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# Short communication: *age2exhume* – a MATLAB/Python script to calculate steady-state vertical exhumation rates from thermochronometric ages and application to the Himalaya

*age2exhume*– a MATLAB/Python script to calculate steady-state vertical exhumation rates from thermochronometric ages and application to the Himalaya Short communication:

*age2exhume*– a MATLAB/Python script to calculate steady-state vertical... Peter van der Beek and Taylor F. Schildgen

### Peter van der Beek

### Taylor F. Schildgen

Interpreting cooling ages from multiple thermochronometric systems and/or
from steep elevation transects with the help of a thermal model can provide
unique insights into the spatial and temporal patterns of rock exhumation.
Although several well-established thermal models allow for a detailed
exploration of how cooling or exhumation rates evolved in a limited area or
along a transect, integrating large, regional datasets in such models
remains challenging. Here, we present *age2exhume*, a thermal model in the
form of a MATLAB or Python script, which can be used to rapidly obtain a
synoptic overview of exhumation rates from large, regional
thermochronometric datasets. The model incorporates surface temperature
based on a defined lapse rate and a local relief correction that is
dependent on the thermochronometric system of interest. Other inputs include
sample cooling age, uncertainty, and an initial (unperturbed) geothermal
gradient. The model is simplified in that it assumes steady, vertical
rock uplift and unchanging topography when calculating exhumation rates. For
this reason, it does not replace more powerful and versatile
thermal–kinematic models, but it has the advantage of simple implementation
and rapidly calculated results. We also provide plots of predicted
exhumation rates as a function of thermochronometric age and the local
relief correction, which can be used to simply look up a first-order
estimate of exhumation rate. In our example dataset, we show exhumation
rates calculated from 1785 cooling ages from the Himalaya associated with
five different thermochronometric systems. Despite the synoptic nature of
the results, they reflect known segmentation patterns and changing
exhumation rates in areas that have undergone structural reorganization.
Moreover, the rapid calculations enable an exploration of the sensitivity of
the results to various input parameters and an illustration of the
importance of explicit modeling of thermal fields when calculating
exhumation rates from thermochronometric data.

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*age2exhume*– a MATLAB/Python script to calculate steady-state vertical exhumation rates from thermochronometric ages and application to the Himalaya, Geochronology, 5, 35–49, https://doi.org/10.5194/gchron-5-35-2023, 2023.

The steady accumulation of thermochronometric data from around the world provides an opportunity to constrain spatial patterns of long-term (million-year timescale) exhumation with high granularity over vast swaths of the Earth's surface. This information can, in turn, provide clues to the driving mechanisms of orogen development and landscape evolution. Several well-established thermal models can be used to extract detailed cooling histories or exhumation rates from input cooling ages spread over a limited area or along an elevation transect. However, integrating information from large datasets, comprising cooling ages from multiple thermochronometers spread over a wide region, remains challenging due to the lack of easy-to-use tools that will handle such vast, multi-system datasets.

The most advanced modeling tools in common use by the thermochronology
community include Pecube (Braun et al., 2012), HeFTy (Ketcham, 2005), QTQt
(Gallagher, 2012), and GLIDE (Fox et al., 2014). Pecube is unique in its
ability to handle forward and inverse thermal–kinematic modeling of
spatially distributed data, including the options for time-varying
topography as well as spatially and temporally variable rock uplift driven
by defined fault geometries and kinematics. This complexity, however,
entails substantial setup requirements and relatively high computational
demands, which tend to limit the spatial extent of modeled datasets to ∼10^{2}–10^{3} km^{2}. HeFTy and QTQt, in contrast, model thermal
histories only, for individual samples or samples that are assumed to fall
into a pseudo-vertical alignment. GLIDE (Fox et al., 2014) was developed
with the aim of extracting exhumation histories from regional datasets.
While powerful, the temporally and spatially continuous coverage of
calculated exhumation rates that the model produces requires interpolations
that can be challenging to interpret without careful consideration of the
spatial and temporal distribution of the input data (Fox et al., 2014;
Schildgen et al., 2018).

Here we present a simple thermal model, *age2exhume*, which is optimized to
provide a synoptic overview of exhumation rates from large regional
datasets. This model, inspired by the original *age2edot* code (Brandon et al.,
1998), takes the form of a MATLAB or Python script that solves for
steady-state exhumation rates from input thermochronometric ages, assuming
vertical exhumation pathways and unchanging topography. A key difference
between *age2edot* and *age2exhume* is that the former (despite its name) solves
for ages given input exhumation rates, whereas our new model solves for
exhumation rates given input ages. This difference makes *age2exhume* more
suitable for calculating exhumation rates from regional datasets, since
individual sample characteristics (e.g., an elevation-dependent surface
temperature and local relief correction), included together with age in an
input file, can be used to calculate an exhumation rate for each sample. A
preliminary version of this code was used to visualize regional
thermochronometric datasets in Schildgen et al. (2018); here, we provide
more detailed background to the model and incorporate the individual sample
characteristics mentioned above into the revised model.

