Two MATLAB programs for computing paleo-elevations and burial ages from paired-cosmogenic nuclides

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Two MATLAB programs for computing paleo-elevations

and burial ages from paired-cosmogenic nuclides

Pierre-Henri Blard, Maarten Lupker, Moïse Rousseau, Jim Tesson

To cite this version:

Pierre-Henri Blard, Maarten Lupker, Moïse Rousseau, Jim Tesson. Two MATLAB programs for

computing paleo-elevations and burial ages from paired-cosmogenic nuclides. MethodsX, Elsevier,

2019, 6, pp.1547-1556. �10.1016/j.mex.2019.05.017�. �hal-02377566�

Method

article

Two

MATLAB

programs

for

computing

paleo-elevations

and

burial

ages

from

paired-cosmogenic

nuclides

Pierre-Henri

Blard

a,b,

*

,

Maarten

Lupker

c

,

Moïse

Rousseau

d

,

Jim

Tesson

a

a

CentredeRecherchesPétrographiquesetGéochimiques(CRPG),UMR7358,CNRS-UniversitédeLorraine, 15rueNotreDamedesPauvres,54500Vandoeuvre-lès-Nancy,France

b_{LaboratoiredeGlaciologie,DGES-IGEOS,UniversitéLibredeBruxelles,1050Bruxelles,Belgium} c

ETHZürich-GeologicalInstitute,Sonnegstrasse5,8092Zürich,Switzerland

d

ResearchInstituteonMinesandEnvironment(RIME)UQAT-Polytechnique,Montréal,Canada ABSTRACT

Methodsbasedoncosmic-rayproducednuclidesarekeytoimproveourunderstandingoftheEarthsurface dynamic.Measuringmultiplecosmogenicnuclidesinthesamerocksamplehasagreatpotential,butdata interpretationrequiresrigorousandoftencomplexmathematicaltreatments.Inordertomakeprogressonthis topic,thispaperpresentstwoeasy-to-useMATLAB©_{programspermittingtoderiveinformationfrompairsof}

cosmogenicnuclides(26_Al-10_Beor10_Be-21_{Ne)measuredinrocksamplesthathavebeenexposedtocosmicraysin}

thepast:“Paleoaltitude.m”and“Burial.m”Codesavailablehereassupplementarymaterial.

“Paleoaltitude.m” computes paleoelevations from a sample whose burial age is known. This new paleoaltimetrymethod is presented in detail in Blard et al.[1]. The present article also develops the mathematicalapproach.

Sincetheelevationofexposuremayaffecttheaccuracyofaburialage[1],thesecondMATLAB©script“Burial. m”isdesignedtocomputeburialagesfrom26_Al-10_Beor10_Be-21_{Nepairs,takingintoaccountthepositionofa}

sample(elevationandlatitude)duringitspreburialexposurehistory.

©2019PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense(http:// creativecommons.org/licenses/by-nc-nd/4.0/).

ARTICLE INFO

Methodname:Cosmogenicnuclides,Paleoaltimetry,Burialage Keywords:Cosmogenicnuclides,10

Be,26

Al,21

Ne,MATLAB,Paleoelevation,Continuousexposure,Steadyerosion,Burialages

Articlehistory:Received19March2019;Accepted16May2019;Availableonline25June2019

*Correspondingauthorat:CentredeRecherchesPétrographiquesetGéochimiques(CRPG),UMR7358,CNRS-Universitéde Lorraine,15 rueNotreDamedesPauvres,54500Vandoeuvre-lès-Nancy,France.

E-mailaddress:pierre-henri.blard@univ-lorraine.fr(P.-H.Blard). https://doi.org/10.1016/j.mex.2019.05.017

2215-0161/© 2019 Published by ElsevierB.V. This is an open access articleunder the CC BY-NC-ND license(http:// creativecommons.org/licenses/by-nc-nd/4.0/).

MethodsX6(2019)1547–1556

ContentslistsavailableatScienceDirect

MethodsX

SpecificationsTable

SubjectArea: EarthandPlanetarySciences

Morespecificsubjectarea: geochemistry;geochronology;cosmogenicnuclides;26

Al;10

Be;21

Ne Methodname: cosmogenicnuclides;paleoaltimetry;burialage

Nameandreferenceof originalmethod

P.-H.Blard,M.Lupker,M.Rousseau-Paired-cosmogenicnuclidepaleoaltimetry–Earthand PlanetaryScienceLetters,2019.doi.org/https://doi.org/10.1016/j.epsl.2019.03.005. Resourceavailability 2Matlabscriptswiththispaper:“Paleoaltitude.m”and“Burial.m”Codesavailablehereas

supplementarymaterial.

