Cosmogenic radionuclides (CRNs) are the standard tool to derive
centennial-to-millennial timescale denudation rates; however, it has been
demonstrated that chemical weathering in some settings can bias CRNs as a
proxy for landscape denudation. Currently, studies investigating CRN
weathering biases have mostly focused on the largely insoluble target
mineral quartz in felsic lithologies. Here, we examine the response of CRN
build-up for both soluble and insoluble target minerals under different
weathering scenarios. We assume a simple box model in which bedrock is
converted to a well-mixed regolith at a constant rate, and denudation occurs by regolith erosion and weathering either in the regolith or along the regolith–bedrock interface, as is common in carbonate bedrock. We show that weathering along the regolith–bedrock interface increases CRN concentrations compared to a no-weathering case and how independently derived weathering rates or degrees can be used to correct for this bias. If weathering is concentrated within the regolith, insoluble target minerals will have a longer regolith residence time and higher nuclide concentration than soluble
target minerals. This bias can be identified and corrected using paired-nuclide measurements of minerals with different solubility coupled with
knowledge of either the bedrock or regolith mineralogy to derive denudation
and long-term weathering rates. Similarly, single-nuclide measurements on
soluble or insoluble minerals can be corrected to determine denudation rates if a weathering rate and compositional data are available. Our model
highlights that for soluble target minerals, the relationship between
nuclide accumulation and denudation is not monotonic. We use this
understanding to map the conditions of regolith mass, weathering, and
denudation rates at which weathering corrections for cosmogenic nuclides
become large and ambiguous, as well as identify environments in which the
bias is mostly negligible and CRN concentrations reliably reflect landscape denudation. We highlight how measurements of CRNs from soluble target minerals, coupled with bedrock and regolith mineralogy, can help to expand the range of landscapes for which centennial-to-millennial timescale
denudation and weathering rates can be obtained.
Introduction
Denudation, the sum of physical erosion and chemical weathering, is a
critical parameter in understanding landscape evolution
(Darwin, 1859; McLennan, 1993). Cosmogenic
radionuclides (CRNs) have become the standard method to quantify denudation
on centennial-to-millennial timescales and on spatial scales from outcrop to
river basin (von Blanckenburg, 2005; Portenga and Bierman, 2011). The build-up of CRNs in minerals from eroding landscapes is sensitive to the rate of mass removal from above (Lal, 1991). This makes
the concentration of CRNs in the Earth's near surface sensitive to physical
erosion as well as chemical weathering (Dixon et al., 2009; Riebe and
Granger, 2013). However, weathering at depths below the attenuation length
of CRNs, or measuring CRN concentrations of minerals with a weatherability
that differs from the bulk rock, can lead to biases in the calculation of a
denudation rate from a measured CRN concentration (Dixon
et al., 2009; Riebe et al., 2001; Riebe and Granger, 2013; Small et al.,
1999).
Previous studies investigating the effects of chemical weathering on CRNs,
hereafter just weathering, mostly focused on 10Be measured on quartz in
granitic landscapes (Dixon
et al., 2009; Riebe et al., 2001, 2003; Riebe and Granger, 2013; Small et
al., 1999). These studies show that quartz as a quasi-insoluble mineral is
enriched in the saprolite and mobile regolith, whereas more weatherable
minerals are depleted. This implies that insoluble minerals reside longer in
the mobile regolith compared to soluble minerals and the bulk rock. The
longer residence time in the production zone leads to an increase in the
quartz 10Be concentration that, without correction, results in an
underestimation of the total denudation rate. However, the enrichment of the
insoluble mineral, commonly quantified by the ratio of zirconium in the
regolith/saprolite compared to the bedrock, allows correction of the
denudation rates for weathering (Dixon
et al., 2009; Riebe et al., 2001; Riebe and Granger, 2013). This
conceptualization of weathering biases has been developed and applied in
granitic landscapes but has not been expanded to CRN measurements on the
soluble target mineral or other weathering processes as they might occur
in other regions subjected to weathering, e.g. limestone areas.
Carbonate regions are underrepresented in tectonic geomorphology studies
despite representing ∼ 18 % of Earth's surface
(Dürr et al., 2005), partially because measuring 10Be
in quartz-rich lithologies has become the standard for catchment-average
denudation rate measurements (von Blanckenburg, 2005).
Yet, carbonate regions are interesting because their topography is highly
sensitive to the interplay between tectonics and climate, owing to their
greater chemical reactivity relative to more silica-rich rocks, and they
exert a significant influence on flora and fauna distributions
(Ott, 2020). The number of studies measuring 36Cl on
calcite in carbonate catchments for denudation rates is increasing (Avni
et al., 2018; Ott et al., 2019; Ryb et al., 2014a, b; Thomas et al.,
2017), and therefore the effect of weathering on soluble target mineral CRN
concentrations needs to be assessed. This effort is worthwhile because it
will allow for the extension of tectonic geomorphology and
centennial-to-millennial timescale denudation rate studies in a significant
portion of the globe that has been traditionally understudied. Moreover, the
sensitivity of cosmogenic nuclide measurements to the weathering rate and
depth will, in theory, allow for measurements from minerals with different
weatherability to constrain long-term weathering rates and depths in various
geologic settings.
In this study, we investigate the theoretical effects of weathering on CRN
concentrations in regions of substantial weathering, e.g. limestone areas,
as well as how paired-nuclide measurements can be exploited to constrain
denudation and weathering rates on millennial timescales. We investigate the
effects of weathering within the regolith, as well as preferential
weathering along the regolith–bedrock interface, on the CRN concentrations
of soluble and insoluble target minerals. We derive the equations for the
different weathering corrections and outline which practical measurements,
for instance, water chemistry or paired-nuclide measurements, can be used
for the corrections. We highlight the non-monotonic relationship between
nuclide concentration and denudation rate for near-surface weathering of
soluble target minerals. Carbonate landscapes can have a high ratio of
weathering to erosion, which can lead to large correction factors.
