Paired-cosmogenic nuclide paleoaltimetry

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Paired-cosmogenic nuclide paleoaltimetry

Pierre-Henri Blard, M Lupker, Moïse Rousseau

To cite this version:

Pierre-Henri Blard, M Lupker, Moïse Rousseau. Paired-cosmogenic nuclide paleoaltimetry. Earth

and Planetary Science Letters, Elsevier, 2019, 515, pp.271-282. �10.1016/j.epsl.2019.03.005�.

�hal-02377568�

Contents lists available atScienceDirect

Earth

and

Planetary

Science

Letters

www.elsevier.com/locate/epsl

Paired-cosmogenic

nuclide

paleoaltimetry

Pierre-Henri Blard

a

,

b

,

∗

,

Maarten Lupker

c

,

Moïse Rousseau

a

,

d

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

b_{LaboratoiredeGlaciologie,DGES-IGEOS,UniversitéLibredeBruxelles,1050Bruxelles,Belgium} c_{ETHZürich- GeologicalInstitute,Sonnegstrasse5,8092Zürich,Switzerland}

d_{ResearchInstituteonMinesandEnvironment(RIME)UQAT-Polytechnique,Montreal,Canada}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received9May2018

Receivedinrevisedform1March2019 Accepted5March2019

Availableonline28March2019 Editor: L.Derry

Keywords:

paleoaltimetry

insitucosmogenicnuclides

10_Be–26_Al–21_Ne

simpleexposurecurves Atacamadesert Andes

Goal – The reconstruction of past topographies remains challenging and only afew methods allow accuratedeterminationofpastsurfaceelevations.Weproposehereanewtechniqueforderiving paleo-elevations,inwhichmultiplecosmogenicnuclidesaremeasuredinthesamegeologicalsampleexposed at the Earth’s surface. This method relies on the altitude dependence of the cosmogenic nuclides’ productionratescombinedwiththeradioactivedecaysofnuclideswithdifferenthalf-lives.

Theory –Thepositionofthetwocosmogenicnuclideexposurecurves(26_Al/10_Bevs10_Beor10_Be/21_Ne vs10_{Be)dependsonthealtitudeofexposure.Ifthestudiedsurfaceshavebeenexposedforsufficiently}

longdurations (>500 ka), orhave beenaffectedby low erosionrates(<1 m Ma−1), measurementof twocosmogenicnuclideswithdifferenthalf-livesthusallowaccurateelevationstobedeterminedwitha reasonableuncertainty(<1000 mat1

σ

).Forshorterexposuredurations,themethodisabletoconstrain minimumelevations.Themainadvantage ofthemethodisthatitisonlyslightlysensitive toerosion: even ifthepreservationstateofthesurfaceisunknown,thebiasonthecomputedelevationremains lowerthan1500minmostcases.Theapproachcanalsobeappliedtopreviouslyexposedsurfacesthat havesubsequentlybeenburied,inordertoreconstructthepaleo-elevationofagivensurfaceovertime rangesof∼0to8 Ma(usingthe26_Al–10_{Bepair)and∼0to12 Ma(usingthe}10_Be–21_Nepair).

Datacomparison –Wetestedthemethodusingthemultiplecosmogenicnuclidesdatasetavailablefor the westernaridtropicalAndes.The altitudes computedusingthe cosmogenicnuclideconcentrations agreewithinuncertaintieswiththereportedsamplingaltitudesoverarangeof0tomorethan4000 m, illustrating the applicability ofthe method.Altitudes computedunder the assumptions ofcontinuous exposureorsteadystateerosionyieldsbestfitsthatarestatisticallyinagreementandclosetothe1:1 lineforboththe26_Al–10_Beandthe21_Ne–10_{Bedataset.The}21_Ne–10_{Beinventoriesinsamplesthathave}

been exposed for more than5 Ma yield elevations that are several hundreds of meters below their present-dayelevations(∼1000 m).Thismayresultfromapost10 MaupliftoftheWestAndes,orfrom anunrecognizedexposureunderwater,orbelowasoilcover.

Implications – Thisstudymayalsohaveimplicationsinotherfieldsthatrelyonmultiplecosmogenic nuclidemeasurements.Thesameapproachmightnotablybeusedtocomputethedepthofexposureof sampleslocatedbelowtherocksurfaceorunderwater.Thisstudymayalsohelptoimprovetheaccuracy ofthecommonburialdatingmethodthatusesmultipleradioactivecosmogenicnuclides.Forlong pre-burial exposures(>500 ka), orlow erosionrates (<1 m Ma−1_{), the values ofthe pre-burialnuclides}

ratiosindeeddependstronglyonthealtitudeofexposure.Itmaybeimportanttoconsiderthepre-burial altitudeofexposureinordertocalculateaccurateburialages.

©2019TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

*

Correspondingauthor at:CentredeRecherches Pétrographiqueset Géochim-iques(CRPG),UMR7358,CNRS- UniversitédeLorraine,15rueNotreDamedes Pauvres,54500Vandoeuvre-lès-Nancy,France.

E-mailaddress:[email protected](P.-H. Blard).

1. Introduction

Cosmogenicnuclidetechniquesarepowerfultoolsfor address-ing a wide variety of scientific problems in Earth sciences (e.g. Dunai,2010; Granger etal., 2013). The majority ofthesestudies have been based on the surface exposure dating of geomorphic surfaces (e.g. Gosse et al., 1995) or the determination of

basin-https://doi.org/10.1016/j.epsl.2019.03.005

averaged erosionrates (e.g.Granger etal., 1996). Forthesequite straightforward applications, it is generally sufficient to measure only one nuclide in the suitable rock samples (e.g. Gosse and Phillips,2001; Lal,1991).Measuringtwocosmogenicnuclideswith differenthalf-livesinthesamerockallowsmorecomplexexposure historiestobeaddressed,andnotably,thedeterminationofburial ages, i.e. the time since a geological sample has been shielded fromcosmic rays (e.g. Granger andMuzikar, 2001). This applica-tionhasled tomanybreakthroughpublicationssinceit isone of the mostaccurate andefficient methodsavailable fordating Ho-minidremnantsincontinentaldeposits (e.g.Grangeretal.,2015; Lebatardetal.,2014).

Nevertheless, it iscrucial to keep exploring new applications, in particular to tackle unsolved scientific questions in Earth sci-ences.Developing amethod fordeterminingaccurate paleoeleva-tionsover geologicaltimescalesremains achallenge. Reconstruct-ing past elevations of a mountain range may provide important clues regarding the geodynamic processes involved (e.g. Husson andSempere, 2003; Molnar et al., 1993). Reliefchanges are also thought to have a major impact on atmospheric circulation and globalpaleoclimates(e.g.MolnarandEngland,1990).Existing pa-leoaltimetrymethodsfrequentlysufferfromseverallimitationsin thatmanyofthemoftenrequiringlooselyconstrainedassumptions (Clarketal.,2007).Thishasledtocontrastingresultsinsome re-gionsandleavesopen thedebateontheamplitudeoftheimpact oftectonicsonglobalclimaticchanges(e.g.BoosandKuang,2010; Lichtetal.,2014).

