One of the main purposes of detrital thermochronology is to provide constraints on the regional-scale exhumation rate and its spatial variability in actively eroding mountain ranges. Procedures that use cooling age distributions coupled with hypsometry and thermal models have been developed in order to extract quantitative estimates of erosion rate and its spatial distribution, assuming steady state between tectonic uplift and erosion. This hypothesis precludes the use of these procedures to assess the likely transient response of mountain belts to changes in tectonic or climatic forcing. Other methods are based on an a priori knowledge of the in situ distribution of ages to interpret the detrital age distributions. In this paper, we describe a simple method that, using the observed detrital mineral age distributions collected along a river, allows us to extract information about the relative distribution of erosion rates in an eroding catchment without relying on a steady-state assumption, the value of thermal parameters or an a priori knowledge of in situ age distributions. The model is based on a relatively low number of parameters describing lithological variability among the various sub-catchments and their sizes and only uses the raw ages. The method we propose is tested against synthetic age distributions to demonstrate its accuracy and the optimum conditions for it use. In order to illustrate the method, we invert age distributions collected along the main trunk of the Tsangpo–Siang–Brahmaputra river system in the eastern Himalaya. From the inversion of the cooling age distributions we predict present-day erosion rates of the catchments along the Tsangpo–Siang–Brahmaputra river system, as well as some of its tributaries. We show that detrital age distributions contain dual information about present-day erosion rate, i.e., from the predicted distribution of surface ages within each catchment and from the relative contribution of any given catchment to the river distribution. The method additionally allows comparing modern erosion rates to long-term exhumation rates. We provide a simple implementation of the method in Python code within a Jupyter Notebook that includes the data used in this paper for illustration purposes.

Thermochronometric methods provide us with information pertaining
to the cooling history of a rock. Various systems and minerals provide
information on different parts of that cooling history, i.e., at a given
temperature but more commonly within a range of temperatures. One of the main
geological processes through which rocks experience cooling is exhumation
towards the cold, quasi-isothermal surface

Besides collecting in situ data, one can also collect and date a large number
of mineral grains from a sand sample collected at a given location in a river
draining an actively eroding area. Such detrital thermochronology datasets
provide a proxy for the distribution of surface rock ages in a given
catchment

Methods have been devised to extract quantitative information from such
detrital datasets concerning the erosion history of a tectonically active
area as well as estimates of its spatial variability.

All of these methods rely on a priori knowledge or hypotheses concerning the age distributions in the catchments drained by the river from which the samples have been collected. Here we explore the possibility of deriving first-order information about the spatial variability in erosion rate in the river catchment when no such knowledge exists and without making any assumption concerning the spatial distribution of ages. We propose to regard ages as passive markers (or tracers) that inform us on the proportion in which mixing takes place today, which must be directly proportional to the present-day erosion rate. For example, the most rapid present-day erosion rates should be predicted where the age distributions change the most rapidly along the river, everything else being accounted for, such as the relative size of neighboring sub-catchments or the potential change in lithology between them. This knowledge about the distribution of erosion rates, which is obtained independently of the absolute values of the ages, can be used to estimate the distribution of ages in each sub-catchment drained by the river. This second piece of information provides additional insight into the spatial distribution of past cooling/erosional events.

In this paper, we present a method that relies on the raw age data only. This
avoids any complication or bias that may arise from trying to compare the
data to theoretical probability density distributions that rely on a thermal
model prediction. We recognize the value of doing so, but thermal models
require making assumptions about past geothermal gradients (heat flux) or
rock thermal conductivity and heat production, which introduces additional
uncertainty in interpreting the data. The first part of this paper describes
the method. To demonstrate its accuracy and explore the limits of its
applicability, we have applied the method to synthetic age datasets for which
we know the erosion rate and its spatial variability. This is done in the
second part of the paper. To illustrate the use of our method, we have
applied it to a dataset collected in the Himalaya (along the
Tsangpo–Siang–Brahmaputra river system). This is explained in the third part
of the paper. There we show that the method yields reliable estimates of the
distribution of present-day erosion rates in these areas as well as
independent information on the spatial extent of past geological events. We
conclude by suggesting potential ways in which the method could be improved.
Note that the approach proposed here has been used on a set of detrital age
distributions collected along the Inn River in the Eastern Alps

