MNRAS440,1296–1321(2014) doi:10.1093/mnras/stu144 AdvanceAccesspublication2014March23 Seeing in the dark – I. Multi-epoch alchemy Eric M. Huff,1‹ Christopher M. Hirata,2 Rachel Mandelbaum,3,4 David Schlegel,5 Urosˇ Seljak5,6,7,8 and Robert H. Lupton3 1DepartmentofAstronomy,UniversityofCaliforniaatBerkeley,Berkeley,CA94720,USA 2DepartmentofAstronomy,CaltechM/C350-17,Pasadena,CA91125,USA 3DepartmentofAstrophysicalSciences,PrincetonUniversity,PeytonHall,Princeton,NJ08544,USA 4DepartmentofPhysics,CarnegieMellonUniversity,Pittsburgh,PA15213,USA 5LawrenceBerkeleyNationalLaboratory,Berkeley,CA94720,USA 6SpaceSciencesLab,DepartmentofPhysicsandDepartmentofAstronomy,UniversityofCalifornia,Berkeley,CA94720,USA 7InstituteoftheEarlyUniverse,EwhaWomansUniversity,Seoul120-750,SouthKorea 8InstituteofTheoreticalPhysics,UniversityofZurich,CH-8057Zurich,Switzerland D o w n Accepted2014January17.Received2014January16;inoriginalform2011December15 lo a d e d fro m ABSTRACT h Weak lensing by large-scale structure is an invaluable cosmological tool given that most of ttp the energy density of the concordance cosmology is invisible. Several large ground-based ://m n imaging surveys will attempt to measure this effect over the coming decade, but reliable ras .o control of the spurious lensing signal introduced by atmospheric turbulence and telescope x fo optics remains a challenging problem. We address this challenge with a demonstration that rd jo pointspreadfunction(PSF)effectsonmeasuredgalaxyshapesintheSloanDigitalSkySurvey urn a (SDSS)canbecorrectedwithexistinganalysistechniques.Inthiswork,weco-addexisting ls .o SDSS imaging on the equatorial stripe in order to build a data set with the statistical power rg tomeasurecosmicshear,whileusingaroundingkernelmethodtonullouttheeffectsofthe at C/ anisotropic PSF. We build a galaxy catalogue from the combined imaging, characterize its alifo photometricpropertiesandshowthatthespuriousshearremaininginthiscatalogueafterthe rn ia PSFcorrectionisnegligiblecomparedtotheexpectedcosmicshearsignal.Weidentifyanew In s source of systematic error in the shear–shear autocorrelations arising from selection biases titu relatedtomasking.Finally,wediscussthecircumstancesinwhichthismethodisexpectedto te o be useful for upcoming ground-based surveys that have lensing as one of the science goals, f T e andidentifythesystematicerrorsthatcanreduceitsefficacy. ch n o lo Key words: gravitational lensing: weak–techniques: image processing–surveys– g y cosmology:observations. on M a y 2 9 , 2 0 1 4 namely the properties of the distribution of dark matter on large 1 INTRODUCTION scalesinrelativelylinearstructures–arenotreadilyobservable. Modern cosmologists can simulate the invisible implications of Fortheforeseeablefuture,themostdirectobservationofthese moderncosmologicalmodels(e.g.thosethatcanexplainthecos- darkcomponentsisthemeasurementofthegravitationaleffectsof micmicrowavebackground,includingKomatsuetal.2011)towhat darkstructuresontheimagesofdistantbackgroundgalaxies.These is generally agreed to be a high level of precision (and probably measurements are made almost exclusively via statistical estima- accuracy;cf.Lawrenceetal.2010).Theeasilyobservableconse- tion of the distortions in the ellipticities of background galaxies. quences of these models for observations of galaxies are not so This takes advantage of the fact that galaxies have no preferred easytocalculate(e.g.Rudd,Zentner&Kravtsov2008;Conroy& orientationinahomogeneous,isotropicuniverse. Wechsler2009;Simhaetal.2012),involvingastheydothephysics Lensingmeasurementshaveplayedasignificantroleinobserva- of the familiar but nevertheless stubbornly complicated baryons. tionalastrophysicsinthelasttwodecades,overarangeofscales Most of the precisely calculable components of these models – and physical regimes. Studies of galaxy evolution benefit from the ability to understand the dark matter haloes that host galaxies (e.g. Hoekstra, Yee & Gladders 2004; Hoekstra et al. 2005; Heymansetal.2006;Mandelbaumetal.2006a,b,2009;Leauthaud (cid:2) E-mail:huff.791@osu.edu etal.2012).Cosmologistshavenootherwaytodirectlymapthe (cid:2)C 2014TheAuthors PublishedbyOxfordUniversityPressonbehalfoftheRoyalAstronomicalSociety LensingI 1297 large-scale matter distribution, which is crucial for constraining oflensingbyalarge-scalestructureatroughlytheexpectedlevel. models of dark energy and modified gravity (Zhang et al. 2007; However, they also highlighted some of the systematic errors: in Reyesetal.2010).Onsmallscales,mapsofthematterdistribution particular, B-mode shear (which cannot be produced by lensing canbetieddirectlytotestsofthecolddarkmatterparadigmand at linear order and is thus indicative of systematic effects) was simulationsoftheformationandevolutionofdarkmatterhaloes. present at a sub-dominant but non-negligible level. Since then, Much has been made of the scientific potential of this tech- the weak lensing community has moved in the direction of both nique. Five years ago, weak lensing was identified by the Dark deep/narrowsurveyswiththeHubbleSpaceTelescope(HST)and EnergyTaskForce(Albrechtetal.2006)asthemostpromisingtool wide/shallowsurveysontheground.TheCosmologicalEvolution forconstrainingcosmologicalmodels.Severallargeground-based Survey(COSMOS)isthepremierexampleoftheformer:inaddi- and space-based survey proposals place a weak lensing