The regional (constant) inputs to the model include crustal thermal properties that can be approximated or derived from the literature (an initial, unperturbed geothermal gradient, thermal model thickness, and thermal diffusivity) and kinetic parameters for the relevant thermochronometric systems, for which default values are provided. Sample-specific inputs include a local relief factor that can be extracted using standard GIS functions from a digital elevation model, elevation, thermochronometric system, age, and age uncertainty. From our example dataset of 1785 cooling ages derived from five different thermochronometric systems in the Himalaya, steady-state, vertical exhumation rates with their uncertainties can be calculated within seconds on a standard laptop computer. Despite the synoptic nature of the results, we show how they reflect several fundamental features of the mountain belt, including strong regional differences that reflect known segmentation patterns and changing exhumation rates in areas that have undergone recent structural reorganization.

## 2.1 Existing thermal models: their applications and limitations

Brandon et al. (1998) presented a simple, first-order approach to predict
thermochronologic ages from input exhumation rates, in the form of a code
called *age2edot*. *Age2edot* calculates a steady-state conductive–advective
geotherm and uses the approach of Dodson (1973) to predict the
cooling-rate-dependent closure temperature of a given thermochronometric
system. It then combines the predicted closure temperature and the
steady-state geotherm to find the closure depth and subsequently calculates
a thermochronometric age by dividing the closure depth by the input
exhumation rate. Kinetic parameters required for the Dodson (1973)
calculation of closure temperature (see Sect. 2.2 below) are derived from
diffusion experiments for noble-gas-based systems (i.e., (U–Th)$/$He and
${}^{\mathrm{40}}\mathrm{Ar}{/}^{\mathrm{39}}\mathrm{Ar}$) and from fitting an Arrhenius relation to experimental
annealing data for fission-track systems (see Reiners and Brandon, 2006, for
more detail). Simplifying assumptions in the *age2edot* approach include (1)
thermal steady state, (2) vertical exhumation paths, (3) unchanging
topography, and (4) constant exhumation rates over the modeled time span.
The most recent version of the *age2edot* code was released more than 15 years
ago (Ehlers et al., 2005) and, because it was distributed as a Microsoft
Windows executable, it is now obsolete.

Willett and Brandon (2013) published a modification to the *age2edot*
approach, in which the steady-state geotherm solution was replaced by an
(inherently transient) half-space solution, a correction for the sample
elevation with respect to the regionally averaged elevation was introduced,
and a best-fit exhumation rate is predicted from an input age and a modern
(i.e., final) geothermal gradient. The code was provided as a MATLAB script.
Although it is computationally efficient, two aspects of this model limit
its use for modeling large regional datasets in our view; one is of a
practical nature, whereas the other is more fundamental. The practical
limitation lies in the need to provide a value (or bounding values) for the
modern geotherm for each prediction. Although this requirement makes
conceptual sense, since only the modern geotherm can potentially be
measured, it is of limited practical use because geothermal gradients are
generally not known at more than very coarse spatial resolution,
particularly in mountain belts. Moreover, the requirement is impractical
when dealing with large datasets of widely varying ages, as geothermal
gradients vary strongly in regions of variable exhumation rates. If the
estimated bounding geotherms are poorly estimated (e.g., too low or high for
a given thermochronometric age), no exhumation rate is returned. The more
fundamental issue lies in the choice of a thermal half-space model, which
leads to a strong sensitivity of the geotherm to exhumation rate and the
persistence of transient thermal conditions even after several tens of
millions of years of steady exhumation (Willett and Brandon, 2013). One type
of data that allows assessing if, and how rapidly, thermal steady state
might be achieved in mountain belts is detrital thermochronology from
sedimentary sequences in foreland basins. Several such datasets show
constant lag times (i.e., thermochronometric age minus depositional age),
interpreted as recording establishment of thermal steady state in the source
area after only a few million years, including in the western European Alps
(Bernet et al., 2001, 2009), the central and eastern Himalaya (Bernet et
al., 2006; Chirouze et al., 2013), the eastern Himalayan syntaxis (Bracciali
et al., 2016; Lang et al., 2016; Govin et al., 2020), Taiwan (Kirstein et
al., 2010), and the Southern Alps of New Zealand (Lang et al., 2020). As
argued by Bracciali et al. (2016), modeling these constant lag times using
a thermal half-space model would require *decreasing* exhumation rates through time,
with a rate of decrease that exactly offsets the transient upward advection
of the geotherm, in all the above cases. More probably, these data indicate
that the thermal half-space model is not ideal for representing orogenic
geotherms.

A completely different approach is taken by the thermal-history modeling codes HeFTy (Ketcham, 2005) and QTQt (Gallagher, 2012). These codes aim at predicting a thermal history from thermochronometric ages and additional measurements (in particular fission-track length distributions, but also kinetic indicators) for single samples, although the most recent versions of these codes allow modeling suites of vertically offset samples. The output of these models, when run in inverse mode, is an optimal time–temperature history and its uncertainty. These thermal history results require assumptions about the past geothermal gradient to be translated to a burial/exhumation history. Gallagher and Brown (1999) and Kohn et al. (2002) spatially interpolated thermal histories derived from large numbers of individual samples, using a precursor of the QTQt code, and combined them with heat-flow maps to derive regional- to continental-scale images of denudation over geological time. This labor-intensive approach requires multiple thermochronometric systems and/or track-length data for each included sample in order to resolve meaningful thermal histories.

