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.

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.

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.

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.

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 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.

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. ().

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 .

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.

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.

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.

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. ).

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. ).

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.

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.

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.

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.

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.

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.

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.