Rede2ningU150PbdiscordancePieterVermeesch11DepartmentofEarthScie - PDF document

RedeningU–PbdiscordancePieterVermeesch11DepartmentofEarthSciences,UniversityCollegeLondon,GowerStreet,LondonWC1E6BTCorrespondence:PieterVermeesch(p.vermeesch@ucl.ac.uk)Abstract.ZirconU–Pbgeochronologyisastapleofcrustalevolutionstudiesandsedimentaryprovenanceanalysis.Construct-ing(detrital)U–Pbagespectraisstraightforwardforconcordant206Pb/238U-and207Pb/206Pb-compositions.Butunfortunately,manyU–Pbdatasetscontainasignicantproportionofdiscordantanalyses.ThispaperinvestigatestwodecisionsthatmustbemadewhenanalysingsuchdiscordantU–Pbdata.First,theanalystmustchoosewhethertousethe206Pb/238U-orthe207Pb/206Pb-date.The206Pb/238U-methodismoreprecise5foryoungsamples,whereasthe207Pb/206Pb-methodisbettersuitedforoldsamples.Howeverthereisnoagreementwhich`cutoff'shouldbeusedtoswitchbetweenthetwo.Thissubjectivedecisioncanbeavoidedbyusingsinglegrainconcordiaages.Theserepresentakindofweightedmeanbetweenthe206Pb/238U-and207Pb/206Pb-methods,whichoffersbetterprecisionthaneitherofthelattertwomethods.Asecondsubjectivedecisionishowtodenethediscordancecutoffbetween`good'and`bad'data.Discordanceisusually10denedas(1)therelativeagedifferencebetweenthe206Pb/238Uand207Pb/206Pbdates.However,thispapershowsthatseveralotherdenitionsarepossibleaswell,including(2)theabsoluteagedifference;(3)thecommon-PbfractionaccordingtotheStacey-Kramersmantleevolutionmodel;(4)thep-valueofconcordance;(5)theperpendicularlogratio(or`Aitchison')distancetotheconcordialine;and(6)thelogratiodistancetothemaximumlikelihoodcompositionontheconcordialine.Applyingthesesixdiscordancelterstoa70,869-graindatasetofzirconU–Pbcompositionsrevealsthat:(i)therelativeage15discordanceltertendstosuppresstheyoungagecomponen

tsinU–Pbagespectra,whilstinatingtheolderagecomponents;(ii)theStacey-Kramersdiscordancelterismorelikelytorejectoldgrainsandlesslikelytorejectyoungones;(iii)thep-valuebaseddiscordancelterhastheundesirableeffectofbiasingtheresultstowardstheleastprecisemeasurements;(iv)thelogratio-baseddiscordanceltersaremoststrictforProterozoicgrains,andmorelenientforPhanerozoicandArchaeanagecomponents;(v)ofallthemethods,thelogratiodistancetotheconcordiacompositionproducesthebestresults,inthesense20thatitproducesagespectrathatmostcloselymatchthoseoftheunltereddata:itsharpensagespectrabutdoesnotchangetheirshape.Thepopularrelativeagedenitionfarestheworstaccordingtothiscriterion.AllthemethodspresentedinthispaperhavebeenimplementedintheIsoplotRtoolboxforgeochronology.1IntroductionTheU–Pbmethodconsistsoftwopaireddecaysystems,inwhichtwoisotopesofthesameradioactiveparent(238Uand235U)25decaytotwoisotopesofthesameradiogenicdaughter(206Pband207Pb,respectively).Thispaireddecaysystemprovidesa1 powerfulinternalconsistencycheckforthemethod,whichisabsentfromotherchronometers.By`doubledating'sampleswiththe206Pb/238Uand207Pb/235Umethods(or,equivalently,the206Pb/238Uand207Pb/206Pb-methods)itispossibletoverifywhethertheisotopicsystemisfreeofprimaryorsecondarydisturbances.Themostreliableageconstraintsareobtainedfromsampleswhose206Pb/238U,207Pb/235Uand207Pb/206Pbagesarestatisticallyindistinguishablefromeachother.U–Pb30compositionsthatfullthisrequirementare`concordant'.Thosethatfailtomeetitare`discordant'.Discordancecanbecausedbyanumberofmechanisms,including:(a)thepresenceofnon-radiogenic(`common')lead;(b)initialdisequilibriumbetweentheshort-livednuclidesofthe238U–206Pband235U–207Pbdecaychai

