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 2〈NR〉=N0+〈PR〉×τR, with 3N0=∑iPi(0)ΛiDe-mR/Λi describing the nuclide concentration at the regolith–bedrock interface, and with 4〈PR〉=∑iPi(0)ΛimR1-e-mR/Λi describing the average production rate within the well-mixed regolith, and the regolith residence time: 5τR=mRD.

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 8mg(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 9τg=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 10〈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×ln⁡mgmi. 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: 11Ng=N0+〈PR〉×ln⁡mgmi-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.

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 12N0=∑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 13〈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.

Following Riebe et al. (2003), we can write the mass balance for the mineral X as 14ER×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 15IRIB=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: 161IB=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 17SRSB=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.

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): 220.05+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 %.

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): 23〈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.

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 24〈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 25∂〈NR,X〉∂D=-〈PR〉×mR×SBD2-2WRD+WR2SBD2×D-WR2=0, with the two solutions D1=WR+WR1-SBSB and 26D2=WR-WR1-SBSB, where only D1 is valid because D2<Dmin⁡.

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).