Pecube (Braun et al., 2012) is a three-dimensional thermal–kinematic code that predicts thermochronometric ages for various user-defined tectonic and geomorphic scenarios, taking into account the spatial and temporal perturbation of the geotherm by rock advection and transient topography. Pecube allows modeling both vertical and non-vertical exhumation paths, the latter controlled by a simple fault-kinematic model, and can be coupled to the neighborhood algorithm (Sambridge, 1999a, b) to run in inverse mode. The code has been used in a wide variety of tectonic and geomorphic settings (see Braun et al., 2012, for an overview), including at the scale of a small orogen (Curry et al., 2021). However, the fairly high computational demands of the code, particularly when run in inverse mode, make it best suited for models of more limited spatial extent (i.e., not exceeding several tens of km in length and width), where simple fault kinematics and/or spatially uniform rock uplift can reasonably represent the tectonic deformation patterns.

GLIDE (Fox et al., 2014) comprises a linear inverse method to infer spatial and temporal variations in exhumation rate from spatially distributed thermochronometer datasets. GLIDE uses a numerical thermal model with a flux boundary condition at the base. The inversion assumes vertical exhumation and a smooth spatial variation in exhumation rates that can be described by a spatial correlation function. In this way, it uses exhumation constraints from one sample to help constrain exhumation in nearby regions, producing exhumation histories that are continuous in space and time. However, it has been argued that the code translates abrupt spatial variations in thermochronological ages, such as across faults, into temporal increases in exhumation rates (Schildgen et al., 2018), unless the faults (or other features) are explicitly included in the correlation structure (Fox et al., 2014; Ballato et al., 2015). Willett et al. (2021) argued that such issues occur mainly in areas of insufficient data coverage without, however, quantifying this term; Schildgen et al. (2018) argued that most sampled regions on Earth with sharp spatial variations in exhumation have insufficient data coverage for unbiased prediction of exhumation-rate histories using GLIDE, if those variations are not taken into account.

From
the above abbreviated review, we conclude that a simple, first-order method
to assess large regional datasets in a consistent manner is currently
lacking from the thermochronology toolbox. We aim to provide such a simple
method with the *age2exhume* code.

## 2.2 *age2exhume* method

Figure 1 shows a sketch outline and flowchart for the *age2exhume* model. Input
parameters for the model include the sea-level temperature *T*_{0},
atmospheric lapse rate *H*, the initial, unperturbed geothermal gradient
*G*_{init}, thermal diffusivity *κ*, and model thickness *L*. The latter
can represent the crustal thickness or, more appropriately, the maximum depth
from which rocks have been exhumed, such as the depth to a regional
detachment horizon. Input data for each sample include a thermochronometric
age and its uncertainty at locations *x* and *y*, sample elevation *h*, and local relief
correction Δ*h*. Kinetic parameters for the main low- to intermediate-temperature thermochronometric systems (apatite and zircon (U–Th)/He and
fission track, mica ${}^{\mathrm{40}}\mathrm{Ar}{/}^{\mathrm{39}}\mathrm{Ar}$) are included as default values but
can be modified if desired.

When calculating exhumation rates from thermochronometric ages, a local
relief correction (Δ*h*) is needed to account for the difference in
elevation of a sample (*h*) relative to an average-elevation (*h*_{avg}) surface
that mimics the shape of the closure isotherm (Stüwe et al., 1994;
Braun, 2002). We follow the procedure of Willett and Brandon (2013) in
estimating the shape of that surface by averaging surface topography over a
circle with a radius of *π*×*z*_{c}, where *z*_{c} is an estimated
closure depth for the relevant thermochronometric system. The local relief correction
Δ*h* is then calculated for each sample as

A brief guide for how to calculate this correction using a digital elevation model in ESRI
ArcMap or in QGIS is provided in Appendix A. To predict a steady-state exhumation rate from a thermochronometric age,
surface temperature, and the local relief correction, the model starts with
an initial guess of the closure depth (*z*_{c}) and exhumation rate
($\dot{e}$) from an initial, unperturbed linear geothermal gradient
(*G*_{init}), a nominal closure temperature (*T*_{c}), and a surface
temperature (*T*_{s}):

*T*_{s} is calculated from an input sea-level temperature
(*T*_{0}), the surface-temperature lapse rate (*H*), and the sample elevation at
the position of *h*_{avg}: ${T}_{s\left(h\right)}={T}_{\mathrm{0}}-H{h}_{\text{avg}}$. We use
*h*_{avg}, rather than the actual sample elevation for this
surface-temperature correction to simulate how surface temperature affects
the thermal field at depth. For higher-temperature thermochronometers with
deeper closure depths, *h*_{avg} becomes more smoothed, and the associated
impact of surface temperature on *z*_{c} is reduced. Note that the initial
unperturbed geothermal gradient (*G*_{init}) is only used to calculate an
appropriate basal temperature and to provide an initial estimate of the
exhumation rate using Eqs. (2) and (3).

