Research Paper

Effect of density uncertainties in cosmogenic 10Be depth-profiles: Datinga cemented Pleistocene alluvial fan (Carboneras Fault, SE Iberia)

Ángel Rodés a,*, Raimon Pallàs a, Regis Braucher b, Ximena Moreno a, Eulàlia Masana a, Didier L. Bourlés b

aDepartament de Geodinàmica i Geofísica, Facultat de Geologia, Universitat de Barcelona, Martí i Franquès s/n, 08028 Barcelona, SpainbCEREGE, UMR 6635, Université Paul Cézanne CNRS-Aix Marseille Université BP80, 13545 Aix en Provence, France

a r t i c l e i n f o

Article history:Received 16 March 2010Received in revised form4 October 2010Accepted 20 October 2010Available online 10 November 2010

Keywords:Cosmic ray exposure datingDenudationGeochronologyNumerical modelPaleoseismologyCalcrete

a b s t r a c t

Cosmonuclide depth-profiles can be used to calculate both the age of landforms and the rates at whicherosion has affected them since their formation. Results are heavily dependent on the selection of theappropriate density of the material exposed to cosmic radiation. In materials where density has changedsignificantly through time, as in alluvial sediments affected by post-depositional calcrete cementation,the uncertainties in density must be accurately modelled to produce reliable results. We develop newequations for an accurate account of density uncertainties and to test the effect of density gain due tocementation processes. We apply them to two 10Be depth-profiles measured in an alluvial fan deformedby the Carboneras Fault (SE Iberia). When a linear increase of density through time is considered, modelresults yield an age ranging from 200 ka to 1 Ma within 1s confidence level.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Cosmonuclide exposure dating is a dependable technique inlandscape evolution studies (Gosse and Phillips, 2001). Theconcentration of cosmonuclides (e.g. 10Be) measured at the surfaceof a given landform depends, among other parameters, on theamount of cosmonuclide lost by erosion, and on the time ofexposure to cosmic radiation. If erosion is assumed to be negligiblefor the period of exposure, an exposure age can be deduced fromthe cosmonuclide concentrations found in superficial materials(Lal, 1991). Moreover, where erosion is not negligible, it is possibleto deduce an exposure age and an erosion rate using numericalmodeling of cosmonuclide concentrations from a series of samplestaken along a depth profile (Siame et al., 2004).

Advances in Accelerator Mass Spectrometry (AMS) in recentyears have increased the precision in 10Be concentrationmeasurements. Thus, as a result of more precise measurements andthe increased speed of computer processors there is now room forimprovement in the conceptual models used to interpret concen-tration depth-profiles. Exposure ages and erosion rates for a given

surface have been calculated in the literature by chi-square inversemodeling of the measured cosmonuclide concentration depth-profiles, assuming no variation of the material density (Siame et al.,2004; Braucher et al., 2009; Nissen et al., 2009). However, wherepost-depositional processes produce density variations throughtime (for example by calcrete formation), the assumption ofconstant density may produce significant age inaccuracies. Ameticulous processing of density uncertainties and of densityevolution is therefore necessary to obtain reliable chronologicalresults from cosmonuclide depth-profile modeling.

The aims of this paper are (1) to assess the precision of age anderosion estimation for depth-profiles affected by density uncer-tainties, (2) to develop a model for 10Be depth-profiles, taking intoaccount the variations in sediment density through time, and (3) toapply the new model to constrain the age of the Pleistocene ElPuntal fan (Carboneras Fault).

2. Modeling density variations and uncertainties in 10Be

depth-profiles

To numerically estimate the age and the erosion rate experi-enced at a depositional landform surface, Siame et al. (2004) andBraucher et al. (2009) used the chi-square inverse approach to fitsynthetic 10Be depth-profile models to measured 10Be datasets,

* Corresponding author. Tel.: þ34 93 403 40 29; fax: þ34 93 402 13 40.E-mail address: angelrodes@ub.edu (Á. Rodés).