ns;(c)partiallossofradiogenicleadduringhighgrademetamorphism;and(d)mixingofdifferentagedomainsduringmicro-analysis(Schoene,2014).Thesecomplicatingeffectscanoftenbediagnosedandremediatedwhenmultiplecogeneticcrystalsareavailablefrom35thesamesample.Ifthealiquotsformanisochron(or`discordia')lineinU–Pbisotopespace,thenthislinecanbeusedtorecoverrobustchronologiesfromdiscordantdata(Ludwig,1998).Unfortunately,thisprocedureisrarelyorneverpossiblefordetritalsamples,inwhichcrystalsofdatablemineralsarenotguaranteedtobecogenetic.WithoutauniversalmechanismtoidentifythecauseofU–Pbdiscordanceandremoveitseffects,detritalgeochronologistshavenochoicebuttoacceptsomediscordantanalysesandsomehowincorporatethemintotheir40agespectra.Thereexistsalackofconsensusamongthedetritalzircongeochronologycommunityonhowtodothis.Twooutstandingquestionsare:1.Whichageestimatetouse?Itiswidelyrecognisedthat206Pb/238Uageestimatesoffertheoptimalaccuracyandprecisionattheyoungendoftheagespectrum,whereasthe207Pb/206Pbmethodisbettersuitedforoldersamples.Howeverthecutoffbetweenthetwoclocksvariesbetweenstudies,withvaluesrangingfrom800Mato1.5Ga(Gehrels,2011;45Spenceretal.,2016).2.Howtoquantifydiscordance?Moststudiesdenediscordanceastherelativeagedifferencebetweenthe206Pb/238Uand207Pb/206Pbages,butsomeadvocatetheuseofstatisticalhypothesistestsandp-valuestoquantifydiscordance(Spenceretal.,2016).Andevenwhenadiscordancedenitionhasbeenagreedupon,therearemanywaystochoosethediscordancecutoff.Forexample,therelativeagediscordancethresholdmayvarybetween10%and30%(Gehrels,502011).Thispaperaddressesbothoftheseissues.Section2advocatestheuseofsingle-grainconcordiaages(Ludwig,1998)asawaytoavoidthearbitrarycutoffbetweenthe206Pb/238Uand207Pb/206Pbmeth

ods.Althoughpreviousworkershavearguedfortheuseofsingle-grainconcordiaagesbefore(seeZimmermannetal.,2018,forarecentexample),thisstudyusesasemi-analyticalmodel,ratherthanpurelyempiricalarguments,todemonstratethesuperiorprecisionofthishybridchronometer.55Section3comparesandcontrastsexistingdiscordanceltersbasedonagedisparityandp-values.Itshowsthattherelativeagedenitionstronglyfavoursoldersamplesoveryoungones,andthatthep-valuedenition,whichhasgainedpopularityinrecentyears,hurtsboththeaccuracyandprecisionofdetritalgeochronology.Theagedisparityandp-valuedenitionsareheuristicbynatureandarenotbasedonrmstatisticalorgeologicalarguments.Althoughtheyarethetwomostpopulardenitionsofdiscordanceinusetoday,theyarebynomeanstheonlytwopossibleoptions.602 Figure1.IllustrativeTera-Wasserburgconcordiadiagramwithaconcordantanddiscordantmeasurement.t68marksthe206Pb/238Uage,t76the207Pb/206Pbage,andtctheconcordiaage.Measurement1isconcordantbecauseitsestimatesfort68,t76andtcareidentical.Measurement2isdiscordantbecausethethreeestimatesdisagree.Theconcordiaageisthemostlikelyagegiventheanalyticaluncertainties.Itfallsbetweentheothertwoageestimates,andoffersthebestanalyticalprecisionofthethree.Section4addressestheinherentbiasesoftheexistingdiscordancedenitionsbyproposingthreenewdenitions,whicharebaseddirectlyonU–Pbcompositionsratherthanontheagescalculatedtherefrom.Therstnewdenitionassumesthatthedis-cordanceiscausedbythepresenceofcommonlead.TheothertwonewdenitionstreatU–Pbdiscordanceasacompositionaldataproblem(sensuAitchison,1986).Isotoperatiosarestrictlypositivequantitiesandlog-contrastsarethe`natural'waytoquantify`distances'betweenthem.Section4introducestwologratiodenitionsofd

iscordance,ignoringandaccountingfor65analyticaluncertainty,respectively.Althoughthenewdenitionsarearguablymoreattractivethantheoldonesfromatheoreticalpointofview,thisdoesnotguaranteethattheyproducemoresensibleresults.Totesttheirperformanceonrealdata,Section5appliesthediscordancelterstoacompilationofzirconU–Pbdata.Althoughthetrueagedistributionofthisdatasetisunknowable,theresultssuggestthatthelogratiobaseddiscordanceltersproducethemostaccurate,andmosteasilyinterpretableresults.Therelative70agedenitionfarestheworst.2Whichagetochoose?TheU–Pbmethodisbasedonthreeseparatechronometers:206Pb/238U,207Pb/235Uand207Pb/206Pb.Thehalf-lifeof235Uismorethansixtimesshorterthanthatof238U,and235Uismorethan100timeslessabundantthan238U.Forthesetworeasons,3 little207PbhasbeenproducedduringthelastbillionyearsofEarthhistorycomparedto206Pb.Consequently,the207Pb/235U75and207Pb/206Pbmethodsarelessprecisethanthe206Pb/238UmethodduringthePhanerozoicandNeoproterozoic.However,duringearlierstagesofEarth'shistory,235Uwassignicantlymoreabundantthanitistoday.The238U/235Uratiowas60at1Ga,26at2Ga,11at3Ga,and5at4Ga.Duetothegreaterabundanceof235Uinthispast,andbecauseitdecaysmuchfasterthan238U,theprecisionofthe207Pb/235Uand207Pb/206Pbclocksexceedsthatofthe206Pb/238UmethodduringthePalaeoproterozoicandArchaean.Thegradualshiftinsensitivitybetweenthetwochronometersisvisibleinthe80slopeofaTera-Wasserburgconcordialine,whichissteepatoldages(high207Pb/206Pbgradientw.r.t.time)andshallowatyoungages(low238U/206Pbgradientw.r.t.time).MostpublisheddetritalzirconU–Pbstudiesswitchfrom206Pb/238Uto207Pb/206PbatsomepointduringtheProterozoic.Unfortunatelytherearetwoproblemswithsuchaswitch.First,itrequiresthes

