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**Earth Surface Dynamics**
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**Short communication**
17 Oct 2018

**Short communication** | 17 Oct 2018

Short communication: Increasing vertical attenuation length of cosmogenic nuclide production on steep slopes negates topographic shielding corrections for catchment erosion rates

- Department of Geosciences, Pennsylvania State University, University Park, Pennsylvania, 16802, USA

- Department of Geosciences, Pennsylvania State University, University Park, Pennsylvania, 16802, USA

**Correspondence**: Roman A. DiBiase (rdibiase@psu.edu)

**Correspondence**: Roman A. DiBiase (rdibiase@psu.edu)

Abstract

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Interpreting catchment-mean erosion rates from in situ
produced cosmogenic ^{10}Be concentrations in stream sediments requires
calculating the catchment-mean ^{10}Be surface production rate and
effective mass attenuation length, both of which can vary locally due to
topographic shielding and slope effects. The most common method for
calculating topographic shielding accounts only for the reduction of nuclide
production rates due to shielding at the surface, leading to catchment-mean
corrections of up to 20 % in steep landscapes, and makes the simplifying
assumption that the effective mass attenuation length for a given nuclide
production mechanism is spatially uniform. Here I evaluate the validity of
this assumption using a simplified catchment geometry with mean slopes
ranging from 0 to 80^{∘} to calculate the spatial
variation in surface skyline shielding, effective mass attenuation length,
and the total effective shielding factor, defined as the ratio of the
shielded surface nuclide concentration to that of an unshielded horizontal
surface. For flat catchments (i.e., uniform elevation of bounding
ridgelines), the effect of increasing vertical attenuation length as a
function of hillslope angle and skyline shielding exactly offsets the effect
of decreasing surface production rate, indicating that no topographic
shielding correction is needed when calculating catchment-mean vertical
erosion rates. For dipping catchments (as characterized by a plane fit to
the bounding ridgelines), the catchment-mean surface nuclide concentrations
are also equal to that of an unshielded horizontal surface, except for cases
of extremely steep range-front catchments, where the surface nuclide
concentrations are counterintuitively higher than the unshielded case due to
added production from oblique cosmic ray paths at depth. These results
indicate that in most cases topographic shielding corrections are
inappropriate for calculating catchment-mean erosion rates, and are only needed
for steep catchments with nonuniform distributions of quartz and/or erosion
rate. By only accounting for shielding of surface production, existing
shielding approaches introduce a slope-dependent systematic error that could
lead to spurious interpretations of relationships between topography and
erosion rate.

How to cite

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How to cite.

DiBiase, R. A.: Short communication: Increasing vertical attenuation length of cosmogenic nuclide production on steep slopes negates topographic shielding corrections for catchment erosion rates, Earth Surf. Dynam., 6, 923–931, https://doi.org/10.5194/esurf-6-923-2018, 2018.

1 Introduction

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Measurement of in situ produced cosmogenic ^{10}Be concentrations in
stream sediments has rapidly become the primary tool for quantifying
catchment-scale erosion rates over timescales of 10^{3}–10^{5} years (Brown
et al., 1995; Granger et al., 1996; von Blanckenburg, 2006; Portenga and
Bierman, 2011; Codilean et al., 2018). Although requiring a number of
simplifying assumptions about the steadiness of erosion and sediment
transport (Bierman and Steig, 1996), erosion rates determined from ^{10}Be
concentrations in stream sediments have yielded insights into a number of key
questions in tectonic geomorphology regarding the sensitivity of erosion
rates to spatiotemporal patterns of climate, tectonics, and rock strength
(e.g., Safran et al., 2005; Binnie et al., 2007; Ouimet et al., 2009;
DiBiase et al., 2010; Bookhagen and Strecker, 2012; Miller et al., 2013;
Scherler et al., 2017).

In contrast to point measurements, where a clear framework exists for
converting ^{10}Be concentrations to either a surface exposure age or
steady erosion rate (e.g., Balco et al., 2008; Marrero et al., 2016), the
interpretation of ^{10}Be concentrations in stream sediment requires
accounting for the spatial variation in elevation, latitude, quartz content,
and erosion rate throughout a watershed (Bierman and Steig, 1996; Granger
and Riebe, 2014). Additionally, topographic shielding corrections that
account for the reduction of cosmic radiation flux on sloped or
skyline-shielded point samples (Dunne et al., 1999) are applied to varying
degrees for determining catchment-mean production rates. These shielding
corrections are either applied at the pixel level (e.g., Codilean, 2006),
catchment level (e.g., Binnie et al., 2006), or not at all (e.g., Portenga
and Bierman, 2011). Although typically small (< 5 %), topographic
corrections can be as large as 20 % for steep catchments (e.g., Norton and
Vanacker, 2009). Because these corrections vary as a function of slope and
relief, any systematic corrections can influence interpretations of
relationships between topography and erosion rate.

