The potential soil production rate, i.e., the upper limit at which bedrock can be converted into transportable material, limits how fast erosion can occur in mountain ranges in the absence of widespread landsliding in bedrock or intact regolith. Traditionally, the potential soil production rate has been considered to be solely dependent on climate and rock characteristics. Data from the San Gabriel Mountains of California, however, suggest that topographic steepness may also influence potential soil production rates. In this paper I test the hypothesis that topographically induced stress opening of preexisting fractures in the bedrock or intact regolith beneath hillslopes of the San Gabriel Mountains increases potential soil production rates in steep portions of the range. A mathematical model for this process predicts a relationship between potential soil production rates and average slope consistent with published data. Once the effects of average slope are accounted for, a small subset of the data suggests that cold temperatures may limit soil production rates at the highest elevations of the range due to the influence of temperature on vegetation growth. These results suggest that climate and rock characteristics may be the sole controls on potential soil production rates as traditionally assumed but that the porosity of bedrock or intact regolith may evolve with topographic steepness in a way that enhances the persistence of soil cover in compressive-stress environments. I develop an empirical equation that relates potential soil production rates in the San Gabriel Mountains to the average slope and a climatic index that accounts for temperature limitations on soil production rates at high elevations. Assuming a balance between soil production and erosion rates on the hillslope scale, I illustrate the interrelationships among potential soil production rates, soil thickness, erosion rates, and topographic steepness that result from the feedbacks among geomorphic, geophysical, and pedogenic processes in the San Gabriel Mountains.

The potential soil production rate (denoted herein by

Geologic map of the central San Gabriel Mountains, California. Potential soil production rates inferred from the data of Heimsath et al. (2012) are also shown. Lithologic units were compiled using Yerkes and Campbell (2005), Morton and Miller (2003), and fig. 3 of Nourse (2002). Faults were mapped from Morton and Miller (2003) and the Quaternary fault and fold database of the United States (U.S. Geological Survey and California Geological Survey, 2006).

Despite its fundamental importance, the geomorphic community has no widely
accepted conceptual or mathematical model for potential soil production
rates. Pelletier and Rasmussen (2009) took an initial step towards developing
such a model by relating

The San Gabriel Mountains (SGM) of California (Fig. 1) have been the focus of many studies of the relationships among tectonic uplift rates, climate, geology, topography, and erosion (e.g., Lifton and Chase, 1992; Spotila et al., 2002; DiBiase et al., 2010, 2012; DiBiase and Whipple, 2011; Heimsath et al., 2012; Dixon et al., 2012). These studies take advantage of a significant west-to-east gradient in exhumation rates in this range. Spotila et al. (2002) documented close associations among exhumation rates, MAP rates, and the locations and densities of active tectonic structures. MAP rates vary by a factor of 2 across the elevation gradient and exhibit a strong correlation with exhumation rates (Spotila et al., 2002, their fig. 10). Lithology, which varies substantially across the range (Fig. 1), also controls exhumation rates. Spotila et al. (2002) demonstrated that exhumation rates are lower, on average, in rocks relatively resistant to weathering (i.e., granite, gabbro, anorthosite, and intrusive rocks) compared to the less resistant schists and gneisses of the range (Spotila et al., 2002, their Fig. 9). This lithologic control on long-term erosion rates can control drainage evolution. For example, Spotila et al. (2002) concluded that the San Gabriel River has exploited the weak Pelona schist to form a rugged canyon between ridges capped by more resistant Cretaceous granodiorite (e.g., Mount Baden Powell). Spotila et al. (2002) concluded that landscape evolution in the SGM was controlled by a combination of tectonics, climate, and rock characteristics.

Heimsath et al. (2012) provided a millennial-timescale perspective on the
geomorphic evolution of the SGM. These authors demonstrated that soil
production rates (

Recent research, stimulated by shallow seismic refraction and drilling
campaigns, has documented the importance of topographically induced stresses
for the development of new fractures (and the opening of preexisting
fractures) in bedrock or intact regolith beneath hillslopes and valleys
(e.g., Miller and Dunne, 1996; Martel, 2006, 2011; Slim et al., 2014;
St. Clair et al., 2015). In this process, the bulk porosity of bedrock and
intact regolith evolves with topographic ruggedness (i.e., topographic slope
or curvature). In a compressive-stress environment such as the SGM,
topographically induced stresses can result in lower compressive stresses, or
even tensile stresses, in rocks near ridgetops. As an elastic solid is
compressed, surface rocks undergo outer-arc stretching where the surface is
convex-outward (i.e., on hillslopes), reducing the horizontal compressive
stress near the surface and eventually inducing tensile stress near ridgetops
in areas of sufficient ruggedness. Such stresses can generate new fractures
or open preexisting fractures in the bedrock or intact regolith, allowing
potential soil production rates to increase. In this paper I test whether
potential soil production rates estimated using the data of Heimsath et
al. (2012) are consistent with the topographically induced stress fracture
opening hypothesis in the SGM. This hypothesis predicts a relationship
between

