Earth and Planetary Science Letters 295 (2010) 511–522

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Earth and Planetary Science Letters

j ourna l homepage: www.e lsev ie r.com/ locate /eps l

Inversion of thermochronological age-elevation profiles to extract independentestimates of denudation and relief history — I: Theory and conceptual model

Pierre G. Valla a,⁎, Frédéric Herman b, Peter A. van der Beek a, Jean Braun a

a Laboratoire de Géodynamique des Chaînes Alpines, Université Joseph Fourier, BP 53, 38041 Grenoble, Franceb Geologisches Institut, ETH Zürich, 8092 Zürich, Switzerland

⁎ Corresponding author. Tel.: +33 476 63 54 64; fax:E-mail address: pierre.valla@e.ujf-grenoble.fr (P.G. V

0012-821X/$ – see front matter © 2010 Elsevier B.V. Adoi:10.1016/j.epsl.2010.04.033

a b s t r a c t

a r t i c l e i n f o

Article history:Received 18 September 2009Received in revised form 15 April 2010Accepted 16 April 2010Available online 18 May 2010

Edited by T.M. Harrison

Keywords:low-temperature thermochronologyage-elevation profilesdenudation raterelief changeinverse modelling

We determine to what extent low-temperature thermochronology data, in particular from age-elevationprofiles, provide independent and quantitative estimates on denudation rates and relief development.Thermochronological age-elevation profiles have been widely used to infer exhumation histories. However,their interpretation has remained inherently one-dimensional, neglecting potential effects of lateral offsetsbetween samples. Furthermore, the potential effects of transient topography on crustal isotherms andconsequently on thermochronological data have not yet been addressed in detail. We investigate thisproblem with the aim of deriving independent estimates of both denudation rates and relief history fromlow-temperature thermochronometers, measuring the relative uncertainties on these parameters and finallyconstraining the timing of potential variations in denudation rate and/or relief development. We adopt anon-linear inversion method combining the three-dimensional thermal-kinematic model Pecube, whichpredicts thermal histories and thermochronological ages from an input denudation and relief history, withan inversion scheme based on the Neighbourhood Algorithm. We use synthetic data predicted from imposeddenudation and relief histories and quantitatively assess the resolution of thermochronological datacollected along an age-elevation profile. Our results show that apatite fission-track (AFT) ages alone do notprovide sufficient quantitative information to independently constrain denudation and relief histories.Multiple thermochronometers (apatite (U–Th)/He (AHe) ages and/or track-length measurements combinedwith AFT ages) are generally successful in constraining denudation rates and timing of rate changes, theoptimum combination of thermochronometers varying with the input scenario (relief change or varyingdenudation rates). However, relief changes can only be quantified and precisely constrained fromthermochronological age-elevation profiles if the rate of relief growth is at least 2–3 times higher thanthe background denudation rate. This limited resolution is due to the depth of the closure isotherm (between∼70 and 110 °C) for the AFT and AHe systems, which only partly record topographic change. Newthermochronometers (such as 4He/3He or OSL) that are sensitive to lower temperatures may be the key forresolving this issue.

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© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Understanding the formation and evolution of orogenic topogra-phy requires a better comprehension of the couplings betweenclimate, tectonics and surface processes (e.g., Beaumont et al., 1992;Willett, 1999; Zeitler et al., 2001). However, direct evidence of thesecouplings remains elusive and quantitative data are needed to betterconstrain the denudation and relief evolution ofmountain belts. Theseare traditionally studied independently and only fewmethods such aslow-temperature thermochronology (e.g., Gallagher et al., 1998;Braun, 2005; Reiners and Brandon, 2006; Reiners, 2007) may enable

us to quantitatively assess both the denudation history and paleo-relief of mountain belts (Braun, 2002a,b).

Here, we explore to what extent low-temperature thermochro-nology (apatite fission-track and (U–Th)/He) data, in particular fromage-elevation profiles, can provide such constraints on denudationhistory and paleo-relief of mountain belts. Apatite fission-track (AFT)age-elevation profiles, i.e. sets of AFT ages collected from differentelevationswithin a spatially restricted domain, have beenwidely usedto infer exhumation histories of specific areas (e.g., Wagner andReimer, 1972; Hurford, 1991; Fitzgerald et al., 1995). More recently,apatite (U–Th)/He (AHe) data have also been used in similar samplingschemes (House et al., 1997; Reiners et al., 2002; Clark et al., 2005).However, age-elevation relationships (AER) have generally consid-ered the problem as one-dimensional, neglecting horizontal offsetsbetween different samples of an elevation profile and therefore thepotential effects of topography (Braun, 2002a; Gallagher et al., 2005;

Fig. 1. Conceptual model showing two end-members exhumation scenarios. Samples(black dots) are collected along an age-elevation profile spanning the ridge-to-valleyrelief; their thermochronological ages reflect, to a first order, the time since they passedthe closure isotherm for the system considered (dashed line). (a) Increase inexhumation rate from an initial rate E1 to a final rate E2, under constant relief. (b)Regional exhumation rates remain constant (E1) but ridge to valley relief increases fromΔh1 to Δh2 through time. We assume relief increase to take place by valley incision; theamount of relief increase is quantified by the parameter R=Δh1/Δh2.

512 P.G. Valla et al. / Earth and Planetary Science Letters 295 (2010) 511–522

Foeken et al., 2007). Topographic effects on thermochronological age-elevation profiles result from the fact that near-surface isotherms arenot horizontal but are deflected by the topography (e.g., House et al.,1998; Braun, 2002a). The intensity of this perturbation and its depthof penetration are governed by the amplitude and wavelength of thetopography. The influence of temporally steady-state topography onthermochronological age-elevation profiles is relatively well under-stood (Stüwe et al., 1994; Manktelow and Grasemann, 1997).However the potential effects of temporally varying topographyhave not yet been addressed, although Braun (2002a) has shownconceptually what kind of patterns can be expected in AER and hashighlighted the potential misinterpretation of AER slopes in terms oflong-term denudation rates.