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Quaternary Geochronology

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

1871-1014/$ e see front matter � 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.quageo.2010.10.004

Quaternary Geochronology 6 (2011) 186e194

assuming no variation in the material density. In this work, we usethe same approach but we introduce the modifications required toaccount for density variations through time. Moreover, we presenta way of considering the density data as a gaussian distribution inthe chi-square inverse approach.

2.1. Constant-density model

The 10Be concentration (C) at a depth x (g cm�2) from the topsurface of a deposit, which has been eroding at a constant rate3 (g cm�2 a�1) since its formation (t years ago), was described by Lal(1991) as:

dCdt

¼ Pe�xL þ 3

dCdx

� lC (1)

where P is the surface production rate in at g�1 a�1, which can becalculated using the CRONUS-Earth online calculator (Balco et al.,2008), L is the attenuation length of the cosmic radiation(g cm�2), and l is the 10Be decay constant (a�1). Eq. (1) may besolved as:

Cðx;3;tÞ ¼P

3Lþ l

e�xL

�

1� e�tðlþ 3LÞ�

(2)

Depth can also be expressed as x ¼ r$z , where z is the currentprofile depth in cm and r is the mean density of the rock or sedi-ment in g cm�3. Considering that the production of 10Be is due tospallation, stopping muons and fast muons, which have differentattenuation lengths and production rates, the 10Be concentrationcan be expressed as:

CðCInher:;x;3;tÞ ¼ CInher: þPspal:3

Lspal:þ l

e� x

Lspal:

�

1� e�t

�

lþ 3Lspal:

�

�

þPstop3

Lstopþ l

e� x

Lstop

�

1� e�t�

lþ 3Lstop

�

�

þPfast3

Lfastþ l

e�x

Lfast

�

1� e�t�

lþ 3Lfast

�

�

(3)

where the first addend is the 10Be inherited from the exposure ofmatter to cosmic rays before its deposition:

CInher: ¼ Cðx;0Þ$e�lt (4)

In the case of alluvial fan deposits, this inherited 10Be may beformed (1) during the deposition of the quartz grains in the sourcearea, (2) during the exhumation of the source area, and (3) duringthe transportation of the quartz grains from the source area to thedeposition site.

2.2. Variable-density model

If we consider that the density of the sediment decreases orincreases constantly with time due to diagenetic processes, the 10Beconcentration can be described by:

dCdt

¼ Pe�xL þ ð3þ kxÞ

dCdx

� lC (5)

where k is the rate of sediment density loss in g cm�3 a�1. Thisfactor can be expressed as:

k ¼r0 � r1tr1

(6)

where r1 is the current density (g cm�3), r0 is the initial density andt is the age of the depositional surface in years. Therefore, negative

values of k correspond to a constant-density gain. Eq. (5) can besolved as:

Cðx;3;tÞ ¼Pke�

3kL

kL

�

ðkx� 3Þ

kL

��lk

G

"

l

k;e�ktðkx� 3Þ

kL;ðkx� 3Þ

kL

#

¼Pke�

3kL

kL

�

ðkx� 3Þ

kL

��lkZ

ðkx�3Þ

kL

e�kt ðkx�3Þ

kL

rlk�1e�rdr (7)

Considering inheritance and production of 10Be due to spallation,stopping muons and fast muons, the variable-density 10Beconcentration model will be:

CðCInher:x;3;tÞ ¼ CInher:

þPspal:k

e� 3

kLspal:

kLspal:

ðkx�3Þ

kLspal:

!�lk

G

"

l

k;e�ktðkx�3Þ

kLspal:;ðkx�3Þ

kLspal:

#

þPstopk

e� 3

kLstop

kLstop

�

ðkx�3Þ

kLstop

��lk

G

"

l

k;e�ktðkx�3Þ

kLstop;ðkx�3Þ

kLstop

#

þPfastk

e�3

kLfast

kLfast

ðkx�3Þ

kLfast

!�lk

G

"

l

k;e�ktðkx�3Þ

kLfast;ðkx�3Þ

kLfast

#

(8)