electionofadiscretediscordancecutoffbetweenthetwomethods.Ifthiscutoffdiffersbetweentwostudies(whichitoftendoes),thenthiscomplicatestheintercomparisonof85theirrespectiveagespectra.Second,thesuddenswitchbetweenthe206Pb/238Uand207Pb/206Pbclocksisoftenmarkedbyadiscretestepintheagespectrum(Puetzetal.,2018).Thisstepisentirelyarticialandobscuresanygeologicallysignicanteventsthatmightoccuraroundthesametime.Bothoftheseproblemscanbesolvedbyusing`hybrid'concordiaagesinsteadof`pure'206Pb/238Uand207Pb/206Pbages.ConcordiaagesaredenedbyLudwig(1998)asthe`mostlikely'(inastatisticalsense)U–Pbagegiventheisotopicratio90compositionanditsanalyticaluncertainty(Figure1).Letr86andr76bethemeasured238U/206Pband207Pb/206Pbratios,respectively;let86]2,76]2and86;r76]betheir(co)variances;andletR68(t)=exp(238t)�1,R75(t)=exp(235t)�1,andR58=235U/238U.ThentheconcordiaagetcisobtainedbynumericallyminimisingthesumofsquaresS1:S=24r86�1=R68(tc)r76�R58R75(tc)=R68(tc)35T2486]286;r76]86;r76]76]235�124r86�1=R68(tc)r76�R58R75(tc)=R68(tc)35(1)Thesinglegrainconcordiaagecombinesthechronometricpowerofthe206Pb/238Uand207Pb/206Pbsystems.Foryoung95(1Ga)samples,theconcordiaageisnearlyidenticaltothe206Pb/238Uage.Foroldsamples(&#x]TJ/;ས ; .96;& T; 7.;݉ ;� Td;&#x [00;2Ga)itapproachesthe207Pb/206Pbage.Usingconcordiaagesremovestheneedforanarbitrarycutoffbetweenthetwochronometers.Anaddi-tionaladvantageisthattheconcordiaageoffersbetterprecisionthanthe206Pb/238Uandthe207Pb/206Pbchronometer(orthe207Pb/235Uforthatmatter).Figure2quantiesthiseffectusingasemi-analyticalmassspectrometrysimulationwhosealgorithmisprovidedinAppendixA.1003Discordance

lters:olddenitionsThemostcommondenitionofdiscordanceusestherelativedifferencebetweenthe206Pb/238Uand207Pb/206Pbageestimate(Gehrels,2011):dr=1�t68=t76(2) 1ThesamecalculationcanalsobeperformedinWetherillspace,andisactuallyeasierthere.4 Figure2.Predicteduncertaintiesofthe206Pb/238U(t68),207Pb/235U(t75),207Pb/206Pb(t76)andconcordia(tc)agesforasyntheticdatasetwithaconstanturaniumconcentration.Dwelltimesanddetectorsensitivitieswerechosensoastoyieldresultsthataresimilartothoseobtainedfromrealdata.Theconcordiaage(solidline)alwaysoffersthebestprecision.SeeAppendixAforfurtherdetailsaboutthecalculationsbehindthisgure.Howeverotherdenitionsarepossibleaswell.Forexample,onecouldalsodenediscordanceintermsofabsoluteage105differences(Puetzetal.,2018):dt=t76�t68(3)AthirdoptionistodenediscordanceintermsofU–Pbcompositionsratherthanages.Spenceretal.(2016)advocateusingp-valuestoassessconcordance.Inthecontextofsinglegrainconcordiaages,thep-valueistheprobabilitythatthesumofsquaresS(Equation1)exceedstheobservedvalueunderachi-squaredistributionwithtwodegreesoffreedom:110dp=Prob�s�SjS22
(4)ZirconU–Pbdatacanbelteredbyremovingallmeasurementswhosediscordancevaluesexceedacertainthresholdvalue.Typicalcutoffvaluesfordrare10–30%(Gehrels,2011),whereasdpisgenerallysetto5%(Spenceretal.,2016).DifferentdiscordancecriteriaproducedifferentU–Pbagespectra.Forexample,arelativeagecutoffwillpreferentiallyremoveyounggrainswhereasanabsoluteagecutoffiscomparativelymorelikelytoremoveoldgrains(Figure3).115Thep-valuedenitionaffectsgrainsdifferentlydependingontheiranalyticalprecision(NemchinandCawood,2005).Forexample,considera1.5Gazirconthatisdr=1%discordant.Ifthisgrainwe

