Introduction
In situ cosmogenic nuclides, such as 10Be and
26Al, are widely used to determine both exposure ages and
denudation rates
e.g.,.
A denudation rate is the sum of the chemical weathering rate and physical
erosion rate. Since the publication of the seminal papers by
, and
, dozens of studies have used concentrations
of cosmogenic nuclides in stream sediments to quantify denudation rates that
are spatially averaged over eroding drainage basins. There are now more than
1000 published catchment-averaged denudation rates
e.g.,,
with many new studies published each year.
Several authors have provided standardized methods for calculating denudation
rates from cosmogenic nuclide concentrations, notably the COSMOCALC package
and the CRONUS-Earth online calculator
. Here we make comparisons with the CRONUS
calculator version 2.2, so we refer to it as CRONUS-2.2 for clarity. These
calculators have been widely adopted by the cosmogenic, quaternary science
and geomorphic communities, in large part because they are easily accessible
and their methods are transparent (i.e., the source files are available
online). These previously published calculators are ideal for calculating
denudation rates or ages from a particular site (e.g., an exposed surface or
a glacial moraine). Existing calculators rely on the principle that there is
an inverse relationship between denudation rate and the concentration of a
nuclide, because slower denudation results in more exposure to cosmic rays.
In addition, these calculators make use of the fact that the concentration of
a nuclide can be inverted for denudation rate if one estimates the production
of the nuclide.
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
cf.,.
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
e.g.,.
Production rates vary spatially, thus users of online calculators must
calculate the effective production rate within a catchment using a weighted
mean of nuclide production in individual pixels. The manner in which these
are provided to existing calculators vary. For example, one must feed a
single weighted mean production, after shielding corrections, to COSMOCALC.
In contrast, one must calculate weighted mean shielding corrections and pass
them to CRONUS-2.2, and in addition must calculate a pressure or elevation
that reproduces the mean production rate before shielding.
Many authors use an averaging scheme for production wherein production is
calculated in each pixel which is then passed to a calculator
e.g.,.
In addition, nuclide concentrations can be affected by partial shielding
caused by snow cover, surrounding topography, and overlying layers of
sediment e.g.,. These again are spatially
distributed and so authors reporting catchment-averaged denudation rates
frequently report averaged shielding values. Although software packages do
exist for calculating spatially averaged topographic shielding
e.g., and snow shielding
e.g.,, results from these models are not
integrated with spatially varying production rates. Finally, in landslide
dominated terrain, removal of thick layers of sediment can dilute cosmogenic
nuclide concentrations in river sediment
. This
factor is often not included in denudation calculations. For these reasons,
specifically urged development of tools dedicated
to the calculation of catchment-averaged denudation rates from cosmogenic
nuclide concentrations.
Here we present software that estimates production and shielding of the
cosmogenic nuclides 10Be and 26Al on a
pixel-by-pixel basis, and propagates uncertainty in AMS measurement and
cosmogenic nuclide production. Based on these calculations the software can
then calculate the expected cosmogenic nuclide concentration from a basin
given a spatially homogenous denudation rate. Finally, the software uses
Newton iteration to calculate the denudation rate that best reproduces the
measured cosmogenic nuclide concentration. We have made this software
available through an open-source platform at
https://github.com/LSDtopotools/LSDTopoTools_CRNBasinwide to allow
community modification and scrutiny, with the goal of enabling users to
report denudation rates that can be easily reproduced by other scientists.
The software distribution includes instructions for building the software on
a virtual machine that can function on common operating systems.
Quantifying denudation rates at a single location
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 , but results are
equivalent to those of more widely used derivations
e.g.,.
We begin by conserving the concentration of cosmogenic nuclide i through
time t:
dCidt=Pi-λiCi,
where Ci is the concentration of cosmogenic nuclide i (Ci is
typically reported in atoms g-1; i could be 10Be or 26Al,
for example), Pi is the local production rate of cosmogenic nuclide i
(in atoms g-1 yr-1) and λi (yr-1) is the decay
constant of cosmogenic nuclide i. Production can be a function of latitude,
altitude (or atmospheric pressure), magnetic field strength and shielding by
rock, soil, water or snow e.g.,.
Cosmogenic nuclides can be produced by both neutrons and muons
e.g.,. Production by neutrons is widely
modeled using a simple function in which production decays exponentially
with depth e.g.,. Muons, on the other hand, are
modeled using a variety of schemes. The CRONUS-2.2 calculator
implements the scheme of
, which requires
computationally expensive integration of muon stopping over a depth profile.
Field-based estimates of muon production demonstrate that
significantly overestimate production by
muons
.
Other authors have used empirical fits of cosmogenic profiles from the field,
typically using a sum of exponential functions, to describe muon production
e.g.,.
The advantage of the scheme is that it
tries to capture the physics of muon passage through the near surface, and
specifically the scheme models how the mean energy of muons increases as one
moves to greater depths in the subsurface. This affects muon production at
depth in a way that is not captured by exponential approximations. Recent
work by has updated the scheme of
to reflect
the muon production rates inferred from field studies. This method still has
the disadvantage that it is computationally expensive, to the extent that
this computational cost is prohibitive if one is to calculate muon production
in numerous pixels across a catchment.
Our approach is to approximate muon production using a sum of exponential
functions
e.g.,.