measure- tiontotwo-pointstatistics(Masseyetal.2007a;Schrabbacketal. mentamongtheirprimarysciencedrivers,includingthePanoramic 2010), it has also produced three-dimensional maps of the mat- Survey Telescope and Rapid Response System (Pan-STARRS),1 ter distribution (Massey et al. 2007a) and the lensing three-point the Dark Energy Survey (DES),2 the Hyper Suprime-Cam (HSC; correlationfunction(Sembolonietal.2011).Excellentcontrolof Miyazakietal.2006)survey,theLargeSynopticSurveyTelescope lensingsystematicsinCOSMOSwasalsoachievedthankstothe (LSST),3 Euclid4 and the Wide-Field Infrared Survey Telescope small number of degrees of freedom controlling the point spread (WFIRST).5 function (PSF; mostly focus variation; Rhodes et al. 2007 ) and D o For all the promise, the technical challenges for these future detailed modelling of charge transfer inefficiency (Massey et al. w n experimentsremainformidable.Anorder-unitydistortiontoback- 2010). However, COSMOS covers only 1.6 deg2, and the small lo a groundgalaxyimagesisproducedbyaphysical,projectedmatter fieldofviewofHSTinstrumentsmakessignificantlylargersurveys de d overdensityof impractical.Theprincipalrecentground-basedcosmicshearpro- fro gramme has been the Canada–France–Hawaii Telescope Legacy m (cid:3)crit= 4πc2GdLddSLS, (1) Sfruormvedyif(fCerFeHntTsLuSb)s.etTshoefrethearCeFnHoTwLsSevdeartaal(Hcoosemkisctrasheetaarl.r2es0u0l6ts; http://m where dL, dS and dLS are the angular-diameter distance from the Sembolonietal.2006;Benjaminetal.2007;Fuetal.2008),and nra observer to the lens and source, and from the lens to the source, the CFHT lensing team is completing a reanalysis using recent s.o x respectively. For characteristic distances of approximately a Gpc, advancesinPSFdeterminationandgalaxyshapemeasurement. fo the critical surface density is 0.1gcm−2. Typical fluctuations in Inlightoftheeffortsshortlytobemadebylarge,expensivesur- rdjo thematterdensityfieldprojectedovercosmologicaldistancesare veystomeasurecosmicshear,weconsideritimperativetoshowthat urn athousandtimessmallerthanthis,soorder10-Mpc-scaledensity suchameasurementcanbeperformedaccurately,withoutsignifi- als fluctuationsintheuniversewilltypicallyproducechangesingalaxy cantcontaminatingsystematicerrors,fromaground-basedobser- .org ellipticities of the order of e ≈ 10−3–10−2 in magnitude. In the vatory.Thisgoalincludesdoingacosmicshearmeasurementwith a/ shot-noise-dominatedregime,theleading-ordercontributiontothe eachofthewide-angleopticalsurveysthatpresentlyexist.Tothis t C a varianceinthecorrelationfunctionoftheellipticitydistortionsis end,wehavere-coaddedtherepeatobservationsontheequatorial lifo stripe (Stripe 82) of the Sloan Digital Sky Survey (SDSS), using rn σ4 ia Var(ξ(cid:5))= Np2(cid:5)air. (2) msheetahromdoelaosguyretmhaetntws.ilOl ouprtgimoaizleisthtoeserednuewcectoh-eadsydsstefmoratpirceceirsrioorns Institu Forashallow((cid:4)z(cid:5)=0.5)galaxysurveywithshapenoisedueto arisingfromuncorrectedPSFanisotropiesbelowthestatisticaler- te o randomgalaxyellipticitiesσ(cid:5)≈0.3and100deg2ofskycoverage, rors. We begin by specifying our requirements in Section 2, and f T reducingtheshot-noisecontributionbelowtheexpectedcosmolog- describingthedatathatweuseinSection3.Adescriptionofthe ech ilceaalstsiagfneawlrpeeqrusirqeusaaresuarrfcamceindueten.sityofusablesourcegalaxiesofat cdoe-sacdridbietioonuranmdecthaotadlofgouree-smtimakaitnigngpitpweoli-npeofinotllfouwnsctiinoSnsecotfiosnta4r.aWnde nology Worse, for ground-based imaging surveys, the observed shape galaxyshapesinSection5.Demonstrationsofthedataqualityand on suitabilityforsensitiveweaklensingmeasurementsaredescribed M distortionsarisingfromatmosphericturbulenceandopticaldistor- a in Section 6. We conclude with lessons forfuture experiments in y tions from the telescope are typically of the order of several per 2 cent,withcoherenceoverangularscalescomparabletothatofthe Section7. 9, 2 0 lensingshapedistortions.Acompetitivemeasurementoftheampli- 1 4 tudeofmatterfluctuationsrequiressuppressingormodellingthese 2 DESIGN REQUIREMENTS coherentspuriousdistortionstooftheorderofonepartin103,and futuresurveyswillneedtodoafactorofseveralbetter. Weaklensingmeasurementsonlargescalesarevulnerabletoava- Achieving both the statistical precision and control of system- rietyofsystematicmeasurementerrors.Inordertomeasurecosmic atic errors that is required for such a measurement has proved shearonthescalesdescribedabove,wemustfirsthaveaclearidea to be a challenge. The early detections (Bacon, Refregier & of what the possible sources of these systematic errors are, and Ellis2000;VanWaerbekeetal.2000;Rhodes,Refregier&Groth towhatlevel(quantitatively)theymustbesuppressed.Thissection 2001;Hoekstraetal.2002;Brownetal.2003;Jarvisetal.2003) describesinturnthemostcommongenericsourcesofmeasurement showed the promise of the method and confirmed the existence errorrelevantforweaklensing,andlaysoutquantitativemethods fordetectingtheirpresenceinourfinalcatalogue. ThePSF6 oftheSDSSexhibitssignificantspatialandtemporal 1http://pan-starrs.ifa.hawaii.edu/public/ variations across the entire survey. We model these effects as a 2http://www.darkenergysurvey.org/ 3http://www.lsst.org/ 4http://sci.esa.int/euclid/ 6Hereweusetheterm‘PSF’todenoteallcontributions:theatmosphere, 5http://wfirst.gsfc.nasa.gov/ optics,trackingerrors,chargediffusionandpixelization. MNRAS440,1296–1321(2014) 1298 E.M.Huffetal. spatiallyvaryingconvolutionkernelG.TheobservedimageI(x)at 