Therefore, we identify the conditions under which denudation and weathering
rates can be reliably estimated with acceptable uncertainty and the
environments where detailed data are needed to correct the chemical
weathering bias. This paper also includes a MATLAB™ software package “WeCode” (Weathering Corrections for
denudation rates) (Ott, 2022) integrated within CRONUScalc v2.1 (Marrero et al., 2016) to perform all weathering corrections and calculations, as well as offering pixel-by-pixel catchment production rate estimates for alluvial samples.
Weathering can be focused at different depths below surface. (a) Australia. Weathering focused mainly at the sharp regolith–bedrock interface. The soil consists mostly of residuum from the limestone bedrock together with aeolian dust. (b) Kansas, USA. Clasts released from the limestone bedrock at Flint Hills gradually decrease in size towards the surface. No clear regolith–bedrock interface is visible, indicating that weathering likely concentrates within the regolith.
Summary of previous work
Small et al. (1999) noted the
effect of near-surface weathering on the concentration of 10Be in
quartz through the enrichment of weathering-resistant quartz in the regolith
compared to the bedrock. Riebe et al. (2001)
showed that the concentration ratio of immobile elements such as [Zr] in the
regolith and bedrock could be used to account for these weathering biases,
assuming that the enrichment of [Zr] is the same as for quartz. Dixon et al. (2009) highlighted that weathering below the CRN attenuation
length would also underestimate the total denudational flux because the CRNs
are only sensitive to near-surface mass removal. They show that total
denudation rates from granitic areas in the Sierra Nevada underestimate the
total denudational flux due to deep weathering by up to a factor of 3.
Riebe and Granger (2013) integrate the
effects of weathering above and below the attenuation length with a box
model of bedrock, saprolite, and soil and show that denudation rates in
Puerto Rico were biased by up to 70 % in comparison to a no-weathering
assumption.
In this study, we follow Riebe and Granger (2013) and use a simple box model
to quantify and account for the weathering bias on CRN concentrations.
Because we focus on the application to carbonate terrains, we will neglect
saprolite and instead apply a model of bedrock overlain by a layer of mobile
regolith. Riebe and Granger (2013) show
that in such cases, the target mineral, depending on its weatherability,
gets either depleted or enriched in the regolith compared to the bulk rock;
this will alter the regolith residence time and thereby the average nuclide
concentration as follows:
〈NR〉=N0+NR,nW×XRXB,
where 〈NR〉 is the average CRN concentration in a well-mixed regolith, N0 is the nuclide concentration entering the regolith from the bedrock below, NR,nW is the nuclide concentration accumulated in the regolith in a no-weathering scenario, and XR/XB is the ratio of the
target mineral in the regolith compared to the bedrock that quantifies the
change of mineral residence time compared to the bulk regolith. The
aforementioned studies focused on the enrichment of quartz as an insoluble
mineral. In this study, we also explore the effects on soluble target
minerals, such as calcite, for application to carbonate landscapes.
Theory and model
Some carbonate regions exhibit weathering mostly along the regolith–bedrock
interface (Fig. 1a), whereas in other regions, weathering is focused within
the regolith (Fig. 1b). Therefore, we investigate weathering both within the
regolith and along the regolith–bedrock interface. We refer to regolith as
the entire mobile layer above the coherent bedrock. We neglect weathering
below the attenuation length of CRNs because experiments have shown that the
majority of dissolution in limestone regions occurs close to the surface or
at the top of bedrock (Gunn, 1981) and that deep mass removal is
comparatively negligible in relation to near-surface weathering
(Worthington and Smart, 2004). First, we show the equations
for the CRN concentration of a target mineral in the standard case of no
weathering; second, we show the effect of regolith–bedrock interface
weathering; and finally, we highlight the impact of regolith weathering on CRN
concentrations.
Model conceptualization and no-weathering case
Analogous to previous studies, we conceptualize denudational processes with
a box model comprising a layer of mobile regolith on top of unweathered
bedrock (Fig. 2). The bedrock either weathers at the regolith–bedrock
interface with the bedrock weathering rate WB or gets converted to regolith at the bedrock erosion rate EB. The regolith is assumed to be well-mixed and therefore has an average production rate 〈PR〉 and nuclide concentration 〈NR〉. Material is exported from the regolith either by
regolith erosion (ER) or by weathering within the regolith (WR). For simplicity, we assume a flux steady state between denudation (D) and the outgoing fluxes (WR, ER, WB), which results in a constant regolith
mass (mR) through time. Table 1 lists and defines all model variables.
Conceptualization of denudation as a simple box model, where the
regolith–bedrock interface lowers at the denudation rate D, by weathering
(WB) and erosion (EB). CRNs are being produced within the regolith of thickness (mR) at an average rate of 〈PR〉, resulting in an average nuclide concentration 〈NR〉, while mass is leaving the regolith by regolith erosion (ER) and weathering (WR).
Variable definition.
SymbolNameUnitNCRN concentrationat. g-1DDenudation rate (total mass loss rate per unit area)g cm-2 ka-1EBBedrock erosion rateg cm-2 ka-1ERRegolith erosion rateg cm-2 ka-1WBRegolith–bedrock interface weathering rateg cm-2 ka-1WRRegolith weathering rateg cm-2 ka-1ΛAttenuation lengthg cm-2ρDensityg cm-3mRRegolith mass per unit areag cm-2mgGrain mass per unit areag cm-2miInitial grain mass (after release from bedrock) per unit areag cm-2kGrain mass weathering constantka-1PProduction rateat. g-1 ka-1iSubscript for production pathwayXB, XRMineral fractions in bedrock and regolithunitlessCaB, CaRFraction of calcite in bedrock and regolithunitlessXB, XRFraction of non-quartz insolubles in bedrock and regolith (e.g. clay)unitlessQB, QRFraction of quartz in bedrock and regolithunitlesstTimekaτRAverage regolith residence timeka
The production rate of cosmogenic nuclides declines with depth, and the
individual production profiles of different production pathways (spallation,
fast muons, negative muons, etc.) are commonly approximated by exponentials
(Braucher et al., 2011, 2013; Granger and Smith, 2000). Exponential production profiles for muons can result in significant biases for certain denudation rate calculations (Balco, 2017) but have the advantage
of being simple in their mathematical formulation. The WeCode software
package utilizes a full depth integration of the muon flux
(Marrero et al., 2016); however, for demonstration purposes, we show equations using exponentials in Sect. 3.1 and 3.2.