This approach has the advantage of being computationally efficient, but it
does not reflect the physics of muon production and therefore does poorly at
capturing muon production at depths beyond a few meters. This is unlikely to
lead to large errors, however, because muon production makes up a very small
percentage of the overall nuclide production at the depths where the
physics-based models
diverge from the exponential models used in CAIRN. We specifically quantify
this difference in Sect. , finding that the
exponential approximation leads to differences between the physics-based
approximation that are relatively small: for a wide range of denudation rates
these differences are less than 2 %.
The exponential approximation for nuclide production used in CAIRN is
Pi(d)=Pi,SLHL∑j=03Si,jFi,je-dΛj,
where Pi,SLHL is the surface production rate
(atoms g-1 yr-1) at sea level and high latitude; Fi,j is a
dimensionless scaling that relates the relative production of neutron
spallation and muon production; Si,j is a dimensionless scaling factor
that lumps the effects of production scaling and shielding of cosmic rays;
d is a mass per unit area which represents the mass overlying a point under
the surface (typically reported in g cm-2), and Λj is the
attenuation length for reaction type j (g cm-2). The reaction types
are j=0 for neutrons and j=1–3 for muons; muons can be either slow
or fast. In general, production from muons relative to neutrons is greater in
landscapes with a high denudation rate or at low elevation
.
The depth d, called shielding depth, is related to depth below the surface
as
d=∫ζ-ηζρ(z)dz,
where ζ (cm) is the elevation of the surface, η (cm) is the depth
in the subsurface of the sample, z (cm) is the elevation in a fixed
reference frame and ρ (g cm-3) is the material density, which may
be a function of depth. For a constant density, d=ρη.
Solving the governing equation
The governing equation (Eq. ) has the following general form:
dCdt+p(t)C=g(t).
In our case, p(t) simply equals λi, which is a constant in this
case, and g(t) is equal to Pi, which is a function of t. Equations of
this form have the solution:
C=1h(t)∫h(t)g(t)dt+const,
where “const” is an integration constant and
h(t)=exp(∫p(t)dt)
which in the case of the governing equation reduces to
h(t)=eλit.
The term g(t) is equal to
g(t)=Pi,SLHL∑j=03Si,jFi,je-dΛj
The shielding depth, d, is a function of time:
d(t)=d0+∫t0tϵ(τ)dτ,
where τ is a dummy variable for time that is replaced by the limits
after integration. Here t0 is the initial time and d0 is the initial
shielding depth. In the case where denudation, denoted ϵ
(g cm-2 yr-1), is steady in time this becomes
d(t)=d0+ϵ(t0-t).
Here denudation is the rate of removal of mass from above the sample per unit
area. If we let the concentration of the cosmogenic nuclide equal C0 at
the initial time, t0, and combine Eqs. (),
(), (), and (), we can solve for
the integration constant (const) and arrive at a solution for cosmogenic
nuclide i at time t:
Ci(t)=C0e-(t-t0)λi+Pi,SLHL∑j=03Si,jFi,jΛi,jϵ+Λi,jλe-d0Λi,jeϵ(t-t0)Λi,j-e-(t-t0)λ.
Equation () is the full governing equation from which
scenario-specific solutions may be derived.
Steady-state solution
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 t0=0 and t=∞. This results in a simplified form:
Ci(d)=Pi,SLHL∑j=03Si,jFi,jΛi,je-d/Λi,jϵ+λiΛi,j,
where ϵ is the denudation rate (g cm-2 yr-1). If we set
d=0 (that is, we solve for material being eroded from the surface, with
no distributed mass loss via chemical weathering),
Eq. () reduces to Eq. (6) from
for denudation only (i.e., no burial or
exposure), and reduces to Eq. (8) of if production is
due exclusively to neutrons. If Eq. () is simplified
to neutron only production, assumes the sample is taken from the surface (d=0), and is solved for erosion rate, one arrives at
ϵ=ΛiPi,SLHLSiCi-λi,
which is equivalent to the widely used Eq. (11) from .
However Eq. () requires adjustment for catchment
averaged estimates of denudation rates because each point in the landscape
from which sediment is derived will have its own local production and
shielding factors. This is why a spatially distributed approach is required.
Snow and self shielding
Equation () is restrictive in that it only considers
material removed from a specific depth, i.e. removed for a single value of
d. In reality samples may come from a zone of finite thickness. This finite
thickness can contribute some shielding to the sample, i.e. the bottom of a
sample is shielded by the mass of the sample that overlies. This shielding is
called self shielding and is generally implemented by assuming that self
shielding can simply be approximated by a reduction in neutron production
e.g.,. Snow can also
reduce production of cosmogenic nuclides
e.g.,. Typically these two forms of
shielding (snow and self) are incorporated in denudation rate calculators as
a scaling coefficient calculated before solving the governing equations
e.g.,, i.e. snow and
self shielding are incorporated into the Si,j term.
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 (ds) to the thickness of
the removed material (dt). Both ds and dt are
shielding thicknesses, therefore they are in units of g cm-2 and thus
differences in material density are taken into account. The depth-averaged
concentration is then
Ci(d)=Pi,SLHLdt∑j=03Si,jFi,jΛi,j2e-ds/Λi,j-e-(ds+dt)/Λi,jϵ+λiΛi,j.
