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Rethinking low-temperature thermochronology data sampling strategies forquantification of denudation and relief histories: A case study in the Frenchwestern Alps
Pierre G. Valla , Peter A. van der Beek, Jean Braun
Institut des Sciences de la Terre, Universit Joseph Fourier, CNRS, BP 53, F-38041 Grenoble, France
a b s t r a c ta r t i c l e i n f o
Article history:Received 10 December 2010
Received in revised form 29 April 2011
Accepted 2 May 2011
Available online 31 May 2011
Editor: T.M. Harrison
Keywords:
low-temperature thermochronology
numerical modeling
inversion
exhumation
relief development
sampling and modeling strategies
We assess the importance of thermochronometric data sampling and modeling strategies for correctlyestimating mountain belt exhumation. Thermochronological age-elevation profiles have been widely used to
infer orogenic exhumation histories; however, recent studies have shown that this sampling strategy may not
be the most pertinent for quantifying both denudation and relief history. Here, we investigate the ability of
combining different thermochronology data sampling schemes with numerical modeling to better constrain
denudation rates and relief changes. We produce synthetic thermochronology datasets for real Alpine
topography under a specific exhumation and relief scenario using the thermal-kinematic model Pecube. We
then adopt an inverse approach based on the Neighborhood Algorithm to quantitatively assess the resolution of
different thermochronology datasets collected following elevation profiles, long transects and valley bottom
sampling. We also test the effect of the modeling approach on denudation and relief predictions, in particular
the influence of the topographic grid resolution and of potential constraints on the geothermal gradient. Our
results show that sampling along a single elevation profile does not allow to quantitatively constrain both
denudation and relief histories. Numerical outputs clearly evidence tradeoffs that limit the capacity of
simultaneously resolving denudation rates and relief change. Quantitative predictions are only slightly
different when combining elevation profiles along different valleys, but are highly improved when using long
transects or valley-bottom samples combined with an elevation profi
le. The resolution with which reliefevolution can be predicted may be increased by a factor of 2 by using spatially distributed datasets. Results of
thermal parameter inversions suggest that the geothermal gradient may be better estimated using elevation
profiles or long-transect sampling rather than using valley bottom samples. Simulations with different model
topography resolutionsshow that degrading the resolution for computational efficiency may result ina loss of
quantitative information on denudation rates and relief history. In summary, we highlight that both
thermochronological sampling strategies and the choice of thermal parameters or model topography resolution
have a significant influence on predicted denudation and relief histories. Ideally,the samplingstrategy should be
designed using preliminary modeling of expected denudation and relief histories, and a sensitivity study on
assumed thermalparameters and model resolution should be performed whenmodeling the data. Although our
modelingis basedon a particular case study of reliefevolution in theFrench westernAlps,we believe that these
inferences have general relevance for thermochronological studies within mountain belts.
2011 Elsevier B.V. All rights reserved.
1. Introduction
Low-temperature thermochronology has been widely used to
quantify the exhumation history of mountain belts (e.g., Braun, 2005;
Gallagher et al., 1998; Reiners and Brandon, 2006). Numerous studies in
the last 20 yr or so have used apatite fission-track (AFT) (e.g., Fitzgerald
et al., 1995; Gallagher et al., 1998; Hurford, 1991) and (U-Th)/He (AHe)
(Ehlers and Farley, 2003; Farley, 2002) datato infer exhumation histories
in specific areas. Moreover, low-temperature thermochronology may
provide useful constraints for assessing paleo-topography and relief
development (e.g., Clark et al., 2005; House et al., 1998; Reiners, 2007).
The effect of steady-state topography on thermochronology data is well
understood (e.g., Mancktelow and Grasemann, 1997; Stwe et al., 1994).
However, potential effects of time-varying topography, leading to
spatially and temporally varying exhumation rates as well as a transient
thermal structure, are more difficult to assess (Braun, 2002a; Valla et al.,
2010).
Thermochronometric sampling strategies have often focused on
obtaining age-elevation profiles, i.e., datasets of AFT and/or AHe ages
collected along a steep elevation transect (e.g., Fitzgerald et al., 1995;
Earth and Planetary Science Letters 307 (2011) 309322
Corresponding author.
E-mail address: [email protected] (P.G. Valla).
0012-821X/$ see front matter 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.epsl.2011.05.003
Contents lists available at ScienceDirect
Earth and Planetary Science Letters
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e p s l
http://dx.doi.org/10.1016/j.epsl.2011.05.003http://dx.doi.org/10.1016/j.epsl.2011.05.003http://dx.doi.org/10.1016/j.epsl.2011.05.003mailto:[email protected]://dx.doi.org/10.1016/j.epsl.2011.05.003http://www.sciencedirect.com/science/journal/0012821Xhttp://www.sciencedirect.com/science/journal/0012821Xhttp://dx.doi.org/10.1016/j.epsl.2011.05.003mailto:[email protected]://dx.doi.org/10.1016/j.epsl.2011.05.0038/6/2019 Rethinking Low-temperature Thermochronology Data Sampling Strategies for Quantification of Denudation and Reli
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Gallagher et al., 2005; Wagnerand Reimer, 1972). Thermochronologic
age-elevation relationships allow inferring apparent exhumation
rates over a specific time window and, in theory, do not require
knowledge of the geothermal gradient. One-dimensional consider-
ation of age-elevation relationships thus remains the most direct way
to interpret thermochronology data, either by simple linear regression
of the data (e.g., Fitzgerald et al., 1995; Wagner and Reimer, 1972) or
by using 1D thermal evolution models (e.g., Gallagher et al., 2005).
However, topographic effects on thermochronology data are noteasily extracted from age-elevation relationships (Braun, 2002a) and
recent modeling studies suggest that constraining mountain belt
paleo-relief from elevation profiles is difficult, even when combining
multiple thermochronometers (Valla et al., 2010; van der Beek et al.,
2010; Vernon et al., 2009).
Recent developments in numerical modeling offer tools for inter-
preting low-temperature thermochronology data, either considering
each sample individually (e.g., Ketcham, 2005) or exploiting the spatial
correlation between samples (Braun et al., 2006; Gallagher et al., 2005;
Stephenson et al., 2006a), to derive thermal histories and eventually
constrain bothdenudation and relief histories. Braun (2002b) proposed
an alternative approach to assess paleo-relief by sampling along a
horizontal transect and using the spatial coherence between the
topographic and thermochronological signals to determine both
regional denudation rate and relief change (Beucher et al., submitted
for publication; Herman et al., 2007, 2010a). However, the main
limitation of this spectral method is that it assumes a spatially and
temporally constant denudation history, and only provides an upper
bound on the timing of denudation or the age of relief change.
Numerical modeling offers the unique possibility to exploit dense
spatial datasets either with or without any specific spatial relationship
(Braun and van der Beek, 2004; Gallagher et al., 2005; Stephenson et al.,
2006a). In this study, we aim to assess under what sampling and
modeling conditionsquantitative constraintson the exhumationhistory
can be extracted from low-temperature thermochronology data. We
define different sampling strategies for a specific topography and test
whether each sampling schemepermits deriving both denudation rates
and relief history. Moreover, we assess to what extent the modeling
approach, especially the spatial resolution of the model grid or anyimposed constraint on the geothermal gradient, can bias predicted
denudation and relief histories. Our approach, although being specific
with respect to the study area and the relief scenario we simulated, has
been designed to more generally assess the potential influence of
specific sampling schemes for constraining orogen exhumation. Our
study thus follows an experimental design approach that has been
widely usedin geophysics(e.g.,Curtis, 2004a,b; Wood andCurtis, 2004),
although for computational simplicity we do not perform a complete
inverse analysis but rather a priori define specific 3D sampling schemes
that are traditionally used in thermochronological studies.