2.5mtelescopeatApachePointObservatoryinSunspot,NewMex- somepositionxisrelatedtothe‘true’imagefby ico(Gunnetal.2006).TheSDSScamera,describedinGunnetal. (cid:2) (1998),imagestheskyinfiveopticalbands(u,g,r,i,z;Fukugitaetal. I(x)= f(y)G(x− y)d2y, (3) 1996; Smith et al. 2002) with the charge-coupled device (CCD) detectorsreadingoutatthesiderealrate.Eachpatchofskypasses whereGistheconvolutionkernelappropriatetotheregionofsky insequencethroughthefivefilters(intheorderr,i,u,g,z)alongone underexamination. ofthesixcolumnsofmosaickedCCDs,andisexposedonceineach OneeffectofaspatiallyvaryingPSFGistoproduceaspurious filterfor54.1s.Thesiteismonitoredforphotometricity(Hoggetal. shearfielddeterminedbytheatmosphereandtelescopethatisstatis- 2001;Tuckeretal.2006).Dataundergoqualityassessment(Ivezic´ ticallyindependentofandsuperposedupontheundistortedgalaxy et al. 2004), and final calibration is done using the ‘ubercalibra- shapepattern.Pointsources(starsandcompletelyunresolvedgalax- tion’procedurebasedonphotometryofstarsinrunoverlapregions ies–wehavenoneed,atpresent,todistinguishthese)sampleonly (Padmanabhanetal.2008).WeusethedatafromtheseventhSDSS thefieldsourcedbyG,andsocanbeusedtoconstrainamodelfor data release (Abazajian et al. 2009), with an updated calibration thesystematicsfield.Anyuncorrectedadditiveshearcontribution fromthesubsequentdatarelease. due to the ellipticity of G will produce a correlation between the Thefootprintofonenight’sobservingissixcolumnsofimaging measuredgalaxyandpointsourceshapes.Thisadditiveshearwill thewidthofoneCCD(13.52arcmin)separatedbyslightlylessthan D bestatisticallyuncorrelatedwiththetruecosmicshearsignal. oneCCDwidth(11.65arcmin).Imagingtakenduringacontinuous ow n Themaskingstepsofthecatalogue-constructionprocedurecan periodoftimeononenightiscollectivelytermedarun;eachsepa- lo a alsoproduceasignificantshapeselectionbias.Forthephotometric ratecolumnofimagingis,sensibly,acameracolumn(or‘camcol’), de d pipelineusedhere,maskedregionsaredefinedassetsofpixels;a andtheimagingalongeachcameracolumnischoppedforprocess- fro galaxy is rejected from the catalogue if the set of pixels making ingpurposesinto8.98arcminlongframesorfields.Successiveruns m up a galaxy intersects the set of masked pixels. On the masked areinterleaved,inordertofillinthegapsbetweencameracolumns. http region boundary, galaxies aligned across the mask boundary are Pairsofinterleavedrunsalongthesamegreatcirclearestripes(each ://m morelikelytoberejectedfromthecataloguethangalaxiesaligned ofwhichhasanorthandasouthstrip). n ra along it producing a spurious shear. This will affect both stars s.o amnudchgallaarxgieerst,hbaunttthhaeteofnfesctatrosn(assptuhreioduisspgearlsaixoyneinllimpteiacsituireesdwsitlelllbaer 3.2 Stripe82 xford jo ellipticities is very small). This mask selection bias produces an MostoftheSDSSimagingdatawereacquiredinthenortherngalac- urn additiveshear,whichwillalsobestatisticallyuncorrelatedwiththe ticcap,withgalacticlatitude|b|>30.Forcommissioning,anddur- als truecosmicshearsignal. ingsiderealtimeswhentheprimarysurveyregionwasunavailable, .org Thesetwoeffectsentertogetherasanadditivetermintheshape thetelescopefrequentlyimageda2◦.5widestripeofskyalongthe a/ clusteringstatistics,as celestialequatorwithrightascension(RA)intheinterval−50◦ < t C a ξmeasured(θ)=ξcosmic(θ)+ξsystematics(θ). (4) RobAseσ1.Thisfunctionalformmanifestlyintegratestounity, kernelK(x)isreal. andhasafractionfw ofthelightinthe‘large’Gaussian.Thetwo Thesecondproblem–thevariationofthePSFacrossthefield Gaussianshavewidthsσ1andσ2,respectively,withσ1<σ2. – is handled by taking the reconstructed PSF on a grid of 8 × 6 The parameters of the double-Gaussian were adjusted by trial points separated by 298 pixels (2 arcmin) in each direction, and anderrorsothatacompactlysupportedkernelK(13×13pixels) constructingagridof48kernelsK.Thekernelsaretheninterpolated canachieveG⊗K≈(cid:8)forawiderangeofrealPSFsGdelivered bilinearlybetweenthefournearestgridpoints,andthenthefinal bytheSDSS.Themostcriticalparameteristhewidthofthecentral imageF(x)isconstructedaccordingto Gaussian,σ .Thisisthemainparametercontrollingtheseeingof (cid:2) 1 thefinalco-addedimage:ifitissettoohigh,thenmanygalaxies F(x)= Kx(y)I(x− y)d2x, (11) D becomeunresolved,whereasifitissettoolow,thenalargenumber ow offieldswithmoderateseeingwillhavetoberejectedbecauseit whereKx isthekernelreconstructedatpositionxinthefield. nlo willbeimpossibletofindakernelKthatachievesthetargetPSF TheconvolutionkernelwillnotcapturePSFmodelfluctuations ad e wiTthhoeutPdSrFamsiazteicdailsltyriabmutpiolinfyinintghethrebnaonisdei.sshowninFig.1. ofunnscctaiolenssboevleorwth2eacrchmipi,nf.eSaitnucreesthaetSthDeSaSrcmmoinduetlePsScFaslearaenqdusamdraallteicr d from arenotcapturedanyway.Weshowbelowthat,evenatθ=1arcmin, h ttp 4.1.2 Theconvolutionkernelanditsapplication tshmearllemcoaminpinagredPStoFtvhaerieaxtipoencstendosthcoatp-ntuoriesdeebryrotrhseinketrhneeltwaore-pvoeirnyt ://mn Equation (6) can formally be solved in Fourier space by taking statisticsatthosescales. ras.o theratio,K˜(k)=(cid:8)˜(k)/G˜(k),wherethetildedenotestheFourier Obviouslytherewillbeinstancesinwhichthekernelreconstruc- xfo tcroamnsefsorwmithantdwokwtheell-wkanvoewvnepcrtoorb.leUmnsf.orOtunneaitseltyh,atthiifstshiemPpSleFidheaas ttthihoeenGriesasuunslotsitinaggno-okwedreenigneholstu.e8gdhInm.Toohrmdeeerernfttosoroecf,oatnhsseterrtueocstfidtchuueatssle(cid:8)mcuu−ststG,bwe⊗eapKcpol,niies.ided.teor rdjourna dtphiovewTifidheeeerlobdsony,lliuyG.