectormusttakeintoaccount matrix.Thus, thedifferentphysicalandinstrumentaleffectsthataffectthesignal D E D E through its path from the source to the recorder, like ionospheric C(cid:48) = e(cid:48)⊗e(cid:48)† = (Je)⊗(Je)† Faraday rotation, parallactic offsets, the geometric delay and in- (9) D E D E strumental gains(and leakages). These effectsare represented by = Je⊗e†J† =J e⊗e† J† =JCJ† theJonesmatrices.Sowecanwritethemeasurementequationin theform(Boonstra2005;Veen,Leshem,&Boonstra2004a) As we are dealing with interferometry, the two J matrices can come from two different telescopes. The effects on the elec- V =A BA† +N ijl il l jl noise tric field vectors in the coherency matrix C(cid:48) can be written wherethedifferenteffectsAinthearrayresponsevectormustbe as an operation of these Jones matrices on the original unaf- introducedintheexactorderinwhichtheyaffectthesignal.The fected coherency matrix C. As we already mentioned, the co- index l is the pixel number and the index i,j represent antenas i herency matrix can also be written as a four vector with c = ((cid:104)e e∗(cid:105),(cid:104)e e∗(cid:105),(cid:104)e e∗(cid:105),(cid:104)e e∗(cid:105)). This vector is related to the andj respectivelyThealgebraofcomplexJonesmatricesisobvi- 1 1 1 2 2 1 2 2 ously non-commutative, i.e. the ordering of the matrices matters. Stokesvectorvia(Parke1948) Thephysicalmeaningofthisisthattheresultsofthedifferentef- 0 1 1 0 0 1 fectsontheincomingelectromagneticwavearenotlinear. B 1 0 0 −1 C s=Lc with L=B C. (10) @ 0 1 1 0 A 4.1 IndividualJonesmatrices 0 −i i 0 Inthissectionwegiveabriefoverviewoftheinstrumentalparame- The matrix has the following property L−1 = 1/2L†. Using the ters(Jonesmatrices)whichareusedforourdatasimulations,their propertiesoftheKroneckerproductwethenfind,intermsofStokes importance,andwhichparametersweuseandperturb. parameters,that D E F:IonosphericFaradayRotation Theionosphereisbirefringent, s(cid:48) =L Je⊗(Je)† =Ns such that one handedness of circular polarization is delayed with N=L(J⊗J)L−1 respecttotheother,introducingadispersivephaseshiftinradians (11) Z with ∆φ≈2.62×10−13λ2 B n ds(SI) || e 1 “ ” N = tr σ Jσ J† . kl 2 k l Itrotatesthelinearpolarizationpositionangleandismoreimpor- tantatlongerwavelengths,attimesclosetothesolarmaximumand whereσ arethePaulimatrices(AppendixA1).AJonesmatrixcan i atSunriseorSunset,whentheionosphereismostactiveandvari- representaphysicallyrealizablestateaslongasthetransmittance able(Eliasson&Thide´ 2008;Norinetal.2008;Eliasson&Thide´ condition(gainorintensitytransmittance)holds;thatis,theratioof theinitialandfinalintensitiesmustbe0(cid:54)g (cid:54)1.Thereciprocity 2007;Thide´ 2007;Spoelstra1996,1995;vanVelthoven&Spoel- stra1992;Spoelstra&Yang1990;Spoelstra&Schilizzi1981).The condition describes the effect when the output signal follows the pathintheinverseorder.ForeveryproperJonesmatrixe(cid:48) =J†e. FaradayrotationJonesmatrixisareal-valuedrotationmatrix: Thisresultdoesnotholdwhenmagneto-opticeffectsarepresent. „ cos∆φ −sin∆φ « F= InthiscasetheMueller–Jonesmatriceshavetobeused.IfaJones sin∆φ cos∆φ matrixrepresentsaphysicallyrealizablestatethereciprocalmatrix TomodeltheTotalElectronContent(TEC)(vanderTol&vander alsorepresentsaphysicaleffect. Veen 2007), we use a Gaussian random field with Kolmogorov- like turbulence. We use a cut-off at scales that correspond to the maximumandminimumbaselinesofWSRT(WesterborkSynthe- sisRadioTelescope),similartothoseofLOFAR.Initialmodelpa- 4 