Because the Earth’s atmosphere significantly attenuates the cosmic-ray flux, the production rate of a cosmogenic nuclide is sensitivetoelevation(Lal,1991).Thispropertyhasbeenexplored asameansforretrievingpaleoelevation,eitherfromcontinuously exposedsurfaces(Brooketal.,1995; Evenstaretal.,2015) orfrom buried paleo-surfacesthat havebeen exposed fora known dura-tion in the past (Blard et al., 2005). However, these exploratory works had to address several important limitations, notably be-cause a method based on the measurement of only one nuclide correspondstoamathematicallyunderdeterminedsystem:this ap-proach thus requires the use of surfaces that have not been af-fected by erosion, andfor which the duration ofpaleo-exposure canbeindependentlyandaccuratelymeasuredwithhighprecision (Blardetal., 2006). Sucha combinationoffavorableconditionsis rarelyfulfilledinnature,makingthemethoddifficulttoapply.

Here we presenta novel approach for using andinterpreting datasets of multiple-cosmogenic nuclides measured in the same rock samples. We show that, when the exposure time is long enough(typically

>

500 ka),theisotopicratiooftwocosmogenic nuclides with contrasting half-lives, along with their respective concentrations, allows calculation of precise and accurate eleva-tions. Under the most favorable conditions, the method enables altitudes to be derived with a precision (

<

500 m) that is useful formostgeologicalapplications.Thestrengthandmainadvantage ofthisnewtechnique isthatitsaccuracyisonlyslightlysensitive (biasfromthisassumptionislessthan1000 m)totheamountof erosion that has affected the exposed paleosurface, a parameter that is generally unknown. We tested the method on an exist-ing dataset of samples collected in the arid partof the Western Andes,usingbedrock anddetrital samplesinwhich pairsof cos-mogenicnuclides (10_Be–21_{Ne and}26_Al–10_{Be) hadbeenmeasured}

in previous studies (Kober et al., 2007; Nishiizumi et al., 2005; Placzeket al.,2010; Ritteret al.,2018a). The calculatedaltitudes areingoodagreementwiththepresent-dayelevationofthe sam-ples, suggesting both that the method is adequate and that the altitude change of the Central Andes has been limited over the last hundreds of thousands of years. We also performed Monte Carlosimulationstoexplorethedetectionlimits, theuncertainties

and the rangeof applicability of the method over the geological timescale, for both the 26Al–10Be and the 10Be–21Ne pairs. This modeling shows that the precision andaccuracy ofthis paleoal-timetry methodbothincrease significantlywithexposuretime or with the low erosion rates of the pre-burial paleo-surface. Sur-faces that have been exposed for longer than 500 ka years (or affected by erosion rates lower than 1 m Ma−1) should be fa-vored, as theseyield accurate elevations witha precision better than1000 m.

In this paper we also make a case that the elevation at the time of initial exposure to cosmic rays should be carefullytaken intoaccountwhencomputingburialages.

2. Generaltheory:determiningaltitudesfrom paired-cosmogenicnuclides

Allofthecalculations(i.e.simulationsanddatainterpretations) presentedherewereperformedusingtheparameterssummarized inTable1.ScalingfactorswerecomputedusingtheCREp calcula-tor(https://crep .otelo .univ-lorraine .fr /#/)(Martinetal.,2017).

In a companionpaper (Blard etal., 2019),we provide a Mat-lab© code, (Paleoaltitude.m), which allows paleoaltitudes to be computed from any dataset of cosmogenic nuclides (either the

26_Al–10_Beorthe10_Be–21_{Nepair),underbothassumptionsof}

con-tinuousexposureanderosion(seeBlardetal.,2019forthe math-ematical description of the used algorithm). This code also plots dataandsimpleexposurecurves(ratiooftwocosmogenicnuclides

vs concentration ofonecosmogenicnuclide)forvariablealtitudes inthesamediagram.

2.1. Theinfluenceofaltitudeonthepositionofthetwo-nuclide exposurecurves

Fig. 1shows theratios oftwo radioactive cosmogenicnuclide (forthepairs26Al–10Beand10Be–21Ne) plottedagainst10Be con-centration atdifferent altitudes. Thesecurves are definedby the equations determining the production of cosmogenic nuclides at the surface for different exposure durations with either (i) zero erosion (higher thick curve) or (ii) steady erosion (lower thin curve)(Fig. 1)(Lal,1991).Insucha diagram,allsamplesthat are exposed atconstant depthandelevationdefinea regionthat has the shape of “banana”.These plots were initially proposed by D. Lal,K.NishiizumiandJ.Klein(https://cosmognosis .wordpress .com / 2017 /03 /30 /upside -down -bananas/).If theexposure durationof a surface isshorter than 500 ka (for the26Al–10Be and 10Be–21Ne pairs, Fig.1),radioactivedecayisnegligible,andtheisotope con-centration ratios equalthe production ratesratio ofthe two nu-clides, meaning that these simple exposure curves overlap each otherontheleftpartofthediagram(Fig.1).Inthatcase,itisonly possible to determine a minimum elevation (Blard et al., 2019). In contrast, when theexposure time is longer (

>

500 ka, Fig. 1), thecurvescomputedfordifferentaltitudesarenolonger superim-posed.Thepositionsofthesetwo-nuclidescurvesarethusaltitude dependent(Fig. 1):forsimilarexposuredurations,samples stand-ing at different elevations have similar 26_Al/10_{Be (or} 10_Be/21_Ne)

ratios,butdifferentcosmogenicnuclidesconcentrations.Although this property has been mentioned in few papers (e.g. Fig. 4 in Stone, 2000),thisis oftenignored intheliterature. The positions of these simple exposure curves correspond to a two-equations-two-unknownsproblem,thetwo unknownsbeingthedurationof exposure(orerosion)andtheelevation.Therefore,measuringtwo nuclidesinasteadilyeroding,ornon-erodingsurface, canin the-ory allow the sample’s exposure altitude to be determined (see section 2.2).The factthatasetofobservationsfromsurface sam-ples should lie in this simple exposure region has already been

Table 1

Parametersusedtoperformcalculations(numericalsimulationanddatainversion)inthisarticle,from(Balcoetal.,2008; Chmeleffetal.,2010; Granger,2006; Kellyetal.,