This study thus aims at: (1) deriving quantitative and independentestimates of both denudation and relief histories (rate and timing)from AER and (2) measuring the relative uncertainties on theseparameters and thus the resolution of the data, which is very difficultto determine with current methods. Recent studies (e.g., Braun andvan der Beek, 2004; Braun and Robert, 2005; Herman et al., 2007)have tackled similar problems by combining the three-dimensionalthermal-kinematic model Pecube (Braun, 2003), which predictsthermal histories and thermochronological ages from an inputdenudation and relief history, with an inversion scheme based onthe Neighbourhood Algorithm (Sambridge, 1999a) to search theparameter space and extract best-fitting scenarios for denudation andrelief histories from the data. We adopt a similar method here but (1)specifically address the problem of interpreting age-elevation profilesand (2) add a model-appraisal stage (Sambridge, 1999a; Herman etal., in press) to fully resolve the inverse problem and derivequantitative measures of the resolution with which we inferdenudation and relief histories.

Here, we use synthetic data predicted from imposed denudationand relief histories, enabling us to quantitatively assess whatconstraints on denudation and relief history we can extract fromlow-temperature thermochronology data. In a companion paper (vander Beek, P.A., P.G. Valla, F. Herman, J. Braun, C. Persano, K.J. Dobson,and E. Labrin, Inversion of thermochronological age-elevation profilesto extract independent estimates of denudation and relief history— II:Application to the FrenchWestern Alps. Submitted to Earth Planet. Sci.Lett., hereafter referred to as van der Beek et al., submitted), we applythe same methodology on a real age-elevation profile in order toinvestigate the potential of AER in constraining the recent evolution ofthewestern Alps in terms of temporally varying denudation rates and/or relief development (e.g., Glotzbach et al., 2008; Vernon et al., 2009).

In the following, we first outline our conceptual model and presentthe synthetic data we will use in subsequent inversions. We thenintroduce our numerical approach and show to what extentpredictions of denudation rates, timing and relief evolution canquantitatively be estimated from AER. We finally discuss theimplications of our results for the use of thermochronology data inconstraining tectonic and climatic controls on the evolution oforogenic topography.

2. Conceptual model

2.1. Modelling approach

In our conceptual model, outlined in Fig. 1, local exhumation ratesat any point in the landscape result from two independent processes:regional denudation (which in the case of steady-state topography isequal to rock uplift as defined by England and Molnar, 1990) andtemporal changes in topography. As thermochronology does notprovide any constraints on surface uplift, topographic changes areexpressed as changes in relief, defined as the difference in elevationΔh between the highest and lowest points in the area underconsideration. A temporal change in local exhumation rate at any

point, as recorded by a thermochronometer, may thus result from anycombination of varying regional denudation rates (E1→E2; Fig. 1a)and/or varying relief (Δh1→Δh2; Fig. 1b).

We use a numerical approachwith simple and imposed exhumationscenarios to assess to what extent both denudation rates and reliefdevelopment can be constrained independently from thermochronolo-gical data. Based on this conceptual model (Fig. 1), we predictthermochronological ages for samples along an age-elevation profilespanning the ridge to valley relief, for three different end-membermodels: (1) both denudation rate and topography are constant throughtime; (2) topography is steady-state but denudation rates increasethrough time; and (3) background denudation rates are constant butrelief increases through time. In the latter case, we assume that reliefincrease results from preferential valley incision; i.e. ridges remain at aconstant elevationwith respect to an exterior reference frame but valleybottoms are lowered, implying spatially varying exhumation rates withhigher rates in the valleys than on the ridges (Fig. 1b).

We compute the thermal structure through time and thermalhistories for material points now at the surface using the three-dimensional thermal-kinematic model Pecube (Braun, 2003); a finiteelement code that solves the heat transfer equation (Carslaw andJaeger, 1959) in 3D (Fig. 2):

ρc∂T∂t + v

∂T∂z

� �=

∂∂x k

∂T∂x +

∂∂y k

∂T∂y +

∂∂z k

∂T∂z + H ð1Þ

where T(x,y,z,t) is temperature (°C), ρ is rock density (kg m−3), c isheat capacity (J kg−1 K−1), v is the vertical velocity of rockswith respect to the base of the model (mm yr−1); k is conductivity(Wm−1 K−1) and H is radioactive heat production (Wm−3). Pecubeis able to solve Eq. (1) for a time-varying surface topography, thuspermitting to model transient relief. Relief evolution is incorporatedhere using the relief factor R, defined as the ratio between the initial(Δh1) and present-day relief (Δh2):

R =Δh1Δh2

: ð2Þ

Fig. 2. Predicted thermal structure and AFT and AHe age-elevation profiles for different denudation and relief history scenarios. (a) Model thermal structure (isotherms annotated in°C) for a steady-state sinusoidal topography with wavelength λ=20 km, amplitude A=4 km and a constant 1 km Myr−1 denudation rate. See Table 1 for other thermo-kinematicparameters used in Pecube modelling. (b)–(c) Predicted AFT (b) and AHe (c) age-elevation profiles for a family of scenarios characterised a change in exhumation rate at 4.8 Ma:E2=1000 mMyr−1 and E1 varying between 0 and 1000 mMyr−1 as indicated in the legend, under steady-state topography (R=1). (d)–(e) Predicted AFT (d) and AHe (e) age-elevation profiles for a family of scenarios characterised by valley carving since 4.8 Ma (R=0) combined with temporally constant regional denudation rates varying between 0 and1000 mMyr−1 as indicated in the legend.

513P.G. Valla et al. / Earth and Planetary Science Letters 295 (2010) 511–522

If R=0, there is no initial relief (i.e. the initial landscape is aplateau at the maximum present-day elevation); if Rb1 paleo-relief islower than today (e.g., Fig. 1b); if R=1 relief is constant through time(e.g., Fig. 1a); finally, if RN1 paleo-relief is higher than at present. Notethat this approach supposes that the planform pattern of ridges andvalleys is fixed, only the relief changes with time.

