Seeing in the dark – I. Multi-epoch alchemy

26 Pages·2014·3.27 MB·English

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
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Accepted2014January17.Received2014January16;inoriginalform2011December15 lo
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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
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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
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Key words: gravitational lensing: weak–techniques: image processing–surveys– g
y
cosmology:observations. on
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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 galax-
ies (e.g. Hoekstra, Yee & Gladders 2004; Hoekstra et al. 2005;
Heymansetal.2006;Mandelbaumetal.2006a,b,2009;Leauthaud
(cid:2)
E-mail:[email protected] 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) RobAse<rve+d5t0h◦i.sTrehgeioSnDmSSa-nIyItsiumpeesrndouvrainpgrothjeecmt(oFnrtihemsoafnSeetpatle.m20b0e8r–) liforn
ia
Thepointsourceandgalaxypopulationshavedifferentsensitiv- November over the years 2005–2007 to collect multicolour light In
s
ities to the ellipticity of the PSF, to optical distortions and to the curvesofTypeIasupernovae.Inthesurveynomenclature,thisre- titu
geometryofthemask.Iftheseareaccountedforthen,asdescribedin gion is Stripe 82. At any given location on the Stripe, there are te
o
e.g.Baconetal.(2003),ameasurementofthepointsource–galaxy onaverage120contributinginterleavedimagingruns,comprising f T
cross-correlationprovidesastraightforwardestimateofthespuri- inaggregatealmostasmuchimagingdataasexistintheremain- ec
h
oussignalsourcedbyuncorrectedPSFvariation.7 Wewillrequire derofthecombinedSDSS-IandSDSS-IIfootprint.Itisherethat no
that the amplitude of this spurious correlation in our final shape significantgainscanbemadefromimageco-addition. log
y
cataloguebesub-dominanttothestatisticalerrors–inparticular, o
n
thattheadditivePSFsystematicsamplitudebeconstrainedtoless M
3.3 Single-epochdataprocessing a
thanthestatisticalerrors. y 2
Theaverageellipticitymeasuredforthegravitationallysheared TherawimagingdataareprocessedbytheautomatedSDSSpho- 9, 2
images of a population of galaxies is proportional to the applied tometric pipeline, PHOTO (Lupton et al. 2001). This pipeline has 01
4
shear; the exact value of this calibration depends on the surface components to handle astrometric and photometric calibration as
brightnessprofilesofthegalaxies.Wewilladdresstheshearcali- well as catalogue construction; it also generates an array of data
brationuncertaintiesinacompanionpaper. quality measurements describing the telescope PSF, the locations
of unreliable pixels and measurements of the photometric qual-
ityofindividualframes.Manyofthesedataqualityindicatorsare
3 DATA
usedduringtheconstructionoftheco-addimaginganditsassoci-
atedcatalogue.Theiruseisdescribedbelow.Adetaileddescription
3.1 TheSloanDigitalSkySurvey
of the image processing pipeline and its outputs can be found in
The Sloan Digital Sky Survey (SDSS; York et al. 2000) and its Stoughtonetal.(2002).Outputscanbefoundinlocationsspecified
successor SDSS-II (Frieman et al. 2008) mapped 10 000 square bytheSDSSdatareleasepapers(Abazajianetal.2003,2004,2005,
degreesacrossthenorthgalacticcapusingadedicatedwide-field 2009;Adelman-McCarthyetal.2006,2007,2008).
PHOTOproducesanumberofintermediateoutputsforthesingle-
epochSDSSimagingthatweuseintheco-additionprocess.Cor-
7Thisstatementistrueforsufficientlylargeareasthatanychancesuperpo- rectedframes,orfpCfiles,areproducedbythepipelinefromthe
sitionsofPSFellipticityandthelensingshearaverageout.Forthisreason, rawCCDimagesofsingleframes;thesearebias-subtractedandflat-
weimposethistestonchunkswitharea≥25deg2. fielded,andanon-linearitycorrectionisappliedwhereappropriate.
MNRAS440,1296–1321(2014)
LensingI 1299
These are the images that are combined during the co-addition
processbelow.
