How Well Can Hillslope Evolution Models “Explain” Topography? Simulating Soil Transport and Production with High-Resolution Topographic Data

How well can hillslope evolution models “explain” topography? Simulating soil transport and production with high-resolution topographic data

Joshua J. Roering Department of Geological Sciences, University of Oregon, Eugene, Oregon 97403-1272, USA

ABSTRACT nonlinear slope- and depth-dependent model form and sediment yield (Gabet et al., 2003; accounts for how soil thickness controls the Roering et al., 2004; Yoo et al., 2005). Quan- The morphology of hillslopes is a direct magnitude of biogenic disturbances that titatively exploring these feedbacks between refl ection of tectonic forcing and climatic and drive transport; this model best preserved hillslope topography and climatic and tectonic biologic processes that drive soil production, the current landscape form, particularly the processes requires that we posit equations to mobilization, and transport. Soil transport narrow, sharply convex hilltops characteris- represent transport on hillslopes, an endeavor on hillslopes affects river incision by provid- tic of the Oregon Coast Range. According to that has occupied geologists for nearly 100 yr. ing tools for channel abrasion and controls our formulation, which provides an explicit In 1909, G.K. Gilbert (Gilbert, 1909) expounded the distribution of sediment that infl uences linkage for relating the distribution of biota upon the ideas of Davis (1892) and proposed a aquatic habitat. Although numerous hillslope to hillslope processes, the degree of hilltop process-based explanation for the convex form transport relationships have been proposed convexity varies nonlinearly with the ratio of “debris”-mantled hillslopes. For a one-dimen- over the past 60+ years, a comprehensive of erosion rate to maximum soil production sional hillslope that erodes at a spatially constant analysis of model predictions for a real land- rate, highlighting the profound infl uence of rate, sediment fl ux must increase linearly with scape has not been performed. Here, we use soil depth on hillslope evolution. distance from the drainage divide (i.e., hilltop). high-resolution topographic data obtained To accommodate the progressive downslope via airborne laser swath mapping (ALSM) to Keywords: hillslope evolution, soil transport, increase in fl ux, slopes become monotonically simulate the long-term evolution of Oregon nonlinear transport, airborne lidar. steeper as increasing gravitational acceleration Coast Range hillslopes and test three pub- facilitates transport via disturbances (such as lished transport models and a new model INTRODUCTION those associated with biogenic activity). The that accounts for nonlinear depth- and slope- progressive downslope increase in slope angle dependent transport. Analysis of one-dimen- In hilly and mountainous landscapes, the defi nes a convex form between hilltops and sional, steady-state solutions for these four downslope movement of soil on hillslopes mod- channels, which is commonly observed for models suggests that plots of gradient-cur- ulates landscape form and sediment delivery to natural slopes. Several decades after Gilbert’s vature may be diagnostic for distinguishing channels. The coupling between rock uplift and work, his conceptual framework was cast into model predictions. To evaluate two-dimen- topographic relief thus depends on the effi cacy equations by Culling (1963) and since the 1960s, sional model predictions for our fi eld site, of diverse climate-related hillslope processes, dozens of quantitative hillslope evolution models we assumed local steady-state erosion for a such as frost heave, bioturbation, and shallow have emerged, virtually all of which incorporate 72,000 m2 sequence of hillslopes and valleys. landsliding. With increasing rates of rock uplift slope angle as the primary control on sediment After calibrating each of the four models, and erosion, for example, mass wasting pro- transport rates (e.g., Ahnert, 1976; Arrowsmith we imposed constant base-level lowering for cesses promote rapid sediment fl uxes on steep et al., 1996; Avouac and Peltzer, 1993; Braun cells within the valley network, simulated slopes and may limit further topographic devel- and Sambridge, 1997; Culling, 1963; Dietrich 500,000 yr of soil production and transport, opment (e.g., Burbank et al., 1996; Montgom- and Perron, 2006; Hanks, 2000; Howard, 1997; and determined which transport model ery and Brandon, 2002). As a result, decipher- Istanbulluoglu et al., 2004; Kirkby, 1971; Kooi best preserved morphologic patterns that ing rates of tectonic forcing from topographic and Beaumont, 1994; Koons, 1989; Tucker and describe the current landscape form. Mod- metrics requires information on hillslope pro- Slingerland, 1994; van der Beek and Braun, els for which fl ux varies proportionally with cess mechanics and rates (e.g., Ahnert, 1970b; 1998; Willett, 1999; Willgoose et al., 1991). hillslope gradient generated broadly convex Ahnert, 1984; Whipple et al., 2005). In the Linear slope-dependent transport models hilltops inconsistent with the sharp-crested, absence of overland fl ow, mass transport on soil- predict convex hillslopes and have a straight- steep-sided slopes of our study site, whereas mantled slopes is often driven by disturbances forward mathematical form, making them the two nonlinear slope-dependent models associated with biological activity (i.e., biotur- simple to implement in analytical and numeri- produced convex-planar slopes consistent bation). This explicit linkage between biology cal models of landscape evolution (Koons, with current hillslope form. Our proposed and sediment transport suggests that ecological 1989). Importantly, linear models predict that changes associated with long-term climatic fl uc- steady-state hillslopes exhibit constant curva- †E-mail: [email protected] tuations may induce fi rst-order shifts in hillslope ture ( Fernandes and Dietrich, 1997). Despite the