Estimates of the maximum or potential soil production rate (i.e., the soil production rate obtained when the buffering effects of soil, if present, are factored out of the measured soil production rate) for the SGM can be made using the residuals obtained from the regression of soil production rates to soil thicknesses reported by Heimsath et al. (2012) (their Fig. 3). The exponential form of the soil production function quantifies the decrease in soil production rates with increasing soil thickness:

Plots of

Heimsath et al. (2012) did not include data points from locations without
soil cover in their regressions because these data points appear (especially
for areas with

Analytic solutions illustrating the perturbation of a regional
compressive-stress field by topography.

The SGM has horizontal compressive stresses of

Savage and Swolfs (1986) studied the role of topography in modifying local
stresses in a model ridge-and-valley geometry that uses a conformal
transformation that includes length scales

In landscapes with

The average slope computed from the model geometry is consistent with the average slope computed by Heimsath et al. (2012). The average slope in the model is computed from the ridgetop to the point of maximum slope in the model geometry. In the SGM, as with any region of narrow, V-shaped valleys, the steepest portion of the hillslope tends to occur at or near the slope base. Heimsath et al. (2012) computed their average slope from hillslope patches over a length scale (approximately 400 m) that included ridgetops and side slopes. As such, the two calculations are consistent.

It is important to note that the local stress modification in the Savage and
Swolfs (1986) model is a function of both local curvature and the slope
averaged over a spatial scale that includes ridgetops and side slopes.
Within an individual hillslope, local curvature controls the sign of stress
modification, with a reduction in compressive stress (and development of
tensile stress in sufficiently rugged terrain) occurring beneath ridgetops
and an increase in compressive stress occurring beneath valley bottoms. The
compressive-stress reduction that occurs beneath ridgetops is the most
important response of the model for the purposes of this paper since the

Figure 3 illustrates the effects of topography on tectonic stresses only,
i.e., gravitational stresses are not included. Gravitational stresses can be
included in the model by superposing the analytic solutions of Savage and
Swolfs (1986) (their Eqs. 34 and 35) with the solutions of Savage
et al. (1985) that quantify the effects of topography on gravitational
stresses (their Eqs. 39 and 40). The result would be a
three-dimensional phase space of solutions corresponding to different values
of the regional tectonic stress

The fit of the solid curve in Fig. 2a to

Climate and vegetation cover of the central San Gabriel Mountains.
Color maps of

In addition to the average slope control associated with the topographically
induced stress fracture opening process, a climatic control on

Figure 4a–c illustrate the mean annual temperature (MAT), MAP, and existing
vegetation height (EVH) for the central portion of the SGM. Above elevations
of approximately 1800 m a.s.l., vegetation height decreases systematically
with increasing elevation (Fig. 4d). This limitation is likely to be
primarily a result of temperature limitations on vegetation growth because
MAP increases with elevation up to and including the highest elevations of
the range. Figure 4e plots the ratio of

Local variability in

The average slope and climatic controls on

The results of this section demonstrate that average slope and possibly
climate exert controls on

Map of the bedrock damage index,

Many studies have proposed a relationship between fracture density and
bedrock weatherability on the basis that fractures provide additional
surface area for chemical weathering and pathways for physical weathering
agents to penetrate into the bedrock or intact regolith (e.g., Molnar, 2004;
Molnar et al., 2007; Goodfellow et al., 2014; Roy et al., 2016a, b). The
difference in erosion rates between the SGM and adjacent San Bernadino
Mountains, for example, has been attributed in part to differences in
fracture density between these ranges (Lifton and Chase, 1992; Spotila et
al., 2002). As such, it is reasonable to hypothesize that differences in