The model then iteratively adapts *T*_{c}, *z*_{c}, and $\dot{e}$ until
convergence to a steady-state solution. Importantly, Δ*h* is not
recalculated after the initial estimate. Given the generally low sensitivity
of Δ*h* to moderate variations in *z*_{c}, we believe this
simplification is worthwhile, considering the consequent reduced
computational demands. At each iterative step, first the advective
perturbation of the geotherm due to exhumation is calculated following
Mancktelow and Grasemann (1997):

where *T*_{(z)} is the temperature at depth *z*, *T*_{L} is the temperature at the
base (*z*=*L*) of the model (${T}_{L}={T}_{\text{avg}}+{G}_{\text{init}}L$, where
*T*_{avg} is the temperature at the average elevation of the whole dataset),
and *κ* is the thermal diffusivity. Equation (4) can be solved for the
closure depth *z*_{c}:

Next, the closure temperature is re-estimated as a function of the cooling rate at the closure depth. First, the depth derivative of Eq. (4) is used to estimate the geothermal gradient:

Equation (6) is evaluated at the closure depth *z*_{c}. Because $\dot{e}=\mathrm{d}z/\mathrm{d}t$, the cooling rate ($\dot{T}$) is

The model then uses the Dodson (1973) equation to relate closure temperature to cooling rate:

where *E*_{a} (activation energy), *D*_{0} (diffusivity at infinite
temperature), and *a* (diffusion domain size) are experimentally determined
kinetic parameters for each thermochronological system, *A* is a geometry
factor, and *τ* (characteristic time) is

Once a new estimate for *T*_{c} is obtained, *z*_{c} is updated using Eq. (5)
and a new estimate for the exhumation rate is obtained with Eq. (3). The
model steps through Eqs. (3)–(9) iteratively (Fig. 1b) until the
change in exhumation rate between successive steps ($\mathrm{\Delta}\dot{e}$)
is smaller than a threshold value; here we use
$|\mathrm{\Delta}\dot{e}/\dot{e}|<{\mathrm{10}}^{-\mathrm{3}}$. To ensure smooth
convergence, the exhumation rate used in each successive step is the average
between the previous and the newly calculated rate.

## 3.1 General model predictions

Figures 2 and 3 show contours of predicted exhumation rates for different
combinations of age and Δ*h*; Fig. 2 shows results for moderate
exhumation rates (<2 km Myr^{−1}) and thermochronometric ages up
to 30 Ma, whereas Fig. 3 zooms in on the youngest ages (<5 Ma) and
shows results for exhumation rates up to 5 km Myr^{−1}. Input parameters
for these models are as in Table 1, except that a constant surface
temperature (*T*_{s}) of 10 ^{∘}C was used, because absolute sample
elevation is not included in these generic models. Kinetic parameters for
the apatite (U–Th)$/$He (AHe) system are derived from Farley (2000), for the
zircon (U–Th)$/$He (ZHe) system from Reiners et al. (2004), and for the
apatite (AFT) and zircon (ZFT) fission-track systems from Reiners and
Brandon (2006). These results can be thought of conceptually as showing age–elevation profiles for different constant exhumation rates, with
elevation measured relative to an average regional elevation as defined in
Sect. 2.2. They can also be used as a plotted lookup table for rapidly
inferring exhumation rate from a given age, Δ*h* combination.

## 3.2 Results from a Himalayan example dataset

Our example dataset from the Himalaya comprises 1785 thermochronologic ages
compiled from papers published through July 2022; data sources are provided
in the Supplement. We have excluded some reported ages from
the Siwaliks (sub-Himalayan fold-thrust belt), as that sedimentary unit
commonly yields unreset ages. We have also excluded the western and eastern
syntaxis regions, where extremely rapid exhumation is driven by processes
that are different from those in the main part of the Himalaya (Zeitler et
al., 2014; Butler, 2019). Finally, we exclude any pre-Himalayan ages
(>60 Ma), as these are not directly linked to exhumation during
Himalayan mountain building. Our dataset comprises 345 white mica
${}^{\mathrm{40}}\mathrm{Ar}{/}^{\mathrm{39}}\mathrm{Ar}$ (MAr) ages, 236 ZFT ages, 783 AFT ages, 281 ZHe ages,
and 140 AHe ages. All ages and sample details are included in a single Excel
file, with columns that include a sample ID number, latitude, longitude,
elevation, Δ*h* value, age, 1*σ* age uncertainty, and a numeric
code for the thermochronologic system (Schildgen and van der Beek, 2022b).
Table 1 shows the parameters we assume for the surface temperature
(*T*_{0}, *H*) and the thermal model (*L*, *G*_{init}, *κ*). Kinetic
parameters used for the AHe, AFT, ZHe, and ZFT systems are the same as for
the general model predictions presented in Sect. 2.1 above; we used the
parameters from Hames and Bowring (1994) for the MAr system.

A map of the calculated exhumation rates for the Himalaya (Fig. 4) shows exhumation plotted such that rates derived from lower-temperature systems plot on top of those from higher-temperature systems. When the symbol for a lower-temperature system is darker than the symbol of a higher-temperature system plotted below it, this implies that exhumation rates have slowed through time. Conversely, a lighter color for the lower-temperature system plotted over a higher-temperature system implies exhumation rates have increased through time. The map reveals patterns in space and time that reflect well-known structural patterns of the range. In general, a band of rapid exhumation rates occurs at the topographic front of the high Himalaya, with slower rates recorded to the north and south. Within this band, the highest rates are generally recorded by the lower-temperature AHe and AFT thermochronometers, suggesting increasing exhumation rates with time. Note that such variable exhumation rates recorded by different co-located thermochronometers formally violate the assumption of constant exhumation rates through time implicit in the model. The rates inferred from the higher-temperature thermochronometers should therefore be considered rough estimates only; they will generally be overestimated in the case of increasing rates through time, and the corresponding rate change will therefore be underestimated. The focused rapid rates at the foot of the high Himalaya together with an increase in exhumation rates for lower-temperature systems are consistent with exhumation being driven by thrusting over a large-scale ramp in the Main Himalayan Thrust (MHT), the interface between the underthrusting Indian continent and the overlying Himalayan units, often associated with duplex development (e.g., Robert et al., 2009; Herman et al., 2010; Coutand et al., 2014; Dal Zilio et al., 2021; van der Beek et al., 2023).