Methoddetails

This article presents two Matlab© _{programs, “Paleoaltitude.m” and “Burial.m”, and their} mathematicaldescriptions.Thesetwogeologicalapplicationsarebasedonthemeasurementofa cosmogenicnuclidespair(26_Al-10_Beor10_Be-21_{Ne)inrocksthathavebeenexposedtocosmicraysin} thegeologicalpast(upto>10millionyears ago)(Fig.1).Thepaleoaltimetrymethodis newand describedin[1].Theburialagemethodisalreadywidelyused(e.g.[2]);themainnoveltyofthescript “Burial.m”istotakeintoaccountthespatialpositionofthesample(latitudeandelevation)inthe calculationoftheburialage[1].Codesavailablehereassupplementarymaterial.

Program1-“Paleoaltitude.m”:Thisscriptcomputestheelevationofarocksamplesinwhicha pairofcosmogenicnuclideswithdifferenthalf-lives(26_Al-10_Beand10_Be-21_{Ne)wasmeasured.Inthe} caseofancientexposures,theburialagehastobeknownandbeaccountedforradioactivedecay.

Program2-“Burial.m”:Thiscodecomputesthedurationsincearockwasburiedfrom cosmic-rays.Altitudeandlatitudeatwhichthepaleo-exposureoccurredhavetobeknown.Botharenecessary inputoftheprogram.

By default, the code includes sea level high latitude production rates computed from the worldwidedatabaseavailableintheCREpcalculator(crep.crpg.cnrs-nancy.fr,ICE-Ddatabase;[3])and theproductionparameterssummarizedin[1].

Theseprogramsweredesignedandoptimizedwiththe2018aversionofMatlab©.Somefunctions couldbeunavailableiftheprogramsarerunwithanolderversionofMatlab©.Alldefaultparameters usedinthecodesarethosedefinedintheTable1ofBlardetal.[1].

Program1:“Paleoaltitude.m”–Computationofpaleoaltitudeofexposure

Thephysicalprinciplesofthisnewpaleoaltimetrymethodarepresentedinthemainarticle[1].

TutorialoftheMatlab©script:_{“Paleoaltitude.m”}

Tousetheprogram,usersmustloadthefunctionfileintheirMatlab©workspaceandrunthe "Paleoaltitude.m"script.Awindowwithclickablebuttonsisthendisplayed.

Necessaryinputdata(intheformof.csvor.xlsxspreadsheets)are: Column1:Samplename

Column2:Latitude(₎

Column3:Presentaltitude(m) Column4:Burialage(Ma)

Columns5and6:Nuclide11s(at.g1₎

Columns7and8:Nuclide21s(at.g1₎

Typesofnuclides(10_Be,26_Alor21_{Ne)mustbedefinedusingtheinteractivebox.Itisimportantto} checkthat thecorrectnuclidesareloadedbelowthecorrespondingheaders. Oncedataarecorrectly uploaded,usersmaypushthe"1-Calculate elevations"buttontostart thecalculations. Itis also

possibletopushthe"2-Plot" buttontodisplaythedatain a twocosmogenicnuclidesdiagram (26_Al/10_Bevs10_Beor10_Be/21_Nevs21_{Ne).Oncethefigureisdisplayed,specificsamplesorsub-dataset} maybeselectedandplotted.Fig.2showsanexampleofsuchaplotthatpermitstocomparedatawith theoreticalexposurecurvesatvariouselevations(bydefault,theprogramplotssurface exposure curvesfor0,2000and4000melevations).

Outputresults(in.csv): Column1:Samplename.

Columns2to4:H(m),H_min(m),H_max(m)-Mean,minimumandmaximumelevationcomputedassumingcontinuous exposure(noerosion).SeeBlardetal.[1]forcomplementaryinformation.

Columns5and6:Tint1s(Ma)-Integrationtimeoverwhichaltitudeiscomputedassumingcontinuousexposure(noerosion).

TintiscomputedfromEq.(7)inBlardetal.[1],usingthenuclidehavingtheshortesthalf-life(26Alor10Be)andthescalingfactor

ofthecomputedelevation.

Columns7to9:H_erosion(m),H_min_Erosion(m),H_max_Erosion(m)-Mean,minimumandmaximumelevationcomputed assumingsteadystateerosion.SeeBlardetal.[1]forcomplementaryinformation.