In this section I invoke a balance between soil production and transport on the hillslope scale in order to illustrate the interrelationships among potential soil production rates, erosion rates, soil thicknesses, and average slopes spatially across the SGM. The conceptual model explored in this section is based on the hypothesis that the average slope depends on the long-term difference between uplift and erosion rates. Uplift rates (assumed for the purposes of this discussion to be equal to exhumation rates) are lower in the western portion of the SGM and higher in the eastern portion (Spotila et al., 2002, their Fig. 7b). As average slope increases in areas with higher uplift rates, erosion rates increase and soils become thinner. Both of these responses represent negative feedback mechanisms that tend to decrease the differences that would otherwise exist between uplift and erosion rates and between erosion rates and soil production rates. If the uplift rate exceeds the potential soil production rate, soil thickness becomes zero and soil production and erosion rates can no longer increase with increasing slope (in the absence of widespread landsliding in bedrock or intact regolith). In such cases, topography with cliffs or steps may form (e.g., Wahrhaftig, 1965; Strudley et al., 2006; Jessup et al., 2010). However, if the potential soil production rate increases with average slope via the topographically induced stress fracture opening process, the transition to bare landscapes can be delayed or prevented as Heimsath et al. (2012) proposed. This represents an additional negative feedback or adjustment mechanism beyond the increase in soil production rates in steep terrain made possible by the exponential form of the soil production function. At the highest elevations of the range, soil production is slower, possibly due to temperature limitations on vegetation growth. The interrelationship between these variables can be quantified without explicit knowledge of the uplift rate since the relationship between soil thickness and average slope implicitly accounts for the uplift rate (i.e., a smaller difference between uplift and erosion rates is characterized by a thinner soil). This conceptual model predicts positive correlations among potential soil production rates, erosion rates, and topographic steepness and negative correlations of all of these variables with soil thickness.

Equation (6), in combination with modified versions of Eqs. (9) and (11)
of Pelletier and Rasmussen (2009), i.e.,

Spatial variations in erosion rates can be estimated using

Using Eq. (10) as a substitution, Eqs. (8) and (9) can be combined
to obtain a single equation for

Plot of soil thickness,

Color maps illustrating the predicted potential soil production rate
from Eq. (6) (

Equations (8) and (9) are the same as Eqs. (9) and (11) of Pelletier and
Rasmussen (2009) except that their Eq. (9) included a term representing
the bedrock–soil density contrast related to a slightly different definition
of

The

The model can be further tested by comparison to the catchment-averaged
erosion rates reported by DiBiase et al. (2010). Figure 8 plots
catchment-averaged erosion rates (unfilled circles) as a function of
catchment-averaged

Plot of the catchment-averaged erosion rates of DiBiase et
al. (2010) (unfilled circles) versus catchment-averaged

This paper adopts a stepwise regression and cluster analysis approach that builds upon the regression analysis that Heimsath et al. (2012) used to characterize the dependence of soil production rate on soil thickness. Stepwise regression is the process of computing the residuals of a regression and testing for additional controls, via additional regression and the calculation of a new set of residuals, until no additional explanatory variable can be identified. Stepwise regression is one method for testing the residuals of a regression for additional controls, which is a recommended step in all regression analyses. I did not apply simultaneous multivariate linear regression (with or without log transformation) because such an approach would have been inconsistent with the complex nonlinear relationships in the data documented by Heimsath et al. (2012) and the analyses presented here.

Estimating

Savage and Swolfs (1986) used a convex–concave geometry, defined by a conformal transformation, in which the slope increases linearly with distance from the divide to the steepest point on the hillslope. In higher-relief portions of the SGM characterized by more planar hillslopes, slopes increase abruptly over a relatively short distance from the ridgetop, then more slowly with increasing distance from the ridgetop. This difference introduces some uncertainty into the application. The model might overestimate the magnitude of topographically induced stress in high-relief portions of the SGM because a more planar slope has a lower curvature than a more parabolic slope and larger curvatures tend to increase extensional stress. On the other hand, more planar hillslopes localize curvature near the ridgetops, which might tend to increase bending stresses that drive extension over and above that predicted by the model for locations near ridgetops.

The effect of topographically induced stresses on the production of intact
regolith or soil is a rapidly evolving field at the boundaries among
geomorphology, geophysics, and structural geology. The results presented
here, based on the Savage and Swolfs (1986) model, represents just one
possible approach to the problem. Miller and Dunne (1996), for example,
modified the Savage and Swolfs (1986) solutions to account for cases with
vertical compressive-stress gradients (their parameter

The results presented here provide a possible process-based understanding of
the dependence of potential soil production rates on topographic steepness
documented by Heimsath et al. (2012) in the SGM. These authors proposed a
negative feedback in which high erosion rates trigger higher potential soil
production rates, with the result that soil cover may be more persistent than
previously thought. The results presented here show that previous models of
topographically induced stresses suggest transitions from compressive to
tensile strength at hillslope angles similar to those at which

Heimsath et al. (2012) argued that

In this paper I estimated spatial variations in the potential soil
production rate,

The dataset analyzed in this paper is available in Table S1 in the Supplement. Any additional information or data will be made available upon request to the author.

The author declares that he has no conflict of interest.

I thank Katherine Guns for drafting Fig. 1. I wish to thank Arjun Heimsath, Kelin Whipple, Simon Mudd, Jean Braun, and three anonymous reviewers for reviews of this paper, and I would also like to express my gratitude to three anonymous reviewers who provided feedback on an earlier version of this paper that was submitted elsewhere. Edited by: Jean Braun Reviewed by: Simon Mudd, Arjun Heimsath, and three anonymous referees