We present the equations in the main text without decay because the
equations with and without decay describe the same general behaviour, but
the equations without decay are easier to follow. The equations with decay
are presented in the Supplement, and WeCode includes radioactive decay in
all calculations.
No-weathering case
In the absence of weathering, the weathering fluxes are zero, such that
WR and WB= 0 and D=EB=ER. The average
nuclide concentration in the regolith is the sum of the nuclide
concentration at the regolith–bedrock interface and the production within
the regolith. This results in
〈NR〉=N0+〈PR〉×τR,
with
N0=∑iPi(0)ΛiDe-mR/Λi
describing the nuclide concentration at the regolith–bedrock interface, and
with
〈PR〉=∑iPi(0)ΛimR1-e-mR/Λi
describing the average production rate within the well-mixed regolith, and
the regolith residence time:
τR=mRD.
The percentage of CRN concentration increase due to weathering at
the regolith–bedrock interface compared to a no-weathering scenario. The
percentage of CRN increase is plotted for different fractions of
regolith–bedrock weathering with respect to denudation and different regolith
masses. Contour lines every 20 %; above 200 % (thick line) the contours are every 100 %.
Weathering at regolith–bedrock interface
If weathering occurs exclusively at the regolith–bedrock interface, the
denudation rate will be D=EB+WB. Sediment comes into the regolith at flux EB with the same concentration N0 as in the no-weathering case. Therefore, the sediment flux into and out of the
regolith goes down for a given D due to weathering at the regolith–bedrock
interface. The modified regolith residence time is
τR=mREB=mRD-WB.
This expression states that τR increases by a factor of
DB/EB compared to the no-weathering case. At steady state, the
regolith CRN concentration is
〈NR〉=N0+〈PR〉×mRD-WB.
Because EB<D, the CRN concentration increases compared to the
no-weathering scenario. Figure 3 shows this behaviour for different regolith
masses and fractions of regolith–bedrock weathering with respect to the
total denudation rate, D. For weathering fractions of 30 %, the bias in
CRN concentration compared to a no-weathering scenario is less than 25 %
for all regolith masses. However, for thick regolith, the corrections become
larger than 100 % (Fig. 3). It is important to note that there is no
difference between the denudation rate predicted by the soluble and
insoluble target minerals in this simplified scenario. The insoluble mineral
would be enriched within the regolith due to dissolution of the soluble
target mineral at the interface, but without further regolith weathering,
the regolith residence time and resulting nuclide concentration would be the
same for both mineral phases.
Conceptual plots showing the effects of regolith weathering on a
homogeneous soluble bedrock for weathering proportional to grain mass. (a) The average grain mass in the regolith 〈mg〉 will decrease with increasing percentage of weathering WR if all other variables are fixed, while the average grain residence time 〈τg〉 and therefore grain CRN concentration would increase. However, the average residence time of regolith mass 〈τR〉 will remain constant. (b) In the absence of weathering, no relationship is predicted between the grain mass and regolith residence time. Weathering will introduce such a relationship and lead to a decrease in grain size with time (blue line). If regolith weathering constitutes 50 % of the denudation, the residence time of a grain will double compared to the no-weathering case; however, the grain will also contribute 50 % less to the total regolith mass. Therefore, no net difference in the average CRN concentration of the
regolith occurs. (c) A CRN sample collected within a certain grain size range can be biased due to weathering. In the case of regolith weathering, larger grain sizes will, on average, have a lower CRN concentration, that is, at minimum, the nuclide concentration entering the regolith N0. Sampling the full regolith grain size range would result in the same CRN concentration compared to the no-weathering case. However, a sample from the coarser end of the grain size range (yellow box) would result in a too low CRN concentration.
Regolith weatheringHomogeneous bedrock case – grain size effects
Consider a homogeneous soluble bedrock where regolith weathering and erosion
occur such that the denudation rate D=EB=ER+WR. The residence time of a parcel of rock within the regolith stays identical to the no-weathering scenario, τR=mRD. The
average regolith nuclide concentration will not be affected by regolith
weathering because no parameter in Eq. (2) would change by partitioning the
outgoing regolith fluxes differently. Despite the average nuclide
concentration staying the same, regolith weathering would affect the size of
grains in the regolith. Grains will lose mass with time in response to
dissolution. A simple way to approximate weathering is as a mass loss
proportional to grain mass such that
mg(t)=mi×e-kt,
where mg is the grain mass at time t, mi is the initial grain mass entering the regolith, and k is a rate constant such that ∂mg∂t=mg×k. The regolith weathering rate WR can be related to k by integrating the mass loss over all grains, and hence,
in practice, requires a known grain size distribution. While it may be
preferable to use an expression that is proportional to surface area rather
than mass, Eq. (8) captures the same general behaviour, but is more convenient because all variables are defined in terms of mass. If weathering is assumed to be proportional to grain mass, based on Eq. (8), grains never fully weather away. In nature, grains eventually disappear at some point; however, this common formulation (e.g. Gabet and Mudd,
2009) captures the general behaviour of older grains contributing less mass,
and generally, the contribution of infinitesimally small grains to the total
concentration is negligible. A grain can only leave the regolith through
regolith erosion ER. Therefore, the average time a grain spends within the regolith becomes
τg=mRER.
It is important to note that this average grain residence time is always
equal to or longer than the average residence time of a rock parcel. This
conceptualization of weathering predicts that with an increasing percentage
of regolith weathering the average regolith grain size reduces and the
grain residence time increases. The average CRN concentration of the
regolith remains constant because older grains with higher concentrations
contribute increasingly less mass to the total regolith, and the average
residence time of mass (τR) stays constant.