THELOFARMEASUREMENTEQUATION rametershavebeenobtainedfromWSRTdata,takenatsimilarfre- LOFARstationsconsistoftilesof4×4crosseddipoleswhichallow quencies,baselinesandtimeintervalstoourplannedLOFARob- forfullStokesmeasurements.IfwehaveN stationseachwith servations. Moreover, the WSRT is located approximately 50 km tel twopolarizationdegreesoffreedom,thenthe2N electricfields fromtheLOFARcoreinExlooandassuchtheWSRTprovidesa tel (cid:13)c 2008RAS,MNRAS000,1–20 8 Labropoulosetal. elementbeamandweassumethatthedipolesareuniformelydis- Figure4.ThedesignofaLOFARHBAdipoleantennaelement. tributedtoformacirculararray.Thus,weignorethetilestructure, forthepurposesofthecurrentpaper.Ifweneedtoformastation- beaminanarbitrarydirectioninsidethedipole/tilebeam,wecan doitbyproperlyweightingthesignalsofeachelementandadding delays.Anothersimplerwaytoproceed,whichissufficientforthe purposesofthispaper,istocalculatethebeamshapeofanaxially symmetric,uniformlydistributedarrayofelementsandthenmul- tiplyitwiththeprimaryelementpatternthatwecalculatedabove. Wealsoassumethatthethin-wireapproximationholds. D:PolarizationLeakageandInstrumentalPolarization Radio– interferometers usually measure the full set of Stokes parameters simultaneously.MeasuringthepolarizationpropertiesofGalactic diffuse emissionas well as the extragalacticsources is an impor- tantaspectoftheEoR-KSP.Thepolarizationfractionofmostas- tronomical sources is typically low, of the order of few per cent. Measuring it with accuracy is thus challenging, but important in ordertoextractscientificinformationfromit.Cross-polarizationor goodtest-bedforforthcomingLOFARobservationsandexpected polarizationleakageisaninstrumentalcontaminationthatcouples ionosphericeffects. orthogonalJonesvectors.Thedipolesarenotideal,soorthogonal polarizationsarenotperfectlyseperated.Aslighttiltbetweenthe P:ParallacticAngle Thismatrixdescribestheorientationofthe dipoles,forexample,canchangethepolarizationreceptionpattern. skyinthetelescope’sfieldofview,orsimilarlytheprojectionof Thismustbetakenintoaccount,especiallywhenweaimtoreacha thedipolesontothesky.Ithasthemathematicalstructureofaro- highdynamicrangeintheimages.Itissupposedtobeoftheorder tation matrix and rotates the position angle of linearly polarized oflessthanafewpercentandbecauseitisageometricalfactor, radiationincidentonthedipoles.Itshouldinprinciplebeanalyti- weexpectthatitisfrequencydependent.TherelevantJonesmatrix callyanddeterministicallyknown,anditsvariationprovideslever- canbewrittenintheformofaunitarymatrix(Heilesetal.2001; agefordeterminingpolarization-dependenteffects.Itcanalsobe Heiles2002;Bhatnagar&Nityananda2001;Reidetal.2008): usedattheinitialstagestodeterminedipoleorientationerrors. „ cosχ −sinχ « „ √1−d −√deiψ « P= D= √ √ (13) sinχ cosχ de−iψ 1−d whereχistheparallacticangle.Weshallincludethismatrixinthe where d is a generally small constant of the order of 10−6 T˙his antennavoltagepatternmatrix. isafirst-orderapproximationtobehaviourthatmightprovemore complex.However,itisveryimportanttoknowthisparameteras E: Antenna Voltage Patterns The LOFAR telescope HBA sta- it can convert unpolarized radiation, such as the EoR signal, into tionsconsistsofcrossedbowtiedipolepairs.Likeeveryantenna, polarizedradiation.Thismatrixhasalmostthestructureofarota- theyhaveadirectionallydependentgainwhichisimportantwhen tionmatrix.Thedifferenceistheinclusionofthephaseψ.When theangularsizeoftheregionontheskyiscomparableto∼λ/D, ψ equals zero, polarization leakage converts