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Quaternary Geochronology xxx (2008) 1–12

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Contents lists avai

Quaternary Geochronology

journal homepage: www.elsevier .com/locate/quageo

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OOF

Research Paper

Determination of both exposure time and denudation rate from an insitu-produced 10Be depth profile: A mathematical proofof uniqueness. Model sensitivity and applications to natural cases

R. Braucher a,*, P. Del Castillo b, L. Siame a, A.J. Hidy c, D.L. Bourles a

a CEREGE UMR6635 Universite Paul Cezanne CNRS BP80 13545 Aix en Provence, Franceb Universite de Picardie Jules Verne, Laboratoire Amienois de mathematiques fondamentales et appliquees, CNRS UMR 6140, 33 r. Saint-Leu, 80039 Amiens Cedex 1, Francec Department of Earth Sciences, Dalhousie University, Edzell Castle Circle, Halifax NS, B3H 4J1 Canada

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a r t i c l e i n f o

Article history:Received 21 June 2007Received in revised form 27 March 2008Accepted 4 June 2008Available online xxx

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Keywords:Cosmogenic nuclide10BeDenudationTime

* Corresponding author. Tel.: þ33 4 42 97 15 09; faE-mail addresses: braucher@cerege.fr (R. Bra

picardie.fr (P. Del Castillo), alanhidy@dal.ca (A.J. Hidy

1871-1014/$ – see front matter � 2008 Published bydoi:10.1016/j.quageo.2008.06.001

Please cite this article in press as: Braucher,depth profile: A mathematical proof..., Quat

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CTEDPRa b s t r a c t

Measurements of radioactive in situ-produced cosmogenic nuclide concentrations in surficial materialexposed to cosmic rays allow either determining the long-term denudation rate assuming that thesurface studied has reached steady-state (where production and losses by denudation and radioactivedecay are in equilibrium) (infinite exposure time), or dating the initiation of exposure to cosmic rays,assuming that the denudation and post-depositional processes are negligible. Criteria for determiningwhether a surface is eroding or undergoing burial as well as quantitative information on denudation orburial rates may be obtained from cosmogenic nuclide depth profiles. With the refinement of thephysical parameters involved in the production of in situ-produced cosmogenic nuclides, a unique well-constrained depth profile now permits determination of both the exposure time and the denudation rateaffecting a surface. In this paper, we first mathematically demonstrate that the exponential decrease ofthe in situ-produced 10Be concentrations observed along a depth profile constrains a unique exposuretime and denudation rate when considering both neutrons and muons. In the second part, an improvedchi-square inversion model is described and tested in the third part with actual measured profiles.

� 2008 Published by Elsevier Ltd.

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UNCORR1. Introduction

Recently, the use of in situ cosmogenic nuclides has revolu-tionized Earth surface studies. They indeed not only allow surficialfeatures such as moraines, ‘‘stoss-and-lee’’ topography and alluvialfans to be dated for long periods as long as several million years, butalso allow distinguishing between both major dynamic processesaffecting surfaces (denudation and burial) and estimating theirrates (Siame et al., 2006). In the Earth’s environment, 10Be is mainlyproduced by interactions of primary cosmic ray particles (a parti-cles and protons) and their secondary particles (neutrons andmuons) with atmospheric nuclei of 14N and 16O. Although most ofthe cosmic ray’s energy is dissipated within the atmosphere,reducing cosmic rays intensity by almost 1000 from the top of theatmosphere to sea level, 10Be is also produced in the lithosphere byspallation of, for example, 16O, 27Al, 28Si, and 56Fe (Lal and Arnold,1995). The flux of the nuclear active particles efficiently dissipatestheir energy through cosmogenic nuclide producing nuclear

x: þ33 4 42 97 15 40.ucher), pierre.delcastillo@u-).

Elsevier Ltd.

R., et al., Determination of boernary Geochronology (2008

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reactions in the lithosphere. Therefore cosmogenic nuclideproduction rates decrease exponentially with the mass of overlyingmaterial with, for each production mechanism, a characteristicattenuation length L (g/cm2).

The evolution of the cosmogenic nuclide production rate (P(x))as a function of depth x (expressed in g/cm2 in order to beindependent from the material density) is given by Eq. (1):

PðxÞ ¼ P eð�x=LÞ; (1)

where P is the surface production rate. However, two maintypes of secondary particles with significantly different atten-uation lengths, neutrons and muons, are involved in litho-spheric in situ-production. The effective production attenuationlength of neutrons, Ln, is indeed shorter (approximately 150 g/cm2) than that of muons: 1500 g/cm2 for negative muons Lm1

and 5300 g/cm2 for fast muons Lm2 (Braucher et al., 2003). Thisimplies that although neutron-induced production is dominantin the near-surface (Brown et al., 1995), muon-induced reactions become dominant below a few meters. Fora surface undergoing denudation (3 in g/cm2/yr) the evolutionof the 10Be concentrations (f) with time (t) and depth (x) is

th exposure time and denudation rate from an in situ-produced 10Be), doi:10.1016/j.quageo.2008.06.001

E

Q1

Table 1Theoretical 10Be concentrations for a 2 m depth profile exposed for 20 ka un-dergoing a denudation rate of 10 m/Ma

Depth (g/cm2) 10Be (at/g)

0 262,88850 190,006100 137,742200 73,352300 40,164400 23,004

These concentrations come from Eq. (3) using a production rate of 15 at/g, a densityof 2 g/cm3; the relative contributions of neutrons and muons (negative and fast) tothe total 10Be production were n¼ 97.85%, m1¼1.5% and m2¼ 0.65% , the relativeattenuation lengths were Ln¼ 150 g/cm2, Lm1¼1500 g/cm2and Lm2¼ 5300 g/cm2

(Braucher et al., 2003).

R. Braucher et al. / Quaternary Geochronology xxx (2008) 1–122

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UNCORRECT

commonly described by the following partial differentialequation (Lal, 1991):

df ¼ P dt eð�x=LÞ þ 3 dtdfdx� lf dt; (2)

with l the radioactive decay constant.Since the half-live of 10Be is short compared to the Earth’s age,

its primordial component has vanished and, in addition, if weassume that the rock studied has undergone a single cosmic rayexposure episode and had no cosmogenic nuclides at the beginningof the present exposure, then the initial concentration of cosmo-genic nuclides equals zero. Thus Eq. (2) may be solved to yield:

fxð3; tÞ ¼nP e�x=Ln

3Lnþ l

�1� e�tð3=LnþlÞ

�þm1P e�x=Lm1

3Lm1þ l

��

1� e�tð3=Lm1þlÞ�þm2P e�x=Lm2

3Lm2þ l

�1� e�tð3=Lm2þlÞ

�;

ð3Þ

where n, m1 and m2 are the relative contributions of neutrons andmuons (negative and fast) to the total 10Be production (m1¼1.5%,m2¼ 0.65% and n¼ 97.85%; Braucher et al., 2003), Ln, Lm1 and Lm2

are the relative attenuation lengths of neutrons and muons (neg-ative and fast).