The pixel-by-pixel skyline-shielding algorithm of Codilean (2006) results in
the largest topographic shielding corrections, and has gained popularity due
to its ease of implementation in the software packages TopoToolbox
(Schwanghart and Scherler, 2014) and CAIRN (Mudd et al., 2016), the latter
of which was used to recalculate published ^{10}Be-derived catchment
erosion rates globally as part of the OCTOPUS compilation project (Codilean
et al., 2018). A key simplification of the Codilean (2006) approach is that
it accounts only for the skyline shielding of surface production, and not
for the change in shielding with depth, which determines the sensitivity of
the effective mass attenuation length for nuclide production as a function
of surface slope and skyline shielding (Dunne et al., 1999; Gosse and
Phillips, 2001). Because a change in the effective mass attenuation length
will directly influence the inferred erosion rate of a sample (Lal, 1991),
the full depth-integrated implications of topographic shielding must be
accounted for when inferring catchment erosion rates from ^{10}Be
concentrations in stream sediments.

Here I model the shielding of incoming cosmic radiation flux responsible for spallogenic production at both the surface and at depth for a simple catchment geometry to evaluate, as a function of catchment slope and relief, the total effect of topographic shielding on surface nuclide concentrations and the partitioning of shielding into surface skyline shielding and changes to the effective mass attenuation length. I then apply this framework to catchments that have a net dip (i.e., dipping plane fit to boundary ridgelines) and compare calculations of total shielding to those from typical pixel-by-pixel skyline-shielding corrections.

2 Theory

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The incoming cosmic ray intensity, *I*(*θ*,*d*), responsible for in situ
cosmogenic nuclide production by neutron spallation can be most simply
described as a function of the incident ray path inclination angle above the
horizon, *θ*, and the mass distance, *d* (g cm^{−2}), traveled along
that pathway:

$$\begin{array}{}\text{(1)}& I(\mathit{\theta},d)={I}_{\mathrm{0}}{\mathrm{sin}}^{m}\mathit{\theta}{e}^{-d/\mathit{\lambda}},\end{array}$$

where *I*_{0} is the maximum cosmic ray intensity at the surface, *m* is an
exponent typically assumed to have a value of 2.3 (e.g., Nishiizumi et al.,
1989), and *λ* is the mass attenuation length (g cm^{−2}) for
unidirectional incoming radiation (Dunne et al., 1999). The mass attenuation
length for unidirectional radiation, *λ*, differs from the nominal
mass attenuation length that describes cosmogenic nuclide production as a
function of depth, Λ, due to the integration of radiation from all
incident angles. Assuming *m*=2.3, a value of *λ*=1.3Λ
results in a close match for horizontal unshielded surfaces with exponential
production profiles typical of spallation reactions (Dunne et al., 1999;
Gosse and Phillips, 2001).

For a horizontal surface sample (*d*=0), the unshielded total cosmic
radiation flux, *F*_{0}, is described by

$$\begin{array}{}\text{(2)}& {F}_{\mathrm{0}}={\int}_{\mathit{\phi}=\mathrm{0}}^{\mathrm{2}\mathit{\pi}}{\int}_{\mathit{\theta}=\mathrm{0}}^{\mathit{\pi}/\mathrm{2}}{I}_{\mathrm{0}}{\mathrm{sin}}^{m}\mathit{\theta}\mathrm{cos}\mathit{\theta}\mathrm{d}\mathit{\theta}\mathrm{d}\mathit{\phi}={\displaystyle \frac{\mathrm{2}\mathit{\pi}{I}_{\mathrm{0}}}{m+\mathrm{1}}},\end{array}$$

where *φ* is the azimuthal angle of incoming radiation, and the term
cos*θ* accounts for the convergence of the spherical coordinate
system. For point samples that are either at depth (*d*>0) or have an
incomplete view of the sky due to topographic shielding by thick (*d*≫*λ*) objects, the total cosmic radiation flux, *F*, is modulated by a
shielding factor, *S*, such that