Columns10and11:Erosion1s(m.Ma1).Erosionisherecomputedconsideringthespallogenicproductiononlyandthe preburial10

Beconcentration,usingEq.(14)inLal[4].

Columns12and13:Tint1s(Ma)-Integrationtimeoverwhichaltitudeiscomputedassumingsteadystateerosion.Tintis

computedfromEq.(7)inBlardetal.[1],usingthenuclidehavingtheshortesthalf-life(26

Alor10

Be)andthespatialscaling factorofthecomputedelevation.

Resultsmaybeexportedintheformofa.csvspreadsheetbyclickingthebutton"Export".

Mathematicaldescriptionoftheprogramusedin“Paleoaltitude.m”tocomputepaleo-elevationsandtheir associateduncertainties

ForanuclideX,

l

Xistheradioactivedecayconstant(yr1),PX(at.g1.yr1)theSLHLproduction rate,NXand

D

NX(at.g1)themeasuredconcentrationwithitsuncertainty.

l

Y,PY,NYand

D

NYarethe variablesspecifictothenuclideY.

Inthecaseofpaleo-exposedmaterial,NXandNYarefirstcorrectedforradioactivedecaysincethe burialinitiation(tburial):

NX ¼NX present



elXtburial ð1aÞ

NY ¼NY present



elYtburial ð1bÞ

Here,itisassumedthatthereisnoinheritanceduetopre-exposureandthataltitudehasremained constantduringexposure.

FortwocosmogenicradioactivenuclidesXandYwithdifferenthalf-lives

l

Xand

l

Y,suchas

l

X<

l

Y: NX¼ f



PX

l

Xþ

me

1eðlXþmeÞt
 ð2aÞ NY¼ f



PY

l

Yþ

me

1_eðlYþmeÞt
 ð2bÞ In Eqs. (2a) and (2b), t(yr) is thepreburial exposure duration, f(dimensionless) the spatial productionratescalingfactor,

m

(cm1)therockabsorptioncoefficient(

m

=

r

/

L

,with

r

thedensityin g.cm-3_and

_L

_{theattenuationlengthing.cm}-2_)and

_e

_(cm.yr1_{)theerosionrate.}

Continuousexposurecase(

e

=0m.Ma1)

In the following, thevariables A, B,

D

A,

D

Band r are defined as: A  ¼   lYNY

PY , B¼  lXNX PX ,

D

A¼ lYDNY PY ,

D

B¼  lXDNX PX andr¼ lX lYwithr<1

Fig.1.Synopticdescriptionofthetwomethodsforthe26

Al-10

Bepair. Case1:Burialageisknownandpaleoaltitudescanbecomputed.

Exactsolution. SolvingtheYnuclideEq.(2b)fortandsubstituteitinXEq.(2a)leadsto:

0¼ff 1A_f  r

B ð3aÞ

Fig.2. Exampleofapairednuclidesplotrealizedusingthe“Plot”functionofthe“Paleoaltitude.m”program.Shownisthe

10_Be-21_{NedatasetfromtheAtacamadesert,Chile,SouthAmerica,reviewedin[1].Dataarefrom([7–11]).}

Fig.3. Graphicaldescriptionofthemathematicalmethodusedtocomputeelevationsandtheirassociatedpositiveand negativeuncertainties.Exampleofatwo-radioactivenuclideinversionwiththe26_Al-10_{Bepairwitha5%analyticalerror(1s)}

(lightorangeellipse).Solidlines:continuousexposure.Dashedlines:steady-statedenudation.Redshadedboxes:upperand lowerforbiddenareas.

t¼1

_l

Y ln 1A f   ð3bÞ

(3b)canalsobewrittenwiththeXnuclide.Bothequationsneedtosatisfytheconditionf_>max(A,B). Eq.(3a)hasnoanalyticalsolutionforfandmustbenumericallysolved.Forthis,thefunctiongthatis decreasingontheinterval]A,1[canbedefinedasfollow:

gðxÞ¼xx 1A_x  r

B ð4aÞ

gðfÞ¼0 ð4bÞ

Toensuretheexistenceofthesolutionf,thefollowingconditionsneedtobesatisfied: lim

A g¼AB>0 ð5aÞ

lim

1 g¼rAB<0 ð5bÞ

(5a)and(5b)conditionsaresatisfiedwhenthesampledoesnotplotinthelowerorupperforbidden areas(Fig.3).