1.1. Former approach

During the 1980s and early 1990s, published works using insitu-produced cosmogenic 10Be (Brown et al., 1994; Lal and Arnold,1995; Nishiizumi et al., 1986, 1991) were only based on the neutronproduced 10Be as physical parameters for muons were not wellconstrained (this was a reasonable approach as the surface pro-duction is dominated by neutrons). As a consequence, assumptionswere required to interpret the measurements. If field observationsor other evidence support the assumption of simple exposurehistory and negligible denudation, then a minimum exposure agemay be calculated using Eq. (4):

tmin ¼ �1l

ln�

1� lf ð0; tÞP

�; (4)

derived from Eq. (3) for 3¼ 0, n¼ 1 and m1¼m2¼ 0. By contrast, iffield evidence indicated an exposure time long enough to reach thesteady-state balance concentration, for example at the surfaces ofstable cratons, maximum denudation rates may be computed usingEq. (5):

3max ¼

Pfð0;NÞ

� l

!Ln; (5)

derived from Eq. (3) for t[½ð3=LnÞ þ l��1 and n¼ 1 andm1¼m2¼ 0.

Provided the steady-state concentration has been reached, thedenudation rate can also be deduced from Eq. (3) using the bestfit curve technique along a 10Be concentration depth profile; thevariable thus being the denudation rate. If the steady-state con-centration has not been reached, there is an infinity of exposuretime–denudation rate pairs that will fit Eq. (3), the 10Be concen-trations measured along a depth profile, when only neutrons areconsidered.

1.2. Use of depth profile

Nowadays, refinement of the physical parameters describing themuon and neutron interactions, in particular their significantlydifferent attenuation lengths (Heisinger et al., 2002a,b; Kim and

Please cite this article in press as: Braucher, R., et al., Determination of bodepth profile: A mathematical proof..., Quaternary Geochronology (2008

DPROOF

Englert, 2004) allow determination of the proportion of 10Be pro-duced by each type of particle. It becomes theoretically possible,using a single cosmogenic nuclide, to estimate both the exposureage and the denudation rate of surfaces affected by relativelyconstant denudation rates (Braucher et al., 2003; Siame et al.,2004). Because 10Be concentrations resulting from reactions withrapidly attenuated neutrons reach steady-state with respect todenudational loss much more rapidly than those resulting fromreactions with the more penetrating muons, 10Be produced at thesurface, which mainly results from interactions with neutrons,might be used to estimate the denudation rate and that produced atseveral meters depth, which mainly results from interactions withmuons to estimate the exposure time.

To illustrate why users have to consider all the particles impliedin the total production of 10Be when studying a depth profile, let usenvisage a 2 m depth profile (six sampling depths) exposed duringthe last 20 ka and undergoing a denudation rate of 10 m/Ma. Theexpected 10Be concentrations are then calculated using Eq. (3),involving both neutrons and muons, with a production rateof 15 at/g/a (Table 1). To model the thus theoretical 10Be depth-produced profile concentrations, the exposure ages implied bydifferent denudation rates (0, 4, 8, 10, 12, 15 m/Ma) were calculatedusing Eq. (3). This was done first considering only neutrons (formerapproach), and, second, considering both neutrons and the muons.For each scenario (time–denudation pair), the best fit wasdetermined by minimizing a chi-square value that is the sum ofeach individual chi-square determined at each depth:

chi-square ¼Xn

i¼1

�Ci � Cðxi; 3; tÞ

si

�2

; (6)

where Ci is the measured (or here the expected) 10Be concentrationat depth xi, C(xi, 3, t) is the modelled 10Be concentration determinedusing Eq. (3), si is the analytical uncertainty at depth i, and n is thetotal number of samples in the profile.

Results of the best fit curve are presented in Fig. 1.Considering that neutrons only yield to a constant chi-square for

any determined exposure age–denudation rate pair, implying thatthey all are a possible solution. In that case, there is no uniquesolution. Considering that both neutrons and muons yield signifi-cantly different chi-square, the minimum chi-square being associ-ated with the exposure age–denudation rate pair (20 ka–10 m/Ma)which corresponds to the input parameters used to construct the10Be concentrations depth profile. In that case, a unique solution isevident. But is this solution unique from a mathematical point ofview? This is what we intend to demonstrate in the following part;then an improved statistical model based on that of Siame et al.(2004) will be presented and tested on natural data sets in a lastsection.

th exposure time and denudation rate from an in situ-produced 10Be), doi:10.1016/j.quageo.2008.06.001

CTEDPROOF

0.10

0.08

0.06

0.00

Ch

i-sq

uare

NeutronsNeutrons + Muons

NeutronsNeutrons + Muons

0 2 4 6 8 10 12 14 16

24000

23000

22000

21000

20000

19000

18000

17000

A

B

Ag

e (a

)

Denudation (m/Ma)

Fig. 1. Case study illustrating the solutions for a theoretically produced exponential down-concentrations profile (exposure age: 20 ka; denudation rate: 10 m/Ma). (A) Chi-squareversus denudation; (B) exposure ages versus denudation. Considering that neutrons only yield to a constant chi-square for any determined exposure age–denudation rate pair,implying that they all are a possible solution. In that case, there is no unique solution. Considering that both neutrons and muons yield to significantly different chi-square, theminimum chi-square being associated with the exposure age-denudation rate pair (20 ka–10 m/Ma; black arrow) which corresponds to the input parameters used to construct the10Be concentrations depth profile. In that case, a unique solution is evident.

R. Braucher et al. / Quaternary Geochronology xxx (2008) 1–12 3

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UNCORRE2. Mathematical approach

Using Eq. (3) we suppose depth parameter x ˛ [0, 1000], and(e, t) ˛ [0, 10�1]� ]0, þN[ and we assume that P ˛; [5, 50]. Tosimplify the notations, we set

ax ¼ nP e�x=Ln ; bx ¼ m1P e�x=Lm1 ;gx ¼ m2P e�x=Lm2 ; (7)

and we introduce the functions

31ðeÞd1

eLnþ l

; 32ðeÞd1

eLm1þ l

; 33ðeÞd1

eLm2þ l

: (8)

Let us remark that the extrema of fx are given by

mðxÞdinf ðe;tÞ˛½0;10�1���0;þN½fxðe; tÞ ¼ 0 ; (9)

and

MðxÞd supðe;tÞ˛½0;10�1���0;þN½

fxðe; tÞ ¼ax þ bx þ gx

l: (10)

(this is achieved for the zero denudation steady-state equilibrium)According to Eqs. (9) and (10), we get

fxð3; tÞ˛�0;MðxÞ½; for all ð3; tÞ˛h0;10�1

i��0;þN½:

We consider two depths x1 and x2 with (x1, x2) ˛ [0, 1000]2. Then,for two 10Be concentrations N1 and N2 with N1 ˛ ]0, M(x1)[ and

Please cite this article in press as: Braucher, R., et al., Determination of bodepth profile: A mathematical proof..., Quaternary Geochronology (2008

N2 ˛ ]0, M(x2)[, N1 s N2 we can introduce the following nonlinearsystem in two unknowns (e, t) given by(

fx1ðe; tÞ ¼ N1;fx2ðe; tÞ ¼ N2:

(11)

Remark 2.1. From Eq. (3), we remark that x 1 fx(e, t) is decreasing.It results that for N1<N2 and x1� x2, system (11) does not admitsolution.