$$\begin{array}{}\text{(3)}& S={\displaystyle \frac{F}{{F}_{\mathrm{0}}}}={\displaystyle \frac{m+\mathrm{1}}{\mathrm{2}\mathit{\pi}}}{\int}_{\mathit{\phi}=\mathrm{0}}^{\mathrm{2}\mathit{\pi}}{\int}_{\mathit{\theta}={\mathit{\theta}}_{\mathrm{0}}\left(\mathit{\phi}\right)}^{\mathit{\pi}/\mathrm{2}}{\mathrm{sin}}^{m}\mathit{\theta}{e}^{-d(\mathit{\theta},\mathit{\phi})/\mathit{\lambda}}\mathrm{cos}\mathit{\theta}\mathrm{d}\mathit{\theta}\mathrm{d}\mathit{\phi},\end{array}$$

where *θ*_{0}(*φ*) is the inclination angle above the
horizon of topographic obstructions in the direction *φ* and
*d*(*θ*,*φ*) varies as a function of both ray path azimuth and
inclination angle (Dunne et al., 1999; Gosse and Phillips, 2001).

Equation (3) has two implications for interpreting exposure ages or erosion
rates from cosmogenic nuclide concentrations of samples partially shielded
by skyline topography (*θ*_{0}(*φ*)>0). First, skyline
shielding will reduce the surface production rate of cosmogenic nuclides by
a factor of *S*_{0}:

$$\begin{array}{}\text{(4)}& {S}_{\mathrm{0}}={\displaystyle \frac{m+\mathrm{1}}{\mathrm{2}\mathit{\pi}}}{\int}_{\mathit{\phi}=\mathrm{0}}^{\mathrm{2}\mathit{\pi}}{\int}_{\mathit{\theta}={\mathit{\theta}}_{\mathrm{0}}\left(\mathit{\phi}\right)}^{\mathit{\pi}/\mathrm{2}}{\mathrm{sin}}^{m}\mathit{\theta}\mathrm{cos}\mathit{\theta}\mathrm{d}\mathit{\theta}\mathrm{d}\mathit{\phi}.\end{array}$$

Second, due to shielding of low-intensity cosmic radiation below incident
angles of *θ*_{0}(φ), the effective mass
attenuation length, Λ_{eff}, will increase relative to the nominal
mass attenuation length for describing cosmogenic nuclide production as a
function of depth, Λ (Dunne et al., 1999; Gosse and Phillips,
2001). For calculating surface exposure ages, only the reduction in surface
production rate due to skyline shielding needs to be taken into account, and Eq. (4) is easily calculated for single points in the landscape (e.g., Balco et
al., 2008). However, for determining erosion rates both the surface
shielding and changing effective attenuation length must be accounted for,
which requires solving Eq. (3) numerically as a function of vertical depth
below the surface, as described in Sect. 3 below.

The importance of accounting for both changes in surface production rate,
*P*, and changes in the effective mass attenuation length, Λ_{eff},
is illustrated by the analytical solution for nuclide concentration, *C*,
measured on a steadily eroding surface for a stable nuclide with an
exponential decrease in production rate with depth:

$$\begin{array}{}\text{(5)}& C\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}P{\mathrm{\Lambda}}_{\mathrm{eff}}/E,\end{array}$$

where *E* is erosion rate (g cm^{−2} yr^{−1}; Lal, 1991). From Eq. (5)
it is clear that increasing Λ_{eff} counters the effect of
decreasing *P* in determining the surface nuclide concentration (or
alternatively for inferring erosion rate).

3 Topographic shielding model for a simplified catchment geometry

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For stream sediment samples that require calculating cosmogenic nuclide
production rates across an entire catchment, solving Eq. (3) as a function
of depth is presently too computationally intensive to be practical.
Consequently, numerical implementations of topographic shielding
calculations at the catchment scale make the simplifying assumption that
Λ_{eff}=Λ, and thus *S*=*S*_{0} (Codilean, 2006; Schwanghart
and Scherler, 2014; Mudd et al., 2016), accounting only for the effect of
decreasing surface production rate, *P*. Here I use a simplified catchment
geometry to solve Eq. (3) and directly calculate the impact of topographic
shielding and surface slope on interpretations of catchment erosion rates
from cosmogenic nuclide concentrations in stream sediments. For simplicity,
I assume that cosmogenic nuclides are only produced by neutron spallation
(i.e., Λ=160 g cm^{−2}) and that the erosion rate is high
enough that radioactive decay is negligible (i.e., *E* > 0.01 g cm^{−2} yr^{−1} for ^{10}Be).