Moreover, weathering of grains will introduce a relation between grain size
and the CRN concentration, where smaller grains will, on average, have
higher concentrations (Fig. 4a), which does not exist for a no-weathering
scenario. Following the assumption that erosion in a well-mixed regolith can
be conceptualized as random plucking of particles, the average regolith
grain mass can be described as
〈mg〉=mi×e-k×mRER=mi×e-k×τg.
This relationship predicts that the CRN concentration is dependent on grain
size, with larger grains having a below-average and smaller grains
an above-average CRN concentration, and it can be solved to determine the
relative residence time of a given initial and final grain mass, i.e. τg=-k-1×lnmgmi. In
practice, this means that a CRN sample would need to be representative of
the grain size distribution in the regolith to accurately represent the
denudation rate. Combining Eqs. (2) and (10), one can define the nuclide
concentration of a grain with respect to its mass:
Ng=N0+〈PR〉×lnmgmi-k.
In practice, the collection of an alluvial sample for CRN measurement
typically involves the selection of a certain grain size range. Depending on
the relation of the sampled grain size to the total grain size distribution
in the regolith, the sample will have either a lower or higher CRN
concentration compared to the average regolith (Fig. 4c). For instance,
sampling the coarser end of the regolith grain size distribution will result
in a lower concentration compared to the average regolith and higher
inferred denudation rates. For cases where grain size decreases
systematically with time, only a sample integrating all grain sizes would
result in a nuclide concentration reflecting the average regolith nuclide
concentration.
A similar reduction in CRN concentration with increasing grain size has been
linked to the supply of deep-seated material, e.g. from landslides (e.g.
Brown et al., 1995; Belmont et al., 2007; Aguilar et al., 2014; Puchol et
al., 2014). Other studies have discussed a grain size reduction during
sediment transport and weathering-facilitated sediment breakdown in the
regolith (e.g.
Carretier et al., 2009; Carretier and Regard, 2011; Lukens et al., 2016;
Sklar et al., 2017; Lupker et al., 2017). The case presented here is more
similar to the latter phenomenon, and in our model, the magnitude of this
grain size bias increases with the weathering intensity. Thick soils and a
high fraction of regolith weathering compared to the total denudation lead
to a large grain size bias. Areas with thin soils or low weathering
intensities have negligible grain size bias.
Regolith weathering in mixed bedrock mineralogy
Most carbonate-bearing rocks contain a mix of soluble and insoluble
minerals. Let us consider the same regolith weathering scenario as in Sect. 3.3.1,
but now with a bedrock containing minerals of heterogeneous weatherability.
In this case, the soluble minerals would be depleted in the regolith because
they are removed by erosion and weathering, whereas the insoluble minerals,
only affected by erosion, would be enriched
(Riebe and Granger, 2013). The
enrichment and depletion of minerals in the regolith involve a change of
regolith residence time for the different minerals by a factor XR/XB
(concentration ratio of mineral X in regolith, XR, and bedrock,
XB). We can rewrite Eq. (1) for the average regolith CRN
concentration measured in the target mineral X using the more accurate
depth integration formulation of the production rates instead of exponential
approximations. In this case, the concentration of nuclides at the
regolith–bedrock interface is the sum of nuclide concentrations produced
through different production pathways i, integrated from the regolith
residence time until infinity (past defined as positive values), such that
N0=∑i∫τ∞PimR+Dtdt.
The number of atoms produced in the regolith is the integral of the average
regolith production rate (〈Pi,R〉) from present to the bulk average regolith residence time, modified by the enrichment/depletion factor (XR/XB), which accounts for a target mineral residence time that differs from the average. Hence, the average nuclide concentration of a target mineral X in the regolith is
〈NR,X〉=∑iXRXB∫0τ〈Pi,R〉dt+N0,
where the first term describes the nuclide accumulation within the regolith,
and the second term is the CRN concentration from the advection of rock to
the regolith–bedrock interface. This expression shows that measuring a
nuclide concentration on a target mineral with a higher weatherability than
the average bedrock will yield a CRN concentration lower than in a
no-weathering scenario, and a less weatherable mineral will accumulate a
greater nuclide concentration (Fig. 5). This occurs because the mean
residence time for a soluble mineral decreases relative to the bulk regolith
residence time, while the residence time for an insoluble mineral increases
relative to that of the bulk soil.
(a) The percentage change in 10Be concentration compared to a no-weathering case for different regolith masses and ratios of regolith and bedrock mineral fractions XRXB at sea level high latitude (SLHL). (b) Conceptual representation of a rock with
carbonate and quartz minerals experiencing regolith weathering. The quartz
grains would be enriched in the regolith compared to the calcite. Therefore,
a 10Be denudation rate computed from quartz without correction would be
lower than the average denudation rate, whereas a 36Cl denudation rate
measured from calcite would be higher.
Following Riebe et al. (2003), we can write the
mass balance for the mineral X as
ER×XR+WR,X=XB×DB,
with WR,X as the regolith weathering rate of mineral X.
Assuming the bedrock is a two-component mix of an insoluble mineral, I, and a soluble mineral, S, the enrichment factor of the insoluble mineral I (WR,X=0) will be
IRIB=DER=11-WRD.
The insoluble mineral cannot be enriched to more than 100 % of the
regolith composition (XR=1). Therefore, we can define a minimum
denudation rate Dmin that needs to be fulfilled to remain at a steady
state:
1IB=11-WRDmin→Dmin=WR1-IB.
The depletion of the soluble mineral S is more complicated. The depletion
factor SR/SB depends on the concentration of the soluble mineral in either the regolith or bedrock. If we assume the bedrock concentration SB is known, we can rewrite Eq. (14) to
SRSB=SBD-WRSBER=SBD-WRSBD-WR.
Equation (15) shows that the enrichment of the insoluble minerals is a
function of the ratio of denudation to erosion, where denudation needs to be
operating at a rate greater than Dmin. In contrast, Eq. (17) shows that
the depletion of the soluble mineral is also a function of the bedrock
composition. We will investigate the effects of this different behaviour in
Sect. 4.