GSA Bulletin; September/October 2008; v. 120; no. 9/10; p. 1248–1262; doi: 10.1130/B26283.1; 12 ﬁ gures; 1 table.

1248 For permission to copy, contact [email protected] © 2008 Geological Society of America How well can hillslope evolution models “explain” topography? visually and computationally appealing nature ing of the valley network. Using the assumption SOIL PRODUCTION MODEL of the linear transport model, the prediction of of locally steady-state topography, they com- constant curvature slopes is inconsistent with pared their results with the initial topographic The presence or absence of soil has a pro- frequently observed convex-planar slopes in form (i.e., current topography) to assess model found infl uence on the mechanics and rates of soil-mantled regions (Carson and Petley, 1970; success. Essentially, they inquired as to how well surfi cial processes as well as feedbacks with Penck, 1953; Strahler, 1950; Young, 1961). To a previously calibrated transport model can pre- ecological processes. Following insights fi rst account for this discrepancy, a raft of trans- serve current landscape form. Here, we adopt a proposed by Gilbert (1877), recent studies have port models emerged in the 1980s and 1990s; similar approach, but instead test a diverse suite demonstrated that the rate of soil production, ε, these models incorporated a nonlinearity in the of representative transport models to systemati- declines exponentially with slope-normal soil fl ux-slope relationship such that transport rates cally determine which best preserves diagnostic depth according to: increase rapidly as slopes approach a critical properties of current topography in the Oregon angle (Anderson, 1994; Anderson and Hum- Coast Range. We also propose a new transport ε −μθ phrey, 1989; Andrews and Bucknam, 1987; model for which fl ux depends nonlinearly on ε = 0 e h cos , (3) cosθ Gabet, 2000; Howard, 1994; Kirkby, 1984; slope and soil depth. Because the depth depen- Roering et al., 1999). Put simply, the fl ux-slope dency depends on the vertical distribution of ε nonlinearity enables nearly steep and planar biological activity in soil, this new formulation where 0 is the maximum production rate asso- (low convexity) sideslopes to erode at rates com- enables us to calibrate and test the model with ciated with zero soil depth, μ is the exponential mensurate with highly convex hilltops. More fi eld-generated biological data. Our results are decay constant, and θ is local slope in degrees recently, several studies have proposed that fl ux useful for predicting sediment yield to channel (Heimsath et al., 2001b). The cosine term in the rates depend on slope angle and soil depth (e.g., networks as well as interpreting and predicting exponential function is included to convert ver- Furbish and Fagherazzi, 2001; Heimsath et al., morphologic response to tectonic and climatic tical soil depth to slope-normal soil depth and 2002; Heimsath et al., 2005; Mudd and Furbish, perturbations in real landscapes. the cosine term in the denominator is included 2007; Yoo et al., 2005), enabling these models to convert soil production measured slope-nor- to be directly coupled with representations of MASS CONSERVATION LAWS FOR mal into a vertical reference frame. Here, we depth-dependent biological processes. HILLSLOPE EVOLUTION express soil depth, h, in a vertical orientation to Despite the widespread availability of com- facilitate presentation and interpretation of our putational resources and high-resolution topo- Landscapes result from the coupled infl uence simulation results. Equation 3 is equivalent to graphic data (via airborne laser altimetry), few of tectonic forcing, which displaces bedrock Equation 5 in Heimsath et al. (2001b). studies have simulated the evolution of “real” both vertically and horizontally, and geomorphic On a steady-state hillslope (i.e., ∂h/∂t = 0) for ∇ ~ landscapes at the process scale to test hillslope processes that both convert substrate into mobile which the erosion rate (E = – · qs) is spatially transport models (Ahnert, 1970a; Braun et al., material and transport that material across the uniform, such that E = ε, steady-state soil depth 2001; Dietrich et al., 2003; Herman and Braun, surface (Dietrich et al., 2003). According to a is given by: 2006; Minasny and McBratney, 2006). In most mass conservation Equation that accounts for geological settings, this endeavor is challenging these processes, the rate of change of the surface ⎛ E ⎞ − ln cosθ because we lack quantitative information on the elevation, z, is given by: ⎝⎜ ε ⎠⎟ original hillslope form necessary for model ini- h = 0 . (4) μθcos tialization. Dietrich et al. (2003) used airborne ∂z ρ ∂h light detection and ranging (LiDAR)-generated =−T r ε + , (1) ∂ ρ ∂ t s t topography of Oregon Coast Range slopes as the Equation 4 indicates that steady-state soil initial condition and applied a nonlinear, slope- depth is not constant, but in fact varies system- dependent transport model and exponential soil where t is time, T is rock uplift rate (L T−1), ε atically with slope angle. To explore the magni- production model to simulate one million years is sediment (or soil) production rate (L T−1), h tude of this effect, we can rewrite Equation 4 in ρ ρ of hillslope evolution driven by base-level lower- is vertical soil depth (L), and r and s are the terms of dimensionless soil depth, h*: bulk densities of rock and soil, (M L−3), respectively (Fig. 1). On hillslopes, the change in the ⎛ E ⎞ − ln cosθ landscape soil thickness with time depends on the balance ⎝⎜ ε ⎠⎟ z 0 surface of erosion by soil transport, q˜ , (L3 L−1 T−1) and hh* ==μ . (5) s cosθ the soil production rate: h ρ s The value of h* decreases nonlinearly with ∂h ρ q =−∇⋅q + r ε. (2) E/ε such that the change in h* along a single ρ s ∂t s ρ 0 r s hillslope is greatest when E/ε is small and the 0 soil slope is steep (Fig. 2). Due to the onset of shal- These equations do not account for mass loss low landsliding, the soil mantle may become bedrock via chemical weathering, which will affect the patchy and tenuous for slope gradients greater T bulk densities of soil and bedrock as well as the than 1.0. Thus, the range of applicability for “erodibility” of bedrock as represented by ε. If this relationship varies with factors that con- ∇ ~ Figure 1. Schematic diagram of mass con- erosion (denoted here as – · qs ) outpaces soil trol shallow landslide susceptibility, including servation components in a soil-mantled land- production, bedrock emerges and erosion is lim- soil properties, root reinforcement via vegeta- scape. See Equations 1 and 2 for explanation. ited by the maximum soil production rate. tion, and rainfall characteristics. Interestingly,