reanalysedbyLA-ICP-MSwithananalyticalprecisionof2%,say,thenitwouldpassthechi-squaretestandbeacceptedasbeingconcordant.However,ifthatsamegrainwereanalysedbyTIMSwithaprecisionof0.2%,thenthep-valuecriterionwouldrejectitasbeingdiscordant(Figure4).Itseemsfundamentallywrongthatanimpreciseanalyticalmethodwouldbefavouredoverapreciseone.Thisisapertinent1205 Figure3.DiscordancecutoffsforfourofthesixdiscordancedenitionsdiscussedinSections3and4.Thedpanddccriteriaarenotshownbecausetheydependontheanalyticaluncertaintyofthemeasurements,whichmayvarybetweenstudies.Thegreyenvelopesmarkcutoffvaluesofdr=20%(relativeagelter),dt=300Ma(absoluteagelter),dsk=2%(Stacey–Kramerslter)andda=15%(perpendicularAitchisondistance)onaTera-Wasserburgconcordiadiagram,whichisplottedinlogarithmicspacetoprovideamorebalancedviewoftheoldandyoungendsofthetimescale.Thedskanddtenvelopesaretruncatedwheretheycrossoverintophysicallyimpossiblenegativeisotoperatiospace.problembecausetechnicalinnovationsareincreasingtheprecisionofallanalyticalapproachestoU–Pbgeochronology.Asprecisionimproves,sodoestheabilitytodetecteversmalldegreesofdiscordance.Usingthep-valuecriterion,theremaycomeatimewhennozirconpassesthislter.Analargumentagainstthep-valuediscordancecriterionisthatitbiasesagainstoldU–Pbages.ThisisbecauseoldzirconcontainsmoreradiogenicPbthanyoungzircondoes.Thereforetheanalyticalprecisionoftheisotopicratiomeasurementstends125tobebetterforoldgrainsthanitisforyoungones.Consequently,thechi-squaretesthasgreaterpower(sensuCohen,1992)torejectthem.Inconclusion,p-valuebaseddiscordanceltersarefundamentallyawed.Despitetheirappealas`objective'toolsforstatisticaldecisionmaking,formalisedhypothesistestssuchaschi-squarea

rerarelyusefulingeology.Forthesamereason,thewidelyusedMSWD(MeanSquareoftheWeightedDeviates,McIntyreetal.,1966)statistic(whichisjustS/2inthiscase)shouldbeusedwithcaution.Thisisbecause,likep-values,alsoMSWDcutoffspunishprecisedatasetsinfavourof130impreciseones.NotethatthiscaveatalsogoesagainsttherecommendationsofSpenceretal.(2016).6 Figure4.Applicationoftheawedp-valuediscordancecriteriontotwosyntheticmeasurementsbyTIMS(left)andLA-ICP-MS(right).ThepreciseTIMSmeasurementislabelledasdiscordanteventhoughitplotsclosertotheconcordialinethantheimpreciseLA-ICP-MSmeasurement,whichislabelledasconcordant.4Discordancelters:newdenitionsSection3reviewedthreeexistingdiscordancedenitions.ThisSectionwillintroducethreenewones.Noneofthedenitionsdiscussedthusfarencodeanyinformationaboutthegeologicalmechanismsbehindthediscordance.AsexplainedinSection1,commonPbisoneofthemostlikelycausesofdiscordance.Usingamantleevolutionmodel(e.g.StaceyandKramers,1975)135toapproximatetheisotopiccompositionofthiscommonPb,discordancecanbedenedas:dsk=1�r86=r86(5)wherer86isthe238U/206Pb-ratiooftheintersectionbetweenconcordiaandastraightlineconnectingthe238U/206Pb–207Pb/206Pbmeasurementtotheinferredmantlecomposition(Figure5).ThecommonPbdenitionofdiscordanceismoreforgivingforyounggrainsthanitisforoldones.Importantly,ifthe140discordanceiscausedbycommonPb,thenthe206Pb/238U,207Pb/206Pbandconcordiaageestimatesareallpositivelybiasedwithrespecttothetrueage.Howeverthisbiascanberemovedbyapplyingacommon-Pbcorrectionafterthedatahavebeenltered.EventhoughEquation5ismathematicallyabletoproducenegativediscordancevalues,suchvaluesceasetohaveageolog-icallymeaningfulinterpretation,becauseitisimpossibleformineralstoinh