We assume that we have collected a series of age datasets measured at

Example of a measured age distribution and the relative heights

The landscape is divided into exclusive contributing areas for each
of the points along the main river where we have measured a dataset and
compiled a distribution from it. We take the convention that Area 1 (of
surface area

Schematic representation of how the landscape is divided into
exclusive contributing areas

The surface areas,

From these simple assumptions, we can write that the number of grains of age

We can now write that the predicted height of bin

We now express Eq. (

We can also write

Considering that we have

Two cases must be considered. First, if there is a noticeable change in
relative bin heights in the detrital record between sites

The second case to consider is when the relative bin heights between two
successive sites do not change , i.e.,

For this, we rewrite Eq. (

From these estimates of the contribution factors,

From the values of the minimum contribution factors,

Age distributions from tributaries can be included to improve the solution
locally, i.e., in the tributary catchment. Let us call

For each bin

Using the method for the main trunk data described in the previous sections,
we know

However, this may lead to unrealistic values of the relative surface
concentrations

We assess the robustness of our estimates of minimum erosion rate

The code is provided as a Jupyter Notebook containing python code and
explanatory notes that refer to the equations given in this paper. The
user must provide a series of input files containing (a) the description of
the sites, i.e., the order in which the sites are located along the river,
whether they drain into the main river stem or into a tributary, the drainage
area

We have assessed the reliability of the erosion rate estimates obtained from
the method by applying it to synthetic age distributions made of

Results of the method applied to the first set of synthetic
datasets. Computed erosion rate distributions for four synthetic datasets.
For each distribution, the box extends from the lower to upper quartile
values, the line corresponds to the median value, and whiskers extend from the
box to show the range of the erosion rate estimates, excluding outliers.
Outliers are indicated by small circles past the end of the whiskers. For
each site, the gray stars correspond to the imposed erosion rates and the
dashed gray line gives the product of the imposed area and fertility factors,

Results of the method applied to the second set of synthetic
datasets with random peak amplitudes. See Fig.

We see that the estimated erosion rates (median value) are in good agreement with the imposed (true) erosion rates, especially when large jumps in erosion rate exists between successive sites/catchments. However, in some cases, there appears to be an artificial increase in estimated erosion rate from site to site. This is clearly seen in the case where the imposed erosion rate is assumed to be uniform at all sites or to decrease from site to site. In both cases, the method predicts an apparent (or spurious) increase in erosion rate at the last site.

To determine what controls the reliability of these estimates, we performed
two other sets of experiments. The first assumes random amplitudes for the
peaks in each of the catchments (Fig.

Results of the method applied to the second set of synthetic
datasets with increasing

Age bins used to construct age distributions shown in
Fig.

Consequently, the accuracy of the estimates of erosion rate obtained from our
method rely on whether the two successive age distributions used to estimate
the

To illustrate the method, we now apply it to a detrital age dataset from the
eastern Himalaya that contains ages obtained using the muscovite

We will investigate the effect of using variable concentrations of
muscovite-bearing rocks in each of the catchments as derived from a
geological map of the area (fifth column in Table

Relative position along the main trunk of the
Tsangpo–Siang–Brahmaputra river system. Negative numbers indicate samples
collected along a tributary. Catchment areas and mineral concentration
factors used to compute the erosion rate reported in Table

Predicted erosion rate in the successive sub-catchments obtained by
assuming uniform values for the

Computed erosion rate distributions obtained by applying the method
to a Himalayan dataset. For each distribution, the box extends from the lower
to upper quartile values, the line corresponds to the median value, and whiskers extend from the box to show the range of the erosion rate estimates,
excluding outliers. Outliers are indicated by small circles past the end of
the whiskers. Erosion rate values are normalized such that the mean is 1. The
gray circles connected by a dashed gray line are the product of the imposed
area and mineral concentration factors,

Observed distributions of ages (light gray bars) in samples
collected at sites shown in Fig.