˜etui.(opkGn)toit=noate0chq.eeurTfiatahtriiesontnowptr(hao6evb)relfneiosumrmtmhibsaatletlhrtyhakedtmeiaPnpxSs,etFtnehdaevdsnaoroiitnfeistsxaskimlaiosnwpgwolayseslsaliibcmalrseoptsyloes. Mαβ = π1σ12 (cid:2) [(cid:8)−G⊗K](x)σx11ααx+2ββe−x2/2σ12d2x. (12) at Cals.org/ ratioinFourierspace,weminimizetheL2normoftheerror, WerequirethatthePSF-inducedellipticitybesmallerthan10−3 lifo (cid:2) inordertoensureacleancosmicshearsignalwiththischaracteristic rn ia E1= |(cid:8)(x)−[G⊗K](x)|2d2x≡(cid:10)(cid:8)−G⊗K(cid:10)2, (8) strength.Thecutsareasfollows. Ins (1) WerejectanentirefieldiftheSDSSsoftwareusedtodeter- titu subjecttoaconstraintontheL2normofthekernel: te (cid:2) amignoeotdhePPSSFFm(tohdeeploisntatgheesstianmgplep-eippeolcinhei,moragPSinP)gf,aoilredwtaosdfeotrecremdintoe of T E2= |K(x)|2d2x≡(cid:10)K(cid:10)2. (9) usealow-orderfittothePSF(PSP_STATUS!=0). ech n If the input noise is white (which is a good approximation), then (2) We reject the cases where the PSF residual is too large re- olog gardlessofthemoments,i.e. y the noise variance on an individual pixel in the convolved image o n isE timesthenoisevarianceintheinputimage.Roughlyspeak- (cid:10)(cid:8)−G⊗K(cid:10)2 M ing,2forkernelsthatattemptto‘deconvolve’theoriginalPSF,and (cid:10)(cid:8)(cid:10)2 >CUT L2. (13) ay consequentlyhavelargepositiveandnegativecontributions,E will 29 comeouttobeverylarge.WeadoptarequirementthatE ≤12.For Thiscutisintendedtoguardagainstpathologicalcases,wherethe , 20 2 PSF that results from the rounding kernel passes the other cuts 1 kernelsthatpoorlyapproximatethetargetPSF(cid:8),E willbevery 4 1 below,butisstillapoorfittothetarget.Thevalueofthiscutwas large.TheproblemofminimizingE subjecttoaconstraintonE is 1 2 determinedbytrialanderror;ontrialco-addsconsistingofasmall mosteasilysolvedbytransformingtotheFourierdomainandthen fractionoftheStripe82footprint,weappliedtheroundingkernel usingthemethodofLagrangemultipliers: andmeasuredtheellipticitiesoftheresultantstars.Thevalueofthis G˜∗(k)(cid:8)˜(k) cutwasadjusteduntiltheresidualstellarellipticitieswerereliably K˜(k)= |G˜(k)|2+(cid:9). (10) lessthan10−3. (3) We reject the cases where the Gaussian-weighted offset is Herethepositiverealnumber(cid:9)istheLagrangemultiplierandits morethanCUT_OFFSETσ ,i.e. value is adjusted until E =1. (cid:9) plays the role of regulating the (cid:3) 1 2 deconvolution;indeedonecanseethatforFouriermodespresent M2 +M2 >CUT OFFSET. (14) 01 10 intheimage,G˜(k)(cid:13)=0,wehavelim(cid:9)→0+K˜(k)=(cid:8)˜(k)/G˜(k). This cut removes the cases where significant astrometric offsets Tosummarize,equation(10)findstheconvolvingkernelKthat makes the final PSF G ⊗ K as close as possible (in the least- were introduced during stacking. The ellipticity errors induced squares sense) to (cid:8) without amplifying the noise. The kernel is truncatedintoa13×13pixelregioncentredattheorigininorder 8ItshouldbenotedthatresidualanisotropiesfromtheLagrangemultiplier to avoid boundary effects and to prevent problems such as bad (cid:9)areaffectedbythequalitycutsdescribedbelow. MNRAS440,1296–1321(2014) LensingI 1301 Table1. ParametersforthePSFrepairindifferentfilters. Parameter u g r i z Units TargetPSFparameters σ1(PSF_SIZE) 1.80 1.40 1.40 1.40 1.40 pixels σ2(PSF_SIZE_WING) 5.10 5.10 5.10 5.10 5.10 pixels fw(FRACWING) 0.035 0.035 0.035 0.035 0.035 pixels FWHMoftargetPSF(cid:8) 1.68 1.31 1.31 1.31 1.31 arcsec 50percentencircledenergyradius 0.86 0.67 0.67 0.67 0.67 arcsec Kernelacceptanceparameters CUT_L2 0.001 0.0025 0.0025 0.0025 0.0025 CUT_OFFSET 0.04 0.01 0.01 0.01 0.01 CUT_ELLIP 0.002 0.0005 0.0005 0.0005 0.0005 CUT_SIZE 0.01 0.0025 0.0025 0.0025 0.0025 CUT_PROF4 0.04 0.01 0.01 0.01 0.01 D Co-additionparameters o w DELTA_SKY_MAX1 0.5 0.25 0.25 0.25 0.25 nmgyarcsec−2 nlo DELTA_SKY_MAX2 0.04 0.02 0.02 0.02 0.02 nmgyarcsec−2 ad e d fro by such offsets scale as the square of the astrometric offset, with 4.2 Noisesymmetrization m h aWnitohrdtheirs-uanssituympprtoipoonr,ttihoenaclhiotyicceoonfsctauntth(eBreerrnesdtuecines&thJeacrevnistr2o0id0i2n)g. Iatniismaagweehlla-sknboewenncfoarcrtecintedwetoakhalevnespinegrfethctaltyecviercnuliafrthcoenPcSenFtriinc ttp://m errorintheellipticitybelowourfidicualceilingof10−3. isophotes, it is possible to produce spurious ellipticity if there is nra (4) We reject the cases where the ellipticity of the final PSF anisotropiccorrelatednoise.Forexample,ifthePSFiselongated s.ox e(cid:4)xceedsCUT_ELLIP,i.e. inthex1 directionandis‘fixed’bysmoothinginthex2 direction, ford Tcaohcm(ierMcpquou02lnaane−rntGittMyaou2Cf0sU)so2Tiua_+rnEPwL(S2LeMFiIgP1(h1iit.)seσ2.a1>fn.oICnn=U-thaTde0aE)a,pLbtaLsinevIndePce.feloloirpfsttimhceiatslylmmraeelslai-dsauumraepldsl,iwt(ut1hid5tihes) mtxgh1aoe.larTrexehysgiuscalelctainanxntgrieolesmidatadhipsatotlhaaa(r1gps)epmrceeaionnrrettarholceiiogdxrni2rneetdglhaabitninioatntshheseesi,xnxi12ntdhtwihereahxcni2ctihxdo1intrhd,eetichrteeuirocsrtnoiyortinheol;andnaitnnhignde ajournals.org/ w (2) biases in which noise fluctuations tend to be elongated in the t C CqchuUo(aT5ns_)etSintWIycZeuwEtor,cueoijl.reedr.cebtsepteohqneudasclattosoeohsuarlwffihtdheuerecaidatalhpeetlilviPpeStimFciotymsizceeenitlieenrllgrioportfice1ixt0yc−;e3eo.dusr x(inw2thdhiiecrehxc2itndiocirnree,castsioeontihtwsatlhiepkroeelsaiihstiovnoeedgnaootfiivsdeeefltfleuucccttituouanat)tiiotoennnssd(owtnohimtcohapkdoeefcitraeaaglsiagelnatxehdye