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. ()
allows some flexibility so that future users can explore different erosion
scenarios, for example removal of sediment through mass wasting. We discuss
this in Sect. , but for the current contribution
we focus on steady-state scenarios.
Topographic shielding
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 . We adopt the
method of , in which both the effect of
dipping sample surfaces and shielding by topography blocking incoming cosmic
rays are computed. The method is spatially
distributed: each pixel in a digital elevation model (DEM) has its own
topographic shielding correction that varies from 0 (completely shielded) to
1 (no topographic shielding). These correction values are calculated by
modeling shadows cast upon each pixel in the DEM from every point in the
sky. This is achieved by modeling shadows incrementally for a range of
zenith (ϕ) values from 0 to 90∘ and azimuth (θ) values
from 0 to 360∘.
As Δθ and Δϕ values decrease, the accuracy with which
the shielding is calculated is expected to increase, as we are modeling
shielding at finer resolutions. However, this benefit is attenuated by
increasing computational cost when these values tend towards (1∘,
1∘). compared the accuracy of
different Δθ and Δϕ by comparing them to a minimum step
size of (5∘, 5∘). Here we exploit the efficiency of our
software and the considerable increase in computing power since 2006 to
explore smaller step sizes. We make the assumption that a step size of
(1∘, 1∘), corresponding to 32 400 iterations of the
shielding algorithm, is an accurate representation of the true shielding
factor to the extent that any further refinement in the measurements would
not yield a significant change in the results of the cosmogenic nuclide
calculations.
Absolute maximum residuals (i.e., greatest residual within the DEM)
for different combinations of Δθ and Δϕ used in
shielding calculations for a high relief basin in the Himalayas.
Δθ (degrees)
Δϕ (degrees)
1
2
3
5
8
10
15
30
45
60
1
0.000
0.002
0.004
0.009
0.010
0.011
0.027
0.053
0.063
0.081
2
0.004
0.004
0.005
0.009
0.010
0.012
0.029
0.057
0.064
0.080
3
0.008
0.008
0.008
0.010
0.011
0.012
0.027
0.053
0.062
0.081
5
0.014
0.015
0.016
0.017
0.018
0.018
0.030
0.056
0.065
0.087
8
0.023
0.023
0.026
0.025
0.027
0.030
0.039
0.064
0.082
0.093
10
0.036
0.037
0.033
0.040
0.035
0.040
0.037
0.063
0.074
0.104
15
0.057
0.059
0.058
0.060
0.060
0.058
0.065
0.084
0.100
0.122
20
0.072
0.071
0.073
0.075
0.077
0.076
0.083
0.111
0.109
0.138
30
0.171
0.172
0.168
0.176
0.167
0.167
0.173
0.188
0.160
0.242
45
0.337
0.340
0.332
0.335
0.346
0.335
0.332
0.393
0.385
0.430
60
0.352
0.352
0.352
0.352
0.352
0.352
0.352
0.352
0.385
0.418
In order to determine the optimal balance between measurement accuracy and
computational efficiency, the full range of (Δθ, Δϕ)
pairs were used to derive shielding values for each cell of a worst-case
scenario: a high-relief section of the Himalaya (650 km2 with a 7000 m
range in elevation). Table presents the maximum absolute
residual value (the error of the pixel with the greatest error) for
topographic shielding of the corresponding step sizes when compared to the
shielding derived for (1∘, 1∘). Using values below
's suggested threshold of (5∘,
5∘) gives increasingly small returns for a larger computational
burden. We suggest that a (Δθ, Δϕ) pair of (8∘,
5∘), requiring 810 iterations, is an optimal value for any high
relief landscape, yielding a maximum absolute error in our test site of
0.018. On lower relief landscapes the (Δθ, Δϕ) values
could be increased to achieve the same level of accuracy. We note that these
data are determined using a 90 m resolution DEM, and errors will be higher
for finer resolution DEMs .
Our topographic shielding calculations rely on two approximations that can
lead to some uncertainty. First, the method of
assumes the horizon attenuates all cosmic
rays, and secondly the production of cosmogenic nuclides obeys a power law
relationship between the cosine of the zenith angle.
have shown these assumptions to be
inaccurate. In addition, the method does
not include changes to the flux penetration distance on the gradient of the
topographic surface e.g.,.
Thus our method, while precise, reflects a simplified model of the true
physics of topographic shielding.
Production scaling
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 , later modified by
, is the simplest and is referred to herein as
Lal/Stone. Later scaling models
have incorporated other parameters such as time-dependent geomagnetic field
variations, solar modulation, and nuclide-specific information, resulting in
a total of seven possible scaling models in the most recent CRONUS calculator
.
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
, including a
low-latitude, high-altitude site in Peru
suggest that the time-independent
Lal/Stone scheme performs similarly to the physics-based schemes presented in
and fits the data better than several other
scaling schemes
. For
these reasons, we scale production rates using the Lal/Stone scheme. This may
lead to some uncertainty because production rates are scaled by the intensity
of the Earth's geomagnetic field e.g.,, and
this intensity has been relatively high over the last 20 kyrs
, meaning that this
approximation could lead to some uncertainty in samples with slow denudation
rates. For example, a rock removal rate of 0.03 mm yr-1 would remove
60 cm in 20 kyrs, and most production of nuclides occurs in the top 60 cm
of rock . However, in cases with faster denudation
rates, the uncertainty introduced by assuming time-invariant production rates
is likely to be much smaller than other sources of uncertainty.
