We report a new program for calculating catchment-averaged denudation
rates from cosmogenic nuclide concentrations. The method (Catchment-Averaged
denudatIon Rates from cosmogenic Nuclides: CAIRN) bundles previously reported
production scaling and topographic shielding algorithms. In addition, it
calculates production and shielding on a pixel-by-pixel basis. We explore the
effect of sampling frequency across both azimuth (

In situ cosmogenic nuclides, such as

Several authors have provided standardized methods for calculating denudation
rates from cosmogenic nuclide concentrations, notably the COSMOCALC package

In the context of catchment-averaged denudation rates, nuclide production
rates will vary in space, and an open-source method of calculating production
and inverting nuclide concentration for denudation rate has yet to emerge.
Due to the lack of an open-source tool, a wide variety of approaches to
calculating catchment-averaged denudation rates are used in the literature,
which makes intercomparison studies challenging

Several factors determine the concentration of a cosmogenic nuclide in a
sample. For instance, elevation and latitude control the production rate of
different cosmogenic nuclides

Many authors use an averaging scheme for production wherein production is
calculated in each pixel which is then passed to a calculator

Here we present software that estimates production and shielding of the
cosmogenic nuclides

We derive a governing equation that tracks the concentration of a cosmogenic
nuclide as it is exposed, exhumed or buried. This approach is adopted because
it is the most general: specific scenarios of both steady and transient
denudation and burial may therefore be derived. Our approach is broadly
similar to that of

We begin by conserving the concentration of cosmogenic nuclide

Cosmogenic nuclides can be produced by both neutrons and muons

The advantage of the

Our approach is to approximate muon production using a sum of exponential
functions

The exponential approximation for nuclide production used in CAIRN is

The depth

The governing equation (Eq.

By convention, we consider the depth profile of cosmogenic nuclide
concentration to be steady in time. This allows analytical solution of the
cosmogenic nuclide concentration at any point in the basin. At steady state,
the particles near the surface have been removed (either through erosion or
chemical weathering) at the same rate for a very long time, so we set

Equation (

Our strategy is slightly different: we calculate snow and self shielding by
integrating the cosmogenic nuclide concentration over a finite depth in
eroded material. For example, if there is no snow, the concentration of
cosmogenic nuclides at a given location is obtained by depth-averaging the
steady concentrations from zero depth (the surface) to the thickness of
eroded material. If snow is present, the concentration is determined by
depth-averaging from the mean snow depth (

In most applications, the thickness of the removed material will be 0, i.e.
the particles from which nuclide concentrations are measured in detrital
sediment are derived from a thin layer removed from the surface of the
catchment. However, the solution described by Eq. (

In addition to snow and self shielding, locations in hilly or mountainous
areas can also receive a reduced flux of cosmic rays because these have been
shielded by surrounding topography

As

Absolute maximum residuals (i.e., greatest residual within the DEM)
for different combinations of

In order to determine the optimal balance between measurement accuracy and
computational efficiency, the full range of (

Our topographic shielding calculations rely on two approximations that can
lead to some uncertainty. First, the method of

Production of cosmogenic nuclides varies as a function of both elevation
(defined via atmospheric pressure) and latitude and these variations are
accounted for by using one of several possible scaling schemes. The classic
scaling model of

These scaling schemes vary in complexity and therefore computational expense.
Time-dependent scaling schemes are far more computationally expensive than
the time-independent scheme of Lal/Stone, which does not consider variations
in geomagnetic field strength. Recent calibration results

The Lal/Stone scaling scheme requires air pressure, whereas most published
studies include only elevation information. We follow the approach of

To calculate the concentration of a cosmogenic nuclide, the scaling factors
for each production pathway (

To determine the scaling terms for the individual production mechanisms
(

So far we have described the calculations that predict the concentration of a
cosmogenic nuclide at one specific location in a basin. All existing
cosmogenic nuclide calculators contain some form of these calculations. A
wide variety of approaches to scale calculations of cosmogenic nuclide
concentrations within a single location to the concentration across entire
catchments have been used in the literature. Some authors have averaged
production rates on a pixel-by-pixel basis but have not considered
topographic shielding

The approach we take in CAIRN differs in that shielding and production rates
are not averaged: these are calculated locally at each pixel. For a given
denudation rate,

Default parameters used in the CAIRN model.

We should note here that the version of CAIRN reported in this contribution
calculates the denudation rate across an entire catchment required to produce
the observed concentration of the target cosmogenic nuclide. That is, CAIRN
assumes denudation rates and target mineral concentrations are the same
everywhere in the catchment. Users can explore the effect of instantaneously
removing mass by modifying

Calculating denudation rates on a pixel-by-pixel basis.