Predicted thermal histories are used to calculate AFT and AHe age-elevation profiles; these are currently the most commonly usedthermochronometers for assessing late-stage exhumation and reliefdevelopment in mountain belts. Thermal histories are translated into

AHe ages using a simple forward model for He production–diffusion–ejection (Farley, 2000). AFT ages are calculated using a forward modelfor AFT annealing (Green et al., 1989) with a new parameter fit to thelaboratory annealing data (MAP3; Stephenson et al., 2006), whichtakes into account uncertainties in laboratory annealing temperaturesas well as independent geological data. More elaborate forwardmodels have recently been developed for both fission-track annealing(e.g., Ketcham, 2005) and He diffusion in apatite (Shuster et al., 2006;Flowers et al., 2009; Gautheron et al., 2009), which would predictslightly different effective closure temperatures depending on

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exhumation scenarios. However, forward models used for AFT andAHe age predictions are similar for both generating the synthetic dataand resolving the inverse problem; the results shown here are thuslargely independent of the age-prediction models we use in Pecube.

2.2. Age-elevation relationships

We use a synthetic relief model characterised by a 2D sinusoidaltopography with a wavelength of 20 km and amplitude of 4 km(Fig. 2a). These values were chosen to allow sufficient denudation ofvalley samples for models with relief increase only; they are alsocharacteristic of the topography observed in the western Alps studyarea of the companion paper. Such topography strongly perturbs low-temperature isotherms and specifically the effective AHe and AFTclosure isotherms (respectively ∼70 °C and ∼110 °C; cf. Fig. 2a). Allmodels are run over 30 Myr, with a late-stage change in exhumationrate or relief (Fig. 2b–e); all samples are exhumed from temperaturesabove the AFT and AHe closure temperature.

Fig. 2b and c shows synthetic AER for scenarios with steady-statetopography and varying denudation rates (Model 1). In all cases,denudation rates are 1000 mMyr−1 since 4.8 Ma; denudation ratesprior to that time vary between 0 and 1000 mMyr−1 for the differentmodels. The two end-member scenarios (i.e. constant denudation rateversus discrete denudation event; respectively Model 1a and Model1e) have widely been discussed previously in the literature (e.g.,Fitzgerald et al., 1995; Gallagher et al., 1998; Braun et al., 2006); wewill start by discussing these end-member cases since they provide aframework in which to discuss the other model results.

The constant-rate denudation case (Fig. 2, Model 1a) leads toperfectly linear AFT and AHe AER (r²=1.00), the slopes of which,however, are 1430 mMyr−1 and 1740 mMyr−1 respectively (i.e.,they overestimate the denudation rate by ∼40 to 70%) due to theperturbation of isotherms under the sinusoidal topography (e.g.,Stüwe et al., 1994; Manktelow and Grasemann, 1997). The oppositecase where denudation only occurs since 4.8 Ma (Fig. 2, Model 1e)leads to AFT AER containing both a convex-up and concave-up break-in-slope (a convex-up break-in-slope only for the AHe AER). Thesebreaks-in-slope represent the base and top, respectively, of the pre-exhumation AFT partial annealing zone or AHe partial retention zoneand provide valuable constraints on the timing of onset and the totalamount of exhumation (Fitzgerald et al., 1995, 1999; Gleadow andBrown, 2000).

Intermediate (and possibly more realistic) cases (Fig. 2, Models 1b,c, d), where denudation rates increase at 4.8 Ma but were non-zerobefore that time, lead tomore-or-less linear to broadly convex-up AERfrom which it would be difficult to intuitively infer a sudden increasein denudation rates, especially when relative errors in AFT or AHe agesof the order of 10% are taken into account. Thus, even a five-foldincrease in denudation rate (from 200 to 1000 mMyr−1, Model 1c,Fig. 2b) may go unnoticed with AFT data only; the AER predicted bythis scenario would most probably be interpreted as indicatingconstant exhumation at a rate ∼550 mMyr−1 (r²=0.95 for a linearfit). This “mean” exhumation rate integrates exhumation at a rate of1000 mMyr−1 during the last 5 Myr after initial exhumation at200 mMyr−1. The AHe AER (Fig. 2c, Model 1c) better reflects therecent denudation history, with a linear slope of ∼1300 mMyr−1 thatoverestimates the true denudation rate by ∼30%, but contains norecord of the previous lower exhumation rates. Of course, additionalconstraints can be placed on the denudation histories by measuringconfined fission-track length distributions (Gallagher et al., 1998;Gleadow and Brown, 2000). The above models, however, predict littlevariation in mean confined track lengths (MTL) for the differentsamples and scenarios; MTL is typically in the range 10.5–12 μmregardless of the pre-4.8 Ma denudation rate, except for very slowrates (i.e., ≤200 mMyr−1). The latter scenarios predict a decrease inMTL with age and elevation, typical of age-elevation profiles contain-

ing an exhumed AFT partial annealing zone (Fitzgerald et al., 1995,1999; Gleadow and Brown, 2000).

Models including relief growth (Model 2) are illustrated in Fig. 2dand e. For these models, valleys are supposed to have been carvedsince 4.8 Ma from a pre-existing horizontal plateau (i.e., R=0, seeEq. (2)). This relief-growth model is combined with constantdenudation, at rates between 0 and 1000 mMyr−1, over the modelled30 Myr timespan (Fig. 2d,e; Model 2a to 2e). Resulting AFT and AHeAER are close to linear for denudation rates ≥200 m Myr−1 (r²N0.95for linear fits to the predicted age-elevation profiles). The AER slopesoverestimate the input regional denudation rates; for instance, theslope is 230 mMyr−1 (AFT) or 310 mMyr−1 (AHe) for an input rate of200 mMyr−1; but 1840 mMyr−1 (AFT) or 2310 mMyr−1 (AHe) foran input rate of 1000 mMyr−1. The reason for this overestimation istwofold. First, exhumation rates are not constant along the profile butvary from the background denudation rate at the summits to that rateaugmented by the rate of valley carving (∼830 mMyr−1) at thevalleys bottoms. On the other hand, since there is no relief before4.8 Ma, isotherms are horizontal until that time and only becomedisturbed as valleys are carved, thus limiting the topographic effectson age-elevation profiles. For moderate to high background denuda-tion rates, isotherms are advected to the surface, allowing to betterrecord post-4.8 Ma relief development. However, for denudation ratesb200 mMyr−1, heat advection is minor and erosion in the valleys isinsufficient to expose samples exhumed from below the AFT partialannealing zone (for rates b100 mMyr−1 the AHe partial retentionzone is not exposed either), leading to distinctive concave-up AER.Such profiles might be interpreted as representing exhumation atrates that diminish through time or, if track-length data are taken intoaccount, as limited and slow long-term denudation (e.g., Glotzbach etal., 2008; Reiners and Brandon, 2006).