Several studies (Glotzbach et al., 2011; Valla et al., 2010; van der
Beek et al., 2010) have recently attempted to extract independent
information on denudation and relief development in the western
European Alps from thermochronology data. This was achieved bycombining the 3D thermal-kinematic model Pecube (Braun, 2003;
Braun et al., submitted for publication), which predicts thermal
histories and thermochronologicalages from an input denudation and
relief history, with an inversion scheme based on the Neighborhood
Algorithm (Sambridge, 1999a,b) to search the parameter space,
extract best-fitting scenarios from the data and derive quantitative
estimates for the parameters (rates, timing and relief ratio) describing
denudation and relief histories.
Weadopt a similarnumericalapproachas Valla et al.(2010); however,
their study specifically focused on the problem of interpreting age-
elevation profiles whereas here we consider various sampling strategies.
Moreover, instead of building a synthetic topography for inverse
modeling (Valla et al., 2010), we use real topography for predicting
synthetic thermochronology data (Fig. 1a). We focus on the Ecrins
Pelvoux massif (van der Beek et al., 2010), one of the External Crystalline
Massifs (ECMs) within the western European Alps (e.g., Dumont et al.,
2008). The Neogene exhumation history of the ECMs has been widely
studied in the last few years, leading to contrasting views regarding
timing and rates of exhumation, and the impact of Pliocene-Quaternary
climate change on late-stage exhumation (Glotzbach et al., 2008, 2010;
Reinecker et al., 2008; van der Beek et al., 2010; Vernon et al., 2009). Our
synthetic approach is motivated by recent predictions forthe exhumation
history of the Ecrins
Pelvoux massif from thermochronology datacollected along either an elevation profile (van der Beek et al., 2010) or
a long transect (Beucher et al., submitted for publication); both studies
providedpreciseconstraints ondenudationhistoriesbut could notresolve
relief evolution quantitatively. More generally, our study has been
motivated by several attempts to quantify both timing and magnitude
of topographicevolution in mountain belts using thermochronology (e.g.,
Braun, 2005; Ehlers and Farley, 2003; Reiners, 2007; Valla et al., 2010).
In the following, we first outline our modeling approach, present
the synthetic data used for the simulations and discuss the inverse
method. We then show inversion results and present to what extent
predictions of denudation rates, relief change and near-surface
thermal structure can be quantitatively determined by various
thermochronological sampling strategies. The potential effects of
varying the model gridresolution are reported in the Supplementary
data. We finally discuss the implications of our findings for
thermochronological sampling strategies to constrain regional
denudation rates and/or topographic evolution of mountain belts.
2. Modeling approach
2.1. Thermal-kinematic modeling and synthetic thermochronology data
The EcrinsPelvoux massif topography is characterized by a
dominant NorthSouth topographic wavelength of ~10 km (Figs. 1a
and 2a) organized around four major valleys (for details and locations
see van der Beek et al., 2010). The relief amplitude is ~3 km, with
summits around 4000 m and valley bottoms at ~1000 m (Fig. 1a). The
EcrinsPelvoux massif has been chosen for our study as it forms a
homogeneous area for the purpose of understanding Neogeneexhumation and relief development, i.e., no major active fault crosses
the massif and consequently its exhumation history can be assumed
to be spatiallyuniform (Delunel et al., 2010; van der Beek et al., 2010).
Based on this topography and an assumed exhumation history, we
predicted synthetic thermochronology data using the Pecube model
(Braun, 2003). Pecube is a 3D thermal-kinematic model that calculates
the thermal structure through time within an exhuming crustal block
and provides thermal histories for rock particles that end up at the
surface of the model, taking into account the effects of evolving
topography (see Valla et al., 2010; Braun et al., submitted for
publication for a detailed discussion). The Pecube finite-element
code permits defining exhumation/relief scenarios that are described
as discrete phases with an imposed regional denudation rate and
paleo-relief. Relief evolution is quantified using the relief ratio R: forR =0, the paleo-relief is a plateau at maximum elevation; for R= 1,
the paleo-relief is equal to present-day conditions; and for R=2, the
paleo-relief is twice as large as the present-day relief. Note that relief
increase operates through preferential valley incision in our model
(Fig. 2a); i.e., ridges remain at a constant elevation but valley bottoms
are lowered (see Valla et al., 2010; Braun et al., submitted for
publication for a complete description), as would be expected for
relief increase by glacial valley carving for instance. Isostatic rebound
in response to this erosional unloading is included. Thus, local
exhumation rates are the sum of: (1) the imposed regional
(background) exhumation; (2) local relief change through valley
incision; (3) spatially varying rock uplift induced by isostatic rebound.
Thermal, mechanical and kinematic parameters used in the forward
model are given in Table 1. The basal crustal temperature (Temp) and
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the crustal heat production (H_prod; here normalized to heat capacity
and density) are fixed to produce a surface geothermal gradient of
~25 C km1.
Predicted thermal histories are then used to calculate different
thermochronometric ages. In this study, we focus on late-stage
exhumation histories and thus only compute low-temperature apatite
fission-track (AFT) and (U-Th)/He (AHe) ages, as well as mean
confined track lengths (MTL; see Valla et al., 2010). Thermal histories
are translated into AHe ages using a simple model for He production
diffusionejection (Farley, 2000); AFT ages are calculated using a
model forAFT annealing(Greenet al., 1989; Stephenson et al., 2006b).
We acknowledge that using these simple algorithms does not
completely reflect the complex annealing and diffusion processes
included in low-temperature thermochronology (Flowers et al., 2009;
Gautheron et al., 2009; Ketcham, 2005). However, as discussed in
Valla et al. (2010), we use the same forward models for AFT and AHe
600' 612' 624'
4448'
4500'
Longitude (E)
La
titude(N)
AHe age (Ma)3 5 7 9 11 13 15
c
600' 612' 624'
4448'
4500'
Longitude (E)
La
titude(N)
AFT age (Ma)3 5 7 9 11 13 15
b
4506
600
612
624
4500
4448500
900
1300
1700
2100
2500
2900
3300
3700
4100
Elevation(m)
Latitude(N)
Longitu
de(
E)
a
1_Profile
3_Profiles
Valleys
Transect
Profile + Valleys
VeneonVeneon
Bonn
e
Bonne
MontontPelvouxelvoux
Barre desarre desEcrinscrinsLa Meijea Meije
Severais
se
Sever
aisse
Rom
anche
Romanc
he
Fig. 1. a) 3D map view of the modeling area (90-m resolution SRTM DEM of the Ecrins Pelvoux massif, Western Alps, France) used to produce AFT and AHe ages maps (b, c). Names
of principal valleys and peaks are indicated for orientation. bc) Apatite fission-track (b) and (U-Th)/He (c) age map produced using Pecube (Braun, 2003; Table 1) and simulating a
scenario of 50% relief increase (Rinput=0.5) during the last 3 Ma (Tinput). The constant background denudation rate is 0.3 km Myr1 (E1input= E2input) and resulting maximum rate of
relief increase over the last 3 Ma is ~0.5 km Myr1 (see text for discussion). Sample locations for the 5 different sampling schemes are shown: (b) Sampling schemes 1_Profile
(yellow circles, based on van der Beek et al., 2010), 3_Profiles (black circles) and Valleys (white circles). (c) Sampling schemes Transect(red circles, partly based on Beucher et al.,
submitted for publication) and Profile+Valleys (yellow circles, partly based on van der Beek et al., 2010).