We calculate uncertainty from both internal (nuclide concentration
uncertainties from accelerator mass spectrometry (AMS) measurements) and
external (shielding and production rate) sources using Gaussian propagation
of uncertainty following

Uncertainties are calculated in terms of the denudation rate,

Three uncertainties are included in the calculation: (i) the uncertainty in
cosmogenic nuclide concentration, (ii) the uncertainty in the production rate
at sea level, high latitude (

The uncertainty on the production rate (

Field studies have shown that muon production based on laboratory experiments

Uncertainties from nuclide concentration, muon production, and production rates are calculated internally by our software. Uncertainties from snow and self shielding rely on user-supplied information and therefore must be estimated separately.

Snow shielding can be supplied as a constant effective snow thickness (in
g cm

To calculate uncertainties, users must supply two scenarios for these
shielding factors. For example, the user could provide two snow thickness
rasters representing variation in snow thickness with 1

To summarize, CAIRN predicts cosmogenic nuclide production from neutrons and
muons using a four exponential approximation of data from

In addition to producing denudation rates, CAIRN also provides
spatially averaged production rates and effective catchment-averaged pressure
(see Sect.

CAIRN iterates on denudation rate until the predicted cosmogenic
concentrations from Eq. (

Self shielding used for spatial averaging is calculated for each pixel

In COSMOCALC's erosion calculator (which calculates denudation), the required
inputs are a combined shielding and scaling term, the cosmogenic nuclide
concentration and the uncertainty in the cosmogenic nuclide concentration.
That is, scaling and shielding are combined in a single, spatially averaged
term. We calculate the scaling factor

The CRONUS calculators (CRONUS-2.2 and CRONUScalc) require a lumped shielding
value and information about either the elevation or pressure of the sample.
Spatial averaging of the lumped shielding value,

The CRONUS calculators then calculate production using either an elevation or
pressure. Production rates are nonlinear with either elevation or pressure,
so we must compute an effective pressure that reproduces the mean production
rate in the catchment. This is because the arithmetic average of either
elevations or pressures within the catchment, when converted to production
rate, will not result in the average production rate due to this
nonlinearity. CAIRN calculates an effective pressure that reproduces the
effective production rate over the catchment. The average production rate is
calculated with

CAIRN provides uncertainty estimates based on uncertainties in the
measurement of nuclide concentrations, and uncertainties in production rates.
It does, however, make an assumption of steady erosion, and also makes
assumptions likely to be violated almost everywhere on Earth due to the long
timescales of geomorphic adjustment, which are on the order of tens of
thousands to millions of years

For the problem of spatially heterogeneous lithology, careful geologic
mapping, such as that done by a handful of recent authors

Ultimately, uncertainties in the spatial distribution of denucation and
source material, and temporal uncertainties in denudation rates, mean that
the uncertainties reported by CAIRN are the minimum uncertainties: they do
not take into account landscape transience, lithology, or variation in snow
shielding. The fact that catchment-averaged denudation rates carry additional
uncertainties is well known, and

Comparison with other methods is difficult because authors reporting
cosmogenic nuclide-derived catchment-averaged denudation rates have not made
their algorithms available as open-source tools. Our spatially averaged
production scaling and shielding estimates are approximations of spatial
averaging reported by other authors. We compare our data to both published
denudation rate estimates, and to estimates of denudation rates generated by
the CRONUS calculator given the spatial averaging described in
Sect.

A schematic drawing of the predicted concentration of a nuclide as a function of denudation rate. If production rates are assumed to be higher, the predicted concentration will be higher for a given denudation rate. If shielding is greater, the predicted concentration is lower for a predicted denudation rate. Thus assumptions about production and shielding will affect the inferred denudation rate given a sample with fixed concentration, shown with the dashed lines.

Data sets used for method comparisons.

It will perhaps aid the reader if we explain how denudation rate estimates
may vary between methods. Firstly, production rates are nonlinearly related
to elevation, and thus spatial averaging of the product of production scaling
and shielding is not the same as the product of the spatial averages of
production scaling and shielding. In addition, previous studies and other
calculators have chosen different parameters for cosmogenic nuclide
production and shielding. For example, past publications have used a wide
variety of methods for estimating topographic shielding (e.g., see
Table

First, we compare results of two methods using the exponential approximation
of muon production (Eq.

Differences between the denudation rate calculated by CAIRN
(

Separating production rate scaling from shielding leads to slightly larger
uncertainty (Fig.