Throughout the analysis below, both the effective mass attenuation length,
Λ_{eff}, and erosion rate, *E*, are defined in the vertical, rather
than slope-normal direction. The vertical (with respect to the geoid)
reference frame was chosen for three reasons. First, most studies report
erosion rate as a vertical lowering rate and assume primarily vertical
exhumation pathways. Second, treatment of slope-normal processes introduces
a grid-scale dependence of erosion and shielding calculations that varies
with topographic roughness (Norton and Vanacker, 2009). Third, for the case
of uniform erosion rate, the resulting shielding calculations do not depend
on the choice of reference frame, as long as the orientation of Λ_{eff} and *E* are defined similarly.

Catchment geometry is simplified as an infinitely long v-shaped valley with
width 2*L*_{h} and uniform hillslope angle *α* (Fig. 1). Because the
ridgelines have uniform elevation, there is no net dip to the catchment; the
effect of valley inclination will be assessed in Sect. 3.3. At a
horizontal distance from the ridgeline *x* and vertical depth below the
surface *z*, the shielding factor, *S*(*x*,*z*), is defined as

$$\begin{array}{ll}{\displaystyle}S\left(x,z\right)& {\displaystyle}={\displaystyle \frac{m+\mathrm{1}}{\mathrm{2}\mathit{\pi}}}{\int}_{\mathit{\phi}=\mathrm{0}}^{\mathrm{2}\mathit{\pi}}{\int}_{\mathit{\theta}={\mathit{\theta}}_{\mathrm{0}}(x,{L}_{\mathrm{h}},z,\mathit{\phi},\mathit{\alpha})}^{\mathit{\pi}/\mathrm{2}}{\mathrm{sin}}^{m}\mathit{\theta}{e}^{-d(z,\mathit{\rho},\mathit{\theta},\mathit{\gamma}\left(\mathit{\alpha},\mathit{\phi}\right))/\mathit{\lambda}}\\ \text{(6)}& {\displaystyle}& {\displaystyle}\mathrm{cos}\mathit{\theta}\mathrm{d}\mathit{\theta}\mathrm{d}\mathit{\phi},\end{array}$$

where *ρ* is rock density and *γ* is the apparent dip of the
hillslope in the azimuthal direction *φ* (Fig. 1b). The inclination
angle integration limit, *θ*_{0}, is a function of
topographic skyline-shielding inclination, and can be determined
geometrically (Fig. 1) as

$$\begin{array}{}\text{(7)}& \mathrm{tan}{\mathit{\theta}}_{\mathrm{0}}=\left\{\begin{array}{l}{\displaystyle \frac{\left(x\mathrm{tan}\mathit{\alpha}+z\right)\mathrm{cos}\mathit{\phi}}{\mathrm{2}{L}_{\mathrm{h}}-x}},\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathrm{0}\le \mathit{\phi}<{\displaystyle \frac{\mathit{\pi}}{\mathrm{2}}}\\ -\mathrm{tan}\mathit{\alpha}\mathrm{cos}\mathit{\phi}-{\displaystyle \frac{z}{x}}\mathrm{cos}\mathit{\phi},\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}{\displaystyle \frac{\mathit{\pi}}{\mathrm{2}}}\le \mathit{\phi}\le \mathit{\pi}\end{array}\right..\end{array}$$

The apparent dip, *γ*, can be derived from the model geometry in Fig. 1 as

$$\begin{array}{}\text{(8)}& \mathrm{tan}\mathit{\gamma}=-\mathrm{tan}\mathit{\alpha}\mathrm{cos}\mathit{\phi},\end{array}$$

and the mass distance traveled through rock by a given incident ray as

$$\begin{array}{}\text{(9)}& d={\displaystyle \frac{\mathit{\rho}z\mathrm{cos}\mathit{\gamma}}{\mathrm{sin}\left(\mathit{\theta}-\mathit{\gamma}\right)}}.\end{array}$$

Equation (5) was numerically solved for a series of hillslopes over a grid
of $(x/{L}_{\mathrm{h}}=[\mathrm{0},\mathrm{1}];\phantom{\rule{0.125em}{0ex}}\mathit{\rho}z/\mathrm{\Lambda}=[\mathrm{0},\mathrm{40}\left]\right)$ with horizontal spacing $\mathrm{d}x=\phantom{\rule{0.125em}{0ex}}{L}_{\mathrm{h}}/\mathrm{500}$ and vertical spacing
$\mathrm{d}z\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{\Lambda}/\mathrm{500}\mathit{\rho}$. To
characterize mean slope controls on the shielding factor, *S*(*x*,z), the above calculation was applied to nine hillslopes
with mean slope, *α*, ranging from 0 to 80^{∘} in 10^{∘}
increments. Because ${L}_{\mathrm{h}}\gg \mathrm{\Lambda}/\mathit{\rho}$ for most natural landscapes, the
resulting distribution of shielding factors is independent of hillslope
scale.