Application of weathering corrections
In the previous section, we laid out the theoretical foundation for CRN
weathering biases. Below we highlight how the above equations can be used
for practical weathering corrections in regions of non-negligible
regolith–bedrock interface or regolith weathering. We outline how paired-nuclide measurements from a soluble and insoluble target mineral can be used
to calculate the landscape denudation rate as well as a long-term weathering
rate. This method could provide a new tool to estimate long-term weathering.
We also demonstrate how single-nuclide measurements can be corrected for
weathering. Previous studies performing regolith weathering corrections
commonly employed the chemical depletion fraction (CDF) integrated over the
entire regolith (Dixon
et al., 2009; Riebe et al., 2001; Riebe and Granger, 2013). Our study
focuses on the application to limestone regions, where carbonate dissolution
rates are commonly calculated from stream water chemistry. Therefore, we
demonstrate how to use carbonate dissolution rates for the weathering
corrections of CRN measurements.
WeCode performs all the calculations mentioned above but also features the
possibility to use the traditional CDF weathering correction. The production
rate and scaling calculations are performed using CRONUScalc, and the input
of CRN data follows the CRONUScalc input scheme. The current version of the
code is written for 10Be in quartz and 36Cl in calcite since these
are the most commonly measured insoluble and soluble target minerals,
respectively. However, the code can easily be expanded to all other nuclides
within CRONUScalc.
WeCode comes with a test data set to illustrate the application. The test
data are supposed to reflect a typical “dirty” limestone composition of
70 % calcite, 5 % quartz, and 25 % clay, where the clay component is regarded as insoluble on the timescale of nuclide build-up. The denudation
rate is 100 mm ka-1, the calcite weathering rate is 50 mm ka-1, the regolith is relatively thick (200 g cm-2 surface area, equivalent to 133 cm regolith thickness assuming a density of 1.5 g cm-3), and we use a mid-latitude low-elevation location
(48.585∘ N, 9.250∘ E, 345 m). The scaling scheme for
the test data is Stone (2000). For consistency and computational
reasons, we use a linear error propagation scheme in line with CRONUScalc,
where every parameter is varied by 10 %, and the difference in the result
is scaled by the uncertainty of the same parameter. For computational
reasons, we only include uncertainty from CRN concentration, the weathering
rate, and the spallation production into the error estimation because we
assume these to be the main sources of uncertainty. However, we acknowledge
that the actual uncertainties might be greater and asymmetrically
distributed and therefore not properly captured by our approach.
Regolith–bedrock interface weathering
Weathering at the regolith–bedrock interface affects the build-up of CRNs in
the soluble and insoluble target mineral in the same way. However, the
mineral phases would be enriched or depleted directly at the interface, and
hence the composition of the regolith would still follow Eqs. (15) and (17). If the weathering rate is known independently, e.g. from stream water
chemistry or the weathering degree from a CDF, one can use Eq. (7) in
Sect. 2.2 to solve for the correct denudation rate. A CDF can be applied
because it is related to the depletion of minerals in the regolith:
CDF=1-XBXR,
and, therefore, Eq. (7) can be adapted for the usage of a CDF to
〈NR〉=N0+〈PR〉×mRD×11-CDF.
This correction requires that the majority of weathering is concentrated at
the regolith–bedrock interface, and the depth of this interface is known.
Calculation of “true” denudation rates for our test data. (a) Correction for weathering along the regolith–bedrock interface for a 10Be measurement on quartz (could also be applied to 36Cl on
calcite). The conventional denudation rate refers to a rate calculated
without weathering correction. (b) Correction for a single-nuclide
measurement and regolith weathering. In this case the weathering rate, e.g.
from stream chemistry, needs to be known. (c) Calculation of denudation and weathering rate from a paired-nuclide measurement, where one target mineral is insoluble and one is soluble. In this case we show the data for a combined measurement of 36Cl on calcite and 10Be on quartz.
Regolith–bedrock interface weathering will lead to an increase in nuclide
concentration and an underestimation of denudation rate if not considered,
independent of a measured target mineral. The CRN concentration of our test
data predicts a conventional 10Be denudation rate of 61 mm ka-1
instead of 100 mm ka-1 (assuming no weathering), but with 50 mm ka-1 weathering at the regolith–bedrock interface, the
weathering-corrected denudation rate is equal to the correct value of 100 mm ka-1 (Fig. 6).
Regolith weatheringPaired-nuclide measurements
The effects of regolith weathering can be corrected with paired-nuclide
measurements on a soluble and insoluble target mineral. Combining Eq. (13) of
the nuclide build-up with the enrichment-depletion factors of Eqs. (15) and (17) demonstrates that the enrichment and depletion of the minerals depend on the total denudation, weathering rate, soil mass, and either the bedrock or
regolith composition. Knowledge of the soluble and insoluble mineral CRN
concentrations, and either the regolith or bedrock composition, results in
an equation system consisting of two CRN build-up equations and one equation
relating the change in composition to weathering (the exact equations would
differ between providing either bedrock or regolith compositional data).
This system of equations can be solved for the three unknowns of total
denudation, weathering rate, and the mineralogy of the regolith or bedrock.
For accuracy, we use the standard CRONUScalc depth integration instead of
the exponential production profiles introduced in Sect. 3.1.1. An
analytical solution to the equation system is not possible, and we use an
optimization algorithm to solve for the correct combination of parameters.
This approach predicts a weathering rate that integrates over the same time
as the CRN measurements.
Carbonates typically contain a large amount of soluble calcite (or some
aragonite or dolomite), a component of clay that can be regarded as
insoluble on the timescale of calcite weathering and varying amounts of
other minerals, such as quartz. We can therefore expand the
two-component system described in Sect. 3.3.2 and add another insoluble
component X, representing clays for example, such that
20QB+XB=QR+XR×ERER+WR=QR+XR×ERDR,21QBXB=QRXR,
with Q representing the quartz fraction. Equation (21) highlights that the ratio of
the insoluble minerals stays constant between bedrock and regolith. Applying
the above equations to our test data concentrations, we find that the
uncorrected 10Be denudation rate would be 61 mm ka-1, whereas the
uncorrected 36Cl denudation rate would be 132 mm ka-1. Combining
both CRN measurements with the bedrock composition yields the correct
denudation rate of 100 mm ka-1, with 50 mm ka-1of weathering,
where the mineralogy of the regolith is predicted from Eq. (20):
0.05+0.25=QR+XR×5050+50,
as there is 10 % quartz, 40 % calcite, and 50 % clay in the regolith (Fig. 6). The erosion-to-denudation ratio of 1/2 predicts that quartz and clay get enriched in the regolith by a factor of 2 compared to the bedrock, whereas calcite gets depleted by 57 %.