Differences between age-elevation profiles predicted by verydifferent scenarios may be subtle and the predicted patternscounter-intuitive. As shown in Fig. 2, scenarios with (Model 2) orwithout (Model 1) topographic change can be interpreted in the sameway, judging from the pattern and slope of the AER. As discussedabove, implicitly one-dimensional interpretation of such profiles islimited because of horizontal offsets between profile samples (up to10 km in our simulations, Fig. 2) and topographic disturbance ofisotherms. For example, Model 1c simulates an increase in denudationrate from 200 mMyr−1 to 1000 mMyr−1 with constant topography;whereas Model 2b combines topographic change with a regionalexhumation rate of 500 mMyr−1. Although these two scenarios arecompletely different, AFT AER (Fig. 2b,d) have indistinguishableslopes of ∼660 mMyr−1 (Model 1c) and ∼700 mMyr−1 (Model 2b)respectively. AHe patterns (Fig. 2c,e) are slightly different(∼1300 mMyr−1 for Model 1c and ∼1000 mMyr−1 for Model 2b);however the difference remains subtle and does not resolve the inputdenudation and relief histories. Straightforward interpretation ofthese two age-elevation profiles will lead to inferred spatially andtemporally averaged exhumation rates that have no simple relation tothe true exhumation history and do not address potential topographicchanges.

3. Inversion method and results

3.1. Methodology: Pecube coupled to the Neighbourhood Algorithm (NA)

We choose two end-member scenarios to test whether nu-merical inversions can fully retrieve denudation and relief historiesfrom AER (Fig. 2, respectively Models 1c and 2c): (1) constanttopography (R=1) but an abrupt change in denudation rates(E1=200 mMyr−1; E2=1000 mMyr−1) 4.8 Ma ago (referred to inthe following as Scenario 1); and (2) a constant denudation rateE1=E2=200 mMyr−1 but 4 km of relief growth (R=0) since 4.8 Ma(Scenario 2). Thermo-kinematic data used as input parameters in

Table 1Thermo-kinematic and elastic parameters used in Pecube. Crustal thickness and basaltemperature are set to obtain a geothermal gradient of 25 °C km−1. Poisson ratio,Young's modulus and equivalent elastic thickness are used for calculating the isostaticrebound in response to relief change. Equivalent elastic thickness is set to a value thatsimulates moderate isostatic rebound.

Parameter (unit) Value

Crustal density (kg m−3) 2700Sublithospheric mantle density (kg m−3) 3200Equivalent elastic thickness (km) 25Young's modulus (Pa) 1.1011

Poisson ratio 0.25Crustal thickness (km) 20Thermal diffusivity (km²Myr−1) 25Basal crustal temperature (°C) 520Sea-level temperature (°C) 15Atmospheric lapse rate (°C km−1) 6Crustal heat production (°C Myr−1) 0

515P.G. Valla et al. / Earth and Planetary Science Letters 295 (2010) 511–522

Pecube are given in Table 1; the simulation was run over 30 Myr toensure that samples are exhumed through their closure isotherm andthat isotherms are at steady state before increasing denudation orrelief. Fig. 3 shows the synthetic datasets used for inverse modelling;they consist of AFT and AHe ages as well as mean fission-track lengths(MTL). AER for AHe and AFT are quite similar for the two scenarios

Fig. 3. Synthetic thermochronological data used for the inverse modelling. (a)–(b) AFT and AAFT and ±0.3 Ma for AHe). (c)–(d) Mean fission-track length (MTL) versus elevation for S

except that the age contrast between ridge and valley samples ishigher for Scenario 2. The main difference between the two scenariosis found in the MTL pattern, with an inverse (Scenario 1) versusnormal (Scenario 2) correlation between MTL and elevation.

Using these predicted “perfect” data, we explore for both Scenarios1 and 2 what quantitative constraints can be set on denudation rates,timing and relief evolution. To do this, we first define for eachparameter a specified range that will be searched by the inversealgorithm (uniform distribution for the priors): for the denudationrate of the first phase (E1): 0–1000 m Myr−1; denudation rate of thesecond phase (E2): 0–1000 mMyr−1; transition time (T): 0–30 Ma;and relief factor (R): 0–2. Since we wish to test to what extent aspecific thermochronological dataset can provide independent esti-mates on those four parameters, we have run four inverse simulations(Table 2) for both Scenarios 1 and 2 using: (1) AFT ages only(inversions 1-A and 2-A); (2) AFT+AHe ages (inversions 1-B and 2-B); (3) AFT ages+MTL (inversions 1-C and 2-C); and finally AFT+AHe+MTL (inversions 1-D and 2-D).

The Neighbourhood Algorithm (NA) is a two-stage numericalapproach to derive Bayesian estimates on input parameters for non-linear inverse problems (Sambridge, 1999a,b). The first or samplingstage is an iterative search method during which sampling graduallyconcentrates on regions of the multidimensional parameter spacewhere the misfit function is optimised (i.e. sets of parameters values

He ages versus elevation for Scenarios 1 and 2, respectively (error bars of ±0.5 Ma forcenarios 1 and 2, respectively (error bars are ±0.2 μm).

Table 2Bayesian estimates after the NA appraisal stage for input parameters E1, E2, T and R (see text for description and discussion). First column defines the parameters for the twoscenarios. Second column gives prior PDF, i.e. the PDF describing the input parameter range for the inversion. Third column reports the parameters used to calculate the inputthermochronological data (Fig. 3). Optimal values and estimated uncertainties of parameters are given in following columns for all inversion simulations (inversions 1-A/2-A, 1-B/2-B, 1-C/2-C and 1-D/2-D).