eritnegativeamountsofcommonPb.Thusitmay145besensibletosetaminimumcutoffofdsk�0whenusingtheStacey-Kramerslter.Eachdiscordiadenitionthatwehavestudiedthusfarisexpressedindifferentunits.Fortheabsoluteagedenition,degreesofdiscordanceareexpressedinunitsoftime(rangingfrom0to4.5Ga).Therelativeagedenitionusesfractionsoftime7 Figure5.UsingtheStaceyandKramers(1975)commonPbmodelasadiscordancecriterion.Thiscriterionassumesthatthediscordanceiscausedbylinearmixing(hence,thelinearscaleofthisTera-Wasserburgplot)betweenradiogenicPb(intersectionsofthemixinglineswithconcordia)andcommonPb(intersectionofthemixinglineswiththeverticalaxis,seeinset).Thedashedlinemarksthe20%(=1=[1+2])discordancecutoff.Thisdiscordancelter,whichmustbeappliedbeforemakinganyactualcommonPbcorrection,ismoreforgivingforyounggrainsthanitisforoldgrains.Inthisrespect,ithastheoppositeeffectoftherelativeageltershowninFigure3.(rangingfrom�1to1).Thep-valuedenitionexpressesdiscordanceintermsofprobability(rangingfrom0to1).AndtheStaceyandKramers(1975)denitionusesfractionsofratios(rangingfrom�1to1).Noneofthesescalesisparticularly150intuitiveornatural.Theycertainlydonotmatchtheusualdenitionofdistanceinthegeographicalsenseoftheword.Toaddressthisissue,itisusefultosubjecttheU–Pbisotopicratiodatatoalogarithmictransformation.SoinsteadofanalysingthedataonaconventionalTera-Wasserburgconcordiadiagram,allcalculationscanbedoneinln(207Pb/206Pb)vs.ln(238U/206Pb)space.Theadvantageofthistransformationisthatitproducesvaluesthatarefreetorangefrom�1to+1.Withinthisinnitedataspace,theEuclideandistancemetriccanbesafelyapplied.155Thereexistsavastbodyofstatisticalliteraturedetailingthetheoreticalandpracticalad

vantagesoflogratioanalysis.Adeeperdiscussionofthistopicfallsoutsidethescopeofthispaper,buttheinterestedreaderisreferredtoAitchison(1986)andPawlowsky-Glahnetal.(2015)forfurtherinformation.TheEuclideandistancebetweenlogratiosisalsoknownasthe`Aitchisondistance'.DiscordancecanberedenedastheAitchisondistancefromthemeasuredlogratiostotheconcordialine.Weintroducetwowaystodosohere.Arstoptionistosimplymeasurethedistancealongaperpendicularlinetothe160concordiacurve(Figure6):da=dx(t76)sinarctandy(t68) dx(t76)(6)8 Figure6.Illustrationofthetwologratiodistancedenitionsofdiscordance.daistheperpendicularAitchisondistancefromthemeasuredlogratiototheconcordialine.dcistheAitchisondistancemeasuredalongalineconnectingthemeasuredvalueandtheconcordiacomposi-tion.wheredx(t)=ln[r86]+ln[R68(t)]anddy(t)=ln[r76]�lnR58R75(t) R68(t)(7)ThisdenitionproducesaparallelbandaroundtheconcordialineinlogarithmicTera-Wasserburgspace.Incontrastwith165dr,dt,dsk,thedacriterionislessstrictatboththeyoungandoldextremesofthegeologicaltimescale,andmorestrictduringtheProterozoicEon,whentheU–Pbmethodismostreliable.TheperpendicularAitchisondistancecriteriondoesnottakeintoaccounttheanalyticalprecisionoftheisotopicmeasure-ments.Toaddressthisissue,wecanalsomeasuretheAitchisondistancealongalineconnectingthemeasuredlogratioandthemaximumlikelihoodcompositionontheconcordialine:170dc=sgn[t76�t68]p dx(tc)2+dy(tc)2(8)wheresgn[]standsfor“thesignof”,whichproducespositivevaluesformeasurementsthatplotabovetheconcordialine,andnegativevaluesformeasurementsthatplotbelowit.9 Figure7.Foursuperimposedkerneldensityestimates(KDEs,usinga50Myrbandwidth)for70,869unlteredzirconU–Pbdates.The207Pb/

235U,206Pb/238Uandconcordiaagespectraaresimilar.HowevertheKDEofthe207Pb/206Pbdatastandsapartfromtheotherthreecurves.Itdeviatesbothattheyoungendoftheagespectrum(whichitsuppresses),andattheoldend(whichitinates).5ApplicationtoacompilationofzirconU–PbdataItisdifculttoascertainthemechanismcausingdiscordanceinanyparticularzircongrain.Therefore,itisunclearwhichofthe175denitionsinSections3and4is`correct'.Allwecandoisapplythemethodstorealsamplesandinvestigatetheiroutcomes.ThisSectionwillapplythediscordancelterstoadatasetof70,869zirconU–PbanalysesthatwereacquiredbySensitiveHighResolutionIonMicro-Probe(SHRIMP)massspectrometeryandcompiledbyDr.S.BodorkosofGeoscienceAustralia.Thedatasetincludes1,665sedimentary,igneousandmetamorphicsamples,mostlyfromAustraliabutincludingsomeotherlocationsaswell.Thedatawereacquiredbyavarietyofinstruments(includingSHRIMP-1,-2and-RG)usingarange180ofdifferentreferencematerials,andprocessedonarangeofdifferentdatareductionsoftware(includingSquid-1,-2andPrawn/Lead).ThedatawerenotsubjectedtoanycommonPbcorrectionorotherlters,andweresavedinaTera-Wasserburgformatwithzeroerrorcorrelationbetweenthe238U/206Pband207Pb/206Pbratiouncertainties.Figure7showsthefrequencydistributionofthecomplete,unltereddatasetasakerneldensityestimate.The207Pb/235U,206Pb/238Uandconcordiaagespectraalllooksimilar.However,the207Pb/206Pbagedistributiondeviatesfromtheotherthree185chronometers.Itreducestheprominenceoftheyoungagecomponents,andinatestheoldendoftheagespectrum.Figure8appliesveofthesixdiscordancelterstothisdatabase(thep-valuelterwasomittedforreasonsgiveninSection3).Inordertoemphasisethedifferencebetweenthediscordancedenitionswhilsttreatingthemonanequalfooting,e