2015; Koberetal.,2011; Lal,1991; Lifton,2016;Martinetal.,2017,2015;Nishiizumietal.,1989,2007;Stone,2000).SLHL:SeaLevelHighLatitudeconditions. Reference Parameter Value 1σ

uncertainty

Comments

Martinetal.,2017 SLHLproductionrateof

10_{Be(at g}−1_yr−1₎ 4.15 0.20 _{sites –}Worldwide_ComputedmeanP_with10from_theLal/Stoneallcalibration_time

dependentscaling,standardatmosphere, atmospheric10_BeVDM

Koberetal.,2011 SLHLproductionrateof

21_{Ne(at g}−1_yr−1₎

17.1 0.9 ComputedfromtheglobalP10anda 21_Ne/10_{Beproductionratioof4.12±0.17}

Nishiizumietal.,1989; Nishiizumietal.,2007; Braucheretal.,2011

SLHLproductionrateof

26_{Al(at g}−1_yr−1₎

27.4 2.5 Froma26_Al/10_{Beproductionratioof}

6.61±0.52 Chmeleffetal.,2010;

Korschineketal.,2010

10_{Behalf-life(yr) 1387000 12000 –}

Granger,2006 26_{Alhalf-life(yr) 717000 17000 –}

– Density(g cm−3_{) 2.7 – –}

Balcoetal.,2008 Attenuationlength (g cm−2₎ 160 – – Lal,1991; Nishiizumietal.,1989; Stone,2000; Martinetal.,2017; Muscheleretal.,2005 Spatialand time-dependentscaling

– – Stonetimedependent –Standard atmosphere –Atmospheric10_BeVDM

Fig. 1. Theoreticalplotsoftheratiosofcosmogenicnuclidesvs10_{Beconcentrationsforthe(A)}26_Al–10_Beand(B)10_Be–21_{Neisotopicsystems.Theratiosareplottedfor}

continuousexposure(thickcurves)andsteadystatedenudation(thincurves)ofthesampledsurfaces,definingthesocalled“simpleexposurecurves”.Thesecurveswere computedat60◦latitudeusingtheproductionparametersdefinedinTable1.

usedtorefineproductionrates(e.g.Koberetal.,2011),butthisis thefirsttimeitisusedtodeterminepaleoelevations.

2.2.Physicalandmathematicalsystematicsofthepaired-cosmogenic nuclidesaltimeter

TheconcentrationofcosmogenicnuclidesatorneartheEarth’s surfaceasafunction oftime,t,anddepthbelowthesurface x is

describedbythefollowingequation(Lal,1991):

N

(

x

,

t

)

=

N

(

x

,

0

)

·

e−λt

+

P

(

x

)

λ

+

με

·



1

−

e−(λ+με)t



(1)

whereN

(

x

,

t

)

isthenuclideconcentration(at g−1_{), N}

₍

_x

_,

₀

₎

_the

ini-tialnuclideconcentration, x (cm)is thedepthbelowthe surface. Thisdepthx canbedefinedforanysortofcoveringmaterial:rock, soil, ice, liquid water. P (at g−1_yr−1_{) is the cosmogenic nuclide}

production rate that is a function of x, and the spatial position ofthe sample,i.e.itsaltitude,latitudeandlongitude.

λ

isthe ra-dioactivedecayconstant(yr−1_),

_μ

₌

_ρ

_/Λ

_{isthecosmicray}

atten-uation constant(cm−1),

Λ

beingtheattenuationlength (g cm−2),

ρ

(g cm−3) the densityofthe covering material,

ε

(cm yr−1) the erosionrateofthesurface.

For surface samples (x

=

0) and under the assumption of no previousexposureorinheritance(N

(

x

,

0

)

=

0),equation(1) canbe simplifiedinto:

N

(

t

)

=

f

·

PSLHL

λ

+

με



1

−

e−(λ+με)t



(2)

wherePSLHListhecosmogenicnuclideproductionratenormalized to SeaLevel High Latitude (SLHL)(at g−1yr−1) and f isthe spa-tial scaling factor, which depends on the position of the sample (altitude, latitude andlongitude). In thistheoretical case, we as-sume that the altitudechange is smallover the exposure time t

(df

/

dt

=

0).Amorecomplicatedcaseofdf

/

dt

=

0 isdiscussedin section2.7.

The scaling factor f accountsfor the spatial variations in the cosmogenicnuclideproductionratesattheEarth’ssurface,dueto the strong influence of the Earth’s environment on cosmic

par-ticles (e.g. Lal, 1991): with increasing elevation the atmospheric depth through which cosmic rays must travel before reaching the surface is smaller, yielding a higher cosmic ray flux. The impact of altitude is significant, since f increases exponentially with elevation: for each 1000 m of elevation gain, the produc-tion rate increases by a factor of two (Lal, 1991). To a lesser extent, f is also controlled by the intensity and the orientation of the Earth’s magnetic field (Dunai, 2001; Lal, 1991), implying that latitude also affectsthe value of f : atthe Equator, produc-tion rates are abouttwo times lower than they are atthe poles (Lal,1991).

f is therefore a function of the sample altitude andlatitude andmay also in some cases includepast variations of the mag-neticfield(e.g.Balcoetal.,2008).Ifthelatitudeandthemagnetic fieldvariationsatasamplesitecanbeindependentlyconstrained, solvingequation (2) for f allowstheelevationofa sample tobe constrained.

Equation(2) hasatotalofthreeunknowns:thescalingfactor f , thesurfaceerosionrate

ε

andtheexposuretime t.Thus, two ex-treme casesmay be considered: i)No erosionor ii) steady-state erosion(t

→ +∞

)(Lal,1991): N

(

t

)

=

f

·

PSLHL

λ



1

−

e−λt



for

ε

=

0 cm yr−1 No erosion (3a) N

=

f

·

PSLHL

λ

+

με

for t

→ ∞

Steady-state erosion (3b) It is possible to reduce the degrees of freedom of equation (3a) and (3b) by combining two isotopic systems with different half-lives and, under favorable circumstances, solve for f . These conditionsaresatisfiedforlongexposuretimes(

>

500 ka)orlow erosion rates (

<

1 m Ma−1) and correspond to the case of non-superimposedsimpleexposurecurves.

Acompanionmethodologypaper(Blardetal.,2019)providesa completedescriptionofthedifferentconditionsthatmustbe con-sideredwhensolvingfor f .

i)Thecaseofacontinuouslyexposed,non-erodedsurface

Inthe zero-erosion case,equation (3a) canbe solved fortime by considering the combinationof two nuclides with production rates PSLHL,1 and PSLHL,2, respective decay constants

λ

1,

λ

2 and

concentrationsN1,N2: fε=0

−

fε=0



1

−

A fε=0



r

=

B (4) withA

=

λ2N2 PSLHL,2, B

=

λ1N1 PSLHL,1,r

=

λ1 λ2.

In this case, there is no analytical solution, and equation (4) mustbenumericallysolvedtodetermine f .Itisimportantto con-sidertheanalyticaluncertainties attachedto N1 andN2 toassess

theabilityto determine f from equation (4) withits appropriate uncertainties(detailedmethodsinBlardetal.,2019).

ii)Caseofsteadystateerosion

Inthecaseofsteadyerosion,equation(3b) leadstothe follow-ingratherstraightforwardanalyticalsolution,forapairof radioac-tivecosmogenicnuclides:

ft→∞

=

_P N1N2

(λ

1

− λ

2

)

SLHL,1N2

−

PSLHL,2N1

(5)

In theory, both equations (4) and (5) can be solved for f . If latitudeandthepastgeomagneticfieldvariationscanbe indepen-dently constrainedwith reasonable confidence, f can inturn be usedtodeterminethesample’saltitudeofexposure.