The highest exhumation rates (>2 km Myr^{−1}) outside of the
Himalayan syntaxes occur in central Nepal (∼84^{∘} E), Sikkim
(∼88^{∘} E), the Kumaun Himalaya (∼80^{∘} E), and the Sutlej Valley
(∼78^{∘} E). High rates (between 1 and 2 km Myr^{−1}) are recorded along the high
Himalayan front throughout northwest India (∼76–80^{∘} E) and more
sporadically in eastern Nepal (∼87^{∘} E) and western Bhutan
(∼89^{∘} E). The lowest exhumation rates along the high Himalayan topographic front
(<0.8 km Myr^{−1}) are found in Kashmir (west of ∼75^{∘} E),
western Nepal (∼81^{∘} E), and from western Bhutan (∼90^{∘} E) to the
east. These lateral variations in exhumation rates have been interpreted as
reflecting lateral variations in the presence/absence and geometry
(location, height, and dip) of the mid-crustal ramp in the MHT, together with
duplex formation and local out-of-sequence thrusting (Hubbard et al., 2021;
Dal Zilio et al., 2021; van der Beek et al., 2023). In some of the more
slowly exhuming regions, in particular in Bhutan, exhumation rates appear to
be decreasing through time, with lower-temperature systems recording lower
exhumation rates than higher-temperature systems. Decreasing exhumation
rates in Bhutan can be linked to slowing convergence across the Bhutan
Himalaya due to transfer of deformation to the Shillong Plateau to the south
(Clark and Bilham, 2008; Coutand et al., 2014, 2016). Similar to the caveats
described above concerning increasing exhumation rates, in areas of
decreasing exhumation rates, the change in rates through time recorded by
different systems will also be underestimated.

The above example illustrates how this method can rapidly provide internally consistent estimates of exhumation rates from multiple thermochronometers from different elevations over a large region. Inferred patterns of exhumation rates can be linked to structural and geophysical observations of orogen segmentation, as above, or to orogen-wide topographic measures for assessing first-order linkages between exhumation rates and morphology (e.g., Clubb et al., 2022).

## 4.1 Importance, uncertainties, and sensitivity

An advantage of the rapid calculations performed by *age2exhume* is that it is
easy to explore the sensitivity of the calculated exhumation rates to
different input parameters (i.e., sample-specific information and
crustal/thermal properties), in addition to evaluating how the iterative
method compares to simpler estimates of exhumation rates. Regarding the
latter, we can compare calculated exhumation rates from *age2exhume* to those
that would be obtained by assuming a simple linear geotherm and fixed
nominal closure temperature, *T*_{c}. Figure 5a compares “initial” exhumation
rates, calculated using Eqs. (2) and (3) (hence, a linear geotherm and fixed
*T*_{c}), with the final exhumation rates predicted by *age2exhume*, which
incorporate perturbations to the geotherm and *T*_{c}. Initial exhumation
rates are calculated using the same thermal parameters of Table 1 and
nominal closure temperatures of 70 ^{∘}C for the AHe system,
120 ^{∘}C for the AFT system, 180 ^{∘}C for the ZHe system,
220 ^{∘}C for the ZFT system, and 350 ^{∘}C for the MAr system. The
comparison shows that for exhumation rates up to ∼0.5 km Myr^{−1},
there is little difference between the two methods (Fig. 5a). At higher
exhumation rates, the methods deviate substantially, with the initial
estimate systematically overestimating the exhumation rate. For example, at
exhumation rates ≥2 km Myr^{−1}, overestimates mostly fall between
100 % and 300 %. These findings can be explained by considering the relative
importance of two competing influences on the closure depth *z*_{c} (Fig. 1a), which directly determines the exhumation rate (Eq. 3). On the one hand,
higher cooling rates – linked to higher exhumation rates – lead to an
increase in *T*_{c}, and hence a deepening of *z*_{c} (Eqs. 8 and 9). On the
other hand, the advective perturbation of the geotherm due to exhumation,
which forces an upward deflection of isotherms, leads to a shallowing of
*z*_{c} for any *T*_{c} (Eq. 5). The degree of advective perturbation of the
geotherm is characterized by the non-dimensional Péclet number: $\mathit{Pe}=\dot{e}L/\mathit{\kappa}$ (e.g., Braun et al., 2006); the
predicted (surface) geothermal gradient thus increases with increasing
exhumation rate (Fig. 6). With higher exhumation rates, the effect of
upward, advective perturbation of isotherms on *z*_{c} dominates over the
effect of the increasing *T*_{c} on *z*_{c}. The scatter in the amount of
overestimation, in particular for the lower-temperature AFT and AHe systems,
is linked to the effect of including Δ*h*, which is more important for
shallower *z*_{c} (Eq. 3).