PHOTOalsogeneratesabitmask(anfpMfile)foreachframede-
scribingpixelsthatareknowntobedefective.Pixelsaremarkedin
thisbitmaskassaturated,cosmicraycontaminated,interpolated(if
acolumnorpixelisknowntobesaturated,orisapriorimarked
as unreliable, PHOTO interpolates over that region). We use these
bitmaskstoexcludebadpixelsfromtheimageco-addition.
Astrometricsolutions(asTransfiles)areproducedbyASTROMfor
eachSDSSframe.Systematicerrorsintheastrometricpositioning
arereportedtolessthan50mas,andtherelativeastrometrybetween
successiveoverlappingframesisapproximately10mas(Pieretal.
2003). The astrometric solution for each run (Pier et al. 2003 ) is
determined by matching against astrometric standard stars from
the United States Naval Observatory (USNO) CCD Astrograph
Catalogue(Zachariasetal.2000).Theco-additionalgorithmrelies D
o
ontheastrometricsolutionsprovided;wehavefounditunnecessary w
n
toresolvetheastrometry. lo
a
For photometric calibration, we use the ‘ubercal’ solutions de- de
d
rivedbyPadmanabhanetal.(2008). fro
TheSDSSpipelineusesbright,isolatedstarswithapparentmag- m
h
nitudesbrighterthan19.5toconstructamodelofthePSFandits ttp
variationacrosseachframe.Foreachframe,thestellarimagesfor ://m
the three neighbouring frames along the scan in both directions n
ra
are used to produce a set of Karhunen–Loe`ve (KL) eigenimages s.o
x
describing the PSF variation (Lupton et al. 2001). A global PSF fo
model for the frame is constructed by allowing the first few KL Figure1. ThedistributionofPSFfullwidthathalf-maximum(FWHM)in rdjo
components to vary up to second order in the image coordinates therbandforallframesonStripe82.Thehalf-widthofthetargetPSFafter urn
across the frame, with the coefficients of the variation being tied roundingisindicatedbythesolidverticalline. als
tfiocitehnetsafoofrethmeeirntsipoanteiadlbvrairgihattisotnarasr.eTshtoerKedLbeyigPeHnOiTmOafgoerseaancdhcboaenfd- 4.1 Fieldsmoothing a.org/
inthepsFieldfiles.Thesearetakenasinputstotheco-addition This section describes the operation of smoothing the map so as t Ca
processandusedforPSFcorrection.Wewilltestthefidelityofthis tomaketheeffectivePSFequaltosometargetPSF.Herewewill lifo
PSFmodelintheco-addedimagesonstarsthatweretoofaintto denote the intrinsic PSF of the telescope by G(x), so that if the rnia
performareliablePSFdeterminationinthesingle-epochdata. intrinsicintensityofanobjectontheskyisf(x),theactualimage In
s
observed(cid:2)is titu
te
4 ALGORITHMS I(x)= G(y)f(x− y)d2y≡[G⊗f](x). (5) of T
e
c
h
Ourgeneralstrategyforcorrectingfortheeffectsofseeingissimi- Ofcourse,thisimageisonlysampledatvaluesofxcorresponding no
lartothatsuggestedinBernstein&Jarvis(2002).Wewillapplya to pixel centres. Our principal objective here is to construct the log
roundingkerneltoeachsingle-epochimagepriortostackingtheen- kernelKsuchthat y o
n
semble.Inthepresenceofaperfectlyunbiasedshape-measurement [I ⊗K](x)=[(cid:8)⊗f](x) or [G⊗K](x)=(cid:8)(x), (6) M
a
method, the application of the rounding kernel will unnecessar- y
islhyadpees-mtroeyasiunrfeomrmenattimone.thAotdtihsekpnroewsennttotiemxies,th(Kowitcehvienr,gneotauln.2b0ia1s2e)d. wtarhgeerteP(cid:8)SiFst(cid:8)heatnadrgthetePnSdFe.teInrmoridneertthoedaoptphriosp,rwiaeteneceodntvoofilurtsitocnhkoeorsneeal 29, 20
1
Assessmentsoftheperformanceofshearmeasurementalgorithms K,whichwilldifferforeveryimagingruncontributingtotheco- 4
onsimulatedimages(e.g.Masseyetal.2007b;Melchioretal.2010) addsatagivenpositiondependingonthefullposition-dependent
havefoundthatmeasurementbiasesfrequentlydependontheform PSFmodelineachrun.ThesearethesubjectsofSections4.1.1and
ofthePSF.AslongastheeffectofthePSFonunsmearedgalaxy 4.1.2,respectively.