alifornia In likelihoodofdetection)makethegalaxyalignedinthex direction. s |M02+M20−M00|>CUT SIZE. (16) For a detailed description of noise-induced ellipticity1biases, see titute Kaiser(2000)orBernstein&Jarvis(2002).Thesephenomenacan o ThiscutlimitstheeffectofPSFdilutioncorrectionerrors(i.e.errors allmimiclensingsignalsandhenceshouldbeeliminatedfromthe f Te inR2;seeSection4.9below).Theeffectofthedilutioncorrection data.Ourmethodofdoingthisistoaddsyntheticnoisetoeachfield chn onthemeasuredellipticityissmallerthanthatofthePSFellipticity soastogivethenoiseproperties4-foldrotationalsymmetry.Tobe olo byafactorofthegalaxyresolution.Thiscutensuresthatdilution precise,wewantthepowerspectrumofthetotalnoise(actualplus gy ocourrraedctoipotnederrreosroslountiothneliemlliitptRici=ty0ar.3e3b3e.low10−3forgalaxiesnear synthetic)tosatisfy on M 2 a (6) We reject the cases where the radial profile of the PSF is PN(k)=PN(e3×k), (18) y 29 severelyinerrorasdeterminedbythefourthmoment,i.e. , 2 |M40+2M22+M04−2M00|>CUT PROF4. (17) pwrhoedruecte3opiseraatvioenctoe3r×norromtaatlestoathveecptolarnbeyo9f0t◦h.eEivmeangeth;otuhgehcritosiss 014 notcircularlysymmetric,thisissufficienttoguaranteezeromean Thevalueofthiscutwasalsodeterminedbytrialanderror,aspart ellipticityforapopulationofrandomlyorientedgalaxiesbecause ofthesameprocedureaswithCUTL2above. ellipticity reverses sign under 90◦ rotations.9 In principle, m-fold Thespecificvaluesoftheparameterschosenforthecutsdepend symmetry for any integer m ≥ 3 would suffice; however, 4-fold onthebandandareshowninTable1.Thetightestconstraintson symmetryistheonlypracticalpossibilityforacamerawithsquare thequalityofthePSFareing,r,iandzbands(randiareused pixels.Forobviousreasons,wewouldliketoachievethisbyadding tomeasuregalaxyshapes).Intheuband,wheretheaverageimage theminimalamountofsyntheticnoisepossible. qualityismuchlowerthanintheotherbands,moreliberalcutscan beappliedbecauseweareinterestedprimarilyinthetotalflux,not theshape.Alsothereismoretogainfromliberalcutsbecausethe signal-to-noiseratiointheubandislower.Nevertheless,aserious error in the size of the PSF will result in erroneous photometry, 9Ingrouptheorylanguage,thenoisepropertiesaresymmetricunderthe andspuriousellipticitycouldintroducecolour/photo-zorselection 4-foldrotationgroupC4,whichisasubgroupofthefullrotationsSO(2). biases that depend on galaxy orientation, so some cuts must be Theconditionforzeromeanellipticityduetonoiseisthatellipticityfall applied. intooneofthenon-trivialrepresentationsofthenoisesymmetrygroup. MNRAS440,1296–1321(2014) 1302 E.M.Huffetal. Thesimplestwaytoachieveequation(18)istodecomposethe sothatthediffractionspikesappearatpositionanglesof45◦,135◦, powerspectrumintoitsactual(‘act’)andsynthetic(‘syn’)compo- 225◦ and 315◦ in the altitude–azimuth coordinate system. There- nents: fore,intheequatorialruns,theorientationofthediffractionspikes relative to equatorial coordinates changes depending on the hour PN(k)=PN(act)(k)+PN(syn)(k). (19) angle of observation. If no correction for this is made, then af- The actual component is the white noise variance v in the input terco-additionofmanyruns,evenmoderatelybrightstarshavea image,smoothedbytheconvolvingkernel: hedgehog-likepatternofdiffractionspikesatmanypositionangles thatcanaffectasignificantfractionofthearea. P(act)(k)=v|K˜(k)|2. (20) N Ourprocedureforremovingdiffractionspikesisasfollows.We SinceKisreal,thispowerspectrumhas2-foldrotationalsymmetry: first identify objects with a PSF flux (i.e. flux defined by a fit to P(act)(k)=P(act)(−k).Theminimalsyntheticnoisepowerspectrum the PSF) exceeding some threshold (corresponding to 9.7 × 105, N N thatsatisfiesequation(18)isthen 8.5×105,2.2×105,1.7×105and1.1×106electronsinfiltersr, (cid:5) (cid:6) i,u,zandg,respectively).Aroundtheseobjects,wemaskacircle PN(syn)(k)=max PN(act)(e3×k)−PN(act)(k),0 . (21) ofradius20pixels(8arcsec)andfourrectanglesofwidth8pixels (3arcsec)andlength60pixels(24arcsec).Therectangleshavethe Gaussian noise with this spectrum can be obtained by taking its objectcentroidatthecentreoftheirshortside,andtheirlongaxis D squareroot, o (cid:3) extendsradiallyfromthecentroidinthedirectionoftheexpected w n T˜(k)= P(syn)(k), (22) diffractionspike. loa N d e d andtransformingtoconfigurationspaceT(x).Thenonegenerates fro whitenoisewithunitvarianceandconvolvesitwithT.Sincethe 4.4 Resampling m h PisSiFntaenrpdohlaetnecdefKromvarthieessaacmroes8st×he6figerlidd,oTfmreufestreanlscoevpaoriyn;tsitassvuasluede Icnomomrdoenr tpoixceoli-zaadtdionim. Iadgeeasl,lyw,ewemuwsotufildrstlikreesathmisplpeixtehleimzatiionntotoa ttp://m forTKh.e Gaussian white noise was generated using Numerical bebothconformal(nolocalshapedistortion)andequal-area(con- nras.o venient for total flux measurements). Unfortunately because the x Recipes gasdev modified to use the ran2 uniform deviate gen- fo eosarnamttoehres(ePreurdenws,scaeastmnaeclv.oe1l,r9ufi9s2eel)dd.Ttnwhuiemcesbeieenrdtahwneadrseficdhlutocestrieotnonsbg,yuaanardfaotnhrtmaeteutlhtaheabstaasmtehdee sHakrooywuinesdvcetuhr,revseeidnq,cueaittooisru(ri|mδa|pn≤oals1ys.si◦bi3sl)e,uwtsoeesaccaahninecavorermobweovthrearonyfgcetlhooessfeedbceyoccnlihdnoiatotiisooinnnsgs. rdjournals saesqeuqeunecnecweiollfb2e04g8en×era1t3e6d1ifGthaeusssoifatnwdareeviiastrees-risung.eFnoerraetaecdh;sfiienlcde, acocnyfloinrmdrailc)aolrpLroajmecbteiortn(;ptehrefeocbtlvyioeuqsuaclh-oairceea)i.sInMoeurcractaosre,(ptheerfMecetlry- a.org/ thereare128rowsofoverlapbetweensuccessivefields,wefillin catorprojectionwouldresultinthepixelscalebeingdifferentby t C tfhroemlatshte1n2e8xrtofiwesldo.fIteaischalsfioeledsswenitthiatlhtehafitrsthte20p4er8io×do1f28thedegveinateers- e(cid:13)rrθo/rθis=tw2.i6ce×th1is0,−o4ra5t.t2he×E1q0u−a4t.o)rTvheersLusamatbδer=tp±ro1j◦e.3c.ti(oTnhweoaurelda aliforn aogterodnreerbraeotfoloarsnf,gesweirn×cthe1a0on1th2th)e,erawtoriestaeqlutnhirueemmsabemenrteowsfyhpnicitxhheeiltssicnino‘nttofhuieslfies’ullrpevdaetybteyr(nomfwatnhilyel paisrsephsaeerartvriceousflhaγarlpy=ess2ea.rt6itoh×uesE,10sqi−un4acteaotreδbituh=tetr±hce1oc◦.uo3l.odrNdifeinintahetceeerssoysfsatrtehymebsweeopcuorolrdrbelhceatmevdes ia Institute inthefluxorshapemeasurements.WehavechosentheMercator o repTehaetiitmsealgfethFro(uxg)haoftuetrtahdedsituirovneoy.fthesyntheticnoiseisakImage. projectionbecausethesmallcosmicshearsignalmeansthatweare f Tec much more sensitive to a given percentage error in shear than in hn flux.Also,afluxerrorof5.2×10−4 isinsignificantcomparedto olo g 4.3 Single-imagemasking theerrorintheflat-fields,sothereisnopointineliminatingitatthe y o Once the kernel-convolved, noise-added image (kImage) is con- expenseofcomplicatingtheshearanalysis. n M Thescaleoftheresampledpixelsmustbesmallerthanthenative a structedforeachrunthatwillcontributetotheco-addsatagiven y position,itisnecessarytoconstructamaskbeforeco-addition.The mpiaxteilosnc.aHleowonevtehre,CitCisDde(s∼ir0a.b3l9e6faorrcistenco)tintoobredetorotosmpraelsle,rsvinecienfthoirs- 29, 2 mask must remove the usual image defects as well as diffraction 0 1 increasesthedatavolumewithnoincreaseininformationcontent. 4 spikes.Itisconstructedasdescribedinthissection,andistermed ItshouldalsonotbenearlyequaltotheCCDscaleinordertoavoid thekMask. WebeginbymaskingoutallpixelsinF(x)forwhichtheconvo- productionofamoire´patternwithlarge-scalepower.Wehaveused 0.36arcsec. lution(equation11)integratesoverabadpixel.SinceKhascompact support–itisnon-zeroonlyina13×13pixelregion–thismeans Theactualresamplingsteprequiresustointerpolatetheimage that for each bad pixel in I(x) we mask out a 13 × 13 block in fromthenativepixelizationontothetargetMercatorpixelization. F(x).Ourdefinitionofa‘badpixel’isonethatisoutofthefield; Thisisdoneby36-pointsecond-orderpolynomialinterpolationon the6×6gridofnativepixelssurroundingthetargetpixel.10Atarget wasinterpolatedbyPHOTO(usuallyduetobeinginabadcolumn); pixelisconsideredmaskedifanyofthe36surroundingpixelsare issaturated;ispotentiallyaffectedbyghosting(viathefpMghost masked. flag);wasnotcheckedforobjectsbyPHOTO;isdeterminedbyPHOTO to be affected by a cosmic ray; or had a model subtracted from it. Note that the first cut means that a six-pixel region is rejected aroundtheedgeofthefield. 10Polynomialinterpolationonanequallyspacedgridofpointsconverges The second and more sophisticated mask is applied to remove tosincinterpolationinthelimitthatthenumberofgridpointsistakento diffraction spikes from stars. The secondary support structure re- infinity.Thisise(cid:7)asilyseenfromtheLagrangeinterpolationformulaandthe sponsibleforthediffractionspikesisonanaltitude–azimuthmount, infiniteproduct, ∞n=1(1−x/n)(1+x/n)=sin(πx)/(πx). MNRAS440,1296–1321(2014) LensingI 1303 4.5 Additionofimages An example of a co-added image, and comparison to a single- epochimage,isshowninFig.2. Afterresamplingtheimages,thenextstepistocombinethemto producetheco-add.Thecombinationproceedsinthreesteps:com- parisonofimagestoreject‘bad’regionsthatwerenotmaskedin earlier stages of the analysis; relative sky estimation; and stack- 4.6 Additionalmasking ing. Note that bad regions must be explicitly rejected: ‘robust’ Beforeconstructingthephotometriccatalogues,wezeroallpixels algorithms such as the median are non-linear and slightly bi- contaminated by bright stars in the Tycho catalogues (Høg et al. ased, and result in a final co-added PSF that depends on ob- 2000),replacingthemwithrandomnoiseofappropriateamplitude. ject flux and morphology, which is not acceptable for lensing Pixelsmaskedinthismannerhavethe‘INTERP’bitsetintheinput studies. fpMfiles,sothatthedownstreamanalysiscanexcludeobjectsthat Rejectionofbadregionsiscriticalbecauseitispossibleforsome incorporatepixelsfromamaskedregion.PixelsthatarekMasked seriousdefectssuchassatellitetrailsto‘leakthrough’earlierstages (according to one of the above criteria) also have ‘INTERP’ bits oftheanalysisandnotbekMasked.Rejectionatthisstageisalso set.Thisfinalstepresultsinacataloguewithacomplexgeometry, thebestwaytoeliminateSolarsystemobjects,mostofwhichwill whichwillbedemonstratedexplicitlyinSection4.9. be known, but which are not easily identified in the single-epoch D fpCs but of course will not show up at the same coordinates in o w successiveruns.Wefirstbineachinputimageinto4×4resampled n lo pixels.Wethencomparethebinnedimagesandrejectthebright- 4.7 Photometriccatalogues a d e est or faintest image if it differs by more than DELTA_SKY_MAX1 d from the mean. When this rejection is done, we actually mask Once each co-added image is constructed, we detect objects us- fro ing the catalogue-construction portion of the SDSS photometric m a 20× 20 resampled pixel region around the affected area. (We h fteonunindcothmatplwetietlhyoumtatshkiesdpbaedcdaiunsgerthegeyiopna,sssaetdeltlhitreoutgrahiltshewceorernoerfs- tpiiopnelainnde,oPbHjOeTcOt-mFReAaMsuErSe.mThenetdpertoacilessosfaFrReAdMeEsSc’rsibceadtamloogrueefcuollnysterlusce-- ttp://m