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age predictions when generating the synthetic data and resolving the
inverse problem; the results shown here are thus largely independent
of the specific age-prediction models.
To produce the synthetic datasets that we will use in the inversion,
wedefine a 2-phase exhumation scenario with steady-statedenudation
(0.3 km Myr1) over 15 Myr and a gradual 50% relief increase (R=0.5,
Fig. 2a) since 3 Ma (T=3 Ma). This scenario simulates moderate relief
increase (~1.5 km over the last 3 Myr, i.e., a maximum relief growth
rate of ~0.5 km Myr1) associated with a relatively low and constant
background denudation rate (0.3 km Myr1). Although this choicefor a
specific denudation and relief scenario renders our analysis somewhat
contextual, we focus on the type of scenario that has been inferred for
the western Alpine massifs (e.g., Glotzbach et al., 2011; van der Beek et
al., 2010; Vernon et al., 2009). Thermochronology data are predicted
following this input scenario on a very high-resolution grid (90-m,
Fig. 1b,c) to ensure that the numerical model properly resolves finite-
amplitude topography effects on the crustal thermal structure. AFT and
AHe age maps(Fig. 1b,c) show a significant age contrast between valley
bottoms (AFT ~7 Ma and AHe ~34 Ma) and ridges (AFT ~15 Ma and
AHe ~1213 Ma). Note that our synthetic AFT and AHe data are similarto measured thermochronological ages in the area (Beucher et al.,
submittedfor publication; van der Beek et al., 2010), suggestingthat the
imposed tectono-geomorphic scenario we define is reasonable.
We sampled both the AFT (and associated MTL) and AHe maps
following five different strategies: (1) a single elevation profile
(Fig. 1b, 1_Profile), partly based on sample locations reported by van
der Beek et al. (2010); (2) three elevation profiles sampling the three
main valleys of the massif (Figs. 1b and 3b, 3_Profiles); (3) valley
bottom samples along the major valleys (Figs. 1b and 3a, Valleys); (4)
an elevation profile combined with valley bottom samples (Figs. 1c
and 3c, Profile+Valleys); and finally (5) a long transect (partly based
on sample locations in Beucher et al., submitted for publication)
across the massif (Figs. 1c and 2, Transect). Each sampling scheme
includes exactly 20 samples for which AFT, AHe and MTL data are
1
2
3
4
Elevation(km)
Distance (km)
Age (Ma) MTL (m)
North South
a
Elevation(km)
Elevation(km)
0 5 10 15 20 25 30 35
4 6 80.5
1.5
2.5
3.5
10 11 12 13 140.5
1.5
2.5
3.5
cb
AFT
AHe
-1
-1
10 12 14
Fig. 2. a) Topographic cross-profile along the sampling scheme Transect(see Fig. 1b for location within the study area). Solid black and dashed red lines represent present-day and
paleo-topography (R =0.5, see text for discussion), respectively. Black circles show sample locations for the sampling scheme Transect, and arrows represent relief evolution along
the topographic profile (background denudation rate is 0.3 km Myr1). (b, c) Synthetic thermochronological ages versuselevation for Transectsamples, including AFT and AHe ages
(b) and mean fission-track length data (MTL, c). Apparent exhumation rates (AER) deduced from linear regression of elevation vs age (b) are 0.34 and 0.33 km Myr 1 for AHe and
AFT data, respectively (r=0.980.99). Note that the MTL pattern shows a normal correlation between MTL and elevation (c).
Table 1
Thermal, kinematic and elastic parameters used in Pecube. Crustal thickness, basal
temperature (Temp) and heat production parameter (H_prod) are set to obtain a
geothermal gradient of ~25 C km1 and a crustal heat production of ~7.5 CMyr1.
Poissonratio,Young's modulusand equivalentelasticthickness areusedfor calculatingthe
isostatic reboundin responseto relief change. Equivalent elasticthickness is set to a value
that simulates moderate isostatic rebound (Valla et al., 2010; van der Beek et al., 2010).
Parameter
(unit)
Inversions value
Crustal thickness (km) 40
Crustal density (kg m3) 2700
Sublithospheric mantle density (kg m3) 3200
Young's modulus (Pa) 1.1011
Poisson ratio 0.25
Equivalent elastic thickness (km) 25
Thermal diffusivity (km2 Myr1) 25
Basal crustal temperature (C) 600
Crustal heat production parameter 0.8
Sea-level temperature (C) 15
Atmospheric lapse rate (C km1) 6
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available. We thus compare the predictions of different sampling
strategies while maintaining identical dataset size. Figs. 2 and 3 show
synthetic AFT and AHe data plotted against elevation for the different
sampling schemes. All AFT and AHe age-elevation relationships suggest
nearly constant apparent exhumation rates (AER) of 0.330.39 and
0.330.47 km Myr1 (see Figs. 2 and 3, correlation coefficients
r0.950.99), respectively. The AER slightly overestimate the input
background denudation rate due to additional local exhumation
following relief growth (most strongly recorded by valley-bottomsamples, Fig. 3a).
2.2. Inverse thermal-kinematic modeling
For each sampling scheme, we run inversions to recover the four
parameters used to describe the imposed tectono-geomorphic
scenario, defining a range in which the inversion procedure will
search for each: (1) denudation rate of the first phase (E1: 0
2 km Myr1); (2) denudation rate of the second phase (E2: 0
2 km Myr1); (3) transition time (T: 0.514.5 Ma); and (4) relief
factor (R: 02). Each inverse simulation is run over 15 Myr. We run
inversions constrained by the full dataset (AFT+AHe+MTL) as well
as subsets of the input thermochronology data (AFT, AHe, AFT+AHe,
AFT+MTL, and AFT+AHe+MTL data) to test what combination of
thermochronometers is most useful to quantify relief and denudation
histories in different sampling schemes.
In a second set of inversions, we test the ability of the different
sampling schemes to constrain not only relief and denudation
histories, but also to retrieve the geothermal gradient (Temp and
H_prod parameters, Table 3). To achieve this, we run five additional
inverse simulations with the full dataset and 6 parameters to invert
(E1, E2, T, and R as described previously; Temp: 500800 C, and
H_prod: 03). We acknowledge that other thermo-physical parame-
ters such as diffusivity or thermal conductivity, as well as the surface
temperature history, are largely unknown and may also vary; butgiven the relatively homogenous lithology in our study area, we
suspect they will do so within much narrower limits than the thermal
parameters we test for and will therefore exert only a second-order
influence on model predictions. Finally, we test the influence of the
grid resolution on relief and denudation predictions (see Supplemen-
tary data for results and discussion).