imagescanbedescribedasaconvolution,theroundingkernelcan
bedesignedtoadjusttheimagePSFtoaformthatisconvenient
4.1.1 ThetargetPSF
fortheshape-measurementmethodathand.Thiscomesatacostin
potentialstatisticalpower,however,asthesizeofthePSFcanonly Here we consider the target PSF (cid:8). It must be constant across
beincreased. different runs in order for the co-add procedure to make sense,
ThelargevariationinSDSSPSFsizes(seeFig.1)willrequire although it need not be the same in different filters. There is a
a trade-off between rejection of a large fraction of the available largeadvantageinhaving(cid:8)becircularlysymmetric.Gaussiansare
imagingandsignificantdilutionofthesignalduetotherounding convenientsincemostgalaxyshape-measurementcodesarebased
convolution. Stacking the images without a kernel, however, will onGaussianmoments,butthisisnotarequirement.InfactthePSF
produceaPSFwithlargevariations–includingstepsatrunbound- Gdeliveredbymosttelescopes,includingtheSDSS,has‘tails’due
ariesortheedgesofregionsmaskedduetoe.g.cosmicrays–that totheatmosphereatlargeradiusthatarefarabovewhatonecould
willbedifficulttomodelaccurately. be expected from a Gaussian. These can in principle be removed
MNRAS440,1296–1321(2014)
1300 E.M.Huffetal.
byaconvolutionkernelKthathasnegativetailsatlargeradius,but columns,saturatedstarsorcosmicraysfrom‘leaking’alloverthe
thereareproblemswhenthesetailsextendtothefieldboundaries field. We also re-scale the resulting kernel to integrate to unity
oracrossbadcolumnsintheCCD.Therefore,wehavechosenthe (K˜(0)=1) but since (cid:9) is small, typically of the order of 10−5,
double-Gaussianformfor(cid:8): this has no practical effect. Note that since G(x) and (cid:8)(x) are
1−f f bothrealfunctions,itfollowsthatinFourierspacetheysatisfythe
(cid:8)(x)= 2πσ2we−x2/2σ12+ 2πwσ2e−x2/2σ22 (7) conditionsG˜(k)=G˜∗(−k)and(cid:8)˜(k)=(cid:8)˜∗(−k),andthenequation
1 2 (10)guaranteesthatasimilarconditionholdsforK:theconvolution
withσ2 >σ1.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.
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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
12<r,i<13 70 than23.5orthei-bandmagnitudeisgreaterthan22.5.
13<r,i<14 50 The magnitude cut was applied to ‘observed’ (at the top of
14<r,i<15 40
the atmosphere) rather than Galactic extinction-corrected magni-
15<r,i<16 30
tudes.Whilethisleadstoanon-uniformgalaxynumberdensity,it
avoids issues with the limiting-S/N varying with position. Using
profileintherband(comparingthelikelihoodsforanexponential theSchlegel,Finkbeiner&Davis(1998)dustmap,withthestan-
andadeVaucouleursmodel),withtheamplitudere-fittedseparately dardextinction-to-reddeningratios(Stoughtonetal.2002,table22),
toeachoftheotherbands.Thisfluxmeasureapproximatesthetrue, alongtheoccupied100◦ lengthofthestripe,ther-bandextinction
totalfluxintherband,andprovidesarobustcolourmeasurement, A has a mean value of 0.141 and a standard deviation of 0.065.
r
which is crucial for photometric estimates of the object redshift (Thei-bandextinctionislowerbyafactorof0.76.)Asimpletest
D
distribution. usingtheCOSMOSMockCatalogue(CMC;Jouveletal.2009)and o
w
ThefinalcrucialoutputofPHOTO-FRAMES,forlensingpurposes,is asizecut15atreff>0.47arcsecpredictsthatthisstandarddeviation 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)