Inpractice,nuclideconcentrationsareonlyknownthrough lab-oratory measurements andare thus affected by analytical uncer-tainties (or biases) that may lead to cases that are not solvable. Inthenextsection2.3,wepresentnumericalsimulationsto com-putetheminimumduration of(paleo)-exposurerequiredto solve the paleoelevation(i.e. thedetectionlimitofthesystem), aswell astheimpactofexposuredurationontheuncertaintyinthe com-putedelevation.

2.3. Thedetectionlimitofthepaleoaltimeterandtheimpactofthe exposuredurationontheuncertainty

The mathematical conditions required to solve equations (4) and (5) are described in detail in the companion publication in MethodsX (Blard et al., 2019). Because of the analytical uncer-tainties attached to measurements of 10_Be, 21_{Ne and} 26_Al

con-centrations, it is possible that cases may be encountered where the system cannot be solved for elevation, or can only be used to determine minimal elevation with a large uncertainty, no-tably in the case of short exposure (

<

500 ka) or erosion rate

>

1 m Ma−1_).

To evaluate the influences of the exposure duration and the erosion rateon the detectionlimit, accuracy and theoverall un-certainty ofthispaleoaltimetrymethod,we performeda numeri-cal MonteCarlosimulation,withanalyticaluncertainties of5% on each cosmogenic nuclide, for the pairs 26_Al–10_{Be and} 10_Be–21_Ne

(Fig.2).

The modeling showsthat both 26_Al–10_{Be and}10_Be–21_Nehave

similarcharacteristics: longerexposureduration(orlowererosion rates) improve both the accuracy andthe precision of the com-puted elevation. Exposure durationslonger than1 Ma orerosion rates lower than 1 m Ma−1 _{guarantee the determination of}

ac-curate elevations,withabsolute1

σ

uncertainties that aresmaller than500 m.Forexposuredurationsshorterthan500 ka,and ero-sion rates higher than 1 m Ma−1, the computed elevations are lower than the real altitudes of exposure. This systematic un-certainty resultsfrom the non-linearityof equations (4) and (5). In other words,for low nuclide concentrations, the two-nuclides curvesare superimposedandthemethodthus onlyallows deter-minationofaminimumelevation(Fig.1and2andmathematical descriptioninBlardetal.,2019).Thisbiasincreaseswhenthe ex-posure duration is shorter–or theerosion ratesishigher –and mayreachseveralhundredsofmeters(Fig.2).Usingnuclidepairs withacosmogenicnuclidehavingashorterhalf-life(e.g.36Cl/10Be) wouldimprovethemethodsensitivityforhighererosionratesand shorterexposuretimes.

In summary, this Monte Carlo simulation shows that the ac-curacy andtheprecision ofthemethodare muchbetter forlong exposuredurations(



100 ka)orlowerosionrates(

1 m Ma−1). However,shorterexposuredurationsmayalsoprovideuseful infor-mationinthatknowingtheminimumelevationatwhichaterrain was exposed couldalsobe ausefuloutcomeincertaingeological contexts.

2.4. Biasduetopoorknowledgeofthedegreeofsurfacepreservation:is thesurfaceerodedornot?

Itisdifficult,oftenimpossible,totellaposteriori whethera pa-leosurface has undergone continuous exposure orsteadyerosion. The unknown state of the paleosurface may induce a systematic uncertainty in the reconstructed elevation of that surface (inde-pendent of analytical uncertainties that are random), since the positions of simple exposure curves differslightly fora continu-ousexposed surfaceandasteadilyeroded surface(Fig.1and3A). Inotherwords,ifapreservedsurfaceiswronglyassumedtohave

Fig. 2. MonteCarlosimulationshowingtheinfluenceoftheexposuretime(AandB)anderosionrate(CandD)ontheaccuracyandtheprecisionofthepaleoaltimeter,for the26_Al–10_Beandthe10_Be–21_{Nepairs.ThesesimulationswereperformedusingtheparameterspresentedinTable1,assuminganalyticaluncertaintiesof5%,atalatitudeof}

60◦,usingtheLal/Stone(Stone,2000) timeindependentscheme,at0,2000and4000m.Foreachtimestep,500randomdraws(5000drawsforeacherosionratestep)were madefornuclides1and2,assumingthat(N1,σ1)and(N2,σ2)follownormaldistributions,σ1andσ2beingtheanalyticaluncertainties(5%here).Next,therandomly pickedcosmogenicnuclidespairsweresolvedforelevationusingequations(4) and(5),inthecaseofcontinuousexposureandsteady-stateerosion,respectively.Central curvesrepresentthemedians(50%)andthemeans,errorenvelopesareboundedbythe0.16and0.84confidenceintervals(1σ).

beeneroded, thiswilllead tooverestimationofthe actual eleva-tionofexposure(Fig.3A).Conversely, assumingcontinuous expo-surewhenthesurface hasactuallyundergoneerosionwillleadto underestimationofelevation.

We estimatedthis bias by modeling the difference in the re-constructedaltitudes for a pair ofsurface nuclideconcentrations byconsidering boththecontinuous exposureandthesteady ero-sion scenarios (Fig. 3B). This bias results from the difference:



f

=

f

(

ε

=

0

)

−

f

(

t

→ ∞)

(equations(4) and(5)).Fig.3Bshows howthissystematicuncertaintyinthereconstructedpaleoaltitude isdirectlylinkedtotheunknownstateofthepaleosurfaceforboth the10_Be–26_Aland10_Be–21_Nepairs.

The modeling highlights that for exposure durations shorter than 100 ka, (or erosion rates higher than 1 m Ma−1), this bias may reach 900 m for a surface exposed at sea level and up to 1900mforexposures at 6000 m (Fig.3B). However, forsurfaces thathavebeenexposedlongenough(



100 ka)(orthathavebeen slowlyeroded

1 m Ma−1) thissystematicuncertaintyinthe re-constructedpaleoaltitudedecreasessignificantly,untilitisnullfor surfaceswithradioactively-saturated10_Be,26_{Al and}21_Ne

concen-trations.

Longerexposures(orlowererosionrates)thussignificantly im-prove both the accuracy andthe precision of the paleoaltimetric method.

2.5. Whymuonscanbe(almost)safelyneglectedhere

In the equations described in section 2.2, PSLHL only refers to the production rate due to spallogenic high-energy neutrons. Thismeansthatthedifferentattenuationlengthsofneutronsand muons (Groom et al., 2001) are ignored here. This is justified because the favorable surfacesfor paleoaltimetry are either non-eroding,andthusthe muogenicproductionatdepthdoesnot af-fect thesurfacenuclideconcentration,orareeroding veryslowly, meaning that the deep production is negligible compared to the totalsurfaceconcentrationsincethemuon-producednuclideswill havedecayedbeforereachingthesurface.