Prior Pdf Input value AFT (inversion 1-A/2-A) AFT+AHe (inversion 1-B/2-B) AFT+MTL (inversion 1-C/2-C) AFT+AHe+MTL (inversion 1-D/2-D)

Scenario 1E1 (km Myr−1) 1.1±0.6 0.2 0.9±0.6 0.9±0.5 0.2±0.1 0.2±0.1E2 (km Myr−1) 1.1±0.6 1.0 0.7±0.3 0.9±0.3 0.9±0.2 0.9±0.3T (Ma) 15.0±8.1 4.8 13.9±8.5 13.6±8.3 4.9±1.6 5.1±1.6R 1.0±0.6 1.0 0.9±0.6 1.1±0.5 0.7±0.4 0.9±0.5

Scenario 2E1 (km Myr−1) 1.1±0.6 0.2 0.2±0.1 0.2±0.05 0.2±0.05 0.2±0.04E2 (km Myr−1) 1.1±0.6 0.2 0.4±0.1 0.4±0.2 0.5±0.2 0.3±0.1T (Ma) 15±8.1 4.8 4.2±3.2 3.6±1.4 3.5±1.3 4.0±1.1R 1.0±0.6 0.0 0.3±0.2 0.2±0.2 0.6±0.4 0.3±0.2

516 P.G. Valla et al. / Earth and Planetary Science Letters 295 (2010) 511–522

that minimize the misfit to the data). The parameter space is dividedinto Voronoi cells, centred on each sampled model, that represent thenearest neighbourhood about each point. A certain amount of best-fitting forward models (here 50%) is used to define new Voronoi cellsand thus to fix a new parameter space. This new parameter space isthen sampled during the next iteration in a random fashion,eventually converging towards one or several sets of parametersthat minimise the misfit function to the data. In our approach, we usea weighted least-squares misfit function ψ:

ψ =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∑N

i=1∑M

j=1

iαj;mod−iαj;dat

� �2

iσ2j

vuuut ð3Þ

where N is the number of datasets (1 for inversions A; 2 for inversionsB and C; 3 for inversions D), M is the number of samples in eachdataset (11), iαj,mod and iαj,dat are predicted and observed values forAFT/AHe ages or MTL, respectively, and iσj is the uncertainty on thedata. Here, we choose to use constant synthetic uncertainties (σj) of±0.5 Ma, ±0.3 Ma and ±0.2 μm for AFT, AHe and MTL, respectively,to provide an equal weight on all samples for misfit calculations. Wechoose to fit the mean track length (MTL) rather than the full track-length distribution for computational convenience and because wecan simply define an uncertainty onMTL that is comparable to the ageuncertainty. Note that we do not use a reduced misfit, i.e. the misfitfunction will be higher when we use a larger number of data. Using ahigh performance cluster, we perform a large amount of forwardmodel runs (∼5000 per inversion), necessary to obtain a statisticallysignificant result given the number of free parameters.

During the second or appraisal stage, the algorithm resamples theentire ensemble of models that were generated during the first stageto provide Bayesian measures of resolution in the form of marginalprobability density functions (PDF) L for each parameter, followingthe equation:

L = ∏N

i=1exp −1

2∑M

j=1

iαj;mod−iαj;dat

� �2

iσ2j

0B@

1CA: ð4Þ

3.2. NA sampling stage results

We first present the results of the sampling stage, i.e. theparameter optimisation, for Scenarios 1 (Fig. 4) and 2 (Fig. 5). Eachforwardmodel is represented by a dot, the colour of which depicts thevalue of the misfit function ψ for the associated model (Eq. (1)),reduced by the number of data. The use of a reducedmisfit function inFigs. 4 and 5 allows us to directly compare inversion runs with

different numbers of data. Parameter estimates are determinedgraphically, arbitrarily considering models represented by misfitvalues b0.3 as “satisfying”. We only represent scatterplots forinversions 1-A/2-A (AFT ages only) and 1-D/2-D (full dataset) sincethey illustrate how coupling different thermochronometers mayprovide better predictions on input parameters. Intermediate inver-sion results combining AFT ages with either MTL or AHe ages will beshown for the appraisal stage.

For Scenario 1 (Fig. 4), using AFT ages only (inversion 1-A, Fig. 4a,b)provides accurate but imprecise predictions for E1 (0.2–0.5 kmMyr−1)and T (7–3 Ma).However, Run1-A leads to erroneous estimates for bothE2 and R, suggesting a minor increase in denudation rate (E2≈0.4–0.7 kmMyr−1) compared to the input value (E2input=1 kmMyr−1) butconverging toward significant relief increase (R≤0.4) instead of aconstant topography (Rinput=1). AddingMTL and AHe ages to AFT ages(inversion 1-D; Fig. 4c,d) leads tomuchbetter predictions. Theprecisionof the estimated E1 and T is improved (E1≈0.15–0.25 kmMyr−1; T≈4–6 Ma). Moreover, estimates on E2 (≈0.7–1 kmMyr−1) and R (≈0.5–0.9) are much closer to the input values for inversion 1-D than forinversion 1-A.

We adopt the samemethodology for Scenario 2 (Fig. 5), presentingfirst inversion results using AFT ages only (inversion 2-A; Fig. 5a,b)and then for AFT ages combined with both MTL and AHe ages(inversion 2-D; Fig. 5c,d). Using AFT ages only in Scenario 2 providesdifferent results than for Scenario 1: both E1 and E2 are predictedcorrectly (E1≈0.15–0.25 km Myr−1 and E2≈0.1–0.4 km Myr−1)whether we use AFT ages alone (inversion 2-A) or the full dataset(inversion 2-D). Themain differences between inversions 2-A and 2-Dlie in the predictions for T, which are much less precise and accuratefor inversion 2-A (T≤5 Ma) than for inversion 2-D (T≈4–5 Ma;Tinput=4.8 Ma). Predictions on R suggest a lower relief increase thanthe input value (R≤0.4; Rinput=0) whether we use AFT ages alone(inversion 2-A) or the full dataset (inversion 2-D).

These results show that the two input scenarios (increase indenudation rate versus relief growth) do not lead to the sameinversion results. Although our graphical interpretation of thesescatterplots (Figs. 4 and 5) remains qualitative, these results stronglysuggest that an AFT AER alone is not sufficient to provide quantitativeindependent estimates on denudation and relief histories; AHe andMTL datasets are needed to constrain the input scenario, leading tomuch better parameter estimates when using all the available dataeven in relatively simple cases with only a single change in relief ordenudation rate.