2.2.1. c2 fit modelingDepth-profile sampling provides a set of data of N 10Be

concentrations (Ci) measured in samples obtained from severalprofile depths (xi). Numerical modeling was performed to calculatethe time (t), inheritance (CInher.) and erosion rate (3) values that fitthe data obtained from the sampled profiles. To fit the models (Eqs.3 and 8) into the data, it is necessary to compute them with aninverse method, especially if uncertainty boundaries need to bedemarcated in the CInher.-3-t space. We used the same c2 fit-basedinverse method employed by Siame et al. (2004) or Braucher et al.(2009). This method defines the solution in the CInher.- 3-t space byminimizing the c2 value:

1 1 σ values values

2 s values2 s values

1 σ values

2 σ values

50 10 15 20

0.02

0.04

0.06

0.08

0.10

Chi-square distributionChi-square distribution

(8 degrees of freedom)(8 degrees of freedom)

Chi-square distribution

(7 degrees of freedom)

χ2 best fit χ2 values

Pro

babili

ty

Forbidden values

68.27 % of allowed values

95.45 % of allowed values

4.55 % of allowed values

Fig. 1. Strategy followed to choose c2 values corresponding to 1s and 2s confidencelevels over a chi-squared distribution. 1s values vary from c2min to a c2 value thatrestricts the distribution area to 68.27% of all possible distribution values (i.e. the partof the distribution beyond c2min). 2s values restrict the distribution area to 95.45% ofallowed values.

Á. Rodés et al. / Quaternary Geochronology 6 (2011) 186e194 187

c2 ¼X

N

i¼0

Ci � CðCInher:;x;3;tÞ

sið3;tÞ

!2

(9)

where Ci are the measured concentrations from the N samples at xidepths, CðCInher: ;x;3;tÞ is the concentration predicted by the model, andsi(3,t) are the uncertainties of the samples, explained below. In thecase of two depth-profiles from the same landform (NþM samples,same age t, same inheritance (CInher.), and two erosion rates 31 and32), solutions are located in the CInher.-31-32-t space. The c2 functionfor two-profile models is:

c2 ¼X

N

i¼0

Ci � CðCInher:;x;31;tÞ

sið31;tÞ

!2

þX

M

j¼0

Cj � CðCInher:;x;32;tÞ

sjð32;tÞ

!2

(10)

As the data are subject to measurement errors and as the sampleshave been affected by biological and diagenetic processes not

considered in the models, they never fully fit the model, but onlyone solution that minimizes the c2 value can be obtained (Braucheret al., 2009). This solution, which is defined by the maximumlikelihood CInher.-3-t combination, is called c2 best fit value ðc2minÞ.To estimate to what degree a model reflects the distribution of thedata, the quality factor of the c2 best fit value is reported. Pugh andWinslow (1966) defined the quality factor of a c2 fit as:

Qf ¼ 1� P

y

2;c2min2

!

(11)

where y is the degrees of freedom of the model (number ofsamples minus the number of modelled parameters). The qualityfactor varies from 0 to 1, with higher values meaning betterfitting (Pugh and Winslow, 1966). The CInher.-3-t values that fitthe data within 1s or 2s confidence level (i.e., 68.27% or 95.45%)are obtained from the c2 distribution, as shown in the exampleof Fig. 1.

Fig. 2. Location of 10Be depth-profiles 1 and 2 in the El Puntal alluvial fan. The sampling area is located on the NW flank of the La Serrata range, a horst-like structure bounded bytwo parallel traces of the left-lateral Carboneras Fault.

Á. Rodés et al. / Quaternary Geochronology 6 (2011) 186e194188

2.3. Uncertainties

When fitting the models Eq. (9) or Eq. (10), we consider not onlythe uncertainties related to concentration measurements (sC) butalso those related to depth estimation (sx), both as independentrandom errors:

sið3;tÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s2C þ s2x

q

(12)

If inheritance is considered as an input parameter, concentrationmeasurement error includes not only errors related to the sampleconcentration measurement, but also the uncertainty in the esti-mation of inheritance:

sC ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s2sample þ s2Inher:

q

(13)

As CInher. is considered constant with depth, the concentrationuncertainty due to x error may be expressed as:

sx ¼dCðx;3;tÞ

dx

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�

sr$zsample

�2þ�

szsample$r�2

r

(14)

where the first addend corresponds to the error in density esti-mation, and the second addend reports the depth rangewhere eachsingle sample was collected (See Appendix A).