Stable 21_{Ne represents a possible exception however, since}

muogenic21Neproducedatdepthdoesnotdecayonitswaytothe surfaceandmaythereforemakeagreatercontributiontothetotal surfaceconcentrationthan10_Beand26_{Aldo.Neglectingmuogenic} 21_{Ne in a slowly eroding surface maytherefore cause}

underesti-Fig. 3. A)Schemeshowingthetheoreticalimpactoftheerosionassumptiononthe computedelevation.B)Maximumabsolutesystematicuncertaintyinthecomputed altitudeofa paleosurface arisingfromthe unknownstate ofthat surface: con-tinuousexposureorsteadyerosion(f=f(ε=0)−f(t→ ∞)).Thissystematic uncertaintyisplottedagainstthe10_{Beconcentration,aproxyofincreasing}

expo-sureduration(ordecreasingerosionrate).Thewrongassumptionthatapreserved surfacehasbeen erodedwillleadtooverestimationoftheactualelevation. Con-versely,assumingcontinuousexposurewhenthesurfacehasactuallyundergone erosionwillleadtounderestimationofelevation.Plotsshowthetwonuclidespairs:

10_Be–26_{Al(solidline)and}10_Be–21_{Ne(dashedline).Theseuncertaintiesareplotted}

forasurfaceexposedatconstantelevationsof0,2000,4000and6000m.a.s.l.

mation of the 10_Be/21_{Ne ratio, and thus underestimation of the}

correctelevation. Thispotential biasmustbe kept inmindwhen using thispair of nuclides asa paleoaltimeter.Ideally, the muo-genicproductiontermshouldbeaddedtotheequationdescribing

21_{Neproduction.}

2.6. Paleoaltimetryusingfossilexposedsurfaces

2.6.1. Approach

Continuouslyexposedsurfacesonlyofferalong-termintegrated signal that includes the most recent elevation changes (Blard et al.,2006).Thealtimetrymethodpresentedinsection2.2mayalso beusedto reconstructthe altitudeofan ancientsurfacethat un-derwent continuous exposure or steady erosion in the past and thathassincebeenburied fora knownperiodoftime.Themain interest ofpaleosurfaces is toprovide a snapshot of the paleoel-evation history of a massif at a particular moment in its uplift, orsubsidence, history. In such a case, the present-day measured concentrationsofradioactivenuclides(10Be,26Al)inacompletely shieldedpaleosurfacemustbe correctedfortheradioactivedecay thatoccurredsinceburial(Fig.4).

A number of requirements need to be fulfilled to derive the most accurate and precise paleoaltitude possible from a paleo-exposed surface: 1) like non-buried surfaces, the paleosur-faces need to have been exposed for long enough during the past (



100 ka), or to have been affected by low erosion rates (

1 m Ma−1_{);2) thesurface musthavebeenrapidly anddeeply}

Fig. 4. A)Schematicdiagramofanidealpaleosurfaceforcosmogenicpaleoaltitude reconstructions,whereapaleosurfaceisrapidlyburiedunderadatable layer.B) Principleofpaleoaltitudeevolutionofasurface:thesurfaceisexposedduringa pe-riodofdifferentelevationandhasbeenburiedsince.Thenuclideconcentrationon thissurfacereflectstheelevationpriortothealtitudechange.C)Simpleexposure curvesevolutionofthepaired26_Aland10_{Beconcentrationsofaburiedsurface.D)}

Simpleexposurecurvesevolutionofthepaired21_Neand10_{Beconcentrationsofa}

buriedsurface.

buried (ideally below tens of meters, Fig. 6; Blard et al., 2006) so that there is negligible post-burial nuclide accumulation; and 3) theburialageshouldbeindependentlyestimated.Underthese conditions, it is possible to correct the cosmogenic nuclide con-centrations measured ona paleosurface, Nmeasured,forradioactive decay that has occurred since burial,



tburial, to reconstruct the initialnuclideconcentration, Ninitial:

Ninitial

=

Nmeasured

·

eλ·tburial (6) Equation(6) canthenbe combinedwithequations(4) and (5) to reconstruct the paleoaltitude of the surface. In the case of sta-ble nuclides, such as 21Ne, the measured concentration remains constant during burial and a radioactive decay correction is not necessary.

Unlikepreviousapproachestocosmogenicnuclide paleoaltime-try (Blard etal., 2005; Libarkin etal., 2002),this paired-nuclides methodisintheoryapplicable forboth steadilyeroding and con-tinuouslyexposedsurfaces.Asthepreservationstateofasurfaceis

sometimesdifficulttoassess(section2.4), thisis thereforea ma-joradvantageoftheapproachproposedinthisarticle.Themethod allows investigation ofa much broaderrange of geomorphic ob-jects that havebeen exposed overa large time-span inthe past. Volcanic environments are likely good candidates for such pale-oaltitudereconstructionsasvolcanicflows providerapidburialof theunderlyingsurfacesandcanbedatedovertimescalesofseveral millionyearsusingabsoluteradiometricmethodssuchasK–Aror Ar–Ar.

2.6.2. Methodrange

Themethodrangedirectlydependsontheanalyticaldetection limitoftheanalyzedcosmogenicnuclides,andthus,onthe com-binedinfluencesofthepaleoexposuredurationandtheageofthe burial ofthefossil exposed surface.Considering thecurrent ana-lyticaldetectionlimitsof10_Be,21_Neand26_Al(

_∼

₅

_×

₁₀3_,

_∼

₁₀5 _and

∼

104 at g−1,respectively), theoldest buriedsurface that may be confidently used to apply this paleoaltrimetry methodis

∼

8 Ma inthecaseofthe 26Al–10Be pairand

∼

12 Mainthecaseofthe

10_Be–21_{Nepair(Fig.4).}

2.7.Integrationtimeandresponsetimeafterrapidupliftorsubsidence scenarios

Giventhat thispaleoaltimetry method requiresrelatively long exposureepisodes(

>

500 ka),itisprobablethattheelevationhas changedduringexposure, meaningthattheassumptionofa time independentscalingfactorisnotvalid:df

/

dt

=

0.

The integration time tint can be defined as the mean age of

a given cosmogenic nuclide contained in a rock sample. tint is

definedbythefollowingequation,whereNmes(at g−1)isthe mea-suredconcentration ofthenuclide N and P (at g−1_yr−1_{) itslocal}

productionrate: tint

=

−

1

λ

·

Nmes N



mes 0 ln



1

−

λ

·

Nmes P



·

dN (7)

tintisdifferentfromtheactualexposuretimetexp thatcanbe

de-rivedfromequation(3a):

texp

=

−

1

λ

·

ln



1

−

λ

·

Nmes P



(8)

Twoextremecasescanbedistinguished:

i)Ifexposuretimetexpismuchshorterthanthehalf-lifeofthe

nuclide(texp

1

/λ

),thentint

=

texp

/

2,

ii)Iftexpismuchlargerthanthehalf-lifeofthenuclide(texp

1

/λ

),thentint

=

1

/λ

.