To perform the inversions, we use the Neighborhood Algorithm
(NA) of Sambridge (1999a,b). NA is a two-stage numerical procedure
for non-linear inverse problems that allows efficient exploration of a
multidimensional parameter space to find best-fitting models (NA
sampling stage) as well as deriving estimates of posterior probability
distribution for inverted parameters from the ensemble of models run
during the sampling stage (NA appraisal stage).
The sampling stage iteratively explores the multi-dimensional
parameter space and progressively concentrates on regions where
parameter values minimize the misfit to the data. NA makes use of
1
2
3
4
1
2
3
4
4 6 8 14
4 6 8 10 12 14 4 6 8 14
1
2
3
4
Elevation(km)
Elevation(km)
Age (Ma) Age (Ma)
Elevatio
n(km)
Age (Ma)
a b
c Profile + Valleys
3_ProfilesValleys
AHe
AFTNorthCenterSouth
ProfileValleys
-1-1
-1
-1
-1
-1
10 12
10 12
Fig. 3. Synthetic thermochronology data (AFT and AHe ages) versus elevation for different sampling schemes; see Fig. 1 for sample locations. a) Synthetic data for the sampling
scheme Valleys. AER deduced from linear regression of elevation vs age are 0.47 and 0.39 km Myr 1 for AHe (white circles) and AFT (black circles) data, respectively (r=0.98). b)
Synthetic data for the sampling scheme 3_Profiles (individual profiles are represented by diamonds, squares and triangles, see Fig. 1b for locations). AER are 0.33 and 0.35 km Myr1
for AHe (white) and AFT (black) data, respectively (r=0.950.99). c) Synthetic data for the sampling scheme Profile+Valleys (profile and valleys samples are represented by
diamonds and circles, respectively). AER are 0.37 and 0.36 km Myr1
for AHe (white) and AFT (black) data, respectively ( r=0.98
0.99).
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simple geometrical concepts by dividing the entire parameter space
into a set ofn-dimensional Voronoi cells, which represent the nearest
neighborhood around each parameter combination. At each step, NA
samples a forward model centered on each Voronoi cell andcalculates
a misfit surface. Subsequent iterations make use of this misfit surface
to sample in regions of the parameter space where the misfit function
is minimized. The performance of the sampling stage is controlled by
tuning parameters determining the number of iterations, the number
of forward models generated at each iteration, and the number ofVoronoi cells to re-sample after each iteration. Varying these
parameters will make the NA method either more explorative of the
multi-dimensional parameter space, or more exploitative and faster in
converging toward potential misfit minima (Sambridge, 1999a).
Based on previous studies (Glotzbach et al., 2011; Valla et al., 2010),
we choose to run ~5000 models distributed in 35 iterations for each
inversion, with 50% of models re-sampled at each iteration.
In our approach, we use a weighted least-squares misfit function
for the NA appraisal stage:
=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
N
i =1
M
j =1
i
j; modi
j;dat
2
ij2
vuuut 1
where Nisthe numberof datasets(13;AFT,AHe,AFT+ AHe,AFT+ MTL,
or AFT+AHe+MTL data), Mis the number of samples in each dataset
(20 samples in total for each sampling scheme), j,modi and j,dat
i are
predicted and observedvalues for AFT/AHeagesor MTLrespectively, and
ji is the uncertainty on the data. Here, we use constant synthetic
uncertainties (ji) of 0.5 Ma and 0.5 m for AFT/AHe ages and MTL,
respectively, to put an equal weight on all samples for misfit calculations
(Valla et al., 2010). The above approach allows us to quantitatively infer
the potential resolution ofperfect thermochronology data in resolving
tectono-geomorphicscenarios(and potentially thermalparameters). We
note, however, that adding random noise in the data could potentially
lead to different model sensitivity in constraining numerical parameters.
In the NA appraisal stage (Sambridge, 1999b), the model ensemblegenerated during the sampling stage is iteratively resampled to
provide optimal estimates in the form of marginal probability-density
functions (1D or 2D PDFs) of parameter values. For all parameters, we
assumea uniform prior distribution withineach parameterrange. The
posterior 1D (or 2D) PDFs are given based on the log-likelihood
function (ln(L)), following the equation:
ln L =0:5 2 =0:5 N
i =1
M
j =1
ij; modi
j;dat
2
ij2
0@
1A 2
3. Inversion results
We first present inversion results for each sampling scheme to
assess the effect of using various combinations of low-temperature
thermochronometers. We then compare the different sampling
schemes to discuss how best to predict denudation and relief
histories. Finally, we show modeling results for inverting the
geothermal gradient. We present inversion results as 1D marginal
posterior PDF's as well as 2D joint PDF's of parameter values and
discuss the reasons for the observed differences in the modeling
outputs (resolution and errors on parameter estimates, and potential
tradeoffs between model parameters) in the discussion section.
Optimal parameter values (mode of the 1D posterior PDF) and 1
uncertainties for the different sampling schemes and data combina-
tions are given in Table 2.
3.1. Sampling strategy
The first sampling scheme (1_Profile, Table 2) considers 20 samples
combined as a single elevation profile. Inversion results yield accurate
predictions for E1 (E1=0.230.28 km Myr1; E1input=0.3 km Myr
1) for
all input thermochronological datasets with uncertainties ~2040%
except when using AHe data alone. Estimates on E2 are less precise
(uncertainties ~3080%) andslightlyoverestimated whenusingAFTdata
alone or AFT+MTL data (E2 ~0.4 km Myr
1
; E2input=0.3 km Myr
1
).AHe data are required (alone or combined with AFT ages) to provide
accurate predictions for E2 (respectively E2=0.330.11 and 0.28
0.18 km Myr1). Adding MTL data, in contrast, does not improve
inversion predictions (Table 2). Timing predictions are underestimated
when using AFT alone, AFT+MTL or AFT+AHe+MTL data (T=1
1.4 Ma) and clearly overestimated for AHealone (T=12.2 Ma). Only the
AFT+ AHe inversion (T=2.8 Ma) gives predicted value of Tclose to the
input value of 3 Ma. However, in all cases uncertainties on Tpredictions
are very high (N100% for all inversions). Relief predictions are in
agreement with theinput value (Rinput=0.5)when using AHedata alone
or combined with AFT ages (R=0.530.58 and 0.63 0.55 respective-
ly). In contrast, using AFT ages alone or combined with MTL
measurements respectively under- (R=0.930.59) and over-predicts
(R=0.18 +0.93/0.18) relief increase. Combining both AFT and AHe
ages with MTL does not lead to an accurate estimate on R (R=1.08
0.55; i.e., between ~50% increasing and 50% decreasing topographic
relief).Moreover, as for Tpredictions, uncertainties on R are higher than
100% for all inversions. In summary, inverse simulations using
thermochronology data along a singleelevation profile provide accurate
estimates on denudation histories (E1 and E2) with moderate to high
uncertainties dependingon thedataset (~30 to 80%). In contrast, Tand R
estimates are not in agreement with input values and systematically
present uncertainties N100%. Best estimates are generally obtained
when using AHe ages alone or combined with AFT data; adding MTL
measurements does not improve denudation and relief predictions.