But how important are these differences in the method of calculating
exhumation rates relative to the uncertainties in any calculated rate? The
uncertainties in reported ages are just one component of the total
uncertainty that one can consider in an exhumation-rate calculation, but the
direct propagation of age uncertainty into the uncertainty on an inferred
exhumation rate provides a simple means of comparison (Fig. 5b). Because of
the non-linear relationship between age and exhumation rate, the
uncertainties in exhumation rates are asymmetric, with
${\dot{e}}_{\text{max}}-\dot{e}>\dot{e}-{\dot{e}}_{\text{min}}$. The bulk of the relative
uncertainties in exhumation rates associated with age uncertainty lie
between 10 % and 50 %, and they are not strongly dependent on exhumation
rate. Higher-temperature systems (ZHe, ZFT and MAr) are generally associated
with lower exhumation-rate uncertainties (<10 %) because of the
smaller age uncertainties associated with these systems. In contrast, AFT
data can have uncertainties that exceed 100 %, because low track
counts due to low U-contents and/or young ages yield large age
uncertainties. Some large relative uncertainties in the AHe and ZHe systems
at lower exhumation rates (<1 km Myr^{−1}) are probably
associated with larger inter-grain scatter in ages due to compositional and
grain-size effects that become more important at lower cooling and exhumation
rates (e.g., Whipp et al., 2022, and references therein). Overall, however,
the bulk of the uncertainties in exhumation rate are smaller than the
differences between the initial and final exhumation rates shown in Fig. 5a
for exhumation rates ≳0.5 km Myr^{−1}. This comparison
implies that the thermal effects of exhumation significantly affect inferred
exhumation rates in tectonically active areas.

The importance of including sample-specific information in exhumation-rate
calculations is illustrated in Fig. 7a and b. Our comparison of exhumation
rates calculated with a constant surface temperature, *T*_{s}, versus those
calculated with *T*_{s} dependent on elevation shows a relatively small
effect, with differences mostly less than 10 %. However, for the
low-temperature thermochronometers AHe and AFT at exhumation rates <1 km Myr^{−1}, differences can reach 20 % (Fig. 7a). The effect of the
local relief correction, Δ*h*, for each sample is generally more
important. Although the magnitude of Δ*h* tends to be reduced for the
lower-temperature systems (because their closure isotherms more closely
mimic surface topography), any given Δ*h* has a stronger impact on
exhumation rates for low-temperature systems (with shallower *z*_{c})
compared to high-temperature systems (Fig. 7b and Eq. 3). Moreover, the effects
are asymmetric: negative Δ*h* values lead to a larger correction in
exhumation rates compared to positive Δ*h* values. For example, a
Δ*h* of +1 km will lead to a ca. 20 % change in calculated
exhumation rate for the AFT system, whereas a Δ*h* of −1 km will lead to
a 30 % to 50 % change (Fig. 7b). This asymmetry results from the non-linear
effect of exhumation rate on *z*_{c}: positive Δ*h* values will lead to
increased predicted exhumation rates, but these increases will be partly
offset by the resulting advective perturbation of the geotherm. In contrast,
the decreased predicted exhumation rates for negative Δ*h* values will
be less modified by advective effects. The importance of including Δ*h* when calculating exhumation rates is further emphasized when considering
that samples are more commonly collected from valley bottoms (with negative
Δ*h* values) than ridgetops. Our Himalayan example dataset bears this
out: the histogram of Δ*h* values is skewed toward negative values,
with a median Δ*h* of −0.53 km (Fig. 7b inset).

We next explore the sensitivity of calculated exhumation rates to crustal
parameters, including the model thickness *L* (Fig. 7c) and the initial,
unperturbed geothermal gradient *G*_{init} (Fig. 7d). These plots show the
percent change in predicted exhumation rates when changing these two
parameters to either a higher or a lower value relative to the default
values of *L*=30 km and *G*_{init}=25 ${}^{\circ}\mathrm{C}\phantom{\rule{0.125em}{0ex}}{\mathrm{km}}^{-\mathrm{1}}$. Decreasing *L* from
30 to 20 km leads to higher predicted exhumation rates (by up to ∼40 %), whereas increasing *L* from 30 to 40 km leads to lower predicted
exhumation rates (by up to $\sim -\mathrm{20}$ %), with the magnitude of the
effect increasing with exhumation rate (Fig. 7c). This behavior can be
understood by considering the effect *L* has on the advective perturbation of
the geotherm, through the Péclet number (see above): the Péclet
number is linearly dependent on *L* so that, for constant exhumation rate
$\dot{e}$ and diffusivity *κ*, increasing *L* will lead to a stronger
perturbation of the geotherm and thus a shallower closure depth for any
thermochronometer. The sensitivity of the predictions to *G*_{init} is of
similar magnitude when considering changes from 25 to 30 ${}^{\circ}\mathrm{C}\phantom{\rule{0.125em}{0ex}}{\mathrm{km}}^{-\mathrm{1}}$ or
from 25 to 20 ${}^{\circ}\mathrm{C}\phantom{\rule{0.125em}{0ex}}{\mathrm{km}}^{-\mathrm{1}}$ (Fig. 7d), but in this case, the effect is
strongest for relatively low exhumation rates and thus relatively
unperturbed geothermal gradients.