Inthepracticalcaseofthepaired-nuclidealtimetry,tint is

de-terminedbythecosmogenicnuclidehavingtheshortesthalf-life. This integration time constrains the response time of the methodafteranaltitudinalchange.Weperformedseveral numeri-caltestswithdifferent“staircase”upliftandsubsidencescenarios. ThesesimulationsshownonFig.5constrainthemethodreactivity. Although these extreme staircase scenarios are not encountered innatural settings,they representbenchmarks andprovide limit valuesfortheresponsetimeofthealtimeter.

These numerical simulations (Fig. 5) show that the response time is dependent on the half-lives of the nuclides considered: inthe uplift case, the recorded elevationis

±

10% similar to the correct altitude after 1.5 Ma of exposure for the 26Al–10Be pair (Fig.5A),whileitrequires

∼

3 Maforthe10_Be–21_{Nepair(Fig.5B).}

Thismodelingalsoshowsthattheresponsetimeoftheregistered altitudeisshorterin thecaseofupliftthan inthe caseof subsi-dence(Fig.5).Thisisduetotheexponentialincreaseinproduction ratewith elevation, which gives a larger weight to the nuclides producedathighelevation.

3. Testingthemethodagainsttheexisting10_Be–21_Ne–26_Al

datasetfromAtacama

3.1. Datadescription

In order to evaluate the accuracy of this new paleoaltimet-ric method as well as its overall uncertainty in real conditions, we tested this approach on an existing dataset of samples, in which multiple cosmogenic nuclides have been measured (26Al and10_{Be, or} 21_{Ne and}10_{Be). Forthis, we selected datasets from}

a geological setting that fulfilled the following criteria: i) sam-ples have presumably been exposed at the surface for a suffi-ciently long and continuous duration (



100 ka); ii) rocks have remainedatthe samesampling elevationwithoutanysignificant elevationchangeduringtheexposure history;andiii) thedataset covers a quite large altitudinal range, from sea-level to several thousand of meters. For this initial test, we chose the Atacama desert,averyaridregionoftheWesternTropicalAndesthatmeets the required geological and geomorphological criteria: i) rain-fall is extremely low (

<

10 mm w.e.yr−1_{), making long exposure}

(

>

1 Ma)possible(Dunaietal., 2005; Ritter etal., 2018b),ii) sev-eral lines of evidence suggest that no altitudinal change greater than 1000 m has occurred since 5 Ma (Garzione et al., 2008; Kar et al., 2016), and iii) the steep west-east topographic gradi-entgivesaccesstoanaltitudinalrangefrom0to5000 m.

We consider here all published datasets that have reported measurementsofcosmogenicnuclidespairs(26_Al–10_Beand10_Be– 21_{Ne) from surface samples collected in the Atacama desert:}

(Kober etal., 2007; Nishiizumi et al., 2005; Placzek etal., 2010; Ritteretal.,2018a,2018b).TableS1presentsthemain characteris-ticsofthesesamples(nature,lithology,samplingcoordinates).The datasetincludes atotal of68samplesforthe 26Al–10Be pairand 43samplesforthe10_Be–21_{Nepair.Alloftheobjectssampledhave}

a quartz rich lithology and variable geomorphologic characteris-tics: bedrock and lake shoreline samples, and detrital objects of differentsizesandsources:clasts,cobblesandbouldersdeposited byalluvialorgravitationalprocesses.

3.2. Comparisonbetweencalculatedandpresent-dayelevations

As a first screening, we plottedthe data in the two isotopes diagram,alongwiththesimpleexposurecurvesfordifferent eleva-tions(0,1500,3000and4500 m).InFig.6,thesamplingelevations arerepresentedbythecoloroftheellipses.Toallowaproper com-parison,allcosmogenicnuclidesconcentrationswerescaledtothe samelatitudeof20◦usingthetimeindependentmodelof(Stone,

2000) andthestandardatmospheremodel(N.O.A.A.,1976). The plots indicate a good first order agreement betweenthe sampling altitudes, shown by the colors of the ellipses, and the elevationsdeducedfromthecosmogenicnuclides,whichare rep-resented by the relative position of these ellipses compared to those of the exposure curves. In the graphs, no sample, except one, plots in the upper forbidden zone – the area above andto therightofthecurves–stronglysuggestingthatnoneofthe sam-ples originated froma higher elevationand that the cosmogenic nuclideproductionratesarewell constrained.Ontheother hand, afew samplesplotbeloworontheleft oftheir respective expo-surecurves,suggestingthat thesesampleshavebeen affectedby periodsofburialorrecentlyexhumed.

Importantly, the majority of the samples are located on their corresponding elevation-curves, suggesting that the dataset was not affected by any significant elevation change since the initia-tionofexposure.

Foramoreprecisecomparison,wealsoinvertedallofthedata tocompute paleoelevationsusingthealtimetrymethod,following the mathematical approach described in (Blard et al., 2019).

Ta-Fig. 5. Simulationsoftheresponsetimeofthealtituderecordedbythepaired-cosmogenicnuclidesaltimetrymethodforpositiveandnegativestaircaseelevationchange scenarios.AandB)instantaneousH+ H upliftforthe26_Al–10_Beand10_Be–21_{Nepairs,respectively.CandD)instantaneousH− H elevationdropforthe}26_Al–10_Beand 10_Be–21_{Nepairs,respectively.}

ble S2 displays the detailedresults and Fig. 7 is a plot of these computedelevationsvs samplingelevations.Sincewedonothave any a priori knowledge of the amount of erosion that affected thesesamples,thetwoextremecasesofcontinuousexposureand steady-stateerosion wereconsidered (Fig. 7A andB). Sample ex-posuresthat didnot satisfythe conditionstocompute elevations are not plotted on Fig. 7. Among the 43 21Ne–10Be samples, el-evations could not be calculated foronly 2 samples. In the case of the 68 26Al–10Be samples, mean altitudes could not be cal-culated for 18 samples in the caseof continuous exposure, and 10in the caseofsteady-state erosion.This difference mainly re-sultsfrom theshorter integrationtimes ofthe 10Be–26Al dataset (0

.

55

±

0

.

28 Ma) compared to those of the 21Ne–10Be dataset (1

.

22

±

0

.

60 Ma).

Fig. 7 plot shows that there is a first order good agree-ment between the computed elevations and the sampling ele-vations. The two extreme scenarios of continuous exposure and steady-state erosion yield quite similar results, indicating that the method is robust, whatever our knowledge of the surface preservation state.Best-fit regressionlines were computed, along with their parameter uncertainties at the 2-sigma confidence level.

In the case of the 26Al–10Be dataset, the best-fit regression curveslieslightlybelowthe1:1line,bothforthecontinuous ex-posure ( y

= (

0

.