Q to U or E to B- whereD isthestationdiameter.Atlowfrequencieswhenthera- modes[rotationacrosstheequatorofthePoincare´ sphere(Heiles dio sky is dominated by point sources, wide-field techniques are etal.2001)].Whenthephasetermissignificant,polarizationleak- requiredaswell.Todeterminetheantennavoltagepatterntofirst age leads to mixing Q, U and V. Polarization leakage manifests order,weassumeananalyticmodelofthedipolepairs:thebasic itselfasclosureerrorsinparallelhandvisibilitiesandtheleakage- parametersasshowninFig.4havethefollowingvalues:L=0.366 inducedclosurephaseandthePancharatnamphaseofoptics(Berry m,h=0.45m,α1=50degandα2=80deg. 1987).Thisphaseariseswhenthestateofpolarizationofthelight AstheEarthrotates,thedipolesrotatewithrespecttothesky. istransformedfollowingaclosedpathinthespaceofstatesofpo- Thiscausesthepolarizationcoordinatesystemtorotate(unlikee.g. larization, which is known as the Poincare´ sphere (Born & Wolf theWSRTwhichhasanequatorialmount)andwemustthuscorrect 1999).Anothertypeofdistortionistheinstrumentalpolarization. forthiseffect.ForapairofcrosseddipolesalongtheXandYaxes Thedipoleandstationdesignsaswellastheprocessingdonecan therelevantJonesmatrixEcanbewrittenas: contributetosucheffects.LOFARuseslinearlypolarizeddipoles. „ E `π −θ,φ− π´ E `π −θ,φ− π´ « TheinterferometricmeasurementoftheStokesparametersUandV E= Eθ`π2−θ,φ− 34π´ Eφ`π2−θ,φ− 34π´ usingtwodifferentdipolesismadebyformingthe“cross-handed” θ 2 4 φ 2 4 products of the signals. If for any reason the signals are received whereφandθarethepolarandazimuthalanglesofthebeampat- withdifferentgains,thiswouldleadtoacertainfractionofpolar- tern. We assume that the X and Y dipoles have the same beam ization.Thiscanbemodelledas: pattern. Because of the spatial distribution of the dipoles within „ « d 0 astation,eachdipolehasadifferentdelayforthereceptionofan Dp = 01 d eiψ , (14) electromagneticwavecomingfromasource.Ifweassumeanar- 2 rowbandsystem,onecancorrectforthisbyshiftingthephasefor whered ,d arethedifferentXandYgains.Weassumethatthe 1 2 thesignalofeachdipoleelementofthestations(i.e.introducingan instrumentalpolarizationaxisliesalongtheaxisdefinedbythema- effectivedelay).WehaveananalyticalmodelfortheHBAantenna trixbasistowritethisequation.InsuchcaseQdoesnotleakinto (cid:13)c 2008RAS,MNRAS000,1–20 TheLOFARDataModel 9 U,butstillI leaksontoQ.Withacarefullychosenbasis(seeAp- pendixA)IdoesnotleakintoU. where c = vec(C) is the 4×1 source coherency vector in the l,n,msystem8. G: Complex baseline-based electronic gain This matrix ac- Ifweassumethattheskycanbedescribedasaclosely-packed countsforallamplitude-phase-andfrequency-independenteffects collectionofpoint-sources,thenthemeasurementequationcanbe ofthestationelectronics.Itshowsaslowvariationoftheorderof rewrittenas: 1–2%overaday.Ithastheform: Vobs =XA (l,m)C(l,m)A†(l,m) „ « ij i j g 0 G= X , l,m 0 g Y whereCisthecoherencymatrixofasinglepoint-sourceat(l,m). wheregX andgY arecomplexnumbers.Itisoneofthemostcom- Forasinglepoint-sourcewiththerestoftheskyblankthissimpli- moncalibrationparametersandtheonecommonlysolvedforinthe fiesfurtherto classicalself-calibrationloop. Vobs =A (l,m)C(l,m)A†(l,m). ij i j B:Bandpass Compensatingforchangeofgainwithfrequencyis Wemustnotethatthisequationislinearoverthecoherencymatrix called bandpass calibration. This matrix is similar to G and de- C,butnotovertheJonesmatricesA.Wereiteratethattheinvidual scribesthefrequency-dependenceoftheantennaelectronics.Sta- JonesmatricesthatformthematrixproductsA,arenotingeneral