3. Field example: El Puntal alluvial fan, Carboneras Fault

The left-lateral Carboneras Fault (SE Iberia) displays amplegeomorphological evidence of Quaternary tectonic activity both inits onshore and offshore sections (Bousquet, 1979; Goy and Zazo,1986; Bell et al., 1997; Faulkner et al., 2003; Gràcia et al., 2006;Moreno et al., 2009). Deformed Quaternary sedimentary units ofknown age imaged in offshore seismic profiles suggest vertical sliprates ranging from 40 to 140 m/Ma (Moreno et al., 2009). In itscentral section, the onshore Carboneras Fault splits into twoparallel traces forming the La Serrata SW-NE linear range (Fig. 2).

The El Puntal alluvial fan, which flanks the La Serrata range tothe NW, corresponds to second oldest generation of alluvialdeposits deformed by the Carboneras Fault. This fan generation isconsidered to be of Middle Pleistocene age on the basis of relativechronology (Moreno et al., 2009), calcrete crust development(Dumas, 1969), and soil development (Schulte and Julià, 2001).Diagenetic characteristics of these deposits are common in semi-arid climates. Outcrops in the El Puntal alluvial fan show uniformnodular and laminated calcrete crusts developed at sub-superficialhorizons (down to c. 50 cm depth), whereas at deeper levelssediments range from weakly to strongly cemented (Fig. 3).

The source area of the El Puntal alluvial fan in the La Serratarange mainly consists of deformed Miocene and Pliocene deltaiccalcarenites and conglomerates with a high content of quartz

Fig. 3. Anthropic outcrops excavated in the El Puntal alluvial fan, where depth Profiles 1 (2.8 m deep) and 2 (1.8 m deep) were sampled. Note the regularity of the fan surface in A,and the cementation of deposits associated with calcrete formation in A and B.

Table 1

Location and mean density of profiles studied. 10Be production rates due to spallation, stopping muons and fast muons (Pspal., Pstop and Pfast, respectively) have been calculatedwith the CRONUS-Earth online calculator v. 2.2, according to Balco et al. (2008).

Profile Latitude(N)

Longitude(W)

Elevation(m)

Mean density(g cm�2)

Pspal.(at g�1 a�1)

Pstop(at g�1 a�1)

Pfast(at g�1 a�1)

PUN-1 36.86315� 2.19352� 110 2.2 � 0.3 4.35 0.0985 0.0855PUN-2 36.86425� 2.19150� 117 2.1 � 0.3 4.38 0.0989 0.0857

Á. Rodés et al. / Quaternary Geochronology 6 (2011) 186e194 189

pebbles and granules. Dacites, andesites, volcanic breccia andMiocene marlstones appear as secondary lithologies.

3.1. Methods

Between 50 and 400 quartz pebbles (2e10 cm of diameter) fromeach samplewere crushed and sieved to yield 250e1000 mmgrains.The samples were then put through a Franz magnetic separator toremove any magnetic material. To eliminate carbonate, sampleswere digested with chlorhydric acid for 48 h, using organic oil as anantifoaming agent. After washing the oil residue, 3 to 6 Hexa-fluorosilicic acid digestions were performed for 3e6 days. Remain-ing quartz grains were cleaned using sequential HF dissolutions toremove any potential atmospheric 10Be (Brown et al.,1991; Kohl andNishiizumi, 1992; Cerling and Craig, 1994). Between 15 and 30 g persample of clean quartz cores were then completely dissolved in HFand spiked with 300 mg of 9Be carrier (Bourlès, 1988; Brown et al.,1992). Beryllium was separated by successive solvent extractionsand alkaline precipitations (Bourlès, 1988; Brown et al., 1992).