achofthelterswasadjusteduntilhalfofthedatawereremoved.Thiswasachievedbydiscordancecutoffsof�18:6dt46:0Myr,�1:4dr3:66%,�0:11dsk0:27%,�0:78da1:94%,and�0:91dc2:20%.190Therearenoticeabledifferencesbetweenthedensityestimates.AsexpectedfromthetheoreticalconsiderationslaidoutinSections3and4,therelativeageltergreatlysuppressestheyoungeragecomponents(1:5Ga)relativetotheolderparts10 oftheagespectrum(�1:5Ga).TheStaceyandKramers(1975)lterhastheoppositeeffect.ItsuppressestheArchaeanagecomponentby50%whilstfurtherincreasingtheprominenceoftheNeoproterozoicandPhanerozoicmodes.Thediscordancedenitionsbasedontheabsoluteagedifferenceandlogratiodistanceshaveacomparativelyminoreffecton195theshapeoftheagespectrum.Thechangeinshapebetweentheagespectrumofthefull(unltered)datasetandtheagespectraoftheltereddatasetscanbevisuallyassessedonquantile-quantileplotsandquantiedusingtheKolmogorov-Smirnov(KS)statistic(Vermeesch,2013).IftheKS-mististakenasameasureofsuccess,thentheconcordiadistancelter(dc)isthemosteffectivediscordancecriterion.It`sharpens'thespectrumwithoutchangingtherelativeprominenceofthemodesat400,1200,1800,and2500Ma.200Figure8removed50%ofthedata,inordertoemphasisethedifferencesbetweenthediscordancelters.Inrealapplications,lessstringentdiscordanceltersareusuallyapplied.Asmentionedintheintroduction,mostcurrentdetritalzirconstudiesapplya10%–30%relativeagecutoff.Usingthetestdata,wecanevaluatetheequivalentvaluesforthedt,dsk,daanddccriteria(Table1).Forexample,arelativeagelterof10%removesthesamefractionofthetestdataasanabsoluteagelterwithdt=97Myr,aStacey-Kramerslterwithdsk=0:62%,aperpendicularAitchisonlterwithda=4:1%,oraconcordia205distan

celterwithdc=4:6%.Thep-valuediscordancelterhasbeenomittedfromthiscomparisonfortworeasons.First,theuseofthislterisdiscour-agedforreasonsgiveninSection3.Second,thep-valuecutoffsthatareequivalenttoanygivenrelativeagedifferencearehighlylaboratorydependent,withpreciseequipmentrequiringdifferentdp-cutoffsthanimpreciseinstruments.Theothervediscordanceltersaremoreuniversallyapplicable.Sousingadifferentsetoftestdatashouldonlymakearelativelyminor210differencetothevaluesinTable1.6ConclusionsThispapercomparedfourU–Pbclocksandsixdiscordancelters.1.The206Pb/238Uclockismostpreciseattheyoungendofthegeologictimescale.2.The207Pb/206Pbmethodismoreprecisethanthe206Pb/238UmethodbeforetheNeoproterozoic.2153.The207Pb/235Uclockoffersnoadvantageovertheothertwomethods.4.Thesinglegrainconcordiaageisapplicabletotheentirespanofgeologictimeandalwaysoffersthebestprecision.Itapproachesthe206Pb/238Uageastimeapproacheszero,andthe207Pb/206Pbageastimeapproachesinnity.Thesixdiscordanceltersincludethreeexistingonesandthreenewones(Table2).1.Therelativeagediscordancedristhemostwidelyusedcriteriontoday.Itismorelikelytoremoveyounggrainsthan220oldones,andstronglyskewstheagedistributiontowardsoldagecomponentsasaresult.11 dr dtdskdadc -10 -71-0.55-3-3.5-5 -48-0.33-2-2.3-4 -41-0.29-1.7-2-3 -34-0.22-1.5-1.7-2 -25-0.16-1.1-1.2-1 -14-0.08-0.58-0.670 00001 140.080.610.72 280.161.21.33 400.231.71.94 490.292.12.35 580.352.42.810 970.624.14.615 1400.9666.820 1901.48.19.425 2501.8111230 3202.4141640 4903.6212550 7005.23038Table1.Conversiontableforthedifferentdiscordancelters,constructedusingthetestdata.Alldiscordancevaluesareexpressedas%,exceptfordt,whichisexpressedinMyr.Thistableallowsthereadertoselectadiscordance