3.2.1. NA appraisal stage resultsResults from the NA appraisal stage differ from the sampling stage

since we do not focus on “satisfying” models via graphical inspectionbut investigate the statistical properties of the model ensemble.

Fig. 4. Scatter diagrams showing results of the NA inversion for Scenario 1. Each dot corresponds to a forward model; its colour is proportional to the value of the reduced misfitbetween predictions and the input data (misfit function ψ reduced by the number of data). Each diagram is the projection onto a plane defined by two of the four parameters(denudation rates E1 and E2; transition time T and relief factor R); horizontal and vertical axes define the parameter space. Note that violet/pink corresponds to low misfits and redcorresponds to high misfits. Results are only shown for end-member inversion experiments with AFT ages alone (inversion 1-A, a-b) and AFT+AHe ages combined with MTL(inversion 1-D, c-d). Black stars show the “true” value of parameters used to calculate the input thermochronological data.

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Posterior PDF's of parameter values for Scenarios 1 and 2 are plotted inFigs. 6 and 7, respectively. Table 2 reports both the most probablevalue and the estimated uncertainty, defined respectively by themode and standard deviation of the marginal PDF, for each parameter(E1, E2, T and R) and all inversions (1-A to 1-D and 2-A to 2-D).

For Scenario 1, the different inversions (1-A, 1-B, 1-C and 1-D) donot lead to similar quantitative predictions for the 4 parameters (E1,E2, T and R). E1 and T estimates are not better resolved than the priorestimates (i.e. a uniform distributionwithin the imposed limits) whenusing AFT ages only or AFT+AHe ages (Fig. 6a–c). Adding MTL tothese datasets provides predictions that are closer to the input valuesand have lower uncertainties (inversions 1-C or 1-D). Including MTLthus appears to be more important than AHe ages to optimizepredictions on these two parameters (E1 and T). In contrast, accurateestimates for E2 (Fig. 6b) requires AFT+AHe ages (inversion 1-B),

MTL (inversion 1-C) or both (inversion 1-D), even though theresolution is similar whatever the input data. Finally, Bayesianestimates for R (Fig. 6d) are poor for all runs, with high uncertainties(±50%) even when we combine AFT and AHe ages with MTL(inversion 1-D).

For the relief-growth experiment (Scenario 2; Fig. 7), Bayesianresults show that we provide tight constrains on E1 (Fig. 7a) evenwhenusing AFT ages only (inversion 2-A). Only the resolution of E1 isimproved by adding more information (0.2±0.1 km Myr−1 forinversion 2-A compared to 0.2±0.04 kmMyr−1 for inversion 2-D).Predictions forE2 and T are alsoquite similar for all the inversions (2-A to2-D; Fig. 7b,c): they all underestimate the age of relief change (TbTinput)and overestimate the denudation rate (E2NE2input) of the second phase.Inversion 2-D, which combines AFT+AHe ages with MTL, leads to E2and T predictions (E2=0.3±0.1 kmMyr−1; T=4.0±1.1 Ma) that are

Fig. 5. Same as Fig. 4 for Scenario 2. Results are only shown for end-member inversion experiments with AFT ages alone (inversion 2-A, a-b) and AFT+AHe ages combined with MTL(inversion 2-D, c-d). Note the positions of the black stars that define the “true” parameters used to calculate the input thermochronological data.

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closer to the input values (E2input=0.2 kmMyr−1; Tinput=4.8 Ma). Inthis case, AHe ages seem to be as important as MTL and AFT ages forproviding reliable estimates of E2 and T. Finally, predictions of R (Fig. 7d)with AFT ages only (R=0.3±0.2) or AFT+AHe ages (R=0.2±0.2) areclose to the input value (Rinput=0) even though they consistentlyunderestimate the relief increase by ∼20–30%. Adding MTL to AFT+AHe ages (inversion 2-D) does not improve the prediction (R=0.3±0.2). Using AFT ages and MTL without AHe data (inversion 1-C) clearlyunderestimates the relief increase by ∼60%.

Our results show that inverse numerical modelling of thermo-chronological AER combining various thermochronometers is anefficient tool for extracting quantitative information on denudationand relief histories. In the following, we will discuss our numericalresults and their implications concerning the use of low-temperaturethermochonometers for quantifying denudation histories and reliefchanges.

4. Discussion

4.1. Direct interpretation of AER

Our results show that even simple tectono-morphic scenarios suchas a single increase in denudation rate (Model 1) or relief (Model 2)provide AER thatmay be problematic to interpret at face value (Fig. 2).AER may be powerful tools to detect first-order tectonic orgeomorphic changes, such as a pulse of denudation after a longperiod of quiescence (Model 1e; Fig. 2b,c) or relief carving with a lowbackground denudation rate (b200 mMyr−1; Models 2d and 2e,Fig. 2d,e). This exhumation pulse is recorded by AFT and AHethermochronometers via exhumation of the partial annealing zone(PAZ) for FT and partial retention zone (PRZ) for He, which imprints aclear signal in the AER (e.g., Fitzgerald et al., 1995, 1999). More subtlechanges in denudation rates and/or relief changes do not have such a

Fig. 6. 1D posterior PDF's for Scenario 1 parameters obtained after the NA appraisal stage: (a) denudation rate of the first exhumation phase (E1); (b) denudation rate of the secondexhumation phase (E2); (c) transition time between the two exhumation phases (T); and (d) relief factor between the paleo-relief and the final relief (R). Each line defines aninversion experiment with a given set of thermochronological data (inversions 1-A, 1-B, 1-C and 1-D). Vertical dashed lines and black stars represent the “true” parameter values.