In the following, we establish the proposition:

Proposition 2.2. Let (x1, x2) ˛ [0, 1000]2, x1< x2, N1 ˛ ]0, M(x1)[ and N2 ˛ ]0, M(x2)[. Then, there exists at most a pair (e, t) ˛ [0,10�1]� ]0, þN[ satisfying system (11).

To prove this proposition, we proceed as follows. First, we studythe function fx defined in Eq. (3). We compute vfx=vt and vfx=ve andwe show that for all (e, t) ˛ [0, 10�1]� ]0, þN[, we haveðvfx=vtÞðe; tÞ > 0 and ðvfx=veÞðe; tÞ < 0. Then, we introduce e0(N1)defined by

e0ðN1Þdmaxn

e˛h0;10�1

i���dt > 0 satisfying fx1ðe; tÞ ¼ N1

o:

We establish that, for all e ˛ [0, e0(N1)], there is a unique t(e) suchthat fx1 ðe; tðeÞÞ ¼ N1. To obtain the uniqueness of the solution ofsystem (11), we prove that e1fx2 ðe; tðeÞÞ is strictly increasing.

We have fx ˛ CN([0, 10�1]� ]0, þN[). From Eq. (3), for all(e, t) ˛ [0, 10�1]� ]0, þN[, it derives that

th exposure time and denudation rate from an in situ-produced 10Be), doi:10.1016/j.quageo.2008.06.001

E

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UNCORRECT

vfxvtðe; tÞ ¼ ax e�t=e1ðeÞ þ bx e�t=e2ðeÞ þ gx e�t=e3ðeÞ; (12)

and

vfxveðe; tÞ ¼ �ax31ðeÞ2

Ln

�1� e�t=31ðeÞ

�þ ax31ðeÞ

Lnt e�t=31ðeÞ

� bx32ðeÞ2

Lm1

�1� e�t=31ðeÞ

�þ bx32ðeÞ

Lm1t e�t=31ðeÞ

� gx33ðeÞ2

Lm2

�1� e�t=33ðeÞ

�þ gx33ðeÞ

Lm2t e�t=33ðeÞ: (13)

From Eq. (12), as ax, bx and gx are positive, we have

vfxvtðe; tÞ > 0 for all ðe; tÞ˛

h0;10�1

i��0;þN

: (14)

On the other hand, we have on [0, 10�1]� ]0, þN[

v2fxvevtðe; tÞ ¼ �t

�ax

Lne�t=31ðeÞ þ bx

Lm1e�t=32ðeÞ þ gx

Lm2e�t=33ðeÞ

�< 0:

As ðvfx=veÞðe;0Þ ¼ 0 for all e ˛ [0, 10�1], it results that

vfxveðe; tÞ < 0; for all ðe; tÞ˛

h0;10�1

i��

0;þN

�: (15)

In the proof of Proposition 2.2, we use the following lemma:

Lemma 2.3. Let

ða1;a2;a3Þ�ðLn;Lm1;Lm2Þ; and ði;jÞ˛ð1;2Þ;ð1;3Þ;ð2;3Þ

�:

(16)

Let us consider the functions defined on [0, 10�1]� [0, þN[ by

gi;jðe;tÞ ¼ e�t=3iðeÞ �

3jðeÞ2

aj

�1�e�t=3jðeÞ

�þ

3jðeÞaj

t e�t=3jðeÞ!

�e�t=3jðeÞ �3iðeÞ2

ai

�1�e�t=3iðeÞ

�þ3iðeÞ

ait e�t=3iðeÞ

!: (17)

For all (i, j) satisfying Eq. (16), we have

gi;jðe; tÞ ¼ 0; for all ðe; tÞ˛h0;10�1

i�h0;þN

h:

The proof of Lemma 2.3 follows.

Proof of Lemma 2.3. Let us recall that

Ln < Lm1 < Lm2 : (18)

We set

ui;jðe; tÞ ¼ �e�t=3iðeÞ 3jðeÞ2

ajþ e�t=3jðeÞ 3iðeÞ2

ai; (19)

vi;jðe; tÞ ¼ e�tð1=3iðeÞþ1=3jðeÞÞ

3jðeÞ2

aj� 3iðeÞ2

ai

!; (20)

wi;jðe; tÞ ¼ e�tð1=3iðeÞþ1=3jðeÞÞ

t3jðeÞaj� t3iðeÞ

ai

!: (21)

Thus, gi, j¼ ui, jþ vi, jþwi, j. From Eq. (8), we have

Please cite this article in press as: Braucher, R., et al., Determination of bodepth profile: A mathematical proof..., Quaternary Geochronology (2008

DPROOF

3jðeÞ2

aj� 3iðeÞ2

ai¼�aj � ai

e2 � aiaj

�aj � ai

l2�

eþ ajl 2ðeþ ailÞ

2: (22)

According to Eq. (8), we have

ejðeÞaj� eiðeÞ

ai¼ �

�aj � ai

l�

eþ ajl ðeþ ailÞ

: (23)

From Eqs. (19)–(23), we get

ðeþailÞ2�eþajl

2�ui;jðe;tÞþvi;jðe;tÞþwi;jðe;tÞ

¼ e�t=3jðeÞ��etð1=3jðeÞ�1=3iðeÞÞajðeþailÞ

2þai�eþajl

2�

þe�t=3iðeÞ��l�aj�ai

tðeþailÞ

�eþajl

þ��

aj�ai e2

�aiaj�ai�aj

l2��: (24)