Geological Society of America Bulletin, September/October 2008 1249 Roering

6 tively linking the magnitude of transport with 5 processes that mobilize and transport soil in E/ε = 0.1 real landscapes. 0 Figure 2. Variation of dimensionless soil depth 4 (h*) with hillslope gradient as given in Equa- Nonlinear Depth- and Slope-Dependent h* 3 0.25 tion 5. Soil depth is deﬁ ned as vertical. Soil depth increases nonlinearly downslope and Model 0.5 2 for a given slope angle, soil depth decreases as 0.75 Here, we propose a revision to the nonlin- 1 0.99 the ratio of steady-state erosion rate to maxi- ε ear slope-dependent model by relating PA to mum soil production rate (E/ 0) increases. 0 the vertical distribution of transport-inducing 0 0.25 0.5 0.75 1 1.25 1.5 agents in a column of soil. Analogous to the gradient, tanθ formulation of Yoo et al. (2005), we can parti-

tion PA into the product of power per unit vol- 2 −3 −3 ume of disturbance agents, PV (ML T L ), and the total volume of disturbance agents per ε → 3 2 as E/ 0 1, h* values approach zero near the and Fagherazzi, 2001; Heimsath et al., 2005; unit area, DA (L /L ): hilltop (i.e., where θ = 0). Accordingly, if the Mudd and Furbish, 2007) according to: = rate of baselevel lowering increases and E/ PDPAAV. (10) ε ≥ qKhz =−θ ∇ 0 1, Equation 5 predicts that the soil mantle s 2 cos , (7) will initially be stripped from hilltops rather Equation 10 indicates that for a column of than sideslopes that are more directly coupled where K2 is a transport rate coeffi cient with soil, we can depth integrate the volume of dis- −1 to channels. units L T and h is specifi ed as the active soil turbance agents to estimate PA. Given that bio- depth (L). The active soil depth can presum- logical organisms, frost heave activity, and other SOIL TRANSPORT MODELS ably be estimated from fi eld-based documenta- transport agents tend to vary systematically with

tion of relevant transport processes. depth, we can calculate DA by integrating the On hillslopes, the transport of soil in the distribution of these agents (through the depth absence of overland ﬂ ow has been described by Nonlinear Slope-Dependent Model of available soil measured normal to the surface, a diverse suite of equations (e.g., Dietrich and hcosθ) according to: Perron, 2006). Numerous models have been Given the ubiquity of convex-planar slopes, proposed, but here we focus on four representa- numerous models have proposed that transport h cos θ = ′ tive transport models that encapsulate a range of varies nonlinearly with gradient such that ﬂ uxes DAz∫ Ddz, (11) mechanical behavior. We do not include slope- increase rapidly as slope angle approaches 0 wash models in this analysis because overland a critical value (Anderson, 1994; Howard,

fl ow erosion is not relevant to our fi eld site in the 1994). Initially, this nonlinearity was proposed where Dz is the volume of disturbance agents Oregon Coast Range. to account for the increasing frequency of land- per unit volume of soil (L3/L3) and z′ is the slides as slopes approach the angle of repose. axis oriented normal to the surface. Equa- Linear Slope-Dependent Model Experimental evidence indicates, however, that tions 10 and 11 enable us to link our transport landsliding is not required to generate the non- model with fi eld-based estimates of how dis- The linear, slope-dependent model was fi rst linearity (Roering et al., 2001b), which instead turbance agents are distributed with depth. In proposed by Culling (1960) and has been used may result from the interplay between friction vegetated landscapes, for example, root growth extensively in landscape evolution models. and gravity in a surface layer undergoing dis- causes soil particle activation, mobilization, According to this model, sediment fl ux varies turbance. A physically based model proposed and downslope transport (Gabet et al., 2003), ∇ proportionally with local gradient, z, where K1 by Andrews and Bucknam (1987) and recast by suggesting that profi les of root density may be is a constant of proportionality with units L2 T−1: Roering et al. (1999) indicates that fl ux varies useful for estimating soil transport rates. Com- according to: monly, root density declines exponentially with qKz =− ∇ . (6) depth measured normal to the surface (e.g., s 1 −∇Kz q = 3 , (8) Canadell et al., 1996; Jackson et al., 1996) such s −∇( 2 1 zSc ) As demonstrated below, Equation 6 predicts that Dz can be defi ned according to that hillslopes attain constant curvature given = α −βz ' steady-state erosion. where Sc is the critical gradient at which fl ux Dz e , (12)

becomes infi nite and K3 is a transport coeffi cient Depth-Slope Product Model (L2 T−1) given by: where α is the density of roots at the surface (L3 L−3) and β, which has units L−1, describes 2 Because transporting agents are relegated to K = P , (9) the rate of exponential decline with depth. 3 ρ gS 2 A the thickness of available soil (measured nor- sc Combining Equations 11 and 12 and integrat- mal to slope as hcosθ), studies that account for ing through the normal soil thickness yields an 2 −3 −2 particle activation via disturbances and particle where PA is power per unit area (ML T L ) expression for DA: density variations with depth propose that fl ux generated by disturbances that drive soil trans- can be represented with the depth-slope prod- port and g is acceleration due to gravity (L T−2). h cos θ α D = α e − β z 'dz ' = (1− e–βhcosθ) uct (Ahnert, 1976; Braun et al., 2001; Furbish Equation 9 provides a framework for quantita- A ∫ β . (13) 0

1250 Geological Society of America Bulletin, September/October 2008 How well can hillslope evolution models “explain” topography?