Single-nuclide measurementInsoluble target mineral
Previous studies focused on using an integrated CDF for weathering
corrections on quartz (Dixon
et al., 2009; Riebe et al., 2001; Riebe and Granger, 2013), which we also
include in our code package. However, we highlight how the weathering
correction can also be achieved with a weathering rate, e.g. independently
derived from runoff and water chemistry measurements. This may be desirable
in carbonate regions, where such weathering rate measurements are common.
However, our corrections consider weathering within the cosmogenic nuclide
production zone, and additional corrections are required for deep
weathering, as observed in tropical settings for example
(Campbell et al., 2021; Dixon et al., 2009).
The enrichment of the insoluble target mineral only depends on the ratio of
denudation to erosion, and therefore the weathering rate and the CRN
concentration are sufficient for calculating a denudation rate. This becomes
apparent if we combine the simplified cosmogenic nuclide concentration Eqs. (2) and (15):
〈NR,X〉=N0+〈PR〉×mRD×DER=N0+〈PR〉×mRER=N0+〈PR〉×mRD-WR.
This is the same as Eq. (7) in the regolith–bedrock interface weathering
except for WR replacing WB. In our synthetic data test with 50 mm ka-1 weathering, the actual denudation rate would be 100 mm ka-1 in comparison to the conventional 10Be rate of 61 mm ka-1. Similar to
the paired-nuclide case, the mineralogy of the regolith can be predicted if
the bedrock mineralogy is known and vice versa. It is important to note that
for the insoluble target mineral, an increase in denudation rate, while the
weathering rate remains constant, will always lead to a decrease in CRN
concentration. In Eq. (23), N0 will decrease with increasing
denudation, as well as the second term. The relationship between average
nuclide concentration in the regolith and denudation rate is monotonic.
Soluble target mineral
In addition to the denudation and weathering rate, the depletion of the
soluble target mineral depends on the mineral composition of the bedrock or
regolith. For the soluble target mineral, this leads to the counterintuitive
behaviour where for a constant weathering rate, the CRN concentration does
not necessarily decrease as denudation increases. If we combine the
simplified nuclide Eqs. (2) and (17) in the same manner as above, we get
〈NR,X〉=N0+〈PR〉×mRD×SBD-WRSB(D-WR)=N0+〈PR〉mR×1-WRSBDD-WR.
An increase in denudation will lower the concentration N0, whereas
depending on the composition, the regolith residence time may either become
shorter due to an increase in the denudation flux or increase because
the soluble target mineral gets less depleted. For denudation rates just
above Dmin, the depletion factor XR/XB will increase rapidly (less depletion) and therefore lead to an increase in CRN
concentration until the maximum possible CRN concentration (Nmax) is
reached at denudation rate DNmax (Fig. 7a). At denudation rates above DNmax the increase in the regolith denudational flux outcompetes the lower depletion, and CRN concentration decreases (Fig. 7a). In other words, at Dmin, the soluble target mineral weathers away instantaneously
upon entering the regolith, and an increase in denudation will lead to a
fraction of grains surviving, thereby increasing the residence time and
concentration; a further increase in denudation, however, will lower the
residence time and nuclide concentration again. If we only consider CRN
production in the regolith and neglect N0, which applies to the case of
thick regolith where production in the bedrock is negligible, we can use Eq. (24) to find the denudation rate with the maximum nuclide accumulation. Taking the derivative of Eq. (24) and setting it to zero returns the denudation rate with maximum nuclide concentration by solving
∂〈NR,X〉∂D=-〈PR〉×mR×SBD2-2WRD+WR2SBD2×D-WR2=0,
with the two solutions
D1=WR+WR1-SBSB
and
D2=WR-WR1-SBSB,
where only D1 is valid because D2<Dmin.
Weathering impact on the concentration of 36Cl measured in pure
calcite at SLHL. CaB=0.7, QB= 0.3, and mR= 200 g cm-2. (a)36Cl concentration versus denudation rate for calcite. The nuclide concentration increases above Dmin until DNmax. At denudation rates higher than Dunique, 36Cl measurements would be unambiguous. (b)DNmax for the same sample, and a range of weathering rates and regolith masses. (c) Same as (b) for Dunique.
It follows that for some nuclide concentrations, a soluble target will have
two denudation rate solutions (see Fig. 7a). If the measured nuclide
concentration is below the one expected for Dmin, there will be a unique solution. We name the denudation rate above this threshold Dunique and discuss the implications of this behaviour below.
WeCode checks the input data from soluble target minerals and notifies the
user if there is more than one solution to a nuclide concentration. For
non-unique solutions, the code computes both denudation rates. We also
provide functions to calculate Dunique, DNmax, and Nmax for a
given set of sample parameters. WeCode can be used in advance to determine
whether a sampling location is suitable to resolve the correct denudation
rate from a soluble target mineral CRN measurement (e.g. 36Cl).
Discussion
We investigated the effects of weathering on soluble and insoluble target
mineral CRN concentrations for different weathering scenarios. We found that
potential corrections for CRN denudation rate calculations exist, but
additional data are required (i.e. a second isotopic system or independent
weathering rate or CDF measurements) beyond a single CRN system measurement.
In all cases, one needs an estimate of regolith thickness or mass, and the
magnitude of the corrections increases with increasing regolith mass. We
also show that paired-nuclide measurements offer the possibility to
constrain both denudation and long-term weathering rates.