Sampling along three different elevation profiles (3_Profiles,
Table 2) provides similar predictions for E1, although the mean
uncertainty is slightly reduced (~2040%, except for AFT+AHe data
for which the uncertainty rises to 100%). E2 estimates are accurate forall inversions, but uncertainties remain quite high (~3080%). Similar
to the previous inversions, T is systematically under-predicted
(T=1.02.1 Ma) except for the AHe inversion (T=6.9 3.8 Ma);
uncertainties remain N100% for all inversions. Relief predictions are
much better than in the simulation using a single age-elevation
profile. All inversions predict relief increase (R =0.180.58), but the
inverse simulation combining all thermochronometric systems clearly
overestimates the R input value (R=0.18+ 0.30/0.18). Uncertainties
on R predictions are also reduced (~6590%, Table 2). As for a single
elevation profile, best estimates on E2 and R (recent exhumation
history) are provided using AHe data; however, for both Tand E1, most
accurate predictions are obtained when using the full dataset (AFT+
AHe+MTL data).
Sampling along valley bottoms (Valleys, Table 2) provides accuratepredictions for E1 and E2. Uncertainties on E1 are low (~320%) only
when using at least two thermochronometers, but are much higher
when using AFT or AHe data alone (~100200%). In contrast,
uncertainties on E2 are fairly constant, whatever the dataset used
(~4060%). Similarly, constraints on Tare very poor using AFT or AHe
data alone (T=11.9 +2.6/5.1 and 6.63.0 Ma, respectively) and
are improved when combining thermochronometers (best estimate
T=3.5 2.3 Ma for AFT+ AHe+MTL data). Uncertainties on T are
high and variable depending on the dataset used: N100% for all
inversions except for thefull dataset. Finally, relief increase is strongly
overestimated when using AFTor AHe data alone (R =0.030.08) and
predictions are not precise (uncertainties ~100%); however, combin-
ing the thermochronometers provides more accurate values of R and
reduced uncertainties (R=0.33
0.48 and uncertainties ~ 50%). In
314 P.G. Valla et al. / Earth and Planetary Science Letters 307 (2011) 309322
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summary, this specific sampling scheme provides reasonably accurate
and precise estimates of denudation history (E1 and E2) and accurate
but not very well resolved predictions for T and R; however this
requires combining at least two different thermochronometers (best
estimates for the full AFT+ AHe+ MTL dataset).
Combining valley bottom samples with an elevation profile(Profile+Valleys, Table 2) provides accurate and precise predictions
for both E1 and E2 (E1=0.230.28 0.010.12 km Myr1 and
E2=0.230.33 0.110.14 km Myr1). T estimates are similar for
allinversions andmostly under-predict thetiming of change(T=0.7
3.8 Ma); uncertainties on Tare around or above 100%. Similarly to the
Valleys sampling scheme, relief change R is strongly overestimated
when using AFT or AHe data alone (R =0.03 +0.55/0.03).
Combining thermochronometers still over-predicts relief change
(R =0.280.38) but uncertainties on R are reduced to ~50%.
Combining valley bottom samples with an elevation profile thus
provides reasonably precise and accurate estimates on E1, E2 and Tfor
nearly all thermochronometer combinations, even though best
estimates on inverted parameters, especially on relief change, are
obtained when using two or more thermochronometers.
The last inversion simulates sampling along a transect across the
massif (Transect, Table 2). This sampling scheme leads to accurate
estimates ofE1 withuncertainties around1040%(except forAFT+ AHe
data, for which the uncertainty rises to ~100%); however, E2 is slightly
over-predicted for AHe, AFT+ AHe and AFT+ MTL datasets (E2
~0.4 km Myr1
, associated uncertainties ~60%). The timing parameteris also not well resolved (T=0.76.3 Ma) with prediction uncertainties
of 100200%. Finally, relief estimates are quite accurate (R=0.180.58)
but associated uncertainties are between 75 and 130%. In summary,
sampling a long transect provides accurate predictions on denudation
rates (E1 and E2), however, timing is not well constrained and relief
estimates are accurate but associatedwith highuncertainties.Moreover,
our inversions results show thatin this sampling configuration, using at
least two thermochronometers strongly improves predictions on
denudation rates and relief evolution.
To better explore the potential differences between the sampling
strategies, Fig. 4 presents 1D marginal PDFs of inverted parameters
(E1, E2, Tand R) for each sampling strategy (using the full dataset). All
sampling schemes provide accurate predictions for E1 (Fig. 4a), even
though more precise estimates (uncertaintiesb
10%) are obtained
Table 2
Output estimates after the NA appraisal stage for inputparameters E1, E2, Tand R (see text fordescriptionand discussion). Thesecond line reports parameterranges andinputvalues
to calculate the synthetic thermochronology data (Fig. 1b,c). Optimal parameter values and associated uncertainties (mode 1of the 1D posterior PDF) are given in the following
lines for various sampling schemes and different input thermochronology datasets. Best parameter estimates (and associated input thermochronology dataset) are also reported as
percentage errors and uncertainties compared to input values (%Input).
Parameter (unit) E 1 (km Myr1) E2 (km Myr
1) T (Ma) R
Range/input 02/0.3 02/0.3 0.514.5/3 02/0.5
1_Profile
AFT 0.28 0.13 0.38 0.26 1.4 + 5.3/0.9 0.93 0.59
AHe 0.28 0.23 0.33 0.11 12.2 + 2.3/5.2 0.53 0.58AFT+AHe 0.28 0.06 0.28 0.18 2.8 + 3.9/2.3 0.63 0.55
AFT+MTL 0.23 0.1 0.43 0.24 1.0 + 8/0.5 0.18 +0.93/0.18
AFT+AHe+MTL 0.28 0.06 0.28 0.27 1.0 + 6.4/0.6 1.08 0.55
Best estimate (%Input) 7% 20% +10% 37% 7% 130% +6% 116%
Data AFT + AHe AHe AFT + AHe AHe
3_Profiles
AFT 0.28 0.12 0.28 0.21 2.1 + 5.5/1.6 0.38 0.41
AHe 0.23 0.1 0.28 0.13 6.9 3.8 0.53 0.32
AFT+AHe 0.28 0.3 0.28 0.14 1.0 + 3.9/0.5 0.58 0.29
AFT+MTL 0.28 0.08 0.33 0.19 1.4 + 5.9/0.9 0.58 0.40
AFT+AHe+MTL 0.28 0.05 0.28 0.15 1.7 + 3.8/1.2 0.18 +0.30/0.18
Best estimate (%Input) 7% 17% 7% 43% 43%127% +6% 64%
Data AFT + AHe + MTL AHe AFT + AHe + MTL AHe
Valleys
AFT 0.33 0.29 0.28 0.11 11.9 + 2.6/
5.1 0.08 + 0.37/
0.08AHe 0.33+0.56/0.33 0.23 0.12 6.6 3.0 0.03 + 0.50/0.03
AFT+AHe 0.28 0.05 0.23 0.11 0.7 + 4.8/0.2 0.48 0.25
AFT+MTL 0.28 0.06 0.33 0.10 2.1 + 5.3/1.6 0.33 0.21
AFT+AHe+MTL 0.28 0.01 0.28 0.11 3.5 2.3 0.38 0.22
Best Estimate (%Input) 7%3% 7%37% 17%77% 4%50%
Data AFT + AHe + MTL AFT + AHe+ MTL AFT + AHe + MTL AFT + AHe
Profile+Valleys
AFT 0.23 0.12 0.28 0.13 3.8 + 5.0/3.3 0.03 +0.55/0.03
AHe 0.23 0.11 0.28 0.14 2.8 + 3.7/2.3 0.03 +0.56/0.03
AFT+AHe 0.28 0.04 0.28 0.14 0.7 + 3.6/0.2 0.38 0.24
AFT+MTL 0.28 0.08 0.33 0.12 1.0 + 6.5/0.5 0.28 0.23