1 Equation 13 can be combined with Equa- cannot account for variations in planform cur- 15 tions 9 and 10, producing an expression for vature that are characteristic of many natural 9 0.8 5 a new transport coeffi cient, K4, which varies hillslopes, it does provide a fi rst-order frame- 3 according to: work for exploring model predictions. For a one- 0.6 dimensional, steady-state hillslope (such that the K4 1 η rates of rock uplift and erosion are equivalent, 0.4 2αP −−βθ βθ K ( h )=−V ()11eehhcos=−η( cos ). (14) i.e., ∂z/∂t = 0 and T = −E), Equations 1 and 2 can 4 βρgS 2 β=0.4 cs be combined and simplifi ed as: 0.2 dq ρ Rewriting Equation 8 in terms of K yields: sr= E. (17) 0 4 ρ dx s 0 0.5 1 1.5 2 2.5 normal soil depth, hcosθ (m) By individually substituting one-dimen- −∇Kh() z ˜q = 4 . (15) sional versions of Equations 6, 7, 8, and 15 into Figure 3. Variation of the dimensionless coef- s −∇()2 1 zSc Equation 17, we obtain a series of expressions fi cient of the nonlinear slope- and depth-

for steady-state hillslope curvature as a func- dependent (K4) model with soil depth as given β To illustrate how K4 depends on soil thick- tion of gradient and erosion rate. Because two in Equation 16. For large values of , trans- ness, we normalized Equation 14 as follows: of the fl ux models (Equations 7 and 15) depend port rates increase rapidly with depth. on soil depth, the steady-state soil depth (Equa- tion 4) is substituted into those equations prior K −βθ K *cos==−4 ()1 e h . (16) to differentiation. One-dimensional curvature 4 η η * for the K1, K2, K3, and K4 (or ) models, respec- dimensionless curvature, C , is calculated from * 2 2 2 2 tively, varies according to: the general form, C = –(d z/dx )/(d z/dx )HT: * Equation 16 indicates that K4 increases rapidly with soil depth measured normal to the C* =−1 (19a) dz2 ρ E 1 surface and approaches a constant value such =− r (18a) 2 ρ dx s K1 that subsequent increases in soil depth result in ln()E ε * = 0 negligible increases in fl ux (Fig. 3). Importantly, C2 (19b) ⎡⎛ ()dz dx 2 ⎞ ⎛ ⎞ ⎤ Equation 12 can take on any functional form ρ E ⎢ − E ⎥ −μ r ⎜ ⎟ ln ⎜ ⎟ that refl ects the vertical distribution of relevant 2 ⎢ D ⎝ ε ⎠ ⎥ dz ρ K ⎣⎝ 1 ⎠ 01D ⎦ = s 2 transport agents in a particular fi eld setting. 2 2 (18b) dx ⎛ ()dz dx ⎞ ⎛ E ⎞ Here, we chose an exponential function because ⎜ ⎟ − ln ⎜ ⎟ ε of its ubiquitous use in describing the distribu- ⎝ D1 ⎠ ⎝ 01D ⎠ ⎛ 2 ⎞ 2 tion of biological agents in soil (Canadell et al., ⎛ dz dx ⎞ −−⎜1 ⎟ 1996; Gabet et al., 2003; Jackson et al., 1996). ⎝⎜ ⎠⎟ ⎝ Sc ⎠ 2 * = In the presence of multiple transport agents, where D1 = 1+(dz/dx) , C3 2 (19c) Equation 10 can be used to calculate P as the ⎛ dz dx ⎞ A 1+ ⎝⎜ ⎠⎟ sum of power per unit area for individual agents, Sc ⎛ 2 ⎞ 2 i.e., P = (D P ) +(D P ) +...