Table 2

The El Puntal sample depths and 10Be concentrations. The value of the blanksample was 10�14 atoms of 10Be/Be. Concentration uncertainties include theblank concentration uncertainty, measurement error and spectrometer standarderror.

Sample Number ofpebbles used

Depth(cm)

10Be concentration(at g�1)

Aprox.

PUN01 50 267 � 5 518525 � 91090PUN02 80 195 � 5 184334 � 8525PUN03 90 141 � 5 264741 � 8419PUN04 150 95 � 5 430387 � 19731PUN05 80 46 � 6 732225 � 21527PUN06 90 3 � 3 1069913 � 41572PUN07 150 163 � 4 200185 � 11087PUN08 400 135 � 3 280303 � 12459PUN09 100 97 � 3 335545 � 10311PUN10 150 58 � 3 485063 � 14172PUN11 200 32 � 2 577632 � 26009PUN12 200 3 � 3 820548 � 23676GRA02 100 >1000 26380 � 2949

Fig. 4. Sample concentrations vs. current effective depth expressed in g cm�2. Error bars include concentration and depth uncertainties in A and B; and also density uncertainties inC and D. A, B) Best fits of models that only consider concentration and depth uncertainties. C, D) Best fits of models that consider concentration, depth and density uncertainties. Thebest fits of variable-density models are depicted by dashed lines. Constant-density model results that fit the measured 10Be dataset within 1s confidence level are depicted by grayareas.

Á. Rodés et al. / Quaternary Geochronology 6 (2011) 186e194190

Measurement of 10Be concentrations was performed at theASTER AMS facility at Aix-en-Provence (France) in 2009. All 10Beconcentrations are calibrated against the National Institute ofStandards and Technology standard reference material 4325 byusing an assigned value of 2.79� 0.03�10�11 and using a 10Be half-life of 1.36 � 0.07 � 106 years (Nishiizumi et al., 2007). 10Beconcentrations in quartz were calculated following Balco (2006).

10Be production rates corresponding to both sites were deducedfrom the location data using the CRONUS-earth online calculatorv.2.2.1 (Table 1). All the computations used the same constants asCRONUS-earth online calculators (hess.ess.washington.edu),explained in Balco et al. (2008), Balco (2009, 2010). Therefore, in allour calculations, a decay constant of 4.9975 10�7 a�1 (Chmeleffet al., 2010) and a spallation attenuation length of Lspall. ¼

160 g cm�2 (Balco et al., 2008) were used. The local muon attenu-ation lengths Lstop ¼ 1303 g cm�2, and Lfast ¼ 2193 g cm�2 werecalculated from our field data (location and depths) using CRONUS-earth online calculator v.2.2.1 Matlab code. The computation of 10Beconcentration models was performed using a computer algebrasystem software.

3.2. Data collection and results

Two small anthropic outcrops excavated across the El Puntalalluvial fan (SW Serrata range) were selected for 10Be depth profilesampling (Figs. 2 and 3, Table 2). Both outcrops are located c. 1 kmfrom the fan apex, where the fan surface is well preserved, andwhere there are no significant signs of erosion or incision. They aremainly made up of cross-bedded pebble conglomerates witha variablematrix content (Fig. 3). Laterally, and also at the bottom inoutcrop 1, major erosional discontinuities divide these high energytorrential facies from gravels with a higher matrix content.

Profiles 1 and 2were sampled at different depths (Table 2, Fig.1).Six sampleswere collected fromeach profile: PUN01 to PUN06 fromoutcrop 1, and PUN07 to PUN12 from outcrop 2 (Fig. 4). In addition,a deep sample (GRA02, Table 2) was collected from a quarry in thePliocene conglomerates close to the apex of the alluvial fan.