cutoffthatremovesthesamefractionoftheirdataastherelativeagecutoff(dr)thattheymayhaveappliedinthepast.denitiondescriptioncomment dr=1�t68=t76relativeagedifferencebiasesagainstyoungsamplesdsk=1�r86=r86fractionofcommonPbbiasesagainstoldsamplesdp=Prob�s�SjS22
p-valueofconcordancebiasesagainstprecisemeasurementsdt=t76�t68absoluteagedifferenceallowsnegativeagesda=dx(t68)sinarctanhdy(t76) dx(t68)iAitchisondistancemoststrictfor`middleaged'samplesdc=sgn[t76�t68]p dx(tc)2+dy(tc)2concordiadistanceleastbiasedTable2.Side-by-sidecomparisonofthedifferentdiscordancelters.12 Figure8.Left:lteredU–Pbagespectraforthetestdata,removingthe50%mostdiscordantgrainsaccordingtovediscordanceltersreviewedinthispaper,shownasakerneldensityestimatewith50Myrbandwidth.Thecomplete(unltered)datasetisshowningrey.Right:quantile-quantileplotscomparingthelteredandunltereddatasets.KS=theKolmogorov-Smirnovstatistic.Therelativeagelter(dr)introducesthegreatestandtheconcordiadistance(dc)thesmallestbias,respectively.2.Theabsoluteagediscordancedtisnotwidelyused.Butitillustratesthedramaticeffectthatthediscordancedenitioncanhaveonthelteredagedistibutions.Comparedwiththerelativeagelter,itismorelikelytorejectoldgrains,andlesslikelytorejectyoungones.Itevenallowsphysicallyimpossiblenegative207Pb/206Pbagestopassthroughit.13 3.Thep-valuebaseddiscordancelterdpmayhaveintuitiveappealasan`objective'denition.Butithasanundesirable225negativeeffectontheprecisionandaccuracyofthelteredresults.Itisbestnottousethislter.4.TheStacey-KramersdiscordancelterdskassumesthatdiscordanceissolelycausedbycommonPbcontamination.Ifthisassumptioniscorrect,thenthed

sklterwillproducethemostaccurateagedistributions,providedthataStaceyandKramers(1975)commonPbcorrectionisappliedtotheltereddataafterwards.5.TheperpendicularAitchisondistancedaisausefulvehicletoillustratetheapplicationoflogratiostatisticstodetrital230zirconU–Pbgeochronology.Itproducesaparallelacceptancezonearoundthe(log-transformed)concordialine.Thislterismostlikelytoreject`middleaged'zircongrains,between1000and2000Ma,wheretheageresolvingpoweroftheU–Pbmethodisgreatest.Aboveandbelowthisinterval,thedacriterionismoreforgiving.Thisbehaviourisdesirablebecausenaturalsamplestendtoexhibitmoreagediscordancebelow1000Maandabove2000Mathanbetweenthesedates.2356.TheconcordiadistancedcisamodiedversionofthedacriterionthattakesintoaccounttheuncertaintiesoftheU–Pbisotopiccomposition.ItseffectsontheU–Pbagedistributionsaremoredifculttovisualisebutaresimilartothoseofthedacriterion.Applyingthedcltertothetestdatashowsthatitminimisesthedifferencebetweentheunlteredandlteredagespectra.Itresultsinatighteningofsubpopulationswithoutchangingtheirpositionorrelativesize.Weadvocatethatthiscriterionbeusedasadiscordancelter.240Allthediscordancelterspresentedinthispaper(botholdandnew)havebeenimplementedinIsoplotR(Vermeesch,2018),ageochronologicaltoolboxwrittenintheRlanguage.FurtherdetailsaboutthisimplementationareprovidedinAp-pendixB.Codeanddataavailability.IsoplotRisfreesoftwarereleasedundertheGPL-3license.Thepackageanditssourcecodeareavailablefromhttps://cran.r-project.org/package=IsoplotR.Thetestdatacanbedownloadedfromhttps://github.com/245pvermees/discpaper/.AppendixA:Comparingtheprecisionofthe207Pb/235U,206Pb/238U,207Pb/206PbandconcordiaclocksTheuncertaintyofaU–Pbdatedep

endsonthreefactors:1.theageand,hence,thetrueisotopicratio;2.thesensitivityoftheiondetectorstoUandPb;and2503.thedwelltimesusedtomeasurethedifferentisotopes.14 Thesethreefactorsvarybetweensamples,andbetweenlabs.Inordertoexploretheireffects,letusrstdenethefollowingparameters:–t68,t75andt76:the206Pb/238U,207Pb/235Uand207Pb/206Pbages(inMa);–38and35:thedecayconstantsof238Uand235U(inMa-1);255–R85:thenatural238U/235Uratio;–R68,R75andR76:thetrue206Pb/238U,207Pb/235Uand207Pb/206Pbatomicratios;–r68,r75andr76:themeasured206Pb/238U,207Pb/235Uand207Pb/206Pbsignalratios;–fPbU:thefractionationfactorbetweenPbandU;–d0638:thedwelltimeratioof206Pband238U;260–d0706:thedwelltimeratioof207Pband206Pb;–n06,n07andn38:thenumberof206Pb,207Pband238Uionscountedduringameasurement.Thenthetrueisotoperatiosaregivenby:R68=exp(38t68)�1(A1)265R75=exp(35t75)�1(A2)R76=1 R85R75 R68(A3)andthemeasuredratiosby:r68=d0638fPbUR68(A4)270r75=d0706d0638fPbUR75(A5)r76=d0706R76(A6)15 sothatthepredicted206Pband207Pbioncountscanbewrittenas:275n06=n38d0638fPbUR68(A7)n07=n06d0706R76(A8)AssumingthatalltheionsaremeasuredbySecondaryElectronMultiplier(SEM),withanalyticaluncertaintiesthataregovernedbyPoissonianshotnoise:28068] r682=1 n38+1 n06(A9)75] r752=1 n38+1 n07(A10)76] r762=1 n06+1 n07(A11)285thenthestandarderrorsofthesignalratiosratiosaregivenby:68]=n06 n38r 1 n38+1 n06(A12)75]=R85n07 n38r 1 n38+1 n07(A13)29076]=n07 n06r 1 n06+1 n07(A14)Finally,theuncertaintiesoftheageestimatesaregivenbystandarderrorpropagation:68]=@t68 @r6868](A15)75]=@t75 @r7575](A16)29576]=@t76 @r76