The Lal/Stone scaling scheme requires air pressure, whereas most published
studies include only elevation information. We follow the approach of
and convert latitude and elevation data to
pressure using the NCEP2 climate reanalysis data
. In certain areas, the ERA-40 reanalysis
has been shown to provide more accurate results and due
to CAIRN's open source design new models can be readily incorporated into the
software. Here we retain the NCEP2 reanalysis to better compare our results
with CRONUS-2.2. We note that if users deploy CAIRN as a spatial averaging
front end to online calculators, they should be vigilant to use the same air
pressure conversion method in both CAIRN and the online calculator.
Combining scaling and shielding
To calculate the concentration of a cosmogenic nuclide, the scaling factors
for each production pathway (Si,j) must be computed. Both topographic
shielding and production rate scaling are subsumed within the scaling terms
(Si,j), whereas snow and self shielding are computed separately (see
Sect. ). These scaling terms are not computed
for each production pathway, but rather are lumped into a single value. We
therefore need to compute the values of the individual scaling factors,
Si,j. To do this, we follow the method of
and calculate scaling factors using an
effective attenuation depth. This is necessary because, when considering
multiple production pathways, the scaling terms for individual production
mechanisms may vary depending on elevation, shielding, sample thickness, or
denudation rates. For example, muogenic pathways will contribute relatively
more to production when there is more shielding since muogenic reactions
penetrate deeper than spallation.
To determine the scaling terms for the individual production mechanisms
(Si,j), we first compute the total scaling at a location
(Stot), which we define as the product of the production rate
scaling (Sp) and the topographic shielding (St), that is
Stot = StSp. Production scaling (Sp)
is estimated using the Lal/Stone scaling scheme and St is
calculated using our topographic shielding algorithms. We then derive the
scaling factors for the individual production mechanisms, Si,j, by
employing a virtual attenuation length, Λv, in units of
g cm-2, following the method of :
Si,j=e-ΛvΛi
We must therefore calculate Λv based on Stot. The
individual production mechanisms must be set such that
Stot=∑j=03Si,jFi,j.
In Eq. (), Stot and Fi,j are known,
whereas Si,j are functions of Λv. We thus iterate upon
Λv, calculating Si,j using
Eq. () using Newton's method until
Eq. () converges on a solution for
Λv. Once the virtual attenuation length is solved, the
Si,j terms are then used in Eq. ().
Uncertainty propagation
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 . We do note that some
authors have used a Monte Carlo approach in determining cosmogenic
nuclide-derived denudation rates because parameter uncertainties can have
non-gaussian distributions e.g.,. CAIRN, at
present, does not implement a Monte Carlo uncertainty approach but rather
follows conventional Gaussian propagation of uncertainty.
Gaussian propagation of uncertainty
Uncertainties are calculated in terms of the denudation rate, ϵ, in
units of g cm-2 yr-1, so that no assumption about material
density is necessary. The standard deviation of the denudation rate,
sϵ, is calculated with
sϵ=∂ϵ∂x2sx2+∂ϵ∂y2sy2+…,
where sx is the standard deviation of x, sy is the standard deviation
of y, and so on. The variables x and y can represent any uncertain
parameter, such as the measurement uncertainty or the production rate of the
nuclide. All uncertainties (e.g., nuclide concentration) are assumed to be at
the one sigma level unless otherwise stated. The derivatives in
Eq. () are calculated using the nominal value plus the
associated uncertainty and then recalculating the denudation rate in the
original, pixel-by-pixel fashion.
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 (Pi,SLHL), and (iii) uncertainty in
muon production. Uncertainty in cosmogenic nuclide concentration is reported
by authors alongside concentrations. For the cosmogenic nuclide concentration
uncertainty, the concentration is used directly to determine the denudation
rate uncertainty. For all other parameters, the uncertainty values help to
predict a new concentration in each pixel, which is then used to determine
denudation rate uncertainty. It is important to note here that we do not
calculate uncertainties inherent in the basin-averaging approach which
assumes spatial homogeneity in source material and denudation rates, and
denudation that is steady in time; we address these uncertainties in
Sect. .
The uncertainty on the production rate (Pi,SLHL) is based on that
used in the CRONUS-2.2 calculator : in CRONUS-2.2
the uncertainty is 0.39 atoms cm-2 yr-1 for 10Be based on a
production rate of 4.49 atoms cm-2 yr-1. This means the
uncertainty in CRONUS-2.2 is 8.7 % of Pi,SLHL for 10Be.
We use this uncertainty for both 10Be and 26Al based on our
production rates reported in Table . Although the
recent CRONUS-Earth calibration has produced
new production rates for both 10Be and 26Al, the production rate
uncertainties remain in the same range as those used here
.
Field studies have shown that muon production based on laboratory experiments
overestimate
muon production observed in deep samples
;
there is still some uncertainty over the exact muon production profile. CAIRN
employs the exponential scaling method from
. It then calculates the upper bound of
uncertainty derived from muon models by calculating the difference between
the default CAIRN muon model and those from the
scheme, which approximates the original
Heisinger results
.
Uncertainty from snow shielding
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-2) or spatially distributed information in the form of a raster.