The high energy sediment facies sampled in Profiles 1 and 2 areconsistent with fast transportation and are in agreement with theproximity of the Carboneras fault, the trace of which is located lessthan 1 km upstream (Fig. 2). This suggests that the potential 10Beinheritance component formed during transportation of quartzgrains is probably negligible. However, most of the source area ofthese deposits is formed by Pliocene conglomerates that could havea certain concentration of 10Be due to the exposure of the quartzgrains prior to their burial (Pliocene) and/or during exhumation(Pleistocene). The first source of inheritance is represented by theGRA02 concentration and was introduced into the models asa minimum value of the variable CInher. (Fig. 5 A).

The irregular cementation pattern of sediments in the outcropsstudied implies a variation in sediment density from 1.8 g cm�2

(uncemented gravels) to 2.5 g cm�2 (gravels with well developedcalcrete crust). Bulk density measurements yielded a mean densityof 2.2 g cm�2 for outcrop 1 and 2.1 g cm�2 for outcrop 2, and 1sdensity uncertainties of 0.2 g cm�2 were considered.

Fig. 5. Probability distribution of each variable according to models. Densities areexpressed in g cm�3 x axis correspond to values of each variable CInher., age (t) anderosion rates (31and 32). For each single value of each variable, the best fit of eachmodel was calculated. The y axis represents the cumulative probability of the calcu-lated best fit (c2(variable)min) with respect to the total distribution over the best valuec2min , as shown in Fig. 1. Models not considering density uncertainties are indicated bygray lines and variable-density models are represented by dashed lines. A) Probabilitydistribution of inheritance according to models. A minimum value of CInher. ¼ CGRA02was introduced. CInher. values predicted by variable-density model (dashed) are similarto those predicted by the constant-density models that considers the initial density. B)Probability distribution of age according to models. Behaviour of variable-densitymodel with respect to age probability distribution is similar to the constant-densitymodel that considers the mean density through time (c. 2 g cm�3). C) Probabilitydistributions of erosion rates according to models. As all the models predicted similarprobability distributions, some of them were not represented.

Table 3

Two-profile models results (y ¼ 11 samples - 4 parameters ¼ 7 degrees of freedom).Model in the first 3 rows and model in the penultimate row consider no densityuncertainties. Models in the first 6 rows do not consider variation of densitiesthrough time. r1 is the current mean density, 2.2 g cm�3 for Profile 1 and 2.1 g cm�3

for Profile 2.

Initialdensity(g cm�3)

Finaldensity(g cm�3)

c2min Qf CInher.,best(103 at g�1)

31,best e 32,best(10�4 g cm�2a�1)

tbest(ka)

1.8 1.8 11.6 0.16 44 0.0e3.9 2432.0 2.0 12.6 0.14 45 5.4e8.1 535r1 r1 12.1 0.14 45 5.9e8.9 7741.8 � 0.2 1.8 � 0.2 3.1 0.75 39 2.2e6.1 3072.0 � 0.2 2.0 � 0.2 4.1 0.68 37 5.8e8.8 676r1 � 0.2 r1 � 0.2 4.9 0.67 27 6.3e9.2 11521.8 r1 12.6 0.13 26 6.4e8.9 6511.8 � 0.2 r1 � 0.2 4.3 0.71 73 0.0e4.7 248

Á. Rodés et al. / Quaternary Geochronology 6 (2011) 186e194 191

Given that geomorphological evidence indicates that bothprofiles must reflect the same age, only two-profile models wereconstructed, using 11 samples to compute 4 parameters, hence,implying 7 degrees of freedom. To evaluate the relevance of densitywhen fitting the models to the 10Be depth profile datasets, thefollowing cases were considered: (i) constant densities withoutuncertainty, (ii) constant density with uncertainty, and (iii) variabledensity with current uncertainty.

Constant-density models shown in Fig. 5 and Table 3 considerinitial (1.8 g cm�2 for uncemented gravels), mean (2.0 g cm�2) andcurrent densities. The variable-density model was computed withan initial density r0 of 1.8 � 0.2 g cm�2. Results of variable-densitymodels are also shown in Fig. 6.