7;[r76](A17)16 FigureB1.ThenewdiscordancelterscanbeaccessedfromIsoplotR'sgraphicaluserinterface,shownhereunderitskerneldensityestimationfunction.where@t68 @r68=1 38(1+R68)1 d0638fPbU(A18)300@t75 @r75=1 35(1+R75)1 d0706fPbU(A19)@t76 @r76=R85R268 (@R75=@t75)R68�R75(@R68=@t68)1 d0706(A20)Figure2showstheresultofthesecalculationsusingrealisticvaluesofn38,fPbUandd0706,whichyieldanoutcomethatissimilartothetestdata,andtotheempiricalresultsofZimmermannetal.(2018).305AppendixB:ImplementationinIsoplotRIsoplotRcanbeaccessedeitherfromthecommandline,orviaagraphicaluserinterface(GUI),eitherofineoronline(http://isoplotr.london-geochron.com).Thediscordanceltersareaccessibleviabothmethods.IntheGUI,thediscordancecanbetabulatedviatheAgefunction,andhasalsobeenincorporatedinIsoplotR'sotherfunctions,includingitsconcordia,weightedmeanandkerneldensityestimationalgorithms.FurtherdetailsareprovidedundertheOptionsmenu310(FigureB1).ToaccessthesamefunctionalityfromthecommandlinerequiresinstallationofIsoplotRfromtheComprehensiveRArchiveNetwork(CRAN):install.packages(IsoplotR)Onceinstalled,thepackagemustbeaddedtotheworkingenvironment:315library(IsoplotR)17 Loadingthetestdataintomemory:UPbread.data(data.csv,method=U-Pb,format=2)ThediscordancecanthenbecalculatedusingIsoplotR'sdiscfilterfunction.Forexample,tocomputetherelativeagediscordance(dr):320trage(UPb,discordance=discfilter(option=r))whichproducesa70;8699tablewhosersteightcolumnslistthe207Pb/235U,206Pb/238U,207Pb/206Pbandconcordiaagesandtheiruncertainties,andwhoseninthcolumnliststherelativeagediscordanceaspercentages.Similarly,tocomputetheconcordiadistance(dc):tcage(UPb,discordance=discfilter(option=c

1;))325PlottingaKDEofthesinglegrainconcordiaagesthatpasstheperpendicularAitchisonlterwith�2da6:dfdiscfilter(option=c,cutoff=c(-2,6))kde(UPb,type=5,cutoff.disc=df)ApplyaStacey-KramerscommonPb-correctiontothedataafterapplyingaStacey-Kramersdiscordancelterwith0dsk0:96:330dfdiscfilter(option=sk,cutoff=c(0,0.96))kde(UPb,common.Pb=3,cutoff.disc=df)Ifthedatasetincludes204Pb(whichisnotthecaseforthetestdata),thenwecanalsoapplyadiscordancelterafterthecommonPbcorrection.Forexample:dfdiscfilter(option=r,before=FALSE,cutoff=c(-5,15))335kde(UPb,common.Pb=3,type=4,cutoff.76=1200,cutoff.disc=df)whereoption=rtriggerstherelativeagelter(dr),common.Pb=3appliesaStacey-KramerstypecommonPbcorrec-tion,type=4usesthe206Pb/238U-ageforyounggrainsandthe207Pb/206Pb-ageforoldones,andcutoff.76markstheage(inMa)atwhichtoswitchfromthe206Pb/238Utothe207Pb/206Pbmethod.Furtherinformationaboutthesefunctionscanbeobtainedfromthebuilt-indocumentation:340?IsoplotR?discfilter?kdeNotethattheexamplesshownheremaytakeafewminutestocompleteduetothelargesizeofthetestdataset.18 Authorcontributions.Thisisasingleauthorpublication.345Competinginterests.Theauthorhasnocompetinginterests.Acknowledgements.ThewritingofthispaperwastriggeredbyastimulatingemailconversationwithChrisSpencerandStevePuetz.ThetestdatawerecompiledbySimonBodorkosofGeoscienceAustralia,andthepaperbenettedfromcarefulreviewsbyPingWangandKeithSircombe,withadditionalfeedbackfromChuckMagee.ThisresearchwassupportedbyNERCstandardgrant#NE/T001518/1(`BeyondIsoplot').35019 ReferencesAitchison,J.:Thestatisticalanalysisofcompositionaldata,London,ChapmanandHall,1986.Cohen,J.:Apowerprimer.,Psychologicalbulletin,112,155,1992.Gehrels,G.:Detrit

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