Denudation rates reported in the literature from catchment-averaged
cosmogenic nuclide concentrations are calculated using a wide variety of
methods. The term erosion rate is often substituted for denudation rate
although few studies attempt to account for chemical weathering

The diversity in methods for calculating denudation rates reported in the
literature means that it is difficult to compare denudation rates when they
come from different studies. This problem has been highlighted by previous
data intercomparison studies

Topographic shielding (

Comparison of the topographic shielding for different values of

Studies typically report erosion or denudation rates in dimensions of length
per time, but this requires an assumption about density, which can vary
spatially and is sometimes not reported. Most studies use a rock equivalent
denudation rate (as opposed to a regolith or soil denudation rate) and thus
densities assumed are typically rock densities (see
Table

Of our seven example data sets (Table

The denudation rates predicted by CAIRN are plotted against reported
denudation rates in Fig.

One component of CAIRN that requires caution is that the snapping of cosmogenic samples to channels is automated: if errors in the DEM place the main channel in the wrong location, or GPS coordinates of the sampling location contain large errors (common in older data sets), there is a chance the basin selected by CAIRN will not be the same as the sampled basin. This can result in large errors as production rates vary significantly with elevation. We have provided a tool in the github repository that allows users to check the basins that are associated with cosmogenic nuclide samples. If these do not match the expected basins, then users will need to manually change the latitude and longitude of the samples until they are located near the correct channel.

We wish to emphasize that the relative denudation rates do not change
significantly between CAIRN and reported values (as evidenced by a clustering
about the 1 : 1 line in Fig.

Comparison of denudation rates reported by selected studies plotted
against denudation rates predicted by CAIRN. The denudation rates for
individual studies use their original assumptions of the density of the
surface material, as reported in Table

Differences between the denudation rate calculated by CAIRN
(

Difference between denudation rate calculated by CAIRN
(

The results from CAIRN are compared to results from both CRONUS calculators.
When comparing output from CAIRN with output from the online CRONUS-2.2
calculator, far larger uncertainties (up to 40 % of the denudation
rate) occur. These differences are not controlled by denudation rate
(Fig.

Production rates of

Concentrations as a function of denudation rate

The other CRONUS calculator, CRONUScalc, incorporates new spallation
production rates and muon production is calculated using production rates
based on a deep core from Antarctica

Parameters used for production of

Differences between the denudation rate calculated by CAIRN
(

Differences between the denudation rate calculated by CAIRN using
the parameters in Table

We have used the spatially averaged shielding and scaling outputs from CAIRN
to determine differences between CAIRN and CRONUScalc. We find that there is
a 2.5 to 5 % difference between the denudation rates predicted by CAIRN
and those predicted by CRONUScalc (Fig.

We present an automated, open-source method for calculating catchment-averaged denudation rates based on the concentrations of in situ cosmogenic nuclides collected in stream sediment. Our catchment-averaged denudation rate method (CAIRN) predicts cosmogenic nuclide concentrations based on pixel-by-pixel scaling and shielding. These concentrations are then averaged to predict the catchment-averaged concentration. Newton iteration is then used to find the denudation rate for which the predicted concentration matches the measured concentration and to derive associated uncertainties. In addition, CAIRN provides spatially averaged shielding and scaling values that can be used by other popular calculators (which do not provide spatial averaging, e.g., CRONUS and COSMOCALC). The CAIRN method is provided as open-source software so that reported denudation rates can be easily reproduced.

The CAIRN method is intended to streamline the computation and reporting of
catchment-averaged denudation rates, but it has limitations that may be the
subject of future developments. At the moment CAIRN assumes steady erosion;
there is no facility for incorporating transient erosion rates which might
affect nuclide concentrations in transient landscapes

Our open source framework allows other users to update the algorithms (e.g., a nesting function could be built on top of the current CAIRN architecture) and different atmospheric reanalysis data or new muon scaling schemes can be added as needed in the future. Thus we hope it will provide a platform for more nuanced estimates of denudation rates from cosmogenic nuclides in the future.

The software is available at the LSDTopoTools Github website
(

Simon Marius Mudd, Martin D. Hurst and Stuart W. D. Grieve wrote the software. Marie-Alice Harel, Simon Marius Mudd and Shasta M. Marrero analyzed the data. Simon Marius Mudd wrote the paper with contributions from other authors.

Simon Marius Mudd and Marie-Alice Harel are funded by US Army Research Office contract number W911NF-13-1-0478 and Simon Marius Mudd and Stuart W. D. Grieve are funded by NERC grant NE/J009970/1. Shasta M. Marrero is funded by NERC grant NE/I025840/1. This paper is published with the permission of the Executive Director of the British Geological Survey (NERC), and was supported by the Climate and Landscape Change research programme at the BGS. We would like to thank Associate Editor Josh West for his helpful comments and also for testing the code. We would also like to thank an anonymous reviewer and Greg Balco for their constructive and beneficial comments which significantly improved the paper. Edited by: A. Joshua West Reviewed by: G. Balco and one anonymous referee