Most snow shielding calculations reported in the literature are based on an
effective attenuation estimated by the thickness of snow
e.g.,, but recent field-based
measurements indicate that snow may attenuate fluxes of cosmic rays to a
greater extent than assumed in simple mass-based snow shielding calculations
. However these uncertainties are
small compared to the extreme uncertainties of the thickness, extent and
duration of snow over millennial timescales, which are unlikely to ever be
well constrained. If no snow shielding values are provided, the software
assumes that there is no snow cover.
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 1σ uncertainty
(how an author might calculate this could fill another paper and is beyond
the scope of our study). The denudation rates of these two scenarios would
then be calculated, and the square of the difference in these two denudation
rates would then be inserted into Eq. (). In this way
users can calculate shielding uncertainties manually.
Summary of CAIRN parameters for denudation calculations
To summarize, CAIRN predicts cosmogenic nuclide production from neutrons and
muons using a four exponential approximation of data from
. These production rates are scaled using
Lal/Stone time-independent scaling. Production is calculated at every pixel,
with atmospheric pressure calculated via interpolation from the NCEP2
reanalysis data . Topographic shielding is
calculated using the method of , and scaled
production rates are multiplied by topographic, snow, and self shielding at
each pixel. Decay rates, attenuation lengths, and parameters for production
are reported in Table . Denudation rates are
reported in g cm-2 yr-1 because in these units no assumptions
about density, which is spatially heterogeneous, are required. In addition,
users must report the AMS standard when supplying nuclide concentrations to
CAIRN and the concentrations are then normalized following the same scheme as
. The CAIRN software prints these parameters to a
file so that if they change in the future based on new calibration data sets,
users will be able to both view and report these updated values.
Spatial averaging for ingestion by other denudation rate calculators
In addition to producing denudation rates, CAIRN also provides
spatially averaged production rates and effective catchment-averaged pressure
(see Sect. ), so that users can compute denudation
rates using other available calculators. Programs such as the CRONUS-Earth
calculators, referred to as CRONUS-2.2 for and
CRONUScalc for , and COSMOCALC do not have the
ability to calculate catchment-averaged parameters. CAIRN can be used
independently to determine production rates or in conjunction with these
other calculators, which allows for the possibility of using time-dependent
scaling and other new features in the future.
Conversion of depth-integrated parameters for calculator ingestion
CAIRN iterates on denudation rate until the predicted cosmogenic
concentrations from Eq. () is reached.
Equation () is a depth-integrated approach that is a
direct solution of the production equations. This depth-integrated solution
subsumes both snow and self shielding. This is different from COSMOCALC
and the CRONUS calculators, which take separate values for shielding. Thus to
pass results from CAIRN to calculators we must first calculate equivalent
snow and self shielding values for each pixel. Note that these values are not
used within denudation rate calculation in CAIRN, they are only used when
shielding values are passed to the COSMOCALC and the CRONUS calculators.
Self shielding used for spatial averaging is calculated for each pixel k
with
Sself,k=Λi,0dt,k1-e-dt,kΛi,0,
where Sself,k is the self shielding correction for the kth
pixel, dt,k is the shielding thickness for the kth pixel (in
g cm-2). Equation () is used in both COSMOCALC and
CRONUS. In the CRONUS calculators, snow shielding is lumped with topographic
shielding, therefore the CRONUS calculators presume the user will determine
the product of snow and topographic shielding at a site with a method of
their choice. COSMOCALC includes a snow shielding calculator which assumes
that the equivalent depth of snow (in g cm-2) attenuates neutron
production following the formula:
Ssnow,k=e-ds,kΛi,0,
where Ssnow,k is the snow shielding correction of the kth pixel
and ds,k is the time-averaged depth of snow water equivalent in
g cm-2. We adopt this approximation when performing spatial averaging.
Recent work suggests snow may attenuate spallation to a greater degree than
predicted by Eq. () , and
suggest that the attenuation length for snow is
reduced compared to rock (they report an attenuation length of
109 g cm-2 for snow). However, the uncertainty in historic snow
thickness vastly outweighs uncertainties from the snow shielding equation.
Although there have been methods suggested to model the evolution of snow
thickness through time e.g.,, the averaging
time for eroded particles that accumulate cosmogenic nuclides is on the order
of thousands to tens of thousands of years e.g.,,
and reconstructing snow thickness over this timescale is highly uncertain.
Users wishing to approximate the attenuation lengths
can feed CAIRN snow rasters with a thicker apparent snow layer. Overall, we
therefore recommend that users include a large range of snow thickness in
their uncertainty analysis, guided by historical observations of snow depth.
Spatial averaging for COSMOCALC
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 SCCtot, which is a lumped
shielding and scaling term, with
SCCtot=1N∑k=0NSsnow,kStopo,kSself,kSi,k,
where terms are calculated on a pixel-by-pixel basis. Snow shielding is
calculated from Eq. (), self shielding is calculated from
Eq. (), and topographic shielding is calculated
accounting for the effects of sloping samples and topography blocking cosmic
rays (see Sect. ). We wish to emphasize that CAIRN
reports SCCtot for users that wish to use it in COSMOCALC, whereas
the denudation rates reported by CAIRN use Eq. () for
snow and self shielding. Production scaling for cosmogenic nuclide i at
pixel k, Si,k, is calculated using Eq. () and
Lal/Stone scaling (Sect. ).