AFT+AHe+MTL 0.28 0.01 0.23 0.11 2.1 + 2.6/1.6 0.28 0.21
Best Estimate (%Input) 7%3% 7% 47% 30%87% 24% 48%
Data AFT + AHe + MTL AFT + AHe AFT + AHe + MTL AFT + AHe
Transect
AFT 0.23 0.12 0.28 0.17 2.1 + 7.0/1.6 0.18 + 0.44/0.18AHe 0.28 0.15 0.43 0.18 1.7 + 6.4/1.2 0.48 + 0.65/0.48
AFT+AHe 0.28 0.32 0.43 0.22 1.0 + 3.4/0.5 0.58 0.39
AFT+MTL 0.28 0.07 0.43 0.21 6.3 3.8 0.38 0.62
AFT+AHe+MTL 0.28 0.03 0.28 0.2 0.7 + 4.71/0.2 0.48 0.37
Best estimate (%Input) 7% 10% 7% 57% 67%113% 4%74%
Data AFT + AHe + MTL AFT AFT + AHe AFT + AHe + MTL
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when sampling a transect, valley bottoms or an elevation profile
combined with valley bottoms; uncertainties from sampling along
elevationprofiles aretwiceas high. TheaccuracyofE2 estimates is also
similar for all sampling schemes, with only the resolution differing
(Fig. 4b); lowest uncertainties on E2 predictions are obtained when
sampling along valley bottoms (~40% for Valleys and Profile+Valleys),
the 3_Profiles sampling scheme yields an uncertainty of ~50% while
the Transect and 1_Profile sampling strategies yield uncertainties of
~70
90%. Fig. 4c confi
rms that timing of change (T) is the leastresolved parameter, with uncertainties of 8090% for the Valleys and
Profile+Valleys sampling schemes, and uncertainties N100% for the
other schemes. Finally, the predicted R (Fig. 4d) is reasonably accurate
(slightly overestimated) for the 3_Profiles, Valleys and Profile+Valleys
sampling schemes and is associated with uncertainties of ~5060%.
Transect sampling also leads to an accurate estimate for R but the
associated uncertainty is high (~ 75%) whereas 1_Profile does not
provide any constraint on R (i.e., it predicts either 50% increasing or
decreasing topographic relief).
3.2. Modeling approach
In all previous inverse simulations, our modeling approach assumed
the geothermal gradient was known. We now present results inverting
for both denudation/relief histories (E1, E2, T and R) and thermal
parameters (Temp and H_prod). We run inversions for allfive sampling
schemes and use the full (AFT+AHe+MTL) dataset to compare
output predictions with previous outcomes (Table 3). Basal temper-
ature (Temp) is slightly under-predicted for most sampling schemes
(Temp ~520565 C; Tempinput=600 C), except for 3_Profiles sampling
that leads to an overestimate (Temp=714 80 C). Associated un-
certainties are low and similar for all inversions (~1116%, Table 3).
Constraints on the crustal heat production parameter (H_prod,
Table 3) are less tight. The Profile+Valleys, 1_Profile and 3_Profiles
sampling strategies lead to accurate predictions (H_prod =0.56
0.94; H_prodinput=0.8) but estimated uncertainties are relativelyhigh (~80110%). Valleys sampling clearly does not resolve the heat-
production parameter (H_prod =0.04 +0.65/0.04) whereas best
predictions for H_prod are provided by the Transect inversion
(H_prod =1.16 0.44).
Predictions for E1 are slightly underestimated for the Transect,
3_Profiles and 1_Profile sampling schemes (E1=0.23 0.1 km Myr1)
and all results provide higher uncertainties on E1 (~30%) compared to
inversions that imposed the geothermal gradient (~320%). Tightest
constraints are placed on E1 using the Valleys and Profile+Valleys
sampling schemes. Results for E2 are variable with best predictions for
the3_Profiles inversion (E2=0.280.14 km Myr1). E2 is slightly over-
predictedin the Transectsamplingscheme(E2=0.430.21 km Myr1)
and under-predicted for Valleys and Profile+Valleys sampling schemes
(E2=0.230.19 km Myr1). However, uncertainties on E2 are very
similar to those obtained when imposing thegeothermal gradient (~50
100%). Timing predictions are all underestimated (T ~0.71.7 Ma) and
associated uncertainties are similar to those found during imposed
0 0.4 0.8 1.2 1.6 2
0.05
0.15
0.25
0.35
0
0.04
0.08
0.12
0.16
Probabi
lity
Probability
Probability
Probabi
lity
a
c
b
d
E1(km Myr-1) E
2(km Myr-1)
T(Ma) R
1_Profile3_ProfilesValleysProfile + valleysTransect
Sampling scheme:
2 6 10 14
0.01
0.03
0.05
0.07
0.09
0 2
0 20
0.02
0.04
0.06
0.08
0.1
0.4 0.8 1.2 1.6
0.4 0.8 1.2 1.6
Fig. 4. 1D posterior parameter PDF's obtained after the NA appraisal stage for the five different sampling schemes (AFT+AHe+MTL dataset): (a) denudation rate of the first
exhumation phase (E1); (b) denudation rate of the second exhumation 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 an inversion experiment with a sampling scheme (see figure for color legend and Fig. 1 for sampling locations). Black stars
represent the input parameter values.
316 P.G. Valla et al. / Earth and Planetary Science Letters 307 (2011) 309322
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geothermal gradient inversions (~70230%). Moreover, best predictions
for Tare provided by the Valleys inversion (T=1.7 + 2.0/1.2 Ma), in
agreement with previous results. Finally, setting no constraint on the
geothermal gradient does not have any effect on relief evolution
predictions as R estimates are very close to values obtained with an
imposed geothermal gradient, and optimal predictions are obtained for
3_Profiles, Valleys and Profile+Valleys sampling schemes.
To summarize, numerical inversions without imposing constraints
on the geothermal gradient provide less accurate and precise
estimates for exhumation rates (E1 and E2), but output results for
timingand reliefchange (Tand R) areverycloseto thosefoundwith an
imposed geothermal gradient. Best estimates for thermal parameters
(H_prod and Temp) are given by Transectsampling, while inversions
including at least one elevation profile (Profile+Valleys, 1_Profile and3_Profiles) provide accurate estimates but associated with relatively
high uncertainties, and valley bottom sampling (Valleys) does not
resolve thermal parameters.
4. Discussion
4.1. Limits to the numerical modeling approach
Before interpreting our results, we should highlight potential
shortcomings associated with our numerical modeling approach.
First of all, the exhumation scenario we choose is fairly simple,
with only two exhumation phases, a constant and spatially uniform
regional denudation rate and a single episode of relief growth. This
choice could be considered specific and restrictive, although weconsciously chose uniform background exhumation within the study
area to focus on relief evolution. A similar inverse approach could,
however, be developed to specifically explore spatially varying
exhumation (e.g., Stephenson et al., 2006a). We have tested one
specific geothermal gradient that we also assume to be spatially
uniform, as we apply our inversions to a region of spatially relatively
homogeneous lithology. However, we note that localfluid circulations
along faults can induce large spatial variations in thermal gradient
(Whipp and Ehlers, 2007).