To show that the right-hand-side of Eq. (24) is positive, weintroduce the function hi, j, defined on [0, 10�1]� [0, 107] by

hi;jðe; tÞ ¼ � aj etð1=3jðeÞ�1=3iðeÞÞðeþ ailÞ2þai

�eþ ajl

2

þ e�t=3iðeÞ�� lt�aj � ai

ðeþ ailÞ

�eþ ajl

þ��

aj � ai e2 � aiaj

�ai � aj

l2�: ð25Þ

For (e, t) ˛ [0, 10�1]� [0, 107], we have

vhi;j

vtðe;tÞ�

aj�ai

aiee�ðaj�aiÞte=aiaj

�eþail

�2

þe�t=3iðeÞ�

l�aj�ai

ðeþailÞ

�eþajl

��1þt

�e

aiþl

���

�e�t=3iðeÞ��

aj�ai e2�aiaj

�aj�ai

l2�

e

aiþl

��: ð26Þ

Thus, as tðe=aiþlÞ>0, we get

vhi;j

vtðe;tÞ�e�ðaj�aiÞte=aiaj

�aj�ai

aie3þ2

�aj�ai

le2þ

�aj�ai

ail

2e

�

þe�t=3iðeÞ���aj�ai

e3

ai�2�aj�ai

le2�ai

�aj�ai

l2e

�:

We deduce easily that, for x ˛ [0, 10�1]� [0, þN[, we have

vhi;j

vtðe; tÞ �

�aj � ai

aie3 þ 2

�aj � ai

le2

þ ai�aj � ai

l2e

�e�t=3iðeÞ

�et=3jðeÞ � 1

�:

According to Eqs. (16) and (18), aj> ai. Thus, for t� 0, we get that

vhi;j

vtðe; tÞ � 0; on

h0;10�1

i��

0;þN

�:

It results that hi, j(e, t) is positive if hi, j(e, 0)� 0. From Eq. (25), wehave

hi;jðe;0Þ ¼ 0; for all e˛h0;10�1

i:

The proof of Proposition 2.2 follows.

Proof of Proposition 2.2. Let N1 ˛ ]0, M(x1)[. Let us introducee0(N1) defined by

e0ðN1Þdmaxn

e˛h0;10�1

i��� dt > 0 satisfying fx1ðe; tÞ ¼ N1

o:

(27)

th exposure time and denudation rate from an in situ-produced 10Be), doi:10.1016/j.quageo.2008.06.001

E

R. Braucher et al. / Quaternary Geochronology xxx (2008) 1–12 5

ARTICLE IN PRESS QUAGEO136_proof � 3 July 2008 � 5/12

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UNCORRECT

In the following, we denote simply e0(N1) by e0. Let e ˛ [0, e0].From Eq. (15), e1fx1 ðe; tÞ is continuous and decreasing, then thereexists t1>0 such that fx1 ðe; t1Þ > N1 (we can choose t1¼ t(e0)).Moreover, fx1 ðe;0Þ ¼ 0, and t1fx1 ðe; tÞ is continuous and strictlyincreasing on [0, þN[. It results that there exists a unique t suchthat fx1 ðe; tÞ ¼ N1. We can conclude that there exists a function f

defined on [0, e0] by f(e)¼ t and satisfying

fx1ðe;fðeÞÞ ¼ N1; for all e˛½0; e0�: (28)

We have f ˛ C1([0, e0]). For e ˛ [0, e0(N1)], we consider the map

j : e1fx2ðe;fðeÞÞ � N2:

We want to prove that the solution of fx2 ðe;fðeÞÞ � N2 ¼ 0 isunique if it exists. The solution is unique if j is strictly increasing ordecreasing. For all e ˛ [0, e0], we have

j0ðeÞ ¼ vfx2

vtðe;fðeÞÞf0ðeÞ þ vfx2

veðe;fðeÞÞ:

From Eq. (28), we have

f0ðeÞ ¼ �vfx1

veðe;fðeÞÞ

vfx1

vtðe;fðeÞÞ

: (29)

According to Eq. (29), it results that the sign of j0 depends on thesign of

vfx1

vtðe;fðeÞÞj0ðeÞ ¼ �vfx2

vtðe;fðeÞÞvfx1

veðe;fðeÞÞ

þ vfx1

vtðe;fðeÞÞvfx2

veðe;fðeÞÞ:

To simplify the computations, for i ˛ {1, 2, 3}, we introduce

jiðe; tÞ ¼ e�t=3iðeÞ; ziðe; tÞ ¼ � 3iðeÞ2

ai

�1� e�t=3iðeÞ

�

þ3iðeÞai

t e�t=3iðeÞ; ð30Þ

where ai is defined in Eq. (16). From Eqs. (12), (13) and (30), we get

�vfx2

vtðe;fðeÞÞvfx1

veðe;fðeÞÞ þ vfx1

vtðe;fðeÞÞvfx2

veðe;fðeÞÞ

¼ ��ax2 j1 þ bx2

j2 þ gx2j3 ðe;fðeÞÞ

�ax1 z1 þ bx1

z2 þ gx1z3

�ðe;fðeÞÞ þ�ax1 j1 þ bx1

j2 þ gx1j3 ðe;fðeÞÞ

�ax2 z1 þ bx2

z2

þ gx2z3 ðe;fðeÞÞ

¼�ax1 bx2

� ax2 bx1

ðj1z2 � j2z1Þðe;fðeÞÞ þ

�ax1 gx2

� ax2 gx1

�ðj1z3 � j3z2Þðe;fðeÞÞ þ

�bx1

gx2� gx1

bx2

ðj2z3 � j3z2Þ

� ðe;fðeÞÞ:(31)

As x1< x2, from Eq. (7), we obtain

ax1 bx2� ax2 bx1

¼ nP2m1 e�x1=Ln e�x2=Ln

�e9x2=Lm1 � e9x1=Lm1

�> 0:

Similarly, we get

ax1 gx2� ax2 gx1

> 0; bx1gx2� gx1

bx2> 0: (32)

To get the sign of the right-hand-side of Eq. (31), we shouldstudy the sign of j1z2� j2z1, j1z3� j3z1, j2z3� j3z2. From Eqs. (17)and (30), we have

Please cite this article in press as: Braucher, R., et al., Determination of bodepth profile: A mathematical proof..., Quaternary Geochronology (2008

F

j1z2 � j2z1 ¼ g1;2; j1z3 � j3z1 ¼ g1;3; j2z3 � j3z2 ¼ g2;3

Applying Lemma 2.3 on [0, e0]� ]0, þN[, we deduce that

ðj1z2 � j2z1Þðe; tÞ > 0; ðj1z3 � j3z1Þðe; tÞ> 0; ðj2z3 � j3z2Þðe; tÞ > 0: (33)

According to Eqs. (31)–(33), it results that for all e ˛ [0, e0]

�vfx2

vtðe;fððeÞÞvfx1

veðe;fððeÞÞ þ vfx1

vtðe;fððeÞÞvfx2

veðe;fððeÞÞi0:

Thus

j0ðeÞ > 0; for all e˛½0; e0�: (34)

