#### Definitions and nomenclature

Because the terminology of weathering and erosion varies across disciplines, we begin by defining the terms used in our derivation and discussion of the *CEF*. Table 1 summarizes the nomenclature used in our equations.

Table 1. Nomenclature used in derivationsSymbol | Description | Units |
---|

*CEF* | chemical erosion factor | unitless |

*D* | total denudation rate, equal to the regolith production rate in steady state | t km^{-2} a^{-1} |

*W*_{sap}, *W*_{soil} | chemical erosion rate from saprolite and soil, respectively | t km^{-2} a^{-1} |

*E*_{sap}, *E*_{soil} | rate of production (i.e. input) and physical erosion (i.e. output) of soil | t km^{-2} a^{-1} |

*Zr*_{soil}, *Zr*_{sap}, *Zr*_{rock} | immobile element or mineral concentrations in soil, saprolite and rock | ppm |

*X*_{soil}, *X*_{sap}, *X*_{rock} | concentration of mineral *X* (e.g. quartz) in soil, saprolite and rock | g g^{-1} |

*P*(0) | nuclide production rate at surface due to one of the production pathways | atoms g^{-1} a^{-1} |

*Λ* | attenuation length scale for nuclide production at depth | g cm^{-2} |

<*P*> | average nuclide production rate in host mineral in soil | atoms g^{-1} a^{-1} |

<*N*> | average nuclide concentration in host mineral in soil | atoms g^{-1} |

*N*_{sap} | nuclide concentration in host minerals in top of saprolite | atoms g^{-1} |

*ρ*, *h* | soil density and thickness (note: their product equals soil mass per unit area) | g cm^{-3}, cm |

*τ*_{soil} | turnover timescale of soil, equal to soil mass per unit area divided by soil denudation rate | ka |

*τ*_{nuclide} | radioactive meanlife of a specified nuclide | Ma |

*H* | regolith thickness | cm |

*A* | area | km^{2} |

<*D*> | spatially averaged denudation rate | t km^{-2} a^{-1} |

*MAP* | mean annual precipitation rate | m a^{-1} |

When discussing processes, we use ‘chemical erosion’ to refer to mass loss by alteration and dissolution of minerals during interactions with meteoric water. ‘Physical erosion’ refers to mass loss by physical removal of mineral grains, both when it occurs at the saprolite–soil interface, due to conversion of saprolite to soil, and when it occurs in soil, due to sediment transport (Figure 1). We use ‘denudation’ to refer collectively to physical and chemical erosion.

In defining components of a weathering profile, we use ‘bedrock’ to refer to unweathered rock (i.e. parent material) at the base of the regolith profile. ‘Saprolite’ refers to weathered rock that retains recognizable bedrock structure and has not been physically mobilized, irrespective of its degree of chemical depletion and volumetric strain relative to unweathered rock. Following usage common in the geochemical literature, we use the term ‘soil’ to refer to material that has been physically mobilized, irrespective of its state of weathering or degree of horizonation. Finally, we use ‘regolith’ to refer collectively to the soil and saprolite that blanket unweathered rock at depth.

#### Geochemical mass balance of regolith

When a regolith profile is in steady state, downward propagation of weathering at the bedrock–regolith interface converts fresh rock into saprolite at a pace equal to the total denudation rate (*D*), which is the sum of all the regolith's erosional effluxes, including physical erosion of the soil (*E*_{soil}) and chemical erosion of both the soil (*W*_{soil}) and saprolite (*W*_{sap}) (Figure 1). Similarly, when soil is in steady state, soil production at the saprolite–soil interface (*E*_{sap}) is equal to the sum of all erosional fluxes from the soil (i.e. *E*_{soil} plus *W*_{soil}). We can apply such steady-state, mass-balance principles both to the regolith as a whole and to individual elements and minerals within it as exemplified in Equation (1) for weathering of an element (or mineral) from saprolite:

- (1)

Here *W*_{X,sap} is the chemical erosion rate from saprolite of the element (or mineral), and *X*_{rock} and *X*_{sap} are its concentrations in rock and saprolite respectively (Riebe *et al*., 2003). For insoluble components of the saprolite and soil, chemical erosion rates are zero and Equation (1) can be rewritten and rearranged as shown in Equation (2), after Dixon *et al*. (2009a).

- (2)

Here, *Zr*_{rock} and *Zr*_{sap} are the concentrations in rock and saprolite of an insoluble mineral or element (in this case zirconium). Enrichment of Zr in saprolite relative to rock reflects chemical losses of other, more soluble elements (Nesbitt, 1979; Stallard, 1985).

The mass balance can be recast into similar relationships for soils in steady state, as shown in Equation (3).

