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

1.9 MB·English

by  Panagiotis Labropoulos

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