DPROO3. Model approach

Recently, Siame et al. (2004) showed that measurement of 10Beconcentrations along a depth profile allows estimating both ex-posure time and denudation rate using a chi-square inversionmodel based on Eq. (3). In the proposed model, however, the an-alytical uncertainties (1s) were not accounted for. Assuming thatthe analytical uncertainties have normal distribution centered onthe measured concentrations, we thus propose to improve the chi-square inversion model using Monte Carlo simulations. This wasperformed for 10Be but can be easily adapted for others nuclidesproviding that their production at depth can be easily modeled. (a)At least, 100 depth profiles are generated by randomly selectinga concentration within the concentration ranges defined by themeasured uncertainties (1s) at each sampling depth. (b) Loops onexposure time and denudation rate are performed and for eachtime – denudation pair a chi-square value is determined. Fora given profile and a time – denudation pair, this value is the sumof each individual chi-square determined at each depth (Eq. (6)).At the end, results can be stored in a (102� X) matrix, 102 columnscome from 100 simulated depth profiles plus two columns fordenudation and time, respectively, and X is the total number ofiterations (X ¼ ððemax � eminÞ=deÞððTmax � TminÞ=dTÞ with emax,emin and Tmax, Tmin as the ranges for denudation and time, re-spectively, and dT and de as the increments for time and de-nudation, respectively; these parameters are set by the user).Because the time–denudation pair that yields to the smallest chi-square value, corresponding theoretically to the best fit, may notbe statistically acceptable for all the simulated profiles, we preferto evaluate the median value for each time–denudation pair(contrary to the arithmetic mean, the median value makes itpossible to attenuate the disturbing influence of extreme values).This median value corresponds to the median of the 100 chi-square values determined for a given time–denudation pair. Theminimum median value gives the time–denudation pair solution.When the measured depth profile is close to an ideal exponentialdecrease, the absolute chi-square and median minimum valuesyield to the same time–denudation pair solution. All the pro-cedures described above are summarized in Fig. 2. Moreover, thisprocedure can be applied for different inherited 10Be concentra-tions. In that case, as previously, the solution corresponds to thelowest median value for a given triplet (denudation–exposureage–inheritance).

4. Model sensitivity

To test this model, theoretical depth profiles have been pro-duced along four increasing lengths (2, 4, 9 and 12 m). Thickness of

th exposure time and denudation rate from an in situ-produced 10Be), doi:10.1016/j.quageo.2008.06.001

EDPROOF

Q2

Denudation

εmin

εmin

Tmin χ2

(εmin;Tmin)

Tmin+

dT

εmin+dε

Tmin+

(x+1).dT

+x.dT

Tmin

εmin+dε

χ2

(εmin;Tmin+dT)

χ2

(εmin+dε;Tmin+(x+1)dT)

χ2

(εmin+dε;Tmin+(x+1)dT)

χ2

(εmin+dε;Tmin+(x+1)dT)

(εmin+dε;Tmin+(x+1)dT)

Median

χ2

(εmin+dε;Tmin+xdT)

χ2

(εmin+dε;Tmin+xdT)

χ2

(εmin+dε;Tmin+xdT)

Tmax

Tmax - dTεmax

εmax χ2

(εmax;Tmax)χ2

(εmax;Tmax)χ2

(εmax;Tmax)

χ2

(εmax;Tmax - dT)

χ2

(εmax;Tmax - dT)

χ2

(εmax;Tmax - dT)

(εmax;Tmax)Median

Median(εmax;

Tmax - dT)

(εmin+dε;Tmin+xdT)

Median

Time Profile 1 Profile 100

χ2

(εmin;Tmin)χ2

(εmin;Tmin)

χ2

(εmin;Tmin+dT)χ2

(εmin;Tmin+dT)

Median(εmin;Tmin)

(εmin;Tmin+dT)Median

Profile 99 Median

Fig. 2. Matrix output of the proposed model. Columns 1 and 2 correspond to denudation rate and exposure time, respectively. Columns 3–102 correspond to depth profilesrandomly generated (here 100 profiles); the last column received the median value of the 100 chi-square values determined for a given time–denudation pair. The number of linesdepends on the chosen incremental steps for both denudation Q4(d3) and time (dT).

R. Braucher et al. / Quaternary Geochronology xxx (2008) 1–126

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UNCORRECT

samples is ignored in this model and is considered to be part ofa post-treatment of the field data as well as all the productioncorrections. Each profile affected by a denudation rate of 10 m/Mahas been virtually exposed 20 and 800 ka, the selected productionrate being 15 at/g/a. Inheritance has not been considered here but isthoroughly discussed in Section 5. 10Be concentrations are sum-marized in Table 2.

To simulate the 1s measurement uncertainties, three Gaussiannoise values were applied to these concentrations: 3, 5 and 10%,corresponding, assuming a 2% variability on accelerator massspectrometry, to 2000, 455 and 104 10Be counts, respectively.Because sample preparation is time consuming and acceleratormass spectrometry measurement is costly, one has to consider theappropriate number of samples to be collected in the field. In thisstudy six samples per profile have been considered. Generatingtheoretical profiles with fewer samples will not alter significantlythe conclusions but in natural studies, because nature doesnot always ‘‘behave’’ as theory, sampling too few samples perprofile can be hazardous (possible outliers, several exposurehistories .). Field experience lets us think that sampling a mini-mum of six samples per profile is a reasonable compromise thatgenerally allows constraining the expected exponential decreaseof production with depth for an unperturbed natural marker thathas undergone a simple exposure history. Model outputs for theprofiles exposed 20 ka and 800 ka with a denudation rateof 10 m/Ma are presented in Table 3 and Table 4, respectively(Figs. 3 and 4).

4.1. Denudation rates

For each individual test, the median values for both de-nudation rate and exposure time agree with the inputs. However,

Please cite this article in press as: Braucher, R., et al., Determination of bodepth profile: A mathematical proof..., Quaternary Geochronology (2008

and obviously, as shown in Fig. 5 and by the minimum andmaximum values in Tables 3 and 4, denudation estimates arecloser to the input values when both exposure time and profilelength increase. Because, as demonstrated in Section 2 there isa unique pair (exposure time–denudation) when consideringboth neutrons and muons, minimization of the range of de-nudation rate (minimum and maximum values) is directly linkedto the muon effect. When profile lengths are not deep enough,the production by neutrons in the total 10Be production is pre-dominant and even if muon production is considered in themodels, this yields to a wide range of acceptable solutions. Thepart of muon production increasing with depth yields to a re-duction of acceptable solutions representing more accurately thelong-term denudation rate. The question is now: how deep tosample? The answer has to deal with theory and practical feasi-bility and first depends on the aim of the sampling. To study olderoding surfaces, a 2 m depth profile may be sufficient because inthat case, exposure time, long enough to reach steady-state bal-ance concentrations, can be neglected. On the contrary, whenworking on young surfaces, time usually cannot be neglected(unless denudation rates are so high that steady-state is reachedin few hundred years). In that case, the question of depth iscrucial. Theoretically speaking the deepest is the best. However,from Fig. 5 it appears that denudation rate estimates derivedfrom 9 and 12 m long depth profiles are not significantly differ-ent. But from a practical point of view, access to such a deepprofile may be very difficult. From Fig. 5 and from Table 3, one candeduce that a 4 m depth profile may be suitable and physicallyreachable, providing that the precision of the measurements isbetter than 5% which roughly corresponds to 500 counts in the10Be detector, assuming a 2% variability on accelerator massspectrometry.