After applying Eq. (6) to a hillslope, it is straightforward to calculate
the surface skyline-shielding component, ${S}_{\mathrm{0}}\left(x\right)=S(x,\mathrm{0})$.
This skyline-shielding component should match the topographic shielding
factor determined from the algorithm of Codilean (2006); for comparison
this parameter was calculated at each pixel in the model catchment using
TopoToolbox (Schwanghart and Scherler, 2014). Two additional parameters were
calculated at each slope position using Eq. (5): the effective vertical mass
attenuation length, Λ_{eff}(*x*), and the total effective shielding
factor, *C*_{eff}(*x*).

Although spallogenic production of cosmogenic nuclides following Eq. (1) is
well-described by an exponential decrease with depth for horizontal
unshielded surfaces, this is not true in general for shielded samples (Dunne
et al., 1999). The effective vertical mass attenuation length, Λ_{eff}(*x*), is approximated by the vertical depth below the surface at which
the shielding factor is 5 % of the surface shielding (i.e., 3 *e*-folding
lengths) such that

$$\begin{array}{}\text{(10)}& S\left(x,{\displaystyle \frac{\mathrm{3}{\mathrm{\Lambda}}_{\mathrm{eff}}\left(x\right)}{\mathit{\rho}}}\right)=\mathrm{0.05}S(x,\mathrm{0}).\end{array}$$

If nuclide production as a function of depth deviates from an exponential
decline, it is inaccurate to use the analytical relationship between surface
sample concentration, *C*(*x*) (atoms g^{−1}), and
steady-state vertical erosion rate, *E* (g cm^{−2} yr^{−1}), typically
applied to eroding samples

$$\begin{array}{}\text{(11)}& C\left(x\right)={\displaystyle \frac{S\left(x\right){P}_{\mathrm{0}}\left(x\right){\mathrm{\Lambda}}_{\mathrm{eff}}\left(x\right)}{E}},\end{array}$$

where *P*_{0}(*x*) is the unshielded surface production rate, corrected for
latitude and air pressure (Lal, 1991). Equation (11) derives from
integrating the path history of a particle being exhumed vertically at a
steady rate *E* and emerging at the surface with an accumulated nuclide
concentration *C*(*x*):

$$\begin{array}{}\text{(12)}& C\left(x\right)={P}_{\mathrm{0}}\left(x\right){\int}_{{t}_{\mathrm{0}}}^{{t}_{\mathrm{surface}}}S(x,z(t\left)\right)\mathrm{d}t,\end{array}$$

which can be parameterized in terms of vertical depth below the surface,
*z*, according to

$$\begin{array}{}\text{(13)}& C\left(x\right)={\displaystyle \frac{{P}_{\mathrm{0}}\left(x\right)}{E/\mathit{\rho}}}{\int}_{\mathrm{0}}^{{z}_{\mathrm{0}}}S(x,z)\mathrm{d}z,\end{array}$$

where the depth of a rock parcel below the surface *z*_{0} at time *t*_{0}
is deep enough such that there is no cosmogenic nuclide production
(${z}_{\mathrm{0}}=\mathrm{40}\mathrm{\Lambda}/\mathit{\rho}$ for the calculations below) and
${t}_{\mathrm{surface}}={t}_{\mathrm{0}}+\mathit{\rho}{z}_{\mathrm{0}}/E$ is the time it takes for a rock parcel to
travel from depth *z*_{0} to the surface (assuming a vertical exhumation
pathway). Because there is no analytical solution for Eq. (13), the integral
needs to be solved numerically. A total effective shielding factor,
*C*_{eff}(x), acts as a correction factor to interpret local
erosion rate from a sample concentration, defined by