78

±

0

.

31

)

·

x

+ (

10

±

600

)

, R2

=

0

.

50,p-value

<

6

×

10−6)andthesteady-erosioncase( y

= (

0

.

70

±

0

.

39

)

·

x

+(

670

±

770

)

,R2

=

0

.

39,p-value

<

10−3). Thisslightdivergencewiththe 1:1lineisduetoagroup of6samplesthat todaystand between 2300and3800 m, butthatyielded computedelevations1000to 2000 mbelowtheirpresentposition.Thisbiasmayresultfrom un-recognizedrecentexhumationofsamplesthatspentmostoftheir exposure histories morethan 50 cm below therock surface (see section 4.1 below). However, ifthese 6 points are excluded, the

agreement between the samplingand the computedelevation is excellent, indicatingthatthe methodisrobust andaccurate, even when using alarge dataset that was not specifically sampledfor paleoaltimetricpurposes.

The 21Ne–10Be dataset yields best fits that are statistically in quite good agreement with the 1:1 line: y

= (

1

.

12

±

0

.

21

)

·

x

−

(

800

±

610

)

,R2

=

0

.

83,p-value

<

2

×

10−13inthecaseof contin-uousexposure,and y

= (

1

.

42

±

0

.

21

)

·

x

− (

760

±

540

)

, R2

=

0

.

83, p-value

<

6

×

10−16, in thecase ofsteady-erosion. The samples thatare notalignedonthe1:1linearethosefromthe Quillagua-Llamara Soledad paleoakeshorelines (Ritter et al., 2018a), which arelocatedatlowelevations(

∼

1000m)andyieldunderestimated computedelevations(rangingfrom

−

1500to500 m).Several pos-sibilities might beproposed to explain the factthat manyof the shorelines samplesofRitter etal. (2018a) yielded computed alti-tudeslowerthantheirpresent-dayelevations:i)Asignificantpart oftheexposuremayhavestartedandoccurredbelowthelake sur-face,orbelowsoilcoverthathasbeenremovedduringtheupper Pleistocene; ii)We did not consider the muogenic production in the equation. As muon production is greater than spallation be-low a depth ofseveralmeters,it isplausiblethat neglecting this process has lowered the 10Be/21Ne ratios after the decay of the muogenic10Beproducedatdepth,leadingtoerroneouslylow ele-vations.

Finally, itis interesting to note that the detrital sampleswith the longest exposure (

∼

20 Ma) and integration times (

∼

2 Ma) also yield computed elevations that are almost identical to (or slightly lower than) their present-day sampling elevations (Ta-ble S2).Thisshowsthatthesesamples,despitetheirdetritalorigin, havecosmogenic-nuclides inventoriesthatdo notresultfrom ex-posuresathigherelevations.Hence,theexposuretimescomputed usingthepresent-day elevationsareprobablycorrectandare not overestimated.

Fig. 6. Plotsofcosmogenicnuclidepairs(A–26_Al–10_BeandB–10_Be–21_{Ne)measuredinsurfacesamplesfromtheAtacamaregion(datafromKoberetal.,2007; Nishiizumi}

etal.,2005; Placzeketal.,2010;Ritteretal.,2018a,2018b).Thereare68samplesforthe26_Al–10_{Bepairand43samplesforthe}21_Ne–10_{Bepair.Thisplotwasrealizedusing}

theproductionparametersgiveninTable1.Allcosmogenicnuclidesconcentrationshavebeenscaledtothesamelatitudeof60◦usingthetimeindependentmodelofStone (2000).Ellipsesareplottedfor68%confidenceintervals.Thecolorofeachellipserepresentsthesamplingelevation.Simpleexposurecurvesareplottedforelevationsof0, 1500,3000and4500m.(Forinterpretationofthecolorsinthefigure(s),thereaderisreferredtothewebversionofthisarticle.)

4. Otherimportantconsiderationsaboutthetwo-cosmogenic nuclidecurves

4.1.Impactoftheexposuredepth:theequivalencebetween atmosphericandrockdepth

Thetheorythatdescribeshowthealtitudeofexposurecontrols the positionof the simple exposure curves (Section2, Equations (3a) and(3b))isalsovalidinthecaseofdifferentdepthsof expo-surebelowtherocksurface(GosseandPhillips,2001).Considering thati)theattenuationlengthsoffastneutrons ingasandinsoils aresimilar(GosseandPhillips,2001),andii)thatsoilshavea den-sityabout2000timeshigherthanthedensityofatmosphereatsea level(assumingasoildensityof

∼

2.4 g cm−3_{,andan atmosphere}

densityof

∼

1

.

2

×

10−3_{g cm}−3_{atsealevel),1000 mofairis}

there-foreequivalenttoasoilthicknessof50 cm.Giventhisequivalence, it iscrucialthat thethicknessandthe densityof theoverburden are well known ifa rockis sampled belowa paleo-exposed sur-face for paleoaltimetricinvestigation. If an unknown process has recentlyexhumedtherock,thereisariskthattheactual altitude ofexposurewillbesignificantlyunderestimatedusingthe paleoal-timetry method(Fig. 3). Thisimportant aspectshould be kept in mindduringsamplingandanyevidencethatsuggeststhesporadic presence ofloess,soils oranycovershould be consideredbefore samplingapaleosurfaceonafreshoutcrop.Thepossibilityof land-slidesshouldlikewisebecarefullyassessed.

A potential depth-meter. If the altitude of exposure is well-known, the position of the simple exposure curve may also be turnedintoasensitiveandprecisedepthmeter(GosseandPhillips,

paleoaltime-Fig. 7. SamplingagainstelevationscalculatedwiththepairedcosmogenicnuclidesmethodconsideringA)continuousexposureandB)steady-stateerosion,forthewhole Atacamadataset(datafromKoberetal.,2007; Nishiizumietal.,2005; Placzeketal.,2010;Ritteretal.,2018a,2018b).Sincenosystematicdifferenceisobservedbetween bedrock,bouldersandcobbles(TableS2),thedifferenttypesofsamplearenotdifferentiatedinthisplot.Thebest-fitregressionandthe1:1linesareshown.Integration timesarerepresentedbythecolor-scale.Uncertaintiesofthebest-fitparametersaretwosigmas;uncertaintyenvelopesareshowninred.(Onlythosesamplesforwhichit waspossibletocalculateelevation,positiveandnegativeerrorbarsareshown.)

try method has a 1

σ

uncertainty of

∼

500 m of atmosphere for the26Al–10Beandthe10Be–21Nepairs(Section2.3).Thus,insuch cases,the methodis potentiallysensitive enough to measurethe depthofexposures withaprecisionof

∼

25 cm ofrock/soil.(Hidy et al., 2018) used the 26_Al–10_{Be couple to calculate a thickness}

of

∼

80 cm of loess cover over a paleosol exposed during more than1 MainYukon,Canada.Forexposuresthatoccurbelowwater (densityof 1 g cm−3 forwater),the uncertaintyin themeasured waterdepthwouldbeslightlylarger(