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clear signal in AER, in particular because elevation profiles are notvertical and samples are laterally offset. Furthermore, direct inter-pretation is difficult since AER are often close to linear; the associatedslope only provides a spatially and temporally averaged exhumationrate through the sampled time window. More information can beretrieved when including AHe data; however, we have seen thatexcept in specific cases (i.e. denudation rates b200 mMyr−1), AHeprofiles resemble AFT AER and do not provide sufficient additionalinformation to derive independent estimates on denudation and reliefhistories. The usefulness of track-length measurements for constrain-ing relief growth is limited by their correlationwith elevation (Fig. 3d)that is similar to the correlation observed for denudation scenariosunder steady-state topography, except where denudation ratesincrease by at least a factor of 5. In the latter scenarios, MTL patternsshow a clear inverse correlation with elevation (Fig. 3c).

4.2. Quantitative inversion of thermochronological data: multipledatasets

Using Pecube with an inverse approach (NA algorithm), we haveanalysed to what extent a specific dataset can provide quantitativeestimates on denudation rates, timing and relief history. Beforediscussing the results, we should point out some limitations associatedwithourmodellingapproach. First,wehave implemented relief changesin such a way that the planform topographic pattern does not changewith time. This is not a limitation of the Pecube code, which can handleany arbitrary topographic change, but permits us to simply parameter-ize relief change through a single parameter R. We note, however, that

evidence for relative longevity of drainage patterns exists in manymountain belts and present such evidence for the Western Alps studyarea in the companion paper. Second, the scenarios we have run, whichall involve a recent increase in relief and/or denudation rate, aresomewhat contextual and obviously only address a limited number ofexhumation and relief scenarios from the infinite choice available.However, such recent changes in exhumation rate and/or relief inmountain belts have been investigated by many studies usingthermochronological AER in recent years (e.g. Fitzgerald et al., 1995,1999; Densmore et al., 2007; Gibson et al., 2007; Whipp et al., 2007;Glotzbach et al., 2008; Vernon et al., 2009). Third, our inferences areobviously somewhat dependent on the age-prediction model for AFTand AHe that we have chosen to implement, as well as on the fact thatwe only use MTL instead of the full track-length distribution.

Keeping the above caveats in mind, we focus here on predictionsconcerning denudation history (E1 and E2) and timing (T); quantita-tive estimates on relief evolution (R) will be discussed in the nextsection. In the case of an increase in denudation rate under steady-state topography (Scenario 1), parameter estimates from AFT agesalone are poor (Fig. 6; Table 2). The inability of AFT ages alone toretrieve denudation and timing parameters explains why theinversion algorithm converges towards a parameter set that isdifferent from the input values (Inversion 1-A, Fig. 4a,b). We haveperformed two tests to ascertain that it is the data and not theinversion procedure that leads to this erroneous prediction (SeeSupplementary Figs. 1 and 2). Resampling a much larger proportion ofmodels between different generations leads to slower convergencebut the same predicted optimal parameter values. In contrast, setting

Fig. 7. Same as Fig. 6 for Scenario 2. (a) denudation rate of the first exhumation phase (E1); (b) denudation rate of the second exhumation phase (E2); (c) transition time between thetwo exhumation phases (T); and (d) relief factor between the paleo-relief and the final relief (R). Each line defines an inversion experiment with a given set of thermochronologicaldata (inversions 2-A, 2-B, 2-C and 2-D). Vertical dashed lines and black stars represent the “true” parameter values.

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R=1 and only solving for denudation rates and timing leads to correctparameter predictions. These tests show that AFT data on their owndo not have sufficient resolution to constrain the denudation andrelief history and also that care should be taken when interpreting theoptimisation results from NA on their own, without considering theresolution of the parameter predictions.

In contrast, AFT data alone lead to good predictions for the inputparameters in a relief-growth context (Scenario 2), even thoughresolution is relatively low (Fig. 7). We thus suggest that, in the casesstudied here, AFT data are more useful for studying relief growth thanfor quantifying variations in denudation rate under constant topog-raphy. This may be explained by the shape of the AER, which is lesslinear in the relief-growth case (Fig. 3b) and shows a higher contrastin AFT ages between ridge and valley-bottom samples than in thedenudation increase scenario (Fig. 3a).

In Scenario 1, AHe ages combined with AFT ages do not add moreinformation than AFT ages alone and predictions of all parametersexcept E2 remain poor (Table 2). The AHe system has a lower closuretemperature than AFT (respectively ∼75 °C and ∼110 °C), whichexplains why AHe ages provide better constraints on the recentdenudation history (E2). Adding MTL data leads to better constraintson both exhumation rates and timing. Quantitative estimates of theexhumation history under steady-state topography thus require bothAFT ages and MTL measurements, whereas AHe ages do not seem toimprove parameter predictions. The reason for this lies in the AHe andMTL profiles (Fig. 3): the AHe AER is very similar to the AFT profile

whereas the MTL pattern is strongly contrasted between ridge (shortMTL) and valley-bottom (longer MTL) samples.

Scenario 2 (i.e., relief growth) shows completely different results.In this case, AHe ages combined with AFT ages provide much betterestimates of E1 and T (Table 2 and Fig. 7a–c), E2 being already wellconstrained by AFT ages alone (Fig. 7b). In contrast, using MTLmeasurements with AFT ages leads to both an overestimation of E2and an underestimation of T. This difference with Scenario 1 may bedue the MTL pattern (Fig. 3d), which is characterised by a normalcorrelation between MTL and elevation in all relief-growth scenarios.Moreover, the AHe AER shows a higher contrast between ridge andvalley-bottom samples than the AFT AER (Fig. 3b), since AHe has alower closure temperature and thus more potential to record changesin either exhumation rates or relief close to the surface. We thussuggest than in the case of relief change studied here, AHe dataprovide more information than MTL measurements.

The problem of course is that in a real-world study, one usuallydoes not know a-priori what scenario (varying denudation rates orrelief change) to expect, and this will probably be the objective of thestudy. Therefore, the best approach appears to be to collect as muchindependent thermochronometry data as possible and typically tocombine AFT and AHe ages with track-length measurements alongaltitudinal profiles. The fundamental reason for applying multiplethermochronometers is that AER sample the topography at a singlewavelength (20 km in our case), which is problematic for extractingindependent information on denudation and relief evolution (Braun,

Fig. 8. Scatterplots showing the quality of relief-change predictions (R) by the inversionapproach developed here as a function of both denudation and relief-growth rates. Eachdot results from an inverse model with a specified input denudation rate and reliefevolution scenario. (a) Absolute difference between predicted and “true” R (the latterindicated by Rinput). (b) 1σ uncertainty on predicted R (defined as the standarddeviation of the PDF of R from its mean value). 1:1 (solid line), 1:2 (dashed line) and 1:3(dotted line) ratio of denudation rate versus relief-growth rate are indicated (seeSupplementary data for numerical results).