- (3)

Here, *W*_{X,soil} is the chemical erosion rate from soil of an element (or mineral), *X*_{soil} is its concentration in soil, and *E*_{soil} is the erosion rate of the soil. For insoluble elements, Equation (3) reduces to

- (4)

Here, *Zr*_{soil} is the zirconium concentration in the soil. The mass balance principles embodied in Equations (1)–(4) are crucial to solving for the effects of chemical erosion on cosmogenic nuclide buildup, as shown next.

#### Effects of chemical erosion from deep saprolite and well-mixed soils

To account for the consequences of chemical erosion on cosmogenic nuclide-based estimates of denudation rates, we first write differential Equation (5), which expresses the buildup of cosmogenic nuclides in a host mineral *X* as a function of inputs and outputs of nuclides from a well-mixed soil (i.e. where nuclide concentrations are homogenized over timescales much shorter than that of nuclide accumulation).

- (5)

Here, *d*<N>/*dt* is the rate of change of the average nuclide concentration in the host mineral in a well mixed soil, <*P*> is the depth-averaged nuclide production rate in the soil, *N*_{sap} is the concentration of cosmogenic nuclides in host minerals in the top of saprolite, *ρ* is soil density, and *h* is soil thickness (Figure 1). In writing Equation (5) we assume that soil mass (i.e. the product of soil density and thickness) is in steady state. This assumption is common to all studies of soil production rates (Heimsath *et al*., 1997), although practitioners have imprecisely stated that only soil thickness needs to be in steady state. We refer the reader to the literature for comprehensive discussions of the rationale behind steady-state assumptions in soil production rate studies (see, for example, Dietrich *et al*., 1995 and Heimsath *et al*., 1997).

Equation (5) is valid for nuclides that are either stable or have a radioactive mean life (*τ*_{nuclide}) that is significantly greater than *ρh*/*E*_{sap}, the turnover timescale of the soil (*τ*_{soil}). This is typically the case in mountainous settings for cosmogenic nuclides in quartz. For example, using typical values of *ρ*, *h*, and *E*_{sap} (e.g. as reported in Riebe *et al*., 2004a), we find that *τ*_{soil} generally ranges from 1 to 50 ka, significantly less than either *τ*_{Al-26} = 1.0 Ma or *τ*_{Be-10} = 2.0 Ma (Nishiizumi *et al*., 2007; Chmeleff *et al*., 2010).

Cosmogenic nuclides are produced in mineral grains by neutron spallation, muon capture, and interactions with high-energy muons (Heisinger *et al*., 2002a, 2002b). Because cosmic radiation attenuates with passage through matter, cosmogenic nuclide production rates decrease with depth in soil and rock. The precise relationship between production rates and depth remains a subject of research. However, available evidence indicates that the decline in production rates with depth can be expressed to good approximation as a series of exponentials (Granger and Smith, 2000), with <*P*> computed as shown in Equation (6).

- (6)

Here, *P*_{i}(0) is the production rate at the surface due to mechanism *i*, and Λ_{i} is the corresponding scaling factor for the decline in production with depth beneath the surface (in g cm^{-2}; see Supplemental Table 1 in online Appendix for values used here).

If chemical erosion of saprolite occurs at depths greater than a few meters, where cosmogenic nuclide production is minimal, then, in steady state, the concentration of cosmogenic nuclides in host minerals at the top of saprolite is given by Equation (7) (Lal, 1991; Heimsath *et al*., 1997).

- (7)

Equation (7) is valid for nuclides that are stable or have a radioactive mean life that is significantly greater than Λ_{i}/*E*_{sap}, and thus do not decay significantly over the timescale of nuclide buildup during exhumation to the surface.

In the case of isotopic steady state (i.e. with *d*<N>/*dt* = 0), Equation (5) can be solved as shown in Equation (8) after combination with Equations (2), (3), (6) and (7), and collection of terms.

- (8)

The term in curly brackets in Equation (8) represents a series of *i* = 4 correction factors which correspond (one each) to the exponential decay terms in the relationship between cosmogenic nuclide production rate and depth; each term must be corrected separately because of the dependence on penetration depth (Λ_{i}), which differs among the production rate mechanisms (see Supplemental Online Data).

Equation (8) expresses the concentration of cosmogenic nuclides in sediment collected from a well-mixed soil, which is eroding in steady state. Traditionally, erosion rates have been inferred from a much simpler approximation (Lal, 1991), reproduced here in Equation (9).

- (9)

Equation (9) appears in Equation (8) as a prefix. It is convenient to write Equation (10), which re-expresses Equation (8) as a correction factor (i.e. the *CEF*) that should be applied to denudation rates that have been inferred from Equation (9).