## 4.2 Limitations and recommended use

The assumptions underlying the model limit its use, strictly speaking, to settings where both topography and exhumation rates are temporally constant throughout the time period considered and exhumation is mainly vertical. The requirement for constant exhumation rates is due to both the use of a steady-state solution for the advectively perturbed geothermal gradient and the use of the Dodson (1973) approach to estimate closure temperatures. These assumptions break down in cases where thermal and exhumation histories are more complex, in particular when they include phases of burial and heating. However, as our Himalayan example shows, our approach can provide first-order information on spatiotemporal patterns of exhumation, highlighting regions of accelerating versus decelerating exhumation. This result comes with the caveat that the model will systematically underestimate the change in exhumation rate, as discussed above. The Himalayan example also shows that first-order results can be obtained in a setting where horizontal advection of rocks is important; however, in this case, the interpretation of possible accelerations or decelerations in exhumation rate should take the regional structure and kinematics into account. For instance, accelerated exhumation may be due to rocks moving over a flat-to-ramp transition in a crustal-scale décollement, rather than a temporal change in tectonic or climatic drivers.

Our analysis of the importance of including advective perturbation of the
geotherm in the thermochronometric age predictions shows that this effect is
not significant for exhumation rates ≲0.5 km Myr^{−1}
(Fig. 5a). At these relatively low rates of exhumation and cooling, kinetic
effects also become important in controlling thermochronometric ages (e.g.,
Whipp et al., 2022) and these are not included in our model. The model is
thus best suited for the analysis of regional datasets from rapidly and
continuously exhuming regions, e.g., tectonically active mountain belts. The
relief correction included in the model makes it suitable to handle data
that were collected at widely varying elevations.

The model assumptions of a constant basal temperature together with an input model thickness are unlikely to be valid over long timescales, and in many cases can only be estimated roughly. However, the speed with which exhumation rates can be calculated from our model enables users to easily investigate the sensitivity of their results to these estimated values. Moreover, while these thermal parameters change the absolute values of the predicted exhumation rates, they affect all predictions similarly (if not equally). Therefore, their influence on spatial patterns in exhumation rates or the correlation of exhumation rates with other metrics will be limited.

We provide three different versions of the model in the form of MATLAB
scripts on Zenodo (van der Beek and Schildgen, 2022): (1) a basic version
that takes a single age–Δ*h* pair as input and returns a single
exhumation rate; (2) a version for which a range of thermochronologic ages
and Δ*h* values are provided and that returns a lookup table of
exhumation rates (used in Sect. 3.1 and Figs. 2 and 3); and (3) a version
that reads an input file of sample locations, elevation, thermochronologic
system, age, and uncertainty and returns a table of exhumation rates with
uncertainty, closure depths, and surface steady-state geotherms for each
sample (used in Sect. 3.2 and Fig. 4). A correctly formatted input file is
also included in the Zenodo repository. We anticipate the latter version to
be most useful, and therefore we also provide a Python script with that same
functionality (Schildgen and van der Beek, 2022a). Alternatively, Figs. 2
and 3 of this paper can be used to simply look up appropriate exhumation
rates for a given age–Δ*h* combination, but note that these figures
are plotted for particular values of the input parameters
*G*_{init}, *L*, and *κ*.

## 4.3 Concluding remarks

The model presented here, *age2exhume*, enables a first-order, synoptic view
of spatial and temporal variations in exhumation rates, calculated in a
rapid, self-consistent manner from different thermochronometers. The main
advantage of our approach over the version of *age2edot* presented by Willett
and Brandon (2013) is that our model does not require the final geothermal
gradient as input, but only the initial, unperturbed geotherm. This aspect
of our model makes it easily applicable to regions with strongly varying
exhumation rates, which are expected to have a wide range of modern
geothermal gradients. The modern geothermal gradient, when known, adds an
additional constraint to the model solution. However, for many regions of
the world, particularly for mountain belts, modern geothermal gradients are
essentially unknown. In our entire Himalayan study region, for instance, the
global heat-flow database (https://ihfc-iugg.org, last access:
10 October 2022) does not contain a single data point. Although there are data
both for the Tibetan Plateau to the north and the Ganges foreland basin to
the south, these are not useful for assessing the perturbed geotherm within
the mountain belt. Our model does provide the predicted steady-state surface
geotherm as output, so it can be compared to any potential measurements (Fig. 6).

Our model assumes steady-state exhumation, unchanging topography, and vertical exhumation pathways, so it is only appropriate for obtaining first-order, synoptic overviews of exhumation-rate patterns in regions of relatively rapid, continuous exhumation. Nevertheless, in the case where ages from multiple thermochronometers are available from individual samples or from samples in close proximity to one another, differences in exhumation rates derived from those ages can be used to map out where changes in exhumation rates have likely occurred, and thus highlight regions where more advanced thermal modeling could be used to extract non-steady-state exhumation histories. The rapidity with which our model calculates regional patterns of exhumation rates also allows testing its sensitivity to the different input parameters.