Despite several recent attempts (e.g., Ehlers et al., 2006; Glotzbach
et al., 2008, 2010; van der Beek et al., 2010; Vernon et al., 2009 ),
quantifying both timing and amplitude of topographic relief changes
in mountain belts using low-temperature thermochronology has
proven elusive, motivating the focus of our study on this specific
exhumation scenario. Although our study design might be somewhat
restrictive, we have consciously chosen quite a general synthetic
study regarding the exhumation history and the near-surface thermal
structure. The reliefevolution scenario should also be quite general, as
moderate relief increase since the mid-Pliocene is expected or has
been encountered in many mountain belts (e.g., Molnar and England,
1990; Zhang et al., 2001). We thus are confident that our inferences
regarding thermochronology sampling strategies for quantifying both
exhumation and relief histories, despite the above limitations, might
be of more general interest for studying at least the late-Neogene
exhumation history of mountain belts.
Second, the way we define relief changes in Pecube does not
capture the complexities of landscape evolution processes. Some
limitations include the constant planform drainage pattern assumedin our numerical approach, and a geometric approach to relief
development (Fig. 2a). An alternative approach would be to combine
Pecube with a landscape evolution model to fully simulate landscape
changes with surface process modeling (Herman et al., 2010b).
However, our synthetic approach allows us to address at first order
the effect of transient topography on thermochronology data and
their intrinsic resolution in quantitatively retrieving potential relief
changes.
We finally note that the different sampling scenarios explored
here, although they represent the most commonly used and simplest
ways to sample a given topography, might not be the most adequate
to independently quantify exhumation and relief histories, i.e., we do
not perform a complete experimental design study.
4.2. Parameter resolution and tradeoffs
Our numerical results show that some parameters(mainly Tand R,
Fig. 4) have been poorly constrained by the NA inversion. The lack of
convergence toward the input parameters may result from the poor
resolution of thermochronology data to constrain timing or relief
parameters (Valla et al., 2010), from some random elements, or from
potential tradeoffs between model parameters. To illustrate this
alternative explanation, Fig. 5 shows 2D scatter plots and PDFs for a
single elevation profile (1_Profile). There is no apparent tradeoff
between the initial and final denudation rates (E1 and E2, Fig. 5a), but
significant tradeoffs appear between the recent denudation rate ( E2)
and both the relief change (R, Fig. 5b) and the timing of change (T,
Table 3
Output estimates after the NA appraisal stage for input parameters E1, E2, T, R, Temp and H_prod. The second line reports parameter ranges and input values for the synthetic
thermochronology data (Fig. 1b,c). Optimal parameter values and associated uncertainties (mode 1) are given in the following lines for various sampling schemes (AFT+AHe+
MTL dataset). Inversion results are also reported as percentage errors and uncertainties compared to input values (%Input).
Pa ra meter (u ni t) E 1 (k m Myr1) E2 (km Myr 1) T (Ma) R Temp (C) H_prod
Range/input 02/0.3 02/0.3 0.514.5/3 02/0.5 500800/600 03/0.8
1_Profile
Mode1 0.23 0.1 0.33 0.29 0.7 + 3.1/0.2 1.08 0.52 549 + 98/49 0.64 + 0.82/0.64
%Input (23% 33%) (+10% 97%) (77% 103%) (+ 116% 104%) (9% 16%) (20%103%)
3_Profiles
Mode 1 0.23 0.1 0.28 0.14 0.7 + 6.9/0.3 0.48 0.27 714 80 0.94 0.66
%Input (23% 33%) (7% 47%) (77% 230%) (4% 54%) (+19% 13%) (+18% 83%)
Valleys
Mode 1 0.33 0.1 0.23 0.19 1.7 + 2.0/1.2 0.33 0.21 519 + 74/19 0.04 + 0.65/0.04
%Input (+10% 33%) (23% 63%) (43% 67%) (34% 42%) (14% 12%) (95%81%)
Profile+Valleys
Mode 1 0.28 0.1 0.23 0.19 0.7 + 5.1/0.3 0.48 0.26 534 + 95/34 0.56 + 0.85/0.56
%Input (7% 33%) (23% 63%) (77% 170%) (4% 52%) (11% 16%) (30%106%)
Transect
Mode 1 0.23 0.09 0.43 0.21 1.4 + 4.4/0.9 0.48 0.4 564 66 1.16 0.44
%Input (23% 30%) (+43% 70%) (53% 147%) (4% 80%) (6% 11%) (+ 45% 55%)
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Fig. 5c). These tradeoffs may explain the relatively poor resolution
observed in the 1D marginal PDFs for these parameters ( Fig. 4c,d).
The potential parameters tradeoffs for the other sampling schemes
are shown in Figs. 6 and 7. Fig. 6 reveals that the tradeoff between E2and R is reduced for the 3_Profiles, Valleys, and Profile+Valleys
schemes (Fig. 6ac), but confirms their tendency to over-estimate the
relief increase. In contrast, there is an important tradeoff in the
Transect sampling scheme (Fig. 6d), which explains the high uncer-
tainties associated with R. Fig. 7 confi
rms that the timing parameterT is poorly constrained by our inverse approach; 2D scatter plots
reveal a significant tradeoff between E2 and T, even though the
covariance appears to be slightly reduced for sampling schemes
Valleys and Profile+Valleys (Fig. 7b,c). Interestingly, both Figs. 6 and 7
suggest that parameter tradeoffs, similar to 1D marginal PDFs, are
highly dependent on the sampling scheme and thus might also be
considered when designing sampling strategies.
4.3. Sampling strategy
Inversionresultsfor a singleelevation profile (1_Profile) show that,
although such a sampling approach allows accurately estimating
denudation rates, it does not constrain relief evolution satisfactorily
(uncertainties N100%, Table 2). This finding is similar to the
conclusion of Valla et al. (2010), who also performed a sensitivity
analysis and showed that relief evolution can only be precisely
predicted from thermochronological age-elevation profi
les if the rateof relief change is at least two to three times higher than the regional
background denudation rate. In our case, the regional denudation rate
is 0.3 km Myr1 and the relief scenario induces a maximum relief-
growth rate of ~0.5 km Myr1 for the last 3 Myr (i.e., ~60% higher
than the background denudation rate); under this configuration, our
inverse results are in good agreement with Valla et al. (2010). We
suggest that the inability of a single age-elevation profile to constrain
0 20 40 60 80 100
0
0.5
1
1.5
2
0 0.5 1 1.5 2
0
0.5
1
1.5
2
0 0.5 1 1.5 2
R
0 0.4 0.8 1.2 1.6 2
10
6
2
14
a b
c
E1(km Myr-1) E
2(km Myr-1)
E2(km Myr-1)
E2
(kmM
yr-
1)
T(Ma)
Fig. 5. Scatter plots (each dot represents a forward model color coded by its misfit value ) and 2D posterior marginal probability density functions with 1confidence contours
(black lines)for the sampling scheme 1_Profile: a)denudation rates of thefirst(E1) vsthesecond(E2) exhumation phases;b) denudation rate of thesecond exhumation phase (E2) vs
relief factor (R); and c) denudation rate of the second exhumation phase ( E2) vs transition time between the two exhumation phases ( T). Black stars represent the input parameter
values.