- (10)

It is important to understand the assumptions that have gone into derivation of Equations (8) and (10). First, they are specific to chemical erosion that occurs exclusively in well-mixed soils and in deep saprolite; if significant chemical erosion occurs in shallow saprolite (at overburden depths less than or not too much greater than *Λ*_{i}), then the effects of chemical erosion are more complicated. In that case, the *CEF* would need to be computed with site-specific information on how chemical erosion varies with depth. Second, our derivation does not capture variations over time in either erosion rates or soil mass; instead, as is the case in most cosmogenic-nuclide-based formulations of erosion rates, Equations (8) and (10) require that erosion is steady over a timescale that is long enough for cosmogenic nuclide concentrations to approach steady state (Lal, 1991). Our analysis requires the additional assumption that both soil mass and its degree of chemical depletion are steady over the timescale of erosion (Heimsath *et al*., 1997; Riebe *et al*., 2001b). Estimates of *CEF* from Equations (8) and (10) will be erroneous to the extent that these assumptions are violated. However, according to a recent sensitivity analysis of the mass balance of eroding regolith, chemical weathering rates inferred from solid phase geochemistry are remarkably insensitive to plausible fluctuations in both soil depth and physical erosion rates, differing from the long-term average by less than 15%, even in the worst-case scenarios (Ferrier and Kirchner, 2008). Hence, Equations (8) and (10), which are based on the same mass balance principles that are used in measuring chemical weathering rates (Brimhall and Dietrich, 1987; Riebe *et al*., 2001b, 2003; Anderson *et al*., 2002), should yield estimates of *CEF* that are likewise robust against plausible changes over time in erosion rate and soil mass. A final note of caution in employing Equations (8) and (10) is that they are only valid for erosion rates that are rapid enough that radioactive decay can be ignored. If this is not the case, then erosion rates will be systematically overestimated due to neglect of radioactive losses of nuclides. In all of the cases discussed later, erosion rates are fast enough that radioactive decay can be ignored without introducing significant errors in the estimation of *CEF*s.

#### Simplifications in *CEF* arising from special circumstances

Equation (10) is a generic formulation of *CEF* that accounts for the effects of both deep and near-surface chemical erosion on cosmogenic nuclides such ^{3}He, ^{10}Be, ^{21}Ne, ^{26}Al in target minerals such as olivine, quartz, and magnetite. If quartz is the host mineral for the cosmogenic nuclide of interest, as has been the case in the vast majority of previous work on catchment-scale erosion rates, we can use *Zr*_{soil}/*Zr*_{sap} as a substitute for *X*_{soil}/*X*_{sap} in Equation (9). This requires the generally reasonable assumption that quartz is insoluble and thus enriched to the same degree as zirconium during chemical erosion (Riebe *et al*., 2001a). In that case, Equation (8) simplifies to

- (11)

If weathering in the saprolite is minimal, such that *Zr*_{sap} = *Zr*_{rock}, Equation (11) reduces further to

- (12)

Equation (12) implies a *CEF* that is equivalent to the quartz enrichment factor of Small *et al*. (1999) except here Zr enrichment is used as a proxy for quartz enrichment, as suggested by Riebe *et al*. (2001a).

If chemical erosion from soils is minimal, such that *Zr*_{soil} = *Zr*_{sap}, or if the mass per unit area of soil (i.e. *ρh*) is small compared with the penetration depth for production by cosmogenic nuclides, then Equation (8) reduces to

- (13)

Here, the *CEF* is simply *Zr*_{sap}/*Zr*_{rock}, equivalent to the correction factor of Dixon *et al*. (2009a) for chemical erosion of deep saprolite. Note that this correction factor is also more generally appropriate for correcting cosmogenic nuclides measured in the top of saprolite (i.e. using *N*_{sap} instead of <*N*>) as originally proposed by Dixon *et al*. (2009a).

If the host mineral for cosmogenic nuclides is soluble (e.g. as in the case of olivine), and if chemical mass losses in the saprolite are minimal (i.e. *Zr*_{sap} = *Zr*_{rock}), Equation (8) reduces to

- (14)

Note that if the host mineral weathers at the same rate as the bulk soil, its concentration will be neither enriched nor depleted in soil relative to bedrock (i.e. *X*_{soil} = *X*_{rock}). In that case, Equation (14) shows that *CEF* would be equal to unity. Thus Equation (10) reduces to Equation (9), the classical formulation of Lal (1991), if weathering of saprolite is negligible and if the host mineral is lost from the soil at a rate that is representative of chemical erosion of the soil as a whole (i.e. all of its minerals on average).