Uncertaintypropagation. Since Eqs. (4a) and (4b) cannot be analytically solved,the confidence interval[f,f+_{]canbedeterminedbydefiningthefunction}

_D

_{gthatdescribestheuncertaintyattached} tothegfunction.

D

gisamonotonedecreasingfunction.Thus,f-andf+,thenegativeandpositive boundariesoff,canbecomputedbyfindingtherootsofg

D

gandg+

D

g:

D

gðxÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

@

gðxÞ

@

A

D

A  2 þ

@

g

_@

ðxÞ B

D

B  2 s ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r

D

A 1A x  r1 " #2 þ

D

B2 v u u t _ð6aÞ ðg

D

gÞðf_Þ¼0 ð6bÞ ðgþ

D

gÞf_þ ¼0 ð6cÞ

Maximumelevationandupperforbiddenzone

f+existswhenthefollowingconditionissatisfied.Thisconditionrequiresthattheuppervalueof thesolutionrangeremainsbelowtheupperforbiddenarea(Fig.3):

lim 1ðgþ

D

gÞ¼rABþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2

D

_A2_þ

D

_B2 q <0 ð7Þ

Minimumelevationandlowerforbiddenzone

Theminimumelevationfcanbecomputedifthecondition(5b)issatisfied,whichleadstothe followingcondition: lim 1ðg

D

gÞ¼rAB ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2

D

_A2_þ

D

_B2 q <0 ð8aÞ

MoreoverlimAðg

D

gÞ¼1.Thus,(g-

D

g)needtobenotmonotonictoensuretheexistenceoff:

9

x02A;1; gðx0Þ>0 and 8x>x0:

@

g

D

g

ð Þ

Whenasampleisclosetothelowerforbiddenarea,reachingthelimitsofconditions(8a)and(8b),itis still possibletodefinea limitvaluefor f, correspondingtoa samplethatstands justabovethe forbiddenarea.Thislimitvalueisdeterminedbytheequilibriumbetweenproductionandradioactive decay,anditisherecomputedas:

f_¼maxðA;BÞ ð9Þ

Steady-stateerosioncase(t->1)

Normal case. For long exposure durations (t >> 1/(

l

+

m

e

)), Eqs.(2a) and (2b) become time-independent.SolvingEq.(2b)for

m

e

andsubstituteitinEq.(2a)thenleadsto:

f¼NXNYð

l

X

l

YÞ

PXNYPYNX ð10aÞ

me

¼f



PY

NY 

l

Y ð10bÞ

Becausef>0and

m

e

>0,Eqs.(10a)and(10b)leadtothethreefollowingconditions,where(11a)

and(11b)areequivalenttothe(5a)and(5b)conditions: NY NX> PY

l

X PX

l

Y ð11aÞ PXNYPYNX<0 ð11bÞ f>maxðA;BÞ ð11cÞ

Theclassicaluncertaintypropagationformulacanherebeusedtocomputeanalyticalsolutionsfor f-andf+,[f-,f+]beingtheone-sigmaconfidenceinterval:

D

f_¼

l

Y

l

X ðPXNYPYNXÞ2

ð12aÞ

f_þ¼fþ

D

f ð12bÞ

f_¼f

D

f ð12cÞ

Particularcaseoflargeerosion. f+andfshouldbeconsideredcarefullywhenerosionislargebecause

D

ftendstoinfinityinthiscase.Therefore

D

fcanbelargerthanf,resultinginanegativef-.Sinceitis requiredthatf>maxðA;BÞ,f-canhoweverbedefinedinanycaseby:

f_¼maxðA;B;f

D

fÞ ð13Þ

Noteonsampleslyingintheforbiddenzone. Whenasampleisintheupperforbiddenzone,theupper uncertaintyf+doesnotexist,f<0,andf-canbeestimatedasfollow:

f_¼max_ðA_;B_{Þ ð14Þ}

Whenasampleisinthelowerforbiddenzone,fdoesnotexistand

me

<0.f+couldthusbedefinedas theminimumelevationwhichensureapositiveerosion:

f_þ¼maxðA;BÞ ð15Þ

Program2:_{“Burial.m”–}Calculationofburialages TutorialoftheMatlab©script:“Burial.m”

Tousetheprogram,usersmustloadthefunctionfileintheirMatlab©workspaceandrunthe "Burial.m"script.Awindowwithclickablebuttonsisthendisplayed.