Spatial averaging for the CRONUS calculators
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, SCRshield, is
calculated with
SCRshield=1N∑k=0NSsnow,kStopo,kSself,k.
Note that we fold the self shielding into the lumped shielding term so that
when transferring data to the CRONUS calculator the sample thickness should
be set to 0.
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
Seffp=1N∑k=0NSi,k.
We then use the Newton iteration on the Lal/Stone scaling scheme to find the
pressure which reproduces the basin average production rate
(Seffp). That way, results from our method can be compared to
results from the CRONUS calculator and, if users are so inclined, they can
use time varying production scalings via the CRONUS calculator (which CAIRN
does not include for reasons outlined in Sect. ).
Uncertainties introduced by spatial and temporal variability
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
e.g.,
versus climate oscillations that are tens to hundreds of thousands of years
e.g.,. In addition, spatial
heterogeneity in lithology and target mineral concentrations can lead to
additional uncertainty to denudation rate estimates
e.g.,. Mass wasting can
also perturb the concentration of cosmogenic nuclides
e.g.,, leading to
further uncertainties. Finally, as noted in
Sect. , if snow shielding is to be taken into
account, one must estimate the shielding provided by snow over millennial
timescales, which, to put it mildly, are difficult to constrain.
For the problem of spatially heterogeneous lithology, careful geologic
mapping, such as that done by a handful of recent authors
e.g.,,
can alleviate some of the uncertainty, but such mapping is logistically
challenging. For landsliding, mass removal can be measured in the field,
modeled e.g.,, or
approximated using mapped landslide inventories
e.g.,. These may be
combined with data on landslide area–volume relationships
e.g.,. The main difficulty here is that it
takes some time for the cosmogenic nuclide concentration to readjust after
mass removal
e.g.,
and thus one must make some estimate of not only the spatial distribution of
landslides but their evolution through time .
Simulating nuclide concentrations in settings where denudation rates vary in
space and time is possible , but computationally
intensive and one must have some confidence that one can accurately
reconstruct the temporal evolution of denudation rates. Although recent
progress has been made in deriving time series of denudation rates from
current topography
e.g.,,
these methods still suffer from the fact that we lack devices for time travel
and struggle to test such reconstructions.
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 estimates that
any catchment-averaged denudation rate carries with it a minimum 30 %
uncertainty. Because the uncertainties mentioned in this section are
difficult, if not impossible to constrain, our approach with CAIRN is to
report the uncertainties that can be constrained and caution users that there
are large additional unconstrained uncertainties related to the assumptions
underpinning the method.
Method comparison
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. . In our comparisons we use seven published
cosmogenic data sets (Table ). These data sets
were chosen to span a wide range of locations (i.e., differing latitudes and
elevations) and denudation rates. The parameters used by CAIRN for these
comparisons are reported in Table .
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. 10Be production rate
(Prod rate) is given for sea level, high latitude and in units of
atoms g-1 yr-1. “CR” or “CR muons” refers to the spallation
or muon calculation methods and production rates used in CRONUS-2.2
. The scaling values, production rates,
topographic shielding and notes reported in this table are for the original
studies: CAIRN uses the same settings (see Table )
for its calculations regardless of site location.
Study
Location
Scaling
Prod. rate
Topo. shielding
Other notes
New Mexico, USA
Lal/Stone
5.2
None
ρ=2.7 g cm-3, no muons.
Colorado, USA
Lal/Stone
4.49 (CR)
None
ρ=2.7 g cm-3, fast muons only.
Idaho, USA
Lal/Stone
4.72
, details not given.
Corrections for chemical weathering.
Ladakh, India
Lal magnetic
4.49 (CR)
Pixel-by-pixel, but details not given.
CR muons. Snow and iceshielding considered.
Tibet
5.12
,
Muons using
and
Δϕ, Δθ not reported.
scheme. ρ=2.65 g cm-3.
Bolivia
None
No muons. ρ not reported. Corrections for quartz fraction.
Garwahl Himalaya
Lal magnetic
4.49 (CR)
Pixel-by-pixel, but details not given.
CR muons. Snow and iceshielding considered.
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 ). Choices of spallation and muon
production rates also affect the final denudation rate. Consider a measured
nuclide concentration that one uses to infer a denudation rate. If one
assumes a high production rate (via either muons or spallation), it means
that for a given denudation rate the predicted nuclide concentration is
higher. Thus, for a given nuclide concentration, the inferred denudation rate
is higher if the assumed production rate is higher (see dashed lines in
Fig. ). If the inferred shielding is
higher, then for a given denudation rate the production is lower, and the
inferred denudation for a given concentration will be lower.
Spatial averaging of production and shielding vs. pixel-by-pixel calculations
First, we compare results of two methods using the exponential approximation
of muon production (Eq. ), used in both COSMOCALC and
the CAIRN calculator. The difference in calculating denudation rates by
iterating upon cosmogenic nuclide concentration from all pixels in a basin
(the CAIRN method) and calculating it by using a spatial average of the
production of scaling and production terms
(Eq. ) is virtually zero if snow and self
shielding are spatially homogenous
(Fig. a). Thus we find that combining
all scaling and shielding terms in a single lumped term is adequate for
calculating denudation rates if computational power is limited.