*h*from digital elevation datasets

To calculate Δ*h*, Willett and Brandon (2013) suggest calculating a
mean value of elevation for a circle that has a radius equal to *π*×*z*_{c}, where *z*_{c} is the closure depth of the system. This calculation
can be done with standard operations in a geographic information system
(GIS) or other tools designed to work with continuous raster datasets. The
following instructions can be followed to calculate Δ*h* values within
ArcMap from ESRI (version 10.8.1) or within QGIS (version 3.26). We have not
tested if the instructions are easily applicable to earlier versions of the
software. Nevertheless, small modifications to these procedures can likely
be found by searching on the names of the functions described below.
Importantly, regardless of the software package used to calculate Δ*h*, the spatial extent of the DEM should extend beyond the limits of the
sample points, with a buffer zone at least equal to the highest radius that
will be considered. For example, the DEM should extend at least ca. 40 km
beyond the spatial extent of the sample data to prevent edge effects from
affecting Δ*h* calculations for the MAr system.

## ESRI ArcMap

In ESRI's ArcMap version 10.8.1, the mean elevation can be calculated using
the Focal Statistic function, found within the “Spatial Analyst Tools –
Neighborhood” tools in Arc Toolbox. The Focal Statistic function provides
an option to average values over a moving circular window with a radius
defined by map units or by a number of pixels. For example, for a standard
90 m resolution Shuttle Radar Topography Mission (SRTM) DEM, and for a desired *z*_{c} of 2000 m (e.g., for the
AHe system), the radius of the circle should be 6280 m, which is
approximately equivalent to 70 pixels. To efficiently calculate Δ*h*
for all samples in a large dataset, it is practical to take advantage of the
“Raster Calculator” (Spatial Analyst Tools – Map Algebra) and the
“Extract Values to Points” functions (Spatial Analyst Tools –
Extraction). The Raster Calculator can be used to subtract the smoothed DEM
calculated in the previous step from the modern DEM. This operation will produce a
continuous raster dataset of Δ*h* values. The “Extract Values to
Points” function samples a raster at the position of each sample data
point and adds the extracted value to a new column (“field”) in the
attribute table of the shapefile. Although the exact procedure described
here may differ for other versions of ArcMap, general functions to calculate
focal statistics, perform arithmetic operations on raster datasets, and
automatically extract values from rasters at the location of sample points
can be found in many versions of the software.

## QGIS

In QGIS 3.26, a procedure to find the average elevation over a defined
circular search area can be accomplished with the SAGA plug-in, which can be
installed directly from the “Plug-in” menu and then “Manage and install
plug-ins”. After installation, the SAGA tools can be found within the
“Processing” menu, then “Toolbox”. Within SAGA, go to the “Raster
Filter” options and then select “Simple filter”. The filter option should
be set to “Smooth” (to calculate an average value), and the Kernel type
set to “Circle”. The radius should be set to the number of pixels that
will provide the correct radius length. Like in the example above, for a
standard 90 m resolution DEM and a desired *z*_{c} of 2000 m (for the AHe
system), the radius of the circle (*π*×*z*_{c}) should be 6280 m, which is approximately equal to 70 pixels. Next, the Raster Calculator
within QGIS can be used to calculate Δ*h* values over the extent of the
DEM, by subtracting the smoothed DEM calculated in the previous step from
the original DEM. Finally, to extract the calculated Δ*h* values for
each sample point, within the standard Processing Toolbox under the “Raster
analysis” heading is the function “Sample raster values”. With this tool,
the point layer containing the sample points should be given as the “Input
layer” and the raster of Δ*h* values should be given as the “Raster
layer”. The output point file includes all of the attributes of the
original point layer but adds a column containing the extracted
Δ*h* value
for each point. That file, by default, is only saved to memory. To save it
permanently, the small square-shaped icon to the right of the layer name can
be clicked to bring up a dialog box that allows saving the file to a defined
location.

The MATLAB scripts for three versions of the *age2exhume*
code, together with an input file, are included in the Zenodo repository:
*age2exhume* MATLAB scripts (https://doi.org/10.5281/zenodo.7341603, van der Beek and Schildgen, 2022). The Python version of *age2exhume*,
together with an input file, can be downloaded from the Zenodo repository:
*age2exhume* Python script (https://doi.org/10.5281/zenodo.7341690, Schildgen and van der Beek, 2022a).

Data used in the example dataset were compiled from the sources listed in the Supplement. An Excel file containing the full dataset and calculated exhumation rates is included in the Zenodo repository: Thermochronology dataset for Himalaya (https://doi.org/10.5281/zenodo.7053115, Schildgen and van der Beek, 2022b).

The supplement related to this article is available online at: https://doi.org/10.5194/gchron-5-35-2023-supplement.

Both authors contributed equally to the development of the code, the compilation of the Himalayan dataset, sensitivity analyses, and writing of the manuscript. TS wrote the Python version of the code.

The contact author has declared that neither of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Taylor F. Schildgen acknowledges support from the ERC Consolidator grant no. 863490 GyroSCoPe; Peter van der Beek acknowledges support from the ERC Advanced grant no. 834271 COOLER. We thank reviewers Matthew Fox and David Whipp for constructive comments that improved the quality of the manuscript and allowed us to refine some aspects of the code. We thank Associate Editor Marissa Tremblay for efficient handling of the manuscript. We also thank David Whipp for his freely accessible online “Geo-Python” course, which provided the base training needed for writing the Python version of the code.

This research has been supported by the European Research Council, H2020 European Research Council.

This paper was edited by Marissa Tremblay and reviewed by David Whipp and Matthew Fox.

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*age2exhume*, a thermal model that can be used to rapidly provide a synoptic overview of exhumation rates from large regional thermochronologic datasets.