th exposure time and denudation rate from an in situ-produced 10Be), doi:10.1016/j.quageo.2008.06.001

E

OF

Table 2Depth profile 10Be concentrations used as input in models comprising six samples per profile

Depth (cm) 200 cm 400 cm 900 cm 1200 cm

20 ka 800 ka 20 ka 800 ka 20 ka 800 ka 20 ka 800 ka

0 262,888 1,216,246 262,888 1,216,246 262,888 1,216,246 262,888 1,216,24625 190,006 910,99350 137,742 691,338 137,742 691,338100 73,352 418,744 73,352 418,744150 40,164 275,870 40,164 275,870200 23,004 199,762 23,004 199,762 23,004 199,762300 9388 134,240 9388 134,240400 5494 110,019500 4197 97,433 4197 97,433750 30,95 77,991800 2957 74,902900 2710 69,2241000 2491 64,1301200 2122 55,404

The material density has been set at 2 g/cm3, the denudation rate at 10 m/Ma, the production rate at 15 at/g/a and exposure times at 20 and 800 ka.

R. Braucher et al. / Quaternary Geochronology xxx (2008) 1–12 7

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RRECT

5. Application to natural data sets

To consider our work to be of real value, it must be now shownthat the model is in fact robust with real data. Three publisheddepth profiles have been used: one from a quartz sandstone corecollected in Antarctica (Brown et al., 1992) to illustrate long-termexposure history on an autochthonous bedrock and two othersfrom alluvial terraces studies (Brocard et al., 2003; Hidy et al.,2005).

5.1. Long-term exposure history

In their paper, Brown et al. (1992) have measured cosmogenic10Be and 16Al as a function of depth in a core of quartz bedrockcollected in South Victoria Land, Antarctica. Although this studywas dedicated to calculate the effective attenuation length ofcosmic rays producing 10Be and 16Al, Brown et al. also presentedthe dependence of calculated ages on assumed denudation ratesusing production rate of 24.7 at/g/a for 10Be based on Lal (1991).An updated production rate of 27.4 at/g/a based on Stone (2000)(using Antarctic air pressure) is used here to correct Brown et al.ages. These ages are ranging from 1.02 to 1.38 Ma with denudationrates ranging from 0 to 0.22 m/Ma. Interestingly, the authorsconcluded that the assumption of steady-state denudation maynot be valid for their samples. Using our model, the exposure agesrange from 1.07 to 1.12 Ma with a denudation rate of 0 m/Ma. This

UNCO

Table 3Model results for the profile exposed 20 ka with a denudation rate of 10 m/Ma

Depth(m)

Uncertainty (%) Mediandenudation(m/Ma)

Min and maxdenudation(m/Ma)

Mediantime (a)

Min and maxtime (a)

2 3 9.52 0; 19.32 19,885 17,692; 22,8555 10.33 0; 27.12 20,296 17,392; 26,869

10 11.97 0; 27.23 20,401 17,312; 26,436

4 3 9.98 6.43; 14.64 19,999 19,160; 21,0235 9.83 0.51; 17.54 19,874 18,311; 21,749

10 9.82 0; 24.27 19,886 18,130; 22,985

9 3 9.83 6.74; 13.68 19,965 19,515; 20,5065 9.8 3.4; 16.8 19,945 19,262; 20,811

10 7.63 0; 23.56 19,632 18,092; 21,587

12 3 10.11 5.89; 12.59 20,002 19,401; 20,4595 9.66 4.39; 15.06 19,950 19,189; 20,456

10 11.04 1.19; 20.52 19,991 18,213; 21,544

Please cite this article in press as: Braucher, R., et al., Determination of bodepth profile: A mathematical proof..., Quaternary Geochronology (2008

DPRO

result is in excellent agreement with those of Brown et al. andleads us to conclude that the samples of this core are not yet atsteady-state, which at this locality would require 6 Ma of expo-sure to be reached. This implies that the negligible erosion as-sumption by Brown et al. is valid and that the calculated ages arenot minimum ages but correspond to the effective exposureduration.

5.2. Mid-term exposure history: alluvial terraces dating

Antarctic depth profiles are ideal to assess long-term exposurehistory but most of the applications on alluvial terraces or alluvialfans concern younger exposure ages. In that case, a precise agedetermination of deposits using cosmogenic nuclides may bea problem because of inheritance, bioturbation or anthropogenicactivities. The work of Brocard et al. (2003) is used to test ourmodel on young alluvial terraces and details of the geomorphologyand soils at the site are described in their article. This article isa study of river incision as a response to tectonic and climaticcontrols. The authors estimate terrace ages using a Monte Carlotechnique applied on pebble depth profiles by considering as un-known values, the inheritance, exposure time and soil density.Because field observations such as terrace structures provide evi-dence for the pristine nature of the surface, no denudation factorwas applied in this article. The main difference between the Bro-card et al. model and ours is that in their model Brocard et al.

Table 4Model results for the profiles exposed 800 ka with a denudation rate of 10 m/Ma

Depth(m)

Uncertainty(%)

Mediandenudation(m/Ma)

Min and maxdenudation(m/Ma)

Mediantime (a)

Min and maxtime (a)

2 3 9.98 9.59; 10.43 793,834 711,751; 947,9985 10.06 9.43; 10.6 796,059 681,155; 1,037,320

10 10.01 9.03; 11.21 790,376 547,918; 1,208,807

4 3 10.04 9.63; 10.34 804,771 745,188; 866,0965 10.01 9.4; 10.54 794,978 728,642; 889,965

10 9.95 8.86; 10.93 775,400 672,978; 1,009,216

9 3 10.01 9.62; 10.36 798,295 762,729; 842,0635 10.1 9.43; 10.72 800,089 736,920; 873,999

10 9.91 8.91; 11.42 778,310 689,429; 920,510

12 3 10.02 9.64; 10.38 802,195 771,146; 835,7115 9.99 9.37; 10.57 796,283 746,137; 854,197

10 9.92 8.63; 11.44 793,717 678,399; 897,794

th exposure time and denudation rate from an in situ-produced 10Be), doi:10.1016/j.quageo.2008.06.001

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RRECTEDPROOF

3

4m

Denudation rate (m/Ma)

0 5 10 15 20 25 30

Tim

e (a)

10%5%3%

2m

Denudation rate (m/Ma)

0 5 10 15 20 25 30

Tim

e (a)

16 000

20 000

24 000

28 000

9m

Denudation rate (m/Ma)

0 5 10 15 20 25 30

12m

Denudation rate (m/Ma)

0 105 15 20 25 30

Tim

e (a)

Tim

e (a)

16 000

20 000

24 000

28 000

16 000

20 000

24 000

28 000

16 000

20 000

24 000

28 000

Fig. 3. Model outputs for the profiles exposed 20 ka with a denudation rate of 10 m/Ma. Four profile lengths have been simulated (2, 4, 9 and 12 m depth). Each symbol representsthe best pair (denudation–exposure time) associated to the minimum chi-square for one profile (100 profiles per uncertainty level). Grey dots, black squares, and open trianglesrefer to 3, 5 and 10% uncertainty in measurement, respectively. Median values are reported in Table 3.