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2002b). However, using different thermochronometers characterizedby different closure temperatures can be interpreted as samplingdifferent critical topographic wavelengths (Braun, 2002b) and thusmay allow extracting independent information on denudation rateand relief change in favourable circumstances.

4.3. Relief evolution

The aim of our study was to determine to what extent AER canprovide quantitative estimates on relief evolution. However, we haveshown that estimates for R are typically not as well constrained asestimates for denudation rates or timing. In Scenario 1, estimates of Rhave ∼50% uncertainty (Table 2) and thus lie between 0.5 (i.e., 50%relief increase) and 1.5 (i.e., 50% relief decrease), whether we use AFTages only or a combination of AFT ages with AHe ages and/or MTL(Fig. 6d). We suggest that, since samples record ∼5 km of rapiddenudation during the last 5 Myr in this scenario, they are insensitiveto relief changes. Scenario 2 provides quite different results since itmodels major relief growth (4 km valley carving) with relativelyminor background denudation. Quantitative estimates of R predictrelief increase (Rb0.5 except for inversion 2-C, Table 2), althoughpredictions are 30–50% lower than the input value (Rinput=0).

Adding AHe to AFT ages does not improve the estimate of R; includingMTL measurements leads to higher uncertainties on R (Table 2).

We thus suggest that quantitative estimates on R may depend onthe background denudation rate as well as the amplitude of reliefgrowth and the time of relief change. We have run ∼70 inversions(see Supplementary Table 1 for details) simulating relief-growth likein Scenario 2 and using AFT, AHe andMTL. For each inversion, we varythe input background denudation rate (E: 0.1 to 1 kmMyr−1), thetime of onset of relief change (T: 1 to 4.8 Ma) and the amplitude ofrelief growth (R: 0 to 0.9). We then compare R estimates (optimalvalues and uncertainties) with the ratio between denudation (E)and relief-growth rates (i.e., Δh/T with Δh=(1−R)×Δhpresent andΔhpresent=4 km). Results show that the quality of predictions for Rdepends on the ratio between the background denudation rate andthe relief-growth rate (Fig. 8). Our inversion approach leads torelatively accurate predictions on R (i.e., b50% error between R andRinput) for scenarios where the relief-growth rate is at least 2 timeshigher than the background denudation rate (Fig. 8a). However,obtaining satisfactory resolution on R (i.e., uncertainty b0.4) requiresa relief-growth rate that is at least 3 times higher than the backgrounddenudation rate (Fig. 8b). This sensitivity analysis suggests that reliefdevelopment must be significantly more rapid than the backgrounddenudation rate to be quantitatively extracted from thermochronol-ogy data, which strongly reduces the number of suitable orogenicsettings for this kind of investigations. In effect, we suggest that thisapproach may be most successful in settings with low backgrounddenudation rates and potentially major recent relief increase, such asthe Sierra Nevada (California) or the Pyrenees (Spain, France).Interestingly, in both settings contrasting conclusions have beendrawn concerning recent relief change from the interpretation ofthermochronological AER (e.g., House et al., 2001 versus Clark et al.2005 for the Sierra Nevada; Fitzgerald et al., 1999 versus Gibson et al.,2007 for the Pyrenees). In contrast, it does not appear very well suitedto constrain potential relief changes in high-exhumation-rate settingssuch as the Himalaya (Whipp et al., 2007) or the Southern Alps of NewZealand (Herman et al., 2007). New thermochronometers and modelsbased on the He distribution within single apatite grains (4He/3Hethermochronology; Shuster et al., 2005) or OSL-thermochronometry(Herman et al., in press) may provide more precise information onrelief changes, especially if they occurred during Quaternary times.

5. Conclusions

This study shows that thermochronological age-elevation profilesmay be used to quantitatively define denudation and relief histories incertain cases. We have used the thermal-kinematic 3D model Pecubeto predict AFT and AHe AER and MTL patterns from imposed inputscenarios with varying denudation or/and relief histories. PredictedAFT AERmostly define linear trends that are delicate to interpret, eventhough AHe ages or MTL can help to define denudation and reliefhistories. In most cases, direct 1D analysis of these profiles would leadto infer temporally averaged exhumation estimates and no indepen-dent constraints on relief history. Combining Pecube with an inversemethod (NA), we have run numerical experiments for syntheticdatasets to determine to what extent denudation rates, reliefdevelopment and the timing of change can be independentlyextracted from thermochronological data. Inversion results showthat AFT ages alone do not provide tight constrains on the denudationor relief histories. Depending on the input scenario (relief change orvarying denudation rates), adding other data (AHe ages and/or MTLmeasurements) to the AFT ages leads to generally accurate predic-tions of denudation rates and timing of change. In contrast, reliefchanges can only be precisely constrained in specific settings whererelief growth is at least 2–3 times higher than the backgrounddenudation rate. New thermochronometers (such as 4He/3He or OSL)

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may be the key for resolving this issue as they more precisely recorddenudation rates and/or geomorphic changes close to the surface.

Acknowledgments

This study is supported by INSU-CNRS through the European ScienceFoundation Eurocores Topo-Europe programme 07-TOPO-EUROPE-FP-023 “Coupled climatic/tectonic forcing of European topographyrevealed through thermochronometry (Thermo-Europe)” and theAgence Nationale de la Recherche project No. ANR-08-BLAN-0303-01“Erosion and Relief Development in the Western Alps”. It forms part ofPV's PhD project at Université Joseph Fourier, supported by the FrenchMinistry for Research and Higher Education. Computations wereperformed on Brutus, the high performance computing facilities atETH Zürich. The codes are available at http://svn-geo.ethz.ch afterregistering at this site. Thorough and constructive reviews by MarkBrandon and Richard Ketcham significantly improved the manuscript.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.epsl.2010.04.033.

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