tiondigitalpoly-filters,usedtoselectthefrequencypassband,are commutative.Theirorderfollowsthesignalpath.Theformofthis notperfectlysquare,buttheyaredeterministicallyknownandtheir equationisquitecomplicated.Everyelementinthenewmatrixisa bandpassbehaviorisexpectedtobequitestableoverlargeperiods non-linearfunctionofalltheparametersthatappearinthevarious oftime.Itsgenericformis: Jonesmatrices. „ b (f) 0 « Ithasbeenproposed(Hamaker2006,2000a)thatwecanself- B= x0 b (f) calibratethegenericAmatrix,applypost-calibrationconstraintsto y ensure consistency of the astronomical absolute calibrations, and The b are real numbers. We model both B and G parameters as recover full polarization measurements of the sky. This is an in- slowlytime-varyingfunctionswithadditivewhitenoiseastheini- terestingideaforlow-frequencyarrays,whereisolatedcalibrators tialcomissioningtestsforLOFARsuggest. areunavailable(duetothefactthatsucharraysseethewholesky). Thisjustemphasizesthefactthatcalibrationandsourcestructure K: Fourier kernel This term has traditionally been called the aretiedtogether-onecannothaveonewithouttheother. Fourierkernel.Forlargerfieldsofviewand/orlongerbaselinesthe 2–DFouriertransformrelationshipbetweentheskyandthemea- suredvisibilitiesisnolongeragoodapproximation.Thistermde- 4.3 Additiveerrors scribesaphaseshiftthataccountsforthegeometricaldelayforthe Theultimatesensitivityofareceivingsystemisdeterminedprin- signalsreceivedbytelescopesiandj.Letx,yandzbetheantenna cipallybythesystemnoise.Thediscussionofthenoiseproperties positionsinacoordinatesystemthatpointstotheWest,Southand ofacomplexreceivingsystemlikeLOFARcanbelengthy(Lopez theZenithrespectively.Wecanprovidethedirectionsontheskyin &Fabregas2002;deBruynet.al2007),soweconcentrateforour termsofthedirectionalcosinesl,m,n.Weassumethattheskyisa unitspheresothatl2+m2+n2 =1(Thompson,Moran,&Swen- purposesonsomebasicprinciples.Thetheoreticalrms(rootmean square) noise level in terms of flux density on the final image is son2001).SincetheXandYdipolesareco-located,therelevant givenby(Thompson,Moran,&Swenson2001) Jonesmatrixtakesthefollowingsimpleform: K=e−ihul+vm+w“√1−l2−m2−1”i„ 1 0 « 1 SEFD 0 1 σ = × (16) noise ηs pN ×(N −1)×∆ν×tint Thismatrixshouldalsobewelldetermined,butsolvingforitcan helptoestimatetheantennapositions,theelectronicpathlengths, whereηsisthesystemefficiencythataccountsforelectronic,dig- the clock errors etc. For this the concept of the array manifold ital losses, N is the number of substations, ∆ν is the frequency isuseful(Manikas2004)Italsoprovidesastrometricinformation bandwidth,tint isthetotalintegrationtimeandSEFDistheSys- bysolvingforl,mandn,whicharethedirectioncosinesofthe temEquivalentFluxDensity.Thesystemnoiseweassumetohave sourcesonthesky. twocontributions:thefirstcomesfromtheskyandisfrequencyde- pendent(≈ ν−2.55)(Shaveretal.1999;Jelicetal.2008)andthe secondcomesfromfromthereceivers.ThescalingoftheA ,the eff 4.2 TheFinalFormoftheMeasurementEquation effectivecollectingareaoftheantennaewithfrequency,introduces alsoafrequencydependence.Wealsoassumethatthedistribution All the above effects can be combined in order to form the of noise over the uv–plane at one frequency is Gaussian, in both Hamaker–Bregman–Sault measurement equation. The equation therealandimaginarypartofthevisibilities.Fora24-tileLOFAR lookslike: corestation,theSEFDwillbearound2000Jy(η ∼0.5),depend- ingonthefinaldesign(deBruynet.al2007).Thismeansthatwe Vobs = R dldm“A ⊗A†”c(l,m), with ij i j l,m,n (15) 8 Here we use the Kronecker product identity AXB=C ⇔ Ai =KiBiGiDiEiPiTiFi `BT ⊗A´vec(X) = vec(C). We can then solve for X, when `BT ⊗A´isinvertible(Golub&Loan1989).Thiswaythemeasurent A†=F†T†P†E†D†G†B†K† equationcanbetransformedintoalinearequation. j j j j j j j j j (cid:13)c 2008RAS,MNRAS000,1–20 10 Labropoulosetal. canreachasensitivityof520mKat150MHzwith1MHzband- a single Tesla S870 system. For the rest we used a 16-way SMP width in one night of 4 hours of observation. Accumulating data workstationwith32GBofRAMandsufficientdiskspace. fromahundrednightsofobservationsbringsthisnumberdownto ∼52 mK. We will assume a constant noise estimate of this mag- 5.1 Cosmologicalsignalandforegroundssimulations nitude for each pixel in the remainder of the paper. The noise is uncorrelatedbetweendifferenttelescopesanddifferentsignals,so Westartbydiscussingthecosmologicalsignalmapsthatweused thatwecanwritethenoisematrixintheform: in the simulation. The cosmological 21-cm signal is generated in 0 ˙n n∗¸ 0 1 a simplified manner, but sufficient for our purposes. We generate i j 1 N=BB ... CC (17) a contiguous cube of dark matter density from redshift z ∼ 6 to @ A z ∼ 12(correspondingtofrequenciesofaround110MHzto200 0 ˙n n∗¸ i j Ntel MHz) as described in Thomas et al. (2008), from equally spaced wheren ={nxx,nyy}. outputsintimeofanN-bodysimulation.Weemployacomoving i i i 100h−1 Mpc,2563 particlesdarkmatteronlysimulation,using Wearenowinasituationthatwecanstarttosimulaterealistic GADGET 2.WefurthermoreassumeaflatΛCDMuniverseand data-setsforLOFAR. set the cosmological parameters to Ω = 0.238, Ω = 0.0418, m b Ω =0.762,σ =0.74,n =0.951andh=0.73,inagreement Λ 8 s withtheWMAP3observations(Spergeletal.2007). 5 REALISTICLOFAR-EOR-KSPDATASIMULATIONS Theratiobetweenthebaryons(basicallyatomichydrogenand Inthissection,wewilldescribetheLOFARinstrumentalresponse helium)todarkmatterwassettoΩ /Ω ≈0.2.Atomichydro- b DM simulationsandtheinversionmethodusedtoenhancetheresultof genisassumedtofollowananalyticalionizationhistoryofexpo- the primary calibration. The simulation consists of two steps: (i) nentialnatureas1+exp(z−z )−1,wherez differsforeach ion ion theforwardstep,whereweusearealisticmodeloftheLOFARre- line-of-sight as 10±N(0,1), where N(0,1) is a normal distri- sponsefunction,includingallinstrumentaleffectsandnoisetogen- butionwithmeanzeroanddispersionone.Thisisdonetomimic eratesimulatedLOFAREoRdata,and(ii)aninversionstepwhere apossiblespreadinredshiftduringwhichthereionizationoccurs. weusethedatamodelwithrealisticsolutionsanderrorrangefor Weplantouseamorerealisticcosmologicalsignalinthefuture, thedatamodelparameters,toobtaintheunderlyingvisibilities.In butforourcurrentpurposes(i.e.testingtheinversionprocess),this thissubsectionweshallconsidertheforwardstep.Thedatamodel issufficientlycomplex. is based on the Hamaker–Bregman–Sault measurement equation, For each of the foreground components, a 5◦ × 5◦ map is asdescribedintheprevioussections,andweusean“onion-layer” generatedinthesamefrequencyrange(between110and200MHz) approach to predict the instrumental response, as is suggested by pertaining to the LOFAR–EoR experiment. In this paper, we use theorderoftheJonesmatricesintheMeasurementEquation.First, onlysimulationsoftheGalacticdiffusesynchrotronemissionand we must obtain a representation of the Fourier Transform of the ofradiogalaxies.Morecomplexforegroundrealizationsareunder