R. Braucher et al. / Quaternary Geochronology xxx (2008) 1–128

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UNCOconsider as a possible solution for each profile the triplet (time,

inheritance and soil density) that yields to an absolute minimum.That is the reason why they mention that more than one bestsolution can be found (one by random profile). On the contrary,our model approach is based, for a given inheritance and soildensity, on the best pair (denudation–time) that yields to theminimum median value applied on all the random profiles. Ourmodel thus includes the Brocard et al. model to which one post-statistical treatment (the median value) is applied. When data arewell distributed along an exponential decrease and when analyt-ical uncertainties are low, the solution deduced by the minimummedian value and by the absolute minimum value given by eachprofile are the same within uncertainties. Because the Drac 1 siteof Brocard et al. (2003) presents the longest depth profile(292 cm), it has been chosen to compare the two models. Fromthat site, Brocard et al. estimate an exposure age ranging from 7.2to 14 ka with inheritance ranging from 10 to 24 kat/g for a soildensity of 2 g/cm3. The median model applied on Drac 1 sitepredicts an estimated denudation rate that ranges from 11.4 to14.4 m/Ma, an exposure age ranging from 9 to 11 ka, for an

Please cite this article in press as: Braucher, R., et al., Determination of bodepth profile: A mathematical proof..., Quaternary Geochronology (2008

inheritance of 12 kat/g and a soil density of 1.85 g/cm3. All un-certainties are circumscribed as described in Granger (2006)which Qfollows Bevington and Robinson (2003) by increasing byone the minimum chi-square median value. Our results are in goodagreement with those of Brocard et al. except of course for thedenudation rate that had been set to zero by Brocard.

To test our model on older fill terraces, 10Be concentrationsmeasured within a depth profile from Lees Ferry terraces (Arizona)are used (Hidy et al., 2005). Table 5 presents the available data forthat site. At that location, the terrace top is extremely flat, notdissected by post-depositional gulleying. Desert pavement is ex-tremely well developed at the location of the pit, meaning thatthere are no large clasts that remained unfragmented (into smallpebbles), the pebbles are well interlocked, their tops are extremelyvarnished, and their bottoms are deeply rubefied. The site was wellaway from the edge of the terrace which is visibly influenced bydiffusion along the escarpment rim. This terrace fragment is iso-lated from any flooding or fluvial stripping because there is a deeplyincised tributary just upstream and there is a deep gulley sepa-rating the terrace from the bank of the Grand Canyon. Rainfall is so

th exposure time and denudation rate from an in situ-produced 10Be), doi:10.1016/j.quageo.2008.06.001

ORRECTEDPROOF

2m

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im

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Fig. 4. Model outputs for the profile exposed 800 ka with a denudation rate of 10 m/Ma. Four profile length have been simulated (2, 4, 9 and 12 m depth). Each symbol representsthe best pair (denudation–exposure time) associated to the minimum chi-square for one profile (100 profiles per uncertainty level). Grey dots, black squares, and open trianglesrefer to 3, 5 and 10% uncertainty in measurement, respectively. Median values are reported in Table 4.

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UNCrare and soil moisture so low that there is no opportunity for sur-

face runoff (the surface area of the terrace is too small and thegradient is too low). Pedogenic carbonate can be observed in thesoil but there are no fragments of the carbonate on the soil surface.From their data, Hidy et al. obtain a minimum exposure age of87.3� 2.1 ka and an inheritance of 109 kat/g, assuming no de-nudation. An average soil density of 2.5 g/cm3 was used based onfield measurements (although seemingly high, this value may beexplained by the presence of boulder-sized quartz-rich clasts in thesoil pit). The surface production rate of 10.01 at/g/a is deduced fromStone (2000) and is corrected from topographic shielding. The samedepth profile concentrations were modeled using our model withthe same soil density. The best pair (denudation–exposure age) was6.8 m/Ma and roughly 300 ka with an inherited 10Be concentrationof 105 kat/g (Table 6). This means that the steady-state is almostreached for this surface. However, this amount of denudation is notsupported by field observations and the chronology by OSL

Please cite this article in press as: Braucher, R., et al., Determination of bodepth profile: A mathematical proof..., Quaternary Geochronology (2008

(independent OSL data have constrained the abandonment of theLees Ferry M4 terrace to 77–98 ka). Such a denudation rate over300 ka would have implied a total of 2.1 m of erosion which isimpossible considering fieldwork evidences. Furthermore, U-seriesand OSL chronologies imply that this terrace cannot be three timesolder than the age they measured. Considering the measured 10Beconcentrations, the only way to lower the age and denudation is toconsider that the samples are closer to the surface than they arewhen applying a density of 2.5 g/cm3. The only way to reduce thesample depth is to consider a lower soil density. A second modelforcing the density to 2 g/cm3 was performed yielding to a de-nudation rate of 1 m/Ma an exposure age of 86 ka with an inherited10Be concentration of 91 kat/g. This is in a better agreement withfield observations but evidenced that the density is a crucial pa-rameter when one wants to deduce from the analysis of the 10Beconcentration distribution along a depth profile the exposure timeand the denudation rate affecting a surface deposit. Then, because

th exposure time and denudation rate from an in situ-produced 10Be), doi:10.1016/j.quageo.2008.06.001

RRECTEDPROOF

2m

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0 5 10 15 20

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e (a)

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20 ka800 ka

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Fig. 5. Effect of exposure time (20 ka (black dots and left axis) and 800 ka (grey dots and right axis)) and depth on the data modeled using 3% uncertainty. See text for explanation.

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NCOthe evolution of measured 10Be concentrations with depth is as

expected by the theory, the last choice was to consider the densityas a ‘‘free’’ parameter as denudation, time and inheritance are.Doing this the best fit of all from a statistical point of view wasobtained for a density of 2.2 g/cm3, a denudation rate of 6.1 m/Ma,an exposure age of 153 ka with an inherited 10Be concentration of99 kat/g (Table 6). This model yields an exposure duration that is

UTable 5Lees Ferry sample identification and 10Be concentrations

Sample name Depth (cm) 10Be (at/g)

GC-04-LF-401 0 949,302� 24,097GC-04-LF-404.30s 27.5 629,031� 15,813GC-04-LF-404.60s 57.5 449,824� 11,295GC-04-LF-404.100s 97.5 323,221� 8283GC-04-LF-404.140s 137.5 224,598� 6024GC-04-LF-404.180s 177.5 173,873� 4518GC-04-LF-404.220s 217.5 148,423� 4142

AMS measurement has been performed at Laurence Livermore National Laboratory;corrected production rate deduced from Stone (2000) is 10.0 at/g/a.