$$\begin{array}{}\text{(14)}& {C}_{\mathrm{eff}}\left(x\right)={\displaystyle \frac{{C}_{\mathrm{shielded}}\left(x\right)}{{C}_{\mathrm{unshielded}}\left(x\right)}}={\displaystyle \frac{{\sum}_{z=\mathrm{0}}^{z={z}_{\mathrm{0}}}S(x,z)\phantom{\rule{0.125em}{0ex}}}{{\sum}_{z=\mathrm{0}}^{z={z}_{\mathrm{0}}}S{}^{\prime}(x,z)}},\end{array}$$

where ${\sum}_{z=\mathrm{0}}^{z={z}_{\mathrm{0}}}S{}^{\prime}(x,z)$
is the integrated shielding depth profile for the case *α*=0
(i.e., no slope or skyline shielding), and *C*_{eff}(x) does
not depend on spatial variations in latitude or air pressure corrections.
Finally, a mean effective shielding factor, ${\stackrel{\mathrm{\u203e}}{C}}_{\mathrm{eff}}$, is defined
for the whole hillslope as

$$\begin{array}{}\text{(15)}& {\stackrel{\mathrm{\u203e}}{C}}_{\mathrm{eff}}={\displaystyle \frac{\mathrm{1}}{{L}_{\mathrm{h}}}}{\sum}_{x=\mathrm{0}}^{x={L}_{h}}{C}_{\mathrm{eff}}\left(x\right),\end{array}$$

which is equivalent to the catchment-mean shielding factor for the simplified valley geometry shown in Fig. 1.

Although the above framework accounts for variations in catchment relief and
hillslope angle, *α*, in all cases there is no net dip to the entire
catchment (i.e., ridgeline elevations are uniform), which is not the case for
natural watersheds. To simplify the geometry of a dipping catchment, I use a
similar approach as Binnie et al. (2006) to model the catchment as a plane
fit through the bounding ridgelines with dip *β*. I focus on two
end-member cases, using examples from the San Gabriel Mountains, California,
USA, for illustration (Fig. 2). First, for an “interior” catchment that is
tributary to a larger valley within a mountain range, the catchment will
have a net shielding similar to the geometry of the hillslope in Fig. 1.
Consequently, the shielding geometry can be approximated by Eqs. (6)–(9) with
*α*=*β*. For the case of an “exterior” catchment that has a net
dip *β* but no opposing skyline shielding, Eq. (7) becomes

$$\begin{array}{}\text{(16)}& \mathrm{tan}{\mathit{\theta}}_{\mathrm{0}}=\left\{\begin{array}{l}\mathrm{0},\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathrm{0}\le \mathit{\phi}<{\displaystyle \frac{\mathit{\pi}}{\mathrm{2}}},\\ -\mathrm{tan}\mathit{\alpha}\mathrm{cos}\mathit{\theta}-{\displaystyle \frac{z}{x}}\mathrm{cos}\mathit{\phi},\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}{\displaystyle \frac{\mathit{\pi}}{\mathrm{2}}}\le \mathit{\phi}\le \mathit{\pi}.\end{array}\right.\end{array}$$

For both examples, I compared the catchment-mean shielding factor, ${\stackrel{\mathrm{\u203e}}{C}}_{\mathrm{eff}}$, to the mean surface skyline-shielding factor, $\stackrel{\mathrm{\u203e}}{{S}_{\mathrm{0}}}$, as calculated using the commonly applied topographic shielding algorithm of Codilean (2006) in TopoToolbox (Schwanghart and Scherler, 2014).

4 Model results

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For the catchment geometry shown in Fig. 1, the local shielding factor,
*S*(*x*,z), decreases with increasing depth, *z*;
distance downslope, *x*; and increasing slope, *α* (Fig. 3). The
surface skyline-shielding factor, *S*_{0}(*x*), decreases with distance
downslope, *x*, and increasing hillslope angle, *α*, with the greatest
shielding occurring in the valley bottoms of steep catchments (Fig. 4a). For
the case *α*=80^{∘}, comparison of *S*_{0}(*x*) with the
topographic shielding algorithm of Codilean (2006) shows that the two are
equivalent (Fig. 4a).