Weathering along the regolith–bedrock interface is straightforward to
correct for when an independent weathering rate is known. The practical
problem is to estimate where in the weathering zone the majority of
weathering is taking place. Water chemistry measurements by Gunn (1981) show that most dissolution in carbonates takes place
within the first metres below ground. A qualitative assessment of whether
weathering occurs mostly in the regolith or along the regolith–bedrock
interface could be based on regolith profiles in the field, analogous to
figure 1, where a gradual grain size reduction in the profile indicates a
dominance of regolith weathering.
The insoluble target mineral is blind to the difference between
regolith–bedrock weathering and regolith weathering. Therefore, measuring
the insoluble target mineral would be a simple way of avoiding ambiguity in
regard to the weathering scenario. In contrast, the soluble target minerals
will be affected differently depending on the weathering scenario;
regolith–bedrock interface weathering would increase the nuclide
concentration, and regolith weathering would decrease the nuclide
concentration. Therefore, a paired-nuclide measurement can, in theory,
distinguish between the two scenarios. If the soluble and insoluble target
mineral CRN concentrations result in the same uncorrected denudation rate
despite weathering, the result would suggest weathering occurred mainly
along the regolith–bedrock interface.
Grain size bias
Continuous weathering within the regolith should reduce the grain size of
soluble material released from the bedrock. In Sect. 3.3.1 we highlight
that the procedure of selecting a grain size window for alluvial cosmogenic
nuclide measurements may therefore be problematic in areas with significant
weathering, as the grain size will be a function of regolith residence time.
In practice, this grain size bias would be difficult to correct because it
would require knowledge of the grain size distribution entering or within
the regolith. If, for instance, the grain size distribution entering the
regolith is known and combined with a weathering model (e.g. weathering
proportional to grain surface area), one could calculate the bias introduced
by measuring a certain grain size. A paired-nuclide measurement combined
with either an independently derived weathering rate, or knowledge of the
regolith and bedrock composition, would allow one to evaluate if there is a
grain size bias. A grain size bias would manifest itself through a predicted
weathering rate that is higher than the independently derived one and a
higher enrichment of insoluble minerals in the regolith than measured.
Generally, areas with thin soils and/or low weathering intensity should not
be affected strongly by grain size biases.
Regolith weathering
We have shown that paired-nuclide measurements of soluble and insoluble
target minerals have the potential to resolve a denudation rate, as well as
a weathering rate. The weathering rate derived from such measurements is on
the timescale of the nuclide build-up and hence a long-term estimate of
chemical weathering within the regolith. The grain size bias and the
magnitude of weathering along the regolith–bedrock interface should be
negligible for the determination of correct denudation and weathering rates.
It is worth noting that the findings here can be applied to various regions,
with mineral–nuclide combinations beyond the 10Be–quartz,
36Cl–calcite pair highlighted in this study, e.g. quartz–magnetite,
quartz–feldspar, quartz–pyroxene, or magnetite–olivine.
An important consideration before measuring CRNs for denudation rates in
landscapes with non-negligible weathering is how large the expected
corrections and potential errors would be. Large weathering corrections will
come with considerable uncertainty due to the simplicity of the models used
for correction and should therefore be avoided. But what magnitude of
correction should be considered too large?
We propose using DNmax and Dunique as a quantitative measure of parameter combinations for the soluble target mineral, below which the recovery of a meaningful denudation rate becomes difficult. Ideally, the
denudation rate within a region for a given regolith mass and weathering
rate should be higher than Dunique for the nuclide concentration to result in a unique solution for denudation rate. In our synthetic example
with thick regolith, the depletion factor of calcite for Dunique
(XR/XB) is 0.78, and hence the weathering correction of the nuclide concentration is only ∼ 22 %. The weathering correction for DNmax is ∼ 55 % of the nuclide concentration in our
synthetic data. To avoid ambiguity in the interpretation of calculated
denudation rates as well as large weathering corrections, we advise
measuring soluble target minerals in regions with denudation rates above
DNmax, and ideally above Dunique. The optimal regions for accurate
denudation rate calculations are, therefore, areas with high denudation
rates, thin regolith, and low to medium weathering rates. The settings that
will generally meet these requirements for carbonate rocks are areas of
moderate to high relief and temperate to arid climates.
Single-nuclide measurements can be corrected for weathering using
independent weathering rates derived from water chemistry. Chemical
weathering rates from water chemistry integrate over a substantially shorter
timescale (hours to a few years) compared to cosmogenic nuclides. Hence,
caution needs to be observed when combining the two methods. Climate models
can be used to check if climatic conditions, such as precipitation and
temperature, have changed significantly throughout the cosmogenic nuclide
averaging window, and can help to assess whether water chemistry-derived
weathering rates could be biased (Ott et al., 2019).
A small caveat of WeCode is that it does not compute separate 36Cl
production rates for regolith and bedrock. The change in chemical
composition from bedrock to regolith due to weathering might affect the
production rates for thermal and epithermal neutrons through the change in
the fraction of absorbed neutrons. Most likely, the water content will also
be higher in the regolith compared to the bedrock, which would increase the
fraction of absorbed neutrons and lower production rates. We did not
incorporate separate production rates for bedrock and regolith to avoid
calculating a large set of production rates and slow down the computation
significantly. However, for samples with low natural [Cl] or high water
content, the production of thermal and epithermal neutrons is low, and the
bias in the calculations is likely negligible. Except for cases of thin
regolith, we recommend using the regolith values because that is where the
majority of the 36Cl is produced.
Other considerations and future research needs
For single-nuclide measurements, site-specific parameters should be
evaluated before choosing which target mineral to sample. Measuring the
insoluble target mineral quartz offers the advantage of minimizing the
potential for a grain size bias. However, for lithologies with low quartz
content, the enrichment factor and thereby the weathering correction of
quartz will be substantially higher than the depletion factor for the
soluble minerals. For a rock with 5 % quartz, 95 % calcite, and a
denudation-to-erosion ratio of 2 DER=2, the enrichment factor of quartz in the regolith would also be 2, whereas the
depletion factor of calcite would be 0.95. The enrichment of quartz would
result in a larger weathering correction, and thus it is preferable to
measure the CRN concentration in the soluble target mineral. This being
said, the soluble target mineral may experience an undesired grain size
bias, introducing a relationship between grain size and nuclide
concentration. Hence, for single-nuclide measurements, the choice of target
minerals should therefore be made based on site-specific assessments of
bedrock composition, regolith thickness, and the expected range of
weathering and denudation rates.