+(D P ) . ρ E ⎛ dz dx ⎞ A(Total) A V 1 A V 2 A V N −−r ⎜1 ⎜ ⎟ ⎟ 2 ρ ⎝ ⎠ dz scK3 ⎝ S ⎠ = 2 MODEL IMPLICATIONS FOR ONE- 2 2 (18c) ⎛ 2 ⎞ dx ⎛ ⎞ ()βμ() ε ⎛ dz/ dx ⎞ DIMENSIONAL, STEADY-STATE dz dx −−()11eDln E 0 ⎜ − ⎟ 1+ ⎜ ⎟ 1 ⎝⎜ ⎠⎟ ⎝ ⎠ ⎝ Sc ⎠ MORPHOLOGY Sc * = . (19d) C4 ⎡ 2 ⎛ 2⎞⎤ β ⎛ dz⎞ ⎛dz/ dx⎞ ⎣⎡D 2 ()11 − D ⎦⎤ + ⎢ D ⎜ ⎟ ⎜ − ⎟⎥ The transport models expressed by Equations 1 22⎢μ ⎝ ⎠ ⎝⎜ ⎠⎟ ⎥ ⎣ dx ⎝ Sc ⎠⎦ ρ E 6, 7, 8, and 15 have profound implications for − r (18d) dz2 ρη the evolution of real landscapes. Before embark- = s 2 ing on a series of two-dimensional simulations, dx ⎛ ⎡ 2 ⎛ 2⎞⎤⎞ The negative value is included such that val- β ⎛dz⎞ ⎛dz/ dx⎞ however, it is instructive to explore the morpho- ⎜⎡DD2 (1 − )⎤ + ⎢ D⎜ ⎟ ⎜1− ⎟⎥⎟ ues of C* are consistent with negatively curved ⎣ 1 2 ⎦ μ 2 ⎝ ⎠ ⎝⎜ ⎠⎟ ⎜ ⎢ dx ⎝ Sc ⎠⎥⎟ logic predictions of these models using a one- ⎜ ⎣ ⎦⎟ (i.e., nonconcave) hillslopes. Equations 19a–19d 2 dimensional, steady-state approximation. For ⎜ ⎛ ⎛ ⎞ 2 ⎞ ⎟ provide a straightforward framework for com- ⎜ − dz/ dx ⎟ this comparison, we use the steady-state relation- D1 ⎜1 ⎜ ⎟ ⎟ paring morphological implications of the four ⎜ ⎝ ⎝ S ⎠ ⎠ ⎟ ship between gradient and curvature for each of ⎝ c ⎠ transport models assuming steady-state condi- the models to distinguish their behavior because: tions (Fig. 4). As shown previously (Fernandes (1) analytical expressions for one-dimensional, and Dietrich, 1997; Roering et al., 1999), con- ()βμ()() ε = ln ED01 gradient-curvature relationships can be readily where D2 e . vexity (negative curvature values) is invariant derived, (2) gradient and curvature are easily To directly compare these predictions, we according to the linear, slope-dependent model * estimated from topographic data for real land- respectively normalized values of curvature cal- (Equation 6), and C1 values are everywhere scapes, and (3) gradient-curvature curves appear culated in Equations 18a–18d by the hilltop cur- equal to −1 (Fig. 4A). In contrast, the depth- 2 2 2 2 * to be diagnostic for distinguishing model predic- vature (d z/dx )HT, defi ned as d z/dx evaluated at slope product model predicts that C2 values tions. Although this one- dimensional approach dz/dx = 0. For each of the four transport models, approach zero with increasing gradient. For