The normalized effective attenuation length, Λ_{eff}∕Λ,
decreases as a function of distance downslope and increases with increasing
hillslope angle (Fig. 4b). Although cosmogenic nuclide
production is concentrated at depths of $\mathit{\rho}z/\mathrm{\Lambda}=\phantom{\rule{0.125em}{0ex}}[\mathrm{0},\phantom{\rule{0.125em}{0ex}}\mathrm{3}]$ for low slopes, production rates at depth for very steep slopes can be greater than
those of flat landscapes, despite lower surface production
rates (Fig. 5).
This effect emerges in part due to the increased effective attenuation
length for collimated radiation in skyline-shielded samples (up to a factor
of 1.3 – Dunne et al., 1999; Gosse and Phillips, 2001), but mainly because
on steep slopes a point at vertical depth *z* below the surface is receiving
incident radiation from oblique pathways that can be shorter than those
overhead (Fig. 1c). Consequently, there is an additional radiation flux that
increases the effective vertical mass attenuation length, Λ_{eff},
an effect that is most pronounced near ridgelines ($x/{L}_{\mathrm{h}}<\sim \mathrm{0.4})$ where
skyline shielding is minimized (Figs. 3, 4b).

The combined effect of the decrease in surface production (Fig. 4a) and the
increase in effective attenuation length (Fig. 4b) leads to a pattern
whereby the total effective shielding factor, *C*_{eff}(x), is
greater than 1 along the upper portion of hillslopes and less than 1
along the lower portion of hillslopes near the valley bottom (Fig. 4c).
Although there may be considerable variation in shielding
depending on slope position for steep slopes ($\mathit{\alpha}>\mathrm{60}{}^{\circ})$, the mean effective
shielding parameter, ${\stackrel{\mathrm{\u203e}}{C}}_{\mathrm{eff}}$, is unity for all cases (Fig. 6a).

For the case of dipping catchments (Fig. 2), the sensitivity of the mean
effective shielding parameter to catchment dip, *β*, depends on whether
catchments are “interior” (i.e., shielded by an opposing catchment) or
“exterior” (i.e., no external skyline shielding). For “interior”
catchments, the shielding calculations are identical to the analysis above,
and thus ${\stackrel{\mathrm{\u203e}}{C}}_{\mathrm{eff}}$ is again unity for all cases (Fig. 6a). For
“exterior” catchments, the increase in effective attenuation length at
steep slopes due to shorter oblique radiation pathways (Fig. 1c) is larger
than the decrease in surface production due to skyline shielding, and
${\stackrel{\mathrm{\u203e}}{C}}_{\mathrm{eff}}$ is greater than 1 (Fig. 6b). However, for all but the
most extreme catchment dips (*β*≤40^{∘}), ${\stackrel{\mathrm{\u203e}}{C}}_{\mathrm{eff}}$ is effectively 1 (within 1 %).

For the two example catchments in the San Gabriel Mountains (Fig. 2), the
mean total effective shielding factor, ${\stackrel{\mathrm{\u203e}}{C}}_{\mathrm{eff}}$, is 1.00,
despite steep catchment dips (*β*=17 and 32^{∘})
and high mean surface skyline shielding, $\stackrel{\mathrm{\u203e}}{{S}_{\mathrm{0}}}$ ($\stackrel{\mathrm{\u203e}}{{S}_{\mathrm{0}}}=\mathrm{0.87}$ and 0.84
as calculated by the algorithm of Codilean, 2006; Fig. 6a).

5 Implications for interpreting catchment erosion rates from ^{10}Be concentrations in stream sediment

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The above results indicate that no correction factor for topographic
shielding is needed to infer catchment-mean erosion rates from ^{10}Be
concentrations in stream sediments for most cases, as long as the assumptions of
spatially uniform quartz content and steady uniform erosion rate are valid.
Only in the extreme case of an “exterior” catchment with mean dip *β*
> 40^{∘} will such corrections be necessary. Although the
approach of only calculating the surface skyline-shielding component of the
total effective shielding factor is appropriate for calculating surface
exposure ages, neglecting the slope and shielding controls on the effective
mass attenuation length leads to a systematic underprediction of the actual
erosion rate. The magnitude of this underprediction increases with
increasing catchment mean slope, as highlighted by a compilation of
catchment erosion rates from steep catchments in the Himalaya and eastern
Tibetan Plateau (red data points, Fig. 5a).

For catchments with spatially variable quartz content or erosion rate, a
spatially distributed total effective shielding factor, *C*_{eff}, must be
calculated at each pixel. Although calculating the surface skyline-shielding
component is straightforward (Codilean, 2006), solving Eq. (3) at depth for
arbitrary catchment geometries is presently too computationally intensive to
be practical. However, while not entirely transferable to arbitrarily rough
topography (e.g., Norton and Vanacker, 2009), Fig. 4c suggests that for
slopes less than 40^{∘}, the total effective shielding factor does
not vary significantly across the hillslope. For steep catchments with
spatially variable quartz content or erosion rate, direct calculation of
shielding at depth is likely needed to calculate the spatially distributed
total effective shielding parameter. In particular, shielding calculations
in landscapes dominated by cliff retreat are poorly suited for treatment in
a vertical reference frame (e.g., Ward and Anderson, 2011).