Another consideration should be whether the soluble and insoluble minerals
are physically separated during the evolution of the regolith. If, for
instance, insoluble minerals are part of larger rock fragments of soluble
minerals while sitting in the regolith, both mineral phases should have the
same residence time. If minerals are not physically separated in the
regolith, the denudation rates from soluble and insoluble minerals should be
equal without weathering corrections. The same should apply if no mixing or
transport of the regolith occurs.
For alluvial samples of soluble target minerals such as calcite, it is worth
assessing whether secondary mineral precipitation occurs in the stream
sediment. Perennial streams in limestone regions commonly form substantial
amounts of secondary calcite within the sand fraction (Erlanger et al., 2021). The high solubility of [Cl] would result in low concentrations of [Cl] in the secondary minerals; however, experiments from cave waters suggest that the 36Cl signature of speleothems would be dominated by meteoric 36Cl (Johnston, 2010). Therefore, secondary target mineral
precipitates would need to be identified and, if relevant, removed from the
sample before measurement.
Despite these caveats, weathering corrections of single-nuclide measurements
and paired-nuclide measurements have the potential to expand the range of
landscapes for which we can determine millennial timescale denudation rates.
Several research needs are identified from our study that will ultimately
assess the robustness of such weathering corrections. These research needs
are listed as follows. (1) More measurements of cosmogenic nuclides with
bedrock and regolith mineralogy are needed. Such measurements will help to
assess if the simple enrichment–depletion models for the evolution of the
bedrock to regolith composition hold up in the field. (2) Paired-nuclide
measurements on different target minerals within the same sample are needed
to test the divergence of nuclide concentrations. In particular, measurements
from landscapes with an increasing weathering-to-denudation ratio can be
useful to test the hypothesis that the divergence in nuclide concentrations
would increase, too. (3) Paired-nuclide measurements combined with
independent weathering rates can determine if the weathering rates
calculated from cosmogenic nuclides match other records. (4) More data on
the distribution of weathering with depth, e.g. for carbonate regions, will
be helpful to assess the relative importance of regolith and
regolith–bedrock interface weathering in various landscapes. Cosmogenic
nuclide measurements can contribute to this question because both weathering
scenarios predict a different response of soluble target mineral CRN
concentrations. (5) Potential grain size biases for soluble target minerals
should be assessed. Paired-nuclide measurements can help to gauge grain size
biases and estimate their magnitude. Grain size measurements of the regolith
and along the bedrock regolith interface and careful selection of sample
grain sizes can be a way to quantify this potential bias.
The best conditions to test many of the hypotheses stated in this study will
likely be met in Critical Zone Observatories. Unfortunately, few of them
exist in limestone regions, highlighting the need for more data on regolith
composition and evolution, especially in limestone regions of varying
weathering rates. The same holds true for measurements of soluble target
minerals such as 36Cl in regolith and alluvial sediments. Most studies
measure 36Cl on bedrock exposures (Avni et al., 2018; Godard et al., 2016; Matsushi et al., 2010; Stone et al., 1994; Thomas et al., 2018; Xu et al., 2013) and only a few in the regolith or in alluvial sediments (Ott
et al., 2019; Ryb et al., 2014b, a; Thomas et al., 2017). Studying
limestone regions and other areas with soluble minerals is of particular
interest because the high weatherability makes them more susceptible to the
interplay of tectonics and climate (Ott et al., 2019;
Simms, 2004). More data on the weathering biases for cosmogenic nuclides
would allow expanding the calculation of denudation rates to new regions and
improve our understanding of how the partitioning of denudation into erosion
and weathering depends on tectonics and climate.
Summary and conclusions
We investigated the effects of chemical weathering on the nuclide
concentration of soluble and insoluble target minerals in regolith-covered
landscapes. Our main findings are the following.
In the case of regolith–bedrock interface weathering, independent knowledge
of the weathering rate or degree from a CDF or water chemistry can be used
to correct cosmogenic-nuclide-derived denudation rates independently from
the target mineral weatherability.
In regions where weathering is concentrated within the regolith, paired-nuclide measurements of a soluble and insoluble target mineral offer the
potential to constrain the denudation rate as well as a long-term weathering
rate.
Previous studies have highlighted how single-nuclide measurements on
insoluble target minerals can be corrected using an integrated CDF value. We expand this approach and show that weathering rates from stream chemistry in combination with bedrock or regolith mineralogy can be used in the same way.
We derive equations that show the relationship between denudation and the nuclide concentration in soluble target minerals is non-monotonic. We use this relationship to map the denudation rates DNmax and Dunique,
for various regolith masses and weathering rates; these can be used as
guidelines for which areas to sample and which to avoid.
CRN measurements, especially from soluble target minerals, in combination with weathering rates, and compositional data from bedrock and regolith, can be used to assess the corrections proposed in this study.
Code and data availability
The WeCode software package is available at
10.5880/GFZ.4.6.2022.001 (Ott, 2022). All data used in this study are available within the WeCode software package.
The supplement related to this article is available online at: https://doi.org/10.5194/gchron-4-455-2022-supplement.
Author contributions
RFO, SFG, and DEG designed the study. RFO conducted the
analysis, developed the WeCode software, and wrote the manuscript with input
from all authors.
Competing interests
The contact author has declared that neither they nor their co-authors have any competing interests.
Disclaimer
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Acknowledgements
We thank Jean Braun for discussions about this study which aided our analysis. We are very grateful for the graphical abstract designed by Emma Lodes.
Financial support
Richard F. Ott was supported by a contract from NAGRA (Swiss National Cooperative for the Disposal of Radioactive Waste) and by the Swiss National Science Foundation fellowship, grant number P2EZP2_191866.
Review statement
This paper was edited by Yeong Bae Seong and reviewed by Claire E. Lukens and one anonymous referee.
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