Differences between the denudation rate calculated by CAIRN
(ϵCAIRN) and the denudation rate using the production factor
(SCCtot) (which includes production scaling and shielding) passed
to COSMOCALC (ϵCC) (a), and differences between the
denudation rate calculated by CAIRN (ϵCAIRN) and the
denudation rate using separate spatial averages for shielding and production
scaling that are then averaged (ϵCC-CRONUS) as a function of
production factor (b). In this case the production factor is
calculated by multiplying the separately averaged shielding
(SCRShield) and scaling (Seffp) factors. This approach
emulates the data requirements for CRONUS-2.2, which calculates production
scaling and accepts a single shielding factor (for snow and topography
combined). Although the shielding and scaling emulate data requirements for
CRONUS-2.2, the denudation rate is calculated using the exponential
production method of CAIRN and COSMOCALC.
Separating production rate scaling from shielding leads to slightly larger
uncertainty (Fig. b), but in terms of
the total uncertainty this averaging also leads to small uncertainties (on
the order of 1–2 % compared to 10–20 % from other sources of
uncertainty). We suspect that many users will want to compare rates
determined by our software with the popular CRONUS calculators
. The CRONUS calculators
internally scale production rates while shielding is supplied by the user.
Consequently, the uncertainties plotted in
Fig. b approximate uncertainties arising
from the spatial averaging process that users must pass to the CRONUS
calculators. Some users may wish to calculate denudation rates using
time-dependent scaling schemes, which is not possible in CAIRN, but CAIRN can
be used as a front end to the CRONUS calculators via its spatial averaging
capabilities with the confidence that this will only introduce relatively
small errors.
Comparison with existing denudation rate estimates
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
cf.,. Studies differ
in their strategies for production rate scaling, topographic, snow, and self
shielding, and the manner in which spatial averaging is performed. In many
cases there is insufficient detail reported that might enable other groups to
reproduce reported denudation rates. A primary motivation behind CAIRN is to
provide an open-source means of computing denudation rates that may then be
reproduced by other groups. We have incorporated reported snow shielding from
previous studies by inverting Eq. () for an annual
average snow thickness and then distributing this thickness over the entire
DEM. We acknowledge this is a poor representation of snow thickness but snow
shielding rasters are rarely available and in most cases there is little
reported snow shielding.
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
.
High-latitude production rates under Lal/Stone scaling of 10Be have
changed in the last 10 years due to an ever increasing number of calibration
sites e.g., and changing AMS standards
. In some cases, muons are not considered,
whereas other studies use a variety of different muon production schemes
(e.g., Table ). Topographic shielding is
occasionally not considered (particularly in older studies). In some cases
the horizon elevation is recorded from a limited number of directions (e.g.,
COSMOCALC includes a calculator using 8 directions), and in other instances
the computational method of is used.
Studies also cite for shielding but this paper
lists several methods for calculating shielding: the equations therein depend
on the number and geometry of shielding objects and this information is
seldom reported. Even when the more robust method of
is used, the spacing of azimuth and angle
of elevation is often not reported.
Topographic shielding (St) calculated using Δϕ=5∘, Δθ=8∘ plotted as a function of reported
shielding.
Comparison of the topographic shielding for different values of
Δϕ and Δθ. The Tibetan basin is for sample 07C13 in
. Maps are projected into WGS1984, UTM
zone 47N. The basin is shown in plot (a), whereas the topographic
shielding factor is shown in plots (b) and (c).
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 ). Because denudation rates are
traditionally reported in dimensions of length per time, we do not suggest
future authors cease reporting denudation in these dimensions, but we do
recommend also reporting denudation rates in dimensions of mass per area per
time (e.g., g cm-2 yr-1) because these units allow simpler
comparison between sites as they require no assumptions about spatially
heterogeneous density.
Of our seven example data sets (Table ), only
three of the original authors reported topographic shielding factors. We calculated
shielding using the CAIRN method with Δϕ=5∘,
Δθ=8∘ in these three high relief landscapes using a
90 m resolution DEM. Our small values of Δϕ and Δθ
lead to variations in shielding between CAIRN and reported values
(Fig. ). Authors typically do not give
enough information to reproduce their shielding calculations, but we note
that authors that employ the equations of use a
limited number of horizon measurements to calculate shielding. For example in
COSMOCALC , users are expected to input
horizon values at 45∘ intervals. Our calculations suggest that this can
lead to lower maximum shielding differences between this method and the CAIRN
method (Table ). An example of the potential
underestimates of topographic shielding is shown in
Fig. .
The denudation rates predicted by CAIRN are plotted against reported
denudation rates in Fig. . These data are scattered
about the 1 : 1 line, but for most samples the CAIRN denudation rate is
lower than the reported denudation rate. Reasons for this vary since the
method used to calculate denudation rates vary in each example study, but
differences are likely to be due to the higher production rates used in
previous studies (Table ) and slightly greater
topographic shielding in CAIRN (see
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. ). In addition
previous studies contain elements modulating denudation rates that are not
contained within the current version of CAIRN. For example,
reports true physical erosion rather than
denudation and modified their denudation rates
based on the quartz content of the source areas.
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 . The
results from CAIRN in this plot use a density of 2.65 g cm-2.