where (Stoughton et al. 2002; Lupton et al., in preparation). It is n of some 4 × 4 regions and did not sufficiently affect the mean ra flux.) nevertheless useful to review the important parts of the FRAMES s.ox NiNmeaxgtews.e11cTohmispudtieffethreendcieffmeruesntcbeeindestkeyrmleinveedl aamndonregmaollveodfbthee- algPoHrOitThOm-FsR.AMESrequiresasinputasetoflongintegerimages,and fordjo cause otherwise a masked pixel in an image with below average aconsiderablearrayofinputsdescribingthepropertiesofthetele- urn sthkeyrwelialltiavpepsekayralsevaeblr–igahntsopfotetninntehgeleccot-eadddsteedpiminacgoe-.aWddeitcioomnp–uates ssccorippetioanndotfhteheobteselersvcinogpecoPnSdFi.tiFoonrs.sPinrginlec-ieppaolcahmdoantga,thFeRsAeMiEsSaudsees- als.org a principal-component decomposition of the variation of the PSF a/ follows. For each pair (i, j) of co-added images, we compute the acrossfiveadjacentfields.Thecomponentsofthisdecomposition t C dpilfefderpeinxceelbmloacpkFs.iT−hiFsjisantadketankaesthaneemsetidmiaanteinof1t2h5es×ky1d2i5ffreerseanmce- aargeeacloloowrdeidnattoesvaarcyroasssaeapcohlyfnraommei.aAl(stythpeiccaoll-yadqdueaddirmataicg)eisnhtahveeitmhe- aliforn S − S. From these differences, we obtain the unweighted least- ia aTtshqbiheuseoaitmlrhueestijemasnoaslgokuefytittohhlneeesvfqeoeurllaetnvhcteeailtnssynkiSoys¯td−leebvnSeeoiltsiedn{detSetbeir}yrpm,oS¯uliapn=teetdod(cid:8)atobnNiy=aa1dpStdahiiri/ttsiiNvceup.lroWaofrcfespeeaodtdiun(drtteht)xoe. sppfaoabirpmironmevecleiiontp,afeawrlgtuehceseotseimPstmaSparpFgoldeyniotneuunPesbtSve.leeFFt-rhoGyrereaistfmuauasrslagsttgiiencaetogn,PmtfhfiSrpitoFsutmtotpaaartttrgihhoaeeemntPrPoeoStSfueFFonr;sbdi.ajsiensscgtttohkpriseerordinpsaeestlrhttaeihepeesp,xfilitarehcsdett Institute of Tec byfour-pointinterpolationfromthenearestblockcentres.Anentire h blockismaskedoutif|S¯ −Si|>DELTA_SKY_MAX2andifitisan GaFuRsAsMiaEnSwfiirdstthssmdeosoctrhibesintghteheimPaSgFe.Cwoiltlhectthioennsaorfrocwonenreocftetdhpeixtwelos nolog extremalvalue(eitherthehighestorlowestskyvalue). y greater than seven times the standard deviation of the sky noise o Thestackingoftheimagesworksbytheusualformula n aremarkedasobjects.Eachobjectisgrownbysixpixelsineach M Ftot(x)= (cid:8)Ni=1wwtoit((xx))Fi(x), (23) doifrepcetaiokns.lFesosrethaacnhothbrjeeectt,itmheeslisthteofloccoanlnestcatneddapridxedlesviisatthioenncouflltehde ay 29, 2 (cid:8) sky. 01 wherewtot(x)= Ni=1wi(x)andw i aretheweights.Becausethe Inordertoavoidincludingobjectsthatrepresentrandomnoise 4 noise is correlated, the optimal weights are scale dependent; we fluctuations, catalogue galaxies are required to have statistically havechosentheoptimalweightsinthelimitofsmall k,i.e.large significant (>7σ) detections in both the r and i bands. Note that scales.Thatis,wi =v−1 wherevisthenoisevarianceinimagei. this is a higher threshold than the >5σ cut used at this stage in Forphotometryoflargeobjects,wtotcanbethoughtofasaninverse thestandardsingle-epochSDSSprocessing.Thiswasnecessitated whitenoisevariance,i.e.themeansquarenoisefluxinaregionof bythefactthatthepixelnoiseinthekImagesiscorrelatedbythe area(cid:15)is1/wtot(cid:15).However,forsmallobjects(whicharealwaysour convolutionprocess. concern),thisisnotthecaseandtheerrorbarsmustbecomputed In the standard SDSS pipeline, FRAMES then re-bins the image fromthemeasurednoisepropertiesoftheco-add. and repeats the search. We choose not to use objects found in thismanner,astheshapemeasurementsoftheseverylowsurface brightnessgalaxieswouldnotbereliable. This detection algorithm is repeated in each filter separately. 11While the fpC images generated from single-epoch data by PHOTO are Objectsdetectedinmultiplebandsaremergedtocontaintheunion skysubtracted,inpracticethisinitialskysubtractionwasnotsufficiently smoothtoavoidtheappearanceoflargebackgroundbrightnessvariation of the pixels in each band if they overlap on the sky. The list of intheco-addimages.Thisshouldnotbesurprising,asPHOTOhasknown peakpositionsineachbandispreserved.Thecentreoftheresulting sky-subtractionproblems(Aiharaetal.2011). singleobjectisdeterminedbythelocationofthehighestpeakinthe MNRAS440,1296–1321(2014) 1304 E.M.Huffetal. D o w n lo a d e d fro m h ttp ://m n ra s .o x fo rd jo u rn a ls .o rg a/ t C a lifo rn ia In s titu te o f T e c h n o lo g y o n M a y 2 9 , 2 0 1 4 Figure2. Acomparisonofaco-addedimage(upperpanel),itsinversevariancemap(middlepanel)andasingle-epochinputmap(bottompanel).The co-addedimageiscentredonRA01h56m34s.8,Dec.−01◦10(cid:17)35(cid:17)(cid:17)(J2000).Eastisattop;theimagespans7.7×2.4arcmin.Thetoppanelshowsther-band image(units:nmgyarcsec−2,squarerootstretch),andthecentrepanelshowstheinversevariancemap(units:nmgy−2arcsec2,linearstretch).Notethedark verticalstripesintheinversevariancemapproducedbybadcolumns,andthesquarepatchesduetocosmicrayhitspropagatingthroughthemaskingprocedure. Thespiralgalaxynearthecentreoftheimageisofmagnituder=17.4.Thesingle-epochimageisfromstrip82S,run4263,field310,camcol1(acquiredon 2003November20atairmass1.21).TheimageshownisthefpCimagefromrerun40ontheDataArchiveServer(units:uncalibrated,linearstretch).The samenumberofpixelsareshown,butnotethatthesingle-epochimageisatthenativepixelscale(0.396insteadof0.36arcsec)andhenceshowsaslightly largerarea. rband.Objectswithmultiplepeaksaredeblended:thedeblending Onceacompletelistofdeblendedpeaks(hereafterobjects)has algorithmassignsimagefluxtoeachpeakintheparentobject.12 been constructed, the properties of each peak are measured. For the purposes of this paper, the most important outputs are the MODELFLUXandMODELFLUX_IVARparameters,13 whicharedeter- 12ShortdescriptionsoftheSDSSdeblendingcanbefoundinStoughton minedfromthetotalfluxinthebest-fitting(PSF-convolved)galaxy etal.