Necessaryinputdata(intheformof.csvor.xlsxspreadsheet)are: Column1:Samplename

Column2:Latitude(₎

Column3:Altitude(m)

Columns4and5:Nuclide11s(at.g1) Columns6and7:Nuclide21s(at.g1).

Nuclides(10_Be,26_Alor21_{Ne)mustbedefinedusingtheinteractivebox.Itisimportanttocheckthat} thecorrectnuclidesareloadedbelowthecorrespondingheaders.Oncedataarecorrectlyuploaded,users maypushthe"1-Calculateburialages"buttontostartcalculations.Itisalsopossibletopushthe"2 -Plot"buttontodisplaythedatainatwocosmogenicnuclidesdiagram(26_Al/10_Bevs10_Beor10_Be/21_Nevs 21_{Ne).Notethattheplotfunctionbecomesfunctionalonlyafterthecalculationofburialages.Oncethe} figureisdisplayed,itispossibletoselectspecificsamplesandplotasub-datasetonly.Fig.4showsan exampleofsuchaplot.

Aftercalculations,resultsmaybeexportedintheformofa.csvspreadsheetbyclickingthebutton "Export".

Outputresults(in.csv): Column1:Samplename

Columns2,3and4:T_Burial(Ma),-1s(Ma),+1s(Ma).BurialageiscomputedsolvingEq.(16)forthe26

Al-10

BepairandEq.(17) forthe10

Be-21

Nepair.Theonesigmaconfidenceinterval[-1s,+1s]iscomputedfromthestatisticaldistributionofthesolutions obtainedfromaMonteCarlosimulation(104

draws).

Columns5and6:Preburialerosion1s(m.Ma1_{).ErosioniscalculatedusingEq.(14)in[4],consideringonlythespallogenic}

productionandthepreburial10_{Beconcentration.}

Columns7and8:Preburialexposureduration1s(ka).Equivalentsurfaceexposuretimeiscomputedassumingnoerosion, usingEqs.(8a)and(8b)in[1].Calculationusesthe21

Neconcentrationinthecaseofthe(10

Be-21

Ne)pairand10

Beinthecaseof the(26

Al-10

Be)pair.

Mathematicaldescriptionofthealgorithmusedinthe_{“Burial.m”}code

Inthisversion,the“Burial.m”programassumesthattwogeologicalconditionsaresatisfied:1) beforeburial,thematerialhadreachedsteady-stateerosionand2)theburialwasinstantaneousand deepenoughtoensuretheabsenceofpost-burialproduction.Inthiscase,theequationlinkingthe burialagetburialandthemeasuredcosmogenicnuclideconcentrationsN1andN2is:

P1 N1 el1tburialP2 N2 el2tburial¼

l

1

l

2 f ð16Þ

SinceEq.(16)hasnoanalyticalsolution,ithastobenumericallysolvedtodeterminetburial.In practice,the“Burial.m”programcallsthe“Burial26Al_10Ne.m”functionthatsolvesEq.(16)usinga MonteCarloapproach.Bydefault,104_{randomdrawsarerealized,assuming}10_Beand26_Alfollowthe normal distributions (N10,

s

10) and (N26,

s

26), where N10 and N26 (at.g1) are the measured

concentrations,

s

10and

s

26(at.g1)theironesigmaanalyticaluncertainties.Thisequationishere appliedinthecaseofthe26_Al-10_Bepair.

Fortheparticularcaseofthe10_Be-21_{Nepair,Eq.(16)simplifiesandcanbeanalyticallysolved:} tburial¼1

_{l }

ln N10 P10



P21 N21þ

l

10 f    ð17Þ Thefunction“Burial10Be_21Ne.m”usesEq.(17)tocomputetburial.

Tokeepcalculationssimple,pre-burialerosion(Columns4and5)isherecomputedconsidering thespallogenic production only. Notehoweverthat theneglection of muogenic productionmay underestimatethecorrecterosionratesbyupto30%(e.g.[5,6]).Ifpreburialerosionisfasterthan 100m.Ma1,neglectingthemuogenicproductionmayleadtounderestimatethecomputedburial agesby_~100to200ka.

Acknowledgements

WearegratefultoGregBalco,DerekFabelandtwoanonymousreviewersfortheirconstructive commentsthatpermittedtoimprovetheMatlab©scriptsandtheearlierversionofthemanuscript. ThisisCRPGcontributionn2690.

AppendixA.Supplementarydata

Codescanbedownloadedhere:https://doi.org/10.1016/j.mex.2019.05.017.

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Al-10

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