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E2) and relief change (R); however, the uncertainties on these
parameters are higher than those obtained with Valleys and Profile+
Valleys sampling strategies, due to higher parameters tradeoffs (Figs. 6
and 7). These outcomes are consistent with recent analysis of AFT data
from a transect across the Pelvoux massif (Beucher et al., submitted for
publication): these data constrain exhumation rates well but provide
little insight into relief change. These differences appear to result from
the spatial configuration of the Transectsampling scheme. Although it
covers a larger area than the 1_Profile and 3_Profiles sampling schemes,
Transectdata only sampleone spatial direction(Fig. 2a) andmay notfully
record the 2D topographic perturbations of isotherms compared to data
that are spatially distributed along different valleys (Valleys or Profile+
Valleys sampling scheme). Combining a long transect with at least one
elevationprofile would have allowed to best capture both short and long
topographic wavelength effects on the thermal structure (Braun, 2002a,
b), simultaneously providing accurate and precise information on both
denudation and relief predictions.
4.4. Modeling approach
Inversionsfor which the thermal parameters are supposed unknown
lead to comparable accuracy but somewhat higher uncertainties for
2
6
10
14
0 0.4 0.8 1.2 1.6 20 0.4 0.8 1.2 1.6 2
00.4 0.8 1.2 1.6 2
00.4 0.8 1.2 1.6 2
2
6
10
14
2
6
10
14
2
6
10
14
a b
c d
3_Profiles Valleys
Profile+Valleys Transect
E2(km Myr-1)E
2(km Myr-1)
E2(km Myr-1) E
2(km Myr-1)
T(Ma)
T
(Ma)
T
(Ma)
T(Ma)
Fig. 7. Scatter plots (see Fig. 5 for legend) and 2D posterior marginal probability density functions with 1 confidence contours (black lines) of denudation rate of the secondexhumation phase (E2) vs transition time between the two exhumation phases (T) for different sampling schemes: a) 3_Profiles; b) Valleys; c) Profile+Valleys; and d) Transect. Black
stars represent the input parameter values.
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denudation and relief predictions than when imposing a geothermal
gradient (Table 3). These differences in predicteduncertainties highlight
the classic problem of determining an appropriate dimensionality for a
numericalproblem,i.e.,the number of inverted parameters compared to
the intrinsic data resolution (see Gallagher et al., 2005, for a discussion
on this subject). Moreover, our results show that sampling along valley
bottoms does not allow resolving accurately the thermal parameters.
Sampling along elevation profiles helps constraining the geothermal
gradient, as suggested by Gallagher et al. (2005), pointing out that eventhough numerical modeling allows us to infer exhumation histories
from complex spatial sample distributions, age-elevation relationships
remain useful for interpreting thermochronology data. The ability of
age-elevation profiles to constrain both the exhumation history and
thermal parameters comes from the altitudinal distribution of samples;
the slope of the age-elevation relationship providing an estimate of the
apparent exhumation rate (e.g., Hurford, 1991), while the age-window
of the dataset may help to constrain the near-surface geothermal
gradient (Gallagher et al., 2005). Surprisingly, the Transect sampling
provided the best estimates on thermal parameters, which can be
explained by the simultaneous spatial and altitudinal data coverage
allowed by this sampling scheme(Fig.2a). It thusmightbe usednot only
for spectral analysis (Braun, 2002b) but also, like elevation profiles, to
constrain the geothermal gradient.
Finally, we have tested the effect of the grid resolution on the
predicted estimates of the various parameters. As reported in the
supplementary material section, we show that a minimum grid
spacing of 1 km (or ~1/10th of the valley spacing) is necessary to
properly resolve exhumation rates and relief change in the typical
mountainous terrain we have used here.
5. Conclusions
Our study was initially designed to determine to which extent
different sampling and modeling strategies could allow constraining
relief evolution from low-temperature thermochronology datasets.
Our results confirm the findings ofValla et al. (2010), who suggested
that age-elevation relationships do not resolve relief changes if these
do not induce an additional spatially variable exhumation rate that isat least two times higher than the regional background denudation
rate. Although our approach is slightly different from Valla et al.
(2010), we have shown here that age-elevation relationships do not
allow constraining relief changes. We have demonstrated that
spatially distributed sampling along a transect, several elevation
profiles or along valley bottoms, preferably combined with an
elevation profile, leads to better predictions of relief evolution. The
main drawback of only sampling along valley bottoms is the loss of
information on the near-surface thermal structure. However, using
higher-temperature thermochronometers may provide supplemen-
tary constraints on thermal parameters. Glotzbach et al. (2011)
recently used a dense thermochronology dataset combining both low-
and high-temperature thermochronometers along elevation profiles
and a tunnel transect to constrain both the near-surface thermalstructure and the exhumation history of the Mont-Blanc massif
(western European Alps). Their modeling results resolved a recent
relief increase whereas other recent studies using only data along a
single age-elevation profile were not able to resolve the recent relief
evolution (van der Beek et al., 2010; Vernon et al., 2009).
Based on synthetic data producedunder an imposed exhumation and
relief scenario, we have tested different sampling strategies to explore
how to best constrain denudation rates, timing and relief change. Our
results show that dense spatial sampling combining valley bottom
samples and elevation profiles, as well as long transects may help to
provide quantitative constraints on complex exhumation histories.
Moreover, we have highlighted that the modeling approach (choice of
thermalparametersor model resolution)potentiallyinfluencespredicted
denudation and relief histories. Although age-elevationprofiles may bea
useful and direct way to interpret thermochronology data, sampling
strategies should be rethought in the light of recent numerical modeling
developments. For a region where relief changeis expected to have been
minimal but the emphasis is on deciphering changes in denudation rate
through time, an appropriate sampling scheme would be using one or
several elevation profiles. However, when one has reason to believe that
exhumation rates may have been relatively constant (e.g., in stable
post-orogenic mountain belts) but the aim is to constrain the timing and
amount of relief change, our study would suggest a better choice is tosample valley bottoms, possibly combined with an elevation profile.
Therefore, ideallythesamplingstrategy should be designeda priori,using
preliminary modeling of expected denudation and relief histories and
preferably using multiple working hypotheses; and a sensitivitystudy on
assumed thermal parameters and model resolution should be performed
when modeling the data.
Acknowledgments
This study is supported by the INSU-CNRS through the European
Science Foundation Topo-Europe program 07-TOPO-EUROPE-FP-023Coupled climatic/tectonic forcing of European topography revealed
through thermochronometry (Thermo-Europe) and the Agence
Nationale de la Recherche project No. ANR-08-BLAN-0303-01 Erosionand Relief Development in the Western Alps. It forms part of PV's PhD
project at Universit Joseph Fourier, supported by the French Ministry
for Research and Higher Education. We thank Malcolm Sambridge and
two anonymous reviewers for insightful comments that helped to
improve the clarity of this manuscript. Computations were performed
on Brutus, the high performance computing facilitiesat ETH Zurich. The
codes are available at http://svn-geo.ethz.ch after registering at this
site.
Appendix A. Supplementary data
Supplementary data to this article can be found online at
doi:10.1016/j.epsl.2011.05.003.
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