Differences between the denudation rate calculated by CAIRN
(ϵCAIRN) and the denudation rate calculated with CRONUS-2.2
(ϵCR2.2) as a function of CAIRN denudation
rate (a), and differences between the denudation rate calculated by
CAIRN (ϵCAIRN) and the denudation rate calculated with
CRONUS-2.2 (ϵCR2.2) as a function of the total scaling,
Stot (b).
Difference between denudation rate calculated by CAIRN
(ϵCAIRN) and the denudation rates calculated by CRONUS-2.2
(ϵCR2.2), but with CRONUS-2.2. parameters updated to have
spallation and muon production reflecting production in CAIRN, which is based
on . Data are from the
study.
Comparison with the CRONUS calculators
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. a) but are instead mainly a function of
the production rate (Fig. b). In the previous
section, we found that differences due to spatial averaging and separation of
shielding from production scaling are small. The large difference is
primarily due to the difference in spallation production rates and the
over-production of muons in CRONUS version 2.2, as described by
. According to
, future versions of this CRONUS
calculator will be updated to have significantly reduced muogenic production
consistent with recent studies
.
If production rates in CRONUS are changed to reflect the production rates
from , we find that differences are quite
small (Fig. ). We see from this
figure that in locations with high production rates just under half of these
differences between CAIRN and CRONUS-2.2 are from the different spallation
rates, whereas in locations with low production rates, most of the
differences are due to the higher muon production present in CRONUS-2.2.
Production rates of 10Be as a function of depth for muons
only (a) and total production (b). These production rates
are calculated using the Lal/Stone scaling at 70∘ N and with a
pressure of 1007 hPa (near sea level). Note the logarithmic depth scale:
eroding particles spend a large amount of their exposure history below
100 g cm-2 and so increased muon production at these depths, despite
being a small fraction of the total production, plays a significant role in
determining the total nuclide concentration (see
Fig. ).
Concentrations as a function of denudation rate (a) and the
fractional differences between the predicted concentration from the
approximation used in CAIRN and both
CRONUS-2.2 and CRONUScalc
(b). These concentrations are calculated for a
hypothetical site at 70∘ N and near sea level (1007 hPa). Note that
although the default production scheme in CAIRN is the
scheme, the production from CRONUScalc
can also be used (see
Table ).
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
. In order to examine the
underlying source of discrepancies between the three calculators, we plot the
total and muon production rates for the CAIRN, CRONUS-2.2, and CRONUScalc
calculators in Fig. . The production rates
for CRONUS-2.2 are calculated directly from the MATLAB scripts available
online. The CRONUScalc production rates are approximated as a three
exponential analytical function with parameters shown in
Table . Although total production rates appear
relatively similar, CRONUScalc and CAIRN predict significantly smaller muon
contributions that CRONUS-2.2. The result is that for the same denudation
rate, the CRONUS-2.2 calculator produces significantly more (in some cases
40 % more) atoms than using CAIRN or CRONUScalc
(Fig. ) leading to a large
discrepancy in calculated denudation rates between CRONUS-2.2 and the other
two calculators (CAIRN and CRONUScalc), which both incorporate more recent
muon production rates. The CAIRN outputs of topographic shielding, as well as
the spatial averaging of both production scaling and shielding, are
independent of these calculators and will still provide spatial averaging for
use with future calculator versions, even as production rates and mechanisms
are updated.
Parameters used for production of 10Be which approximate the
scheme in CRONUScalc .
λ10Be values are the same as defaults listed previously.
The Fi values represent spallation and fast and slow muons, respectively.
Parameter
Value
Λi
160; 1460; 11 040 g cm-2
10Be PSLHL
4.075 atoms g-1 yr-1
10Be Fi
0.9837; 0.0137; 0.0025 (dimensionless)
Differences between the denudation rate calculated by CAIRN
(ϵCAIRN) and the denudation rate calculated with CRONUScalc
(ϵCRCalc) as a function of CAIRN denudation rate for
selected studies.
Differences between the denudation rate calculated by CAIRN using
the parameters in Table to approximate
CRONUScalc production (ϵCAIRN-CRCalc) and the denudation
rate calculated with CRONUScalc (ϵCRCalc) as a function of
CAIRN denudation rate for selected studies.
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. ).
Currently CRONUScalc is not able to calculate very high denudation rates (for
rates greater than ∼ 0.06 g cm-2 yr-1 the current version
of CRONUScalc crashes; it was designed for exposure ages and becomes
computationally unstable at high erosion rates) so we cannot compare CAIRN to
CRONUScalc for all of the example data sets. The differences in
Fig. arise from two sources: first, we
must pass the product of the scaling (Seffp) and shielding
(SCRshield) to CRONUScalc rather than calculating pixel by pixel
values. Second, the default muon production in CAIRN is derived from the
scheme, which is slightly different than
the production schemes derived from and
(see Fig. ).
In CAIRN, users can choose the muon production scheme, and we have
implemented an approximation of the muon production scheme from
that uses the exponential form of
Eq. () (see
Table ). It is important to note that the CAIRN
implementation of muons from assumes that
Λ=160 g cm-2 for spallation, whereas in CRONUScalc this
attenuation length can vary as a function of latitude and pressure. We
compare the denudation rates from CAIRN using the production parameters in
Table (ϵCAIRN-CRC) with the
default production scheme of in
Fig. . The differences here are smaller
(mostly less than 2 %) suggesting that much of the difference seen in
Fig. is due to spatial averaging.