4. Discussion

4.1. Variable-density model behaviour

If an initial density lower than the current density is used, thecurvature of the theoretical concentration depth profile increases. A

similar effect can be observed in a buried depth profile. However,unlike the erosion process, the density gain implies a higherproduction of cosmogenic nuclides at depth while keeping thesame production near the surface. Fig. 7A shows that deep nuclideconcentrations are more sensitive to density variations whereasshallow nuclide concentrations are more sensitive to burial orerosion processes.

Fig. 7B shows that concentrations obtained from the variable-density model are only slightly different from those obtained bya constant-density model using the time averaged density. The useof a constant-density model with the time averaged density couldbe a reasonable approach despite the fact that the profile obtainedfrom the variable-density model has a greater curvature.

4.2. El Puntal alluvial fan

The deepest sample PUN01 (Profile 1) shows an abnormallyhigh concentration of 10Be (Fig. 4). Considering the stratigraphicerosional discontinuities observed in outcrop 1, this high concen-tration value can be interpreted as corresponding to an older

BA

C D

Fig. 6. Projections of the best fit result (black dots) and the CInher.-31-32-t hyper-volume corresponding to variable-density model results that fit the data within 1s confidence level.Owing to the simplicity of the relationship between 31 and 32 (which is almost linear, as shown in graph B), only four 2D projections are represented. Graph A indicates that inheritanceand erosion rate behave almost as independent parameters; and graphsC andD showthat the age is sensitive to erosion rate and inheritance. The inset graph inD represents source areaerosion rate vs. CInher. values. CInher. values are calculated considering a Pliocene inherited amount of 10Be (CGRA02) and the 10Be accumulated during exhumation, which depends on thesource area erosion rate. Shaded areas correspond to results that are improbable because of indirect estimates: CInher. in A and D, and erosion rates in B and C.

Á. Rodés et al. / Quaternary Geochronology 6 (2011) 186e194192

alluvial surface. Hence, PUN01 was treated as an outlier and wasnot computed in the models.

Studying the pedogenic carbonate coatings in a semi-aridMediterranean climate during the Pleistocene and the Holocene,Badía et al. (2009) observed that the gain in carbonate in fluvialterraces was regular through time. This suggests a linear increase indensity with time in alluvial sediments, as contemplated in ourvariable-densitymodel. According to the density evolution reflectedin this model, the El Puntal alluvial fanwould have a most probableage of 248 ka (Table 3), with a 1s confidence level ranging between214 and 1625 ka, as indicated by the black dashed line in Fig. 5B.

As shown in Table 3, all models that consider density uncer-tainties obtained a significantlybetterquality factor thanmodelsnotconsidering density uncertainties, suggesting that density uncer-tainty is a relevant factor that must be taken into account whenfitting data and models. Fig. 4 shows that the uncertainty of x(expressed in g cm�2) increaseswith depth due to the uncertainty indensity. For deep samples, x uncertainty is more relevant than theconcentration uncertainty when fitting the model to the data.

Mathematically this implies that, in Eq. (12), sxobtains higher valuesthan sC as depth increases. When this uncertainty is considered,a higher variability of models fit the datawithin 1s confidence levelthanwhen it is not (Fig. 4A and B vs. C and D). As shown in Fig. 5, theincrease in model freedomwhen considering density uncertaintiesaffects the inheritance and ages predicted by the models, but doesnot significantly affect the erosion rate predictions.

Best fits of models that consider a density variation throughtime show similar quality factors (Table 3) but slightly differentprobability distribution ranges of inheritance and age (Fig. 5).However, probability distributions of free parameters in variable-density models are similar to those predicted by constant-densitymodels using a time averaged density. Therefore, to ignore thedensity evolution of these sediments by using the current densitywould result in an overestimation of the age and of the inheritance.