∼

50 cm).Notethatthe mini-mumrequiredexposuretimewillbesignificantlyreduced(andthe precision improved) using a radioactive cosmogenic nuclide that hasa relativelyshorthalf-life,suchas14_{C(half-lifeof}14_C

₌

₅₇₃₀

years).Althoughpresentingthelimitsofthisspecificaspectofthe methodisbeyondthescopeofthisarticle,itisworth notingthat the14C–10Bepairhasthebestpotentialfordepthdeterminations. It has notably been used to constrain the depth of partial snow shieldingintheGotthardPassareaintheSwissAlps(Hippeetal.,

2014).Severalintriguingquestionscanthusbeaddressedwiththis

depthmeter,suchasmeasuringthemeandepthoflandslidesorthe thicknessofpaleocoversofanynature,forexamplesoil,ash,loess, snow, iceorvegetation.Hydrological studies mighteven be con-ducted withthis method,as ittheoretically allows measurement ofthe(paleo)waterdepthofa(paleo)lake.

4.2. Impactoftheelevationofexposureontheaccuracyofburialages

The burial age dating method has beenwidely used to place importantgeochronologicalconstraintsonseveralmajorproblems inEarthsciences(e.g.Grangeretal.,2015; Sartégouetal.,2018). One ofthe mainstrengths ofthismethodis duetothe factthat the preburial 26_Al/10_{Be (or} 10_Be/21_{Ne) ratio can oftenbe}

consid-eredas“independentoflatitudeandaltitude” (GrangerandMuzikar,

2001). However, as illustrated by Fig. 8,in the caseof long pre-burialexposureages(

>

100 ka)orlowerosionrates(

<

1 m Ma−1_),

altitude couldaffectthe initialpreburial 26Al/10Be (or 10Be/21Ne) ratioand,whenitisknown,itshouldthereforebetakeninto ac-countintheburialagecalculation(Equations(9) and(10)).Inthe

Fig. 8. Exampleshowingtheimpactofthealtitudeofexposureontheaccuracyof burialages,inthecaseofthe26_Al–10_{Besystem.Inthegivenexample,assumingan}

initial26_Al/10_{Beratiosimilartotheproductionratioleadstoanoverestimatethe}

actualburialage.Forlongpreburialexposuredurations(>100ka)orlowerosion rates(<1 m Ma−1_{),itisimportanttotakeintoaccounttheelevationofexposureto}

computethepreburial26_Al/10_Be(or10_Be/21_{Ne)ratioand,thus,toobtainaccurate}

burialages.

specific case ofthe burial dating of cave sediments, an accurate calculationshouldusetheelevationofthequartz-richwatershed, ratherthan the altitude of the cave. Fortunately, inthe majority ofcases,preburialratios areinpracticesimilar totheproduction ratios,implyingthatmanyburialagescomputedwiththis approx-imationarenotaffectedbythispotentialelevation-relatedbias.

Foramaterialthathasbeenexposedundersteady-erosion con-ditionsbeforeburial, theequation thatmustbe usedto compute theburialagetburialis:

P1 N1 e−λ1·tburial

−

P2 N2 e−λ2·tburial

=

λ

1

− λ

2 f (9)

Ifnuclides1and2areboth radioactive,equation (9) hasno ana-lyticalsolutionandmustbenumericallysolvedtodeterminetburial. Fortheparticularcaseofacosmogenicnuclidespairwithonly oneradioactiveisotope,suchas10Be–21Ne,equation(9) simplifies andcanbeanalyticallysolved:

tburial

=

−

1

λ

·

ln

N10 P10

·



P21 N21

+

λ

10 f



(10)

AMatlab©code,(Burial.m), computingburial agestakinginto accountthealtitudeofthepreburialexposureepisodeisavailable inaMethodsXcompanionpaper(Blardetal.,2019).

5. Concludingremarks

•

In this article, we propose a new means of determining pa-leoelevations using pairs of insitu cosmogenic nuclides that havedifferenthalf-lives.Positionsofcosmogenicnuclide sim-ple exposure curves are altitude-dependent, a property that may be usedto constrainthe elevationofexposure. In prac-tice,givencurrentanalyticalcapabilities,thebestnuclidepairs forthispurposeare26_Al–10_Beand10_Be–21_{Neinquartz,which}

haverespectiverangesof6Maand12 Ma,respectively.

•

Both 26_Al–10_{Be and} 10_Be–21_{Ne systems require minimum}

equivalent exposure durations that are longer than 100 ka (ideally

>

500 ka, or erosion rates lower than 1 m Ma−1₎

toyield accurate elevations,witha typical1

σ

uncertaintyof

∼

500 m.

•

Iftheexposuredurationisshorterthan500 ka(orifthe ero-sion rateis lessthan

<

1 m Ma−1), themethod allows mini-mumelevationsofexposuretobecalculated.

•

Apoorknowledgeofthepreservationstateofthesurface(i.e. theabsenceorpresenceoferosion)mayinduceabiasonthe computedelevationoflessthan1000 matsealevel.

•

We applied themethod tobedrock anddetrital objects from the Atacama desert (Andes). Results show a goodagreement betweenthecomputedaltitudesandthepresent-daysampling elevations.

•

Giventhat the majorityoferosion rateson Eartharegreater than 1 m Ma−1_{,themethod maynotbe applicable toobtain}

mean elevations in many regions. However, if erosion rates are greater than 1 m Ma−1, the method may be applied to detritalmaterialtodeterminetheminimumbasin-averaged al-titude–aconstraintthatcouldbeusefulinaddressingseveral geodynamics-relatedproblems.

•

Anintriguingdirectapplicationthatcanbe derivedfromthis paleoaltimetrymethodisdepthmetry,i.e. themeasurementof soil,snow,iceorwaterthicknessduringexposure.

•

Since the positions of a two-nuclide curves is altitude de-pendent,then the pre-burial cosmogenicnuclide ratioof the buried material maybe affected by thealtitude ofexposure. This must be considered before applying the burial dating method (Granger and Muzikar, 2001) in order to avoid any potentialinaccuracy.Thisisparticularlytrueinthecaseof wa-tershedsthat experience low erosionrates(

<

1 m Ma−1) and thatspanlargeelevationranges(

>

1000 m).

Acknowledgements

CarefulandconstructivereviewsbyGregBalcoandDerekFabel helpedustoimprovethesubmittedmanuscript.Authorsare grate-fulforstimulatingdiscussionswithVincentRegard,Sébastien Car-retier,RaphaëlPik,JulienCharreauandJérômeLavé.AliceWilliams carefully checked the English quality of the manuscript. This is CRPGcontributionn◦2689.

Appendix A. Supplementarymaterial

Supplementarymaterialrelatedtothisarticlecanbefound on-lineathttps://doi .org /10 .1016 /j .epsl .2019 .03 .005.

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