Please cite this article in press as: Braucher, R., et al., Determination of bodepth profile: A mathematical proof..., Quaternary Geochronology (2008

double that of Hidy et al. and would have implied a 93 cm loss ofsoil by erosion. Fig. 6 is the representation of Lees Ferry data set andthe 10Be concentrations modeled using parameters presented inTable 6.

Table 6Comparison of model outputs for the Lees Ferry terrace depth profile

Hidy et al. Our model

Density (g/cm3) 2.5 2 2.5 2.2Denudation (m/Ma) 0 0.95 6.8 6.1Exposure time 87,300 86,000 299,000 153,000Inher. 10Be at/g 109,000 91,000 105,000 99,000Associated minimum

chi-square52.37 14.87 12.09 8.12

Hidy et al. (2005) obtained a minimum exposure age as they assume no de-nudation. In our two first attempts, density was forced to be 2 g/cm3 and 2.5 g/cm3.The last 2.2 g/cm3 density is the one determined by the model and for which theminimum median chi-square value is obtained overall. It is important to note thatHidy et al. did not use the surficial desert pavement sample (GC-04-LF-401) in theirmodel.

th exposure time and denudation rate from an in situ-produced 10Be), doi:10.1016/j.quageo.2008.06.001

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th

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)

Lees Ferry datasetHidy et al. modelOur model (d=2.5 g/cm3)Our model (d=2 g/cm3)Our model (d=2.2 g/cm3)

Fig. 6. Lees Ferry (Arizona) 10Be concentrations at depth. Black dots refer to measured concentrations with their associated error bars; grey dots refer to Hidy et al. model (adenudation of 0 m/Ma; an exposure time of 87.3 ka, an inheritance of 109 kat/g and a density of 2.5 g/cm3); open dots refer to our model using a density of 2.5 g/cm3, an exposureage of 86 ka, an inhertance of 91 kat/g and a denudation rate of 0.95 m/Ma; open triangle refer to our model using a density of 2 g/cm3, an exposure age of 299 ka, an inheritance of105 kat/g and a denudation rate of 6.8 m/Ma and open squares refer to our best model using a density of 2.2 g/cm3, an exposure age of 153 ka, an inheritance of 99 kat/g anda denudation rate of 6.1 m/Ma. In this last case, density, exposure age, denudation rate and inheritance were considered as free parameters whereas in the other models density wasa fixed value. It is important to note that Hidy et al. did not use the surficial desert pavement sample (GC-04-LF-401) in their model but in this Figure, we put the surficialconcentration implied by Hidy et al. parameters.

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UNCORRECT

6. Conclusion

In this paper, we have mathematically proved that for two mea-sured 10Be concentrations N1 at depth x1 and N2 at depth x2, froma surface undergoing denudation and a single exposure to cosmicrays, only one pair (exposure time–denudation rate) is necessary todefine the system. This is not true considering neutrons only. Thedemonstration has been made using all the particles (neutrons, fastand stop muons) and their associated physical parameters involvedin the production of in situ-produced 10Be. Providing that the profilehas not been perturbed, since all the samples will have the sameexposure history at their specific depth, this demonstration based ontwo concentrations N1 at depth x1 and N2 at depth x2 randomlychosen, has been extended and tested to more than two concentra-tions. To do so, an improved model of the chi-square inversion modelof Siame et al. (2004) has been developed. This Monte Carlo modelrandomly generates a large number of depth profiles. For a givenprofile and a quadruplet (time–denudation–density and in-heritance), the sum of each individual chi-square determined at eachdepth is calculated. Then, considering all the generated depth profilesfor a given quadruplet (time–denudation–density and inheritance),a median value corresponding to the median of all the chi-squarevalues determined with the same quadruplet is calculated. Theminimum median value gives the time–denudation pair solution.When the measured depth profile is close to an ideal exponentialdecrease and analytical uncertainties are better than 5%, the absolutechi-square minimum and median minimum values yield to the samesolution. Although the model can help provide a unique solution forthe quadruplet, it will always be necessary for the user to considerthe geological evidence to evaluate the model output.

Acknowledgements

The authors thank J. Pederson and J. Gosse for having providedunpublished data set from Lees Ferry terraces as well as twoanonymous reviewers. Editorial handiling by (John C. Gosse).

Please cite this article in press as: Braucher, R., et al., Determination of bodepth profile: A mathematical proof..., Quaternary Geochronology (2008

D

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Brown, E.T., Bourles, D.L., Colin, F., Raisbeck, G.M., Yiou, F., Desgarceaux, S., 1995.Evidence for muon-induced production of 10Be in near-surface rocks from theCongo. Geophysical Research Letters 22 (6), 703–706.

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Heisinger, B., Lal, D., Jull, A.J.T., Kubik, P., Ivy-Ochs, S., Knie, K., Nolte, E.,2002b. Production of selected cosmogenic radionuclides by muons; 2.Capture of negative muons. Earth and Planetary Science Letters 200, 357–369.

Hidy, A.J., Pederson, J.L., Cragun, W.S., Gosse, J.C., 2005. Cosmogenic 10Be exposuredating of Colorado river terraces at Lees Ferry, Arizona. Geological Society ofAmerica Abstracts 37–7, 296.

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Lal, D., Arnold, J.R., 1995. Tracing quartz through environment. Earth and PlanetaryScience Letters 91, 1–5.

Nishiizumi, K., Kohl, C.P., Arnold, J.R., Klein, J., Fink, D., Middleton, R., 1991. Cosmicray produced 10Be and 26Al in Antarctic rocks: exposure and erosion history.Earth and planetary Science Letters 104, 440–454.

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Siame, L., Bourles, D.L., Brown, E.T., 2006. In situ-produced cosmogenic nuclides andquantification of geological processes. Geological Society of America SpecialPaper 415, 146.

UNCORRECTE

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Siame, L., Bellier, O., Braucher, R., Sebrier, M., Cushing, M., Bourles, D.L., Hamelin, B.,Baroux, E., de Voogd, B., Raisbeck, G., Yiou, F., 2004. Local erosion rates versusactive tectonics: cosmic ray exposure modelling in Provence (South-EastFrance). Earth and Planetary Science Letters 220 (3–4), 345–364.

Stone, J.O., 2000. Air pressure and cosmogenic isotope production. Journal ofGeophysical Research 105, 23753–23759.

DPROOF

th exposure time and denudation rate from an in situ-produced 10Be), doi:10.1016/j.quageo.2008.06.001