Predictedvalues of inheritance (Fig. 5A) implyerosion rateshigherthan50m/Ma in the source area of the El Puntal fan,which is in the LaSerrata range (inset graph in Fig. 6D). In general, the erosion rates inthe uplifted range of La Serrata are not expected to be much higherthan the dip-slip rates of the Carboneras Fault (c. 40e140 m/Ma,according toMoreno et al. (2009)). Hence, at a conservative estimate,erosion rates higher than 200 m/Ma are not expected in the sourceareaof theEl Puntal fan,which implies a lower limitof c. 50,000 atg�1

for inheritance. According to the variable-densitymodel, these valuesfor inheritance correspond to agesT1,000,000 a (Fig. 6D). Therefore,according to the estimates of the La Serrata erosion rates, the age ofthe El Puntal alluvial fan is probably lower than 1 Ma.

All the results of the models reflect a higher erosion rate inProfile 2 than in Profile 1 (Fig. 6B). The models indicate that sincethe abandonment of the alluvial fan, thicknesses of 0e2.8 m and0.6e4.5 m have been lost by erosion at the tops of Profiles 1 and 2,respectively. This relatively large contrast in denudation within therestricted area of sampling can be ascribed to the proximity of anactive channel to outcrop 1, which since the abandonment of thefan surface could have increased erosion due to diffusion processesin neighboring areas. However, the highest values of surfacelowering predicted by the models (2.8 and 4.5 m) are probably notrealistic considering the smooth surface of the El Puntal fan (Figs. 2and 3A). Fig. 6C shows that the highest erosion rate values, and thusthe greatest lowerings, correspond to the oldest ages in the 10Bemodels. Hence, the regularity of the El Puntal Fan surface suggeststhat the oldest ages predicted by the model may not be realistic inline with the constraints deduced from inheritance estimates.

5. Conclusions

Although inheritance is the most important factor in restrictingthe age of a depositional landform, the density of a deposit is alsoa crucial factor in modeling cosmogenic nuclide depth-profiles tointerpret precise concentration measurements. In cases where thedistribution of density is not well constrained, an adequate prop-agation of the vertical error is needed to obtain realistic confidencelevel boundaries (sx in Eqs. 12 and 14). In landforms made up ofcemented sediments, models that do not take into account theincrease in density due to diagenetic processes are expected tosignificantly overestimate landform ages.

The variable-density model presented here, which considersa uniform density gain, can be used to yield more realistic ageestimates. Comparison of results between variable and constant-densitymodels shows that, if a constant-densitymodel is used to fitdata from cemented sediments, a time averaged density must beconsidered.

Once the 10Be inheritance is constrained by indirect estimationof erosion rates in the source area, 10Be data modeling of two depthprofiles indicates that the El Puntal alluvial fan has an age ranging

Fig. 7. Examples of depth concentrations according to constant and variable-densitymodels, with an exposure age of 500 ka. A. Effects of density gain and erosion intheoretical cosmogenic concentration depth-profiles. B. Comparison between a vari-able-density model considering a density evolution from 1.6 to 2.4 g cm�3 andconstant-density models considering initial, final and time averaged densities. Allmodels are calculated for sea level, high-latitude production rates and attenuationlengths of Pspal. ¼ 4.5 at g�1 a�1, Pstop ¼ 0.096 at g�1 a�1, Pfast ¼ 0.085 at g�1 a�1,Lspal. ¼ 160 g cm�2, Lstop ¼ 1510 g cm�2 and Lfast ¼ 4320 g cm�2.

Á. Rodés et al. / Quaternary Geochronology 6 (2011) 186e194 193

from 214 ka to c. 1 Ma within 1s confidence level. This is the firstnumerical age constraint on the second oldest generation of alluvialfans associated with the Carboneras Fault zone.

Acknowledgements

This study has been sponsored by Spanish National ProjectEVENT (CGL2006-12861-C02-02) and supported by the Consolider-Ingenio 2010 programme, under CSD2006-0004 “Topo-Iberia”. TheFrench AMS national facility ASTER (CEREGE) is supported bythe INSU/CNRS, the French MESR, IRD and CEA. We are gratefulto M. Arnold, G. Aumaître and K. Keddadouche for 10Be AMSmeasurements.

Appendix. Supplementary material

Supplementary data related to this article can be found online atdoi:10.1016/j.quageo.2010.10.004.

Editorial handling by: R. Grun

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