Maps of predicted median erosion rates (central panel) and relative
surface age concentrations from the muscovite detrital data from eastern
Himalaya. See Figs.

Results are shown in Fig.

The most salient result predicted by the method is that the erosion rate in
the eastern Himalayan syntaxis should be at least 5–7

We also note that erosion rates in sub-catchments Y and X do not need to be
noticeably smaller than the estimates of erosion rate in their host
catchments (B and C). As explained earlier, the true erosion rates could be
larger than those of their host catchments. This could be the case for
sub-catchment Y, where we predict an erosion rate identical to that of
catchment B. These estimates are likely to be reliable because their area is
similar to that of their sub-catchment (they occupy a non-negligible portion
of their host catchment) and because they have strikingly different age
distributions than their host catchments (B and C; Fig.

Interestingly, there is a good correspondence between present-day erosion rate and where the youngest ages are being generated (sites B and C), with the notable exception of the most downstream catchment (Z). In other words, where the mixing analysis predicts a high minimum erosion rate to account for a substantial change in the age distribution between two adjacent catchments is also where it predicts the highest concentration of young ages in the surface rocks. At the downstream end of the river (Catchment Z), we predict a relatively high minimum erosion rate from the mixing model but a relatively low concentration of young ages in comparison to the other catchments. This could mean that, in catchment Z, the present-day high erosion rate is relatively recent and has not yet led to a complete resetting of cooling ages which were set during earlier events.

Relative uncertainty in erosion rate scaled by the relative
uncertainty in mineral concentration factor for the estimates obtained at
each of the six sites along the main river trunk. The first site has a fixed
relative erosion rate (

We also note that the difference between the two quartile values (vertical
size of the boxes in Fig.

One of the sources of uncertainty in our estimates of the erosion rate comes
from the assumed value of the mineral concentration factors,

As described in this paper, our methods relies on the existence of age
clusters (or bins) that can be found in the age distributions collected at
various sites along a river. The method could be generalized by constructing
kernel density estimates of the distribution of ages at each sites. These
could be used to estimate the minimum contribution factors,

We have developed a simple method to extract spatially variable erosion rates and surface age distributions from detrital cooling age datasets from modern river sands. The method is based on what we believe are the simplest assumptions necessary to interpret such data and does not rely on a priori knowledge of the distribution of ages in surrounding catchments. In describing the method we have demonstrated that it is suited to extract two apparent sources of information pertaining to the spatial distribution of erosion rate along the river. The first comes from using the age distributions as fingerprints characterizing the areas between two successive sites where detrital samples were collected. This allows us to predict first-order estimates of the relative erosion rate between these areas and the distribution of ages in surficial rocks in each area. These estimates of age distributions can be used as a second independent information on the past and present erosion rate in each area.

By applying the method to an existing dataset from the eastern Himalaya, we show that the method provides estimates of present-day erosion rate patterns in the area that is consistent with previous, independent estimates, potentially evidencing that the fast present-day erosion rates in some parts of the study area are relatively young. We stress, however, that our method can only provide reliable estimates of erosion rate when the age distributions observed at two successive sites are different.

Importantly, the method is limited to providing the spatial distribution of erosion rate; independent information is necessary to transform those into absolute estimates of erosion rate.

We provide a simple implementation of the method in python within a Jupyter Notebook and the data collected from published sources and used in this paper for illustration purposes.

The work leading to the results presented here was supported by the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme FP7/People/2012/ITN, grant agreement number 316966. Edited by: Sebastien Castelltort Reviewed by: Mark Brandon and Marco Giovanni Malusà