The modeling approach above assumes a simplified angular distribution of
cosmic radiation flux (Eq. 1) and only accounts for cosmogenic nuclide
production via spallation. In actuality, the cosmic radiation flux does not
go to zero at the horizon, and becomes increasingly collimated (higher *m*)
with increasing atmospheric depth (Argento et al., 2015). Thus, the
sensitivity of the effective mass attenuation length to shielding will
increase with increasing elevation. However, the magnitude of changes in the
effective mass attenuation length due to shielding-induced collimation is at
most 30 % (Dunne et al., 1999), compared to the potential factor of 3 or
more increase due to shorter oblique radiation pathways on very steep slopes
(Figs. 1c; 4b). For hillslope gradients commonly observed in cosmogenic
nuclide studies of steep landscapes (30–40^{∘}), the increase in
effective mass attenuation length due to shielding-induced collimation and
slope effects are 2 %–5 % and 6 %–15 %, respectively (Dunne et al., 1999;
Fig. 4b). The dependence of Λ on atmospheric depth, which is
typically not accounted for in catchment erosion studies, is minor
(< 10 % for extreme case of catchment with 4 km of relief Marrero
et al., 2016) compared to the above slope effect for most landscapes.
Treatment of cosmogenic nuclide production by muons is less constrained than
spallogenic production, but the angular distribution of production by muons
is likely similar to that for spallation reactions and also sensitive to
latitude and atmospheric depth (Heisinger et al., 2002a, b).

Overall, the effect of topographic shielding corrections on interpreting
catchment erosion rates is small compared to typical assumptions inherent to
detrital cosmogenic nuclide methods. In particular, the assumption of steady
lowering is likely to be increasingly inappropriate for rapidly eroding
landscapes characterized by a significant contribution of muonogenic
production or slowly eroding landscapes where ^{10}Be concentrations
integrate over glacial–interglacial climate cycles. Steep landscapes
characterized by stochastic mass wasting present additional complications
(Niemi et al., 2005; Yanites et al., 2009), requiring the nontrivial
calculation of spatially distributed shielding parameters for an arbitrary
catchment geometry. Nonetheless, in all cases accounting only for surface
skyline shielding (e.g., Codilean, 2006) without including its concurrent
influence on the effective attenuation length yields incorrect results.

6 Conclusions

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The simplified model presented here for catchment-scale topographic shielding of incoming cosmic radiation highlights the two competing effects of slope and skyline shielding. As catchment relief increases, surface production rates are reduced due to increased skyline shielding. However, for shielded samples radiation is increasingly collimated, and for sloped surfaces oblique radiation pathways increase nuclide production at depth. Both of these effects lead to deeper effective vertical mass attenuation lengths, which offset the reduction in surface production when inferring erosion rates from cosmogenic nuclide concentrations. At the catchment scale, the mean total effective shielding factor is 1 for a large range of catchment geometries, suggesting that topographic shielding corrections for catchment samples are generally not needed, and that applying commonly used topographic shielding algorithms leads to underestimation of true erosion rates by up to 20 %. Although these corrections are typically small compared to other methodological uncertainties, they vary systematically with slope and relief. Consequently, misapplication of shielding correction factors could influence interpretations of relationships between topography and erosion rate.

Data availability

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Data availability.

MATLAB codes used to generate Figs. 3, 4, and 6 are included as Supplement and available at: https://github.com/romandibiase/catchment-shielding (last access: 15 October 2018).

Supplement

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Supplement.

The supplement related to this article is available online at: https://doi.org/10.5194/esurf-6-923-2018-supplement.

Competing interests

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Competing interests.

The author declares that there is no conflict of interest.

Acknowledgements

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Acknowledgements.

This project was supported by funding from National Science Foundation grant
EAR-160814, and benefited from discussions with Kelin Whipple, Alexander Neely, Paul Bierman.
Review comments from Greg Balco, Dirk Scherler, and an anonymous reviewer
helped improve the manuscript.

Edited by: Jane Willenbring

Reviewed by: Greg Balco and one anonymous referee

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