sky,asitisperceivedbyaninterferometer.Toachievethiswepre- investigationandwewillbeconsideringtheminfuturework. dictthevaluesoftheKJonesmatrix.Thisincludescalculatingthe Of all components of the foregrounds, galactic diffuse syn- u,v andwtermsaswellasthel,mandndirectioncosines.For chrotron emission(GDSE) isby farthe most dominant and orig- thepurposeofthispaperwecalculatedthemforsixhoursofinte- inatesfromtheinteractionbetweenfreeelectronsintheinterstel- grationandtensecondsofaveraging.Thecentreoftheskymapsis lar medium and the Galactic magnetic field. The intensity of the atδ =52◦andα =12hours. synchrotronemissioncanbeexpressedintermsofthebrightness c c Thenextstepistopredictthe‘true’uncorruptedvisibilitiesfor temperatureT anditsspectrumisclosetoafeaturelesspowerlaw b differentskyrealizations.Thosearethevisibilitiesasseenbyaper- T ∼νβ,whereβisthebrightnesstemperaturespectralindex. b fectlycalibratedandnoiselessinstrument.Inthisstep,thesource At high Galactic latitudes the minimum brightness tempera- structure and the calibration problem are linked together in the tureoftheGDSEisabout20Kat325MHzwithvariationsofthe visibilities (Pearson & Readhead 1984; Hamaker 2006, 2000a,b, orderof2percentonscalesfrom5–30arcminacrossthesky(de 1999),thusweneedtoinvestigatetheperformanceoftheinversion Bruynet.al1998).AtthesameGalacticlatitudes,thetemperature method,usingskymodelswithdifferentcomplexities(seesubsec- spectralindexβoftheGDSEisabout−2.55at100MHz(Shaver tion5.3).Inthispaperweapplythemethodtothegalacticdiffuse etal.1999;Rogers&Bowman2008)andsteepenstowardshigher emissionsuperimposedonthecosmologicalsignal.Inthefuturewe frequencies, but also gradually changes with position on the sky willalsoconsideracollectionofpointsources(thiswillbeuseful (e.g.Reich&Reich1988;Plataniaetal.1998). fortheconstructionandimprovementtheLocalSkyModel). In the four dimensional (three spatial and one frequency) The simulated maps between 200 and 110 MHz were cre- simulations of the GDSE, all its observed characteristics are in- atedatahighfrequencyresolution(100kHz)correspondingtothe cluded:spatialandfrequencyvariationsofbrightnesstemperature LOFAREoRdata-setresolutionandweresubsequentlybinnedin and spectral index, as well as brightness temperature variations frequencyusinga1 MHzmovingaverageboxfunction.Eachsim- along the line of sight. The spatial distribution of the 3D ampli- ulatedvisibilitydata-sethasasizeofapproximately100MB.We tude and brightness temperature spectral index of the GDSE are produceddata-setsfor128frequencychannels(reducedfrom320 generated as Gaussian random fields, while along the frequency thatwillbeusedduringtherealexperiment). direction we assume a power law. The final map of the GDSE at Thesimulation,inversionandsignalextractionalgorithmsare each frequency is obtained by integrating the 4D cube along one implemented in MATLAB, with extensive use of MATLAB exe- spatial direction. All spatial and frequency properties of the sim- cutablesfiles,tospeedupthemostoftenvisitedloops.Duetothe ulatedGDSEarenormalizedtomatchthevaluesofobservations. parallelnatureoftheprocedures,theDistributingComputingTool- Inadditiontothesimulationsofthetotalbrightnesstemperature, boxisused,whilemanyBLASoperationsandtheFFTareimple- polarized Galactic synchrotron emission maps are also produced, mentedusingtherelevantNVIDIACUDA2.0libraries,andrunon thatincludemultipleFaradayscreensalongthelineofsight.Fora (cid:13)c 2008RAS,MNRAS000,1–20

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