(2002,section4.4.3)andontheSDSSwebsiteathttp://www.sdss. org/dr7/algorithms/deblend.html.Adetailedpaperdescribingthedeblender isforthcoming(Luptonetal.,inpreparation). 13http://www.sdss3.org/dr8/algorithms/magnitudes.php MNRAS440,1296–1321(2014) LensingI 1305 Table2. Maskingradiusasafunctionofap- (iv) allobjectswhereabadpixelwasclosetotheobjectcentre parentstellarmagnitude. (INTERP_CENTER)ineitheroftheroribands,orboth (v) allobjectsthatareparentsofblends(i.e.measuredagainin Magnituderange Maskingradius(arcsec) termsoftheindividualchildobjects); (vi) those for which the observed r-band magnitude is greater r,i<12 100 120.47arcsecpredictsthatthisstandarddeviation nlo apostagestampimageforeveryuniqueobjectdetectedinthecat- should result in a 1σ variation of ±3 per cent in the galaxy den- ad alogue,exceptforthoseobjectsforwhichthedeblenderalgorithm sityand±1percentinthemeanredshift(cid:4)z(cid:5).Thesystematicerror ed failed. introduced by non-uniform depth, which should be second order fro m intheamplitudeofvariations,isexpectedtobenegligibleforthe h 4.8 Lensingcatalogueconstruction ptruurepoofsefsutoufrethperoSjDecStSs.analysis.Notehoweverthatthiswillnotbe ttp://m n Acof-taedrdPeHdOTimO-aFgReAsM,EwSehaatstecmopntsttrouecltiemdinanateobspjeucrtiocuastadleotgeucetiofnrosm,sttahres proMceasnsyiosfttohepsreodcuuctseatrweoapspelpieadraitneosnhlaypoenceabtaalnodg.uTehs,eorneseuflotrofetahcihs ras.ox andgalaxiesthatareunsuitableforshapemeasurement.Informa- ofthetwoshape-measurementbands,sothereareasmallnumber ford tionfromtheinputmask(fpM)filesispropagatedthroughtothe ofgalaxieswhichappearinonlyoneofthetwocatalogues. jou catalogue,sothatobjectsthatincorporatebadpixelsidentifiedear- TheSDSSphotometricpipelineisknowntoproducesignificant rna lier in the pipeline can be excluded as needed. Due to the nature skyproximityeffects,whereinthephotometricpropertiesofobjects ls.o of the kImages produced by the image co-addition, many of the detectednearabrightstararesystematicallybiased.Theeffectof arg/ standardSDSSflagswillnotbeused(e.g.,byconstruction,there brightstarsonthemeasuredtangentialshearofnearbygalaxiesin t C athreeknIomsaagtuersataerdepmixareklse)d.aAssinwteerdpeoslactreibde;oabbjoevcets,minatshkeedphroetgoimonestroicf soifntghlee-eefpfoecchtsSsDeeSnStdhaetrae,iswsehomwanskinthFeigr.e3g.ioMnostiavraotuenddbybrtihgehtscsatalerss aliforn catalogueoutputswiththesebitssetareremovedfromthecatalogue withamaskingradiusthatdependsontheapparentr-band(model) ia In aatrethailssosteaxgcel.uAdneyd,gaaslaPxHiOeTsOo-FnRwAMhiEcShwthilelndoebtgleenndeirnagteaulgnoiqriuthempofsatialegde magnitudeofthestarsasgiveninTable2. stitute stampsfortheseobjects. o 4.9 Shapemeasurement f T PHOTO-FRAMESalsoattemptstoclassifyobjectsas‘stars’or‘galax- e c ies’onthebasisoftherelativefluxesinthePSFandgalaxymodel Once the final shape catalogue has been constructed, we use the hn o fits(Luptonetal.,inpreparation).Objectsthatarewelldescribedby re-Gaussianizationshape-measurementmethodofHirata&Seljak lo g aPSFareclassifiedasstars;wedonotincludetheseobjectsinthe (2003)togenerateanellipticitymeasureforeachobject.Thepro- y o shapecatalogue,butsetthemasideasaidsfordetectingsystematic cessingcodeandscriptareamodificationofthoseusedinMandel- n M errors. baumetal.(2005).Re-Gaussianizationisnotanespeciallymodern ay To minimize these effects, we also match against a list of all shape-measurement technique, but we have used it previously on 29 objects classified as stars in the single-epoch SDSS catalogues14 SDSS data, it meets our requirements for shear calibration given , 20 withapparentmagnitudesintheiorrbandbrighterthan15.We theexpectedstatisticalpower,andwehadawell-testedcodethat 14 removeobjectsfromthecataloguewithinanangularseparationof interfacedtoPHOTO-FRAMESoutputsatthetimeofinitiatingthecos- thesebrightstarsthatdependsonthestellarapparentmagnitudeas micshearproject.Therefore,wechosetocontinueusingitforthis describedinTable2. analysis. Inadditiontothesebasiccuts,wecullthefollowingobjectsfrom thelensingcatalogue: 4.9.1 Overviewofre-Gaussianization (i) allobjectswherethemodelfluxorellipticitymomentmea- There-Gaussianizationmethodwasanoutgrowthofpreviouswork surementfailed; byBernstein&Jarvis(2002).Theydefinedtheadaptivemoments (ii) allobjectswithin62pixelsofthebeginningorendofaframe; M of an image I by finding the Gaussian G[I] that minimizes (iii) allobjectsdetectedonlyinthebinnedimages(BINNED2or I BINNED4); 15For an reff of the PSF of 0.67arcsec and a resolution factor cut at R2 >√0.333, we expect the minimum reff of a usable galaxy to be 14Asourskycoverageislesscompletethanthesingle-epochdata,weuse 0.67 0.333/(1−0.333)arcsec. This is of course only a very rough es- thesingle-epochcataloguesinmaskingsoastoremoveobjectsthatarein timate,butthisapplicationoftheCMCprovidesasimpleandfastwayto closeproximitytoastarthatisinoneofourmaskedregions. estimatetheimpactofmarginalchangesinsurveyparameters. MNRAS440,1296–1321(2014)

Seeing in the dark – I. Multi-epoch alchemy

Similar Seeing in the dark – I. Multi-epoch alchemy