Geological Society of America Bulletin, September/October 2008 1251 Roering

planar 0 0 E/ε =0.99 0 E/ε =0.99 -0.25 0 0.9 0.5 -0.25 0.9 β/μ 0.1 =0.1 -0.5 -0.5 0.5 C* C* 2 3 C * C*4 -0.75 0.1 10 -0.75

C*3 Sc=1.0 Sc=1.0 -1 C*4 -1 C* C* β/μ=1 E/ε =0.5 AB1 2 C 0 -1.25 -1.25 convex 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 gradient gradient gradient Figure 4. Variation of dimensionless curvature (C*) with hillslope gradient for one-dimensional, steady-state hillslopes with uniform erosion. * ε (A) The K1 model predicts that C1 is constant, and the K2 model predicts that C2* approaches zero with slope depending on the value of E/ 0. * (B) and (C) The K3 model predicts an invariant, sigmoidal curve where Sc = 1.0, whereas the K4 model predicts that C4 values approach zero in β ε a different fashion depending on the values of and E/ 0.

* ε high ratios of erosion rate to maximum soil pro- (C2 curve in Fig. 4A). When E/ 0 <0.9, increas- Formation has been compressed into a series of ε β μ * duction rate (E/ 0 >0.9), the transition between ing values of / cause the C4 curve to approach low-amplitude, gently dipping folds (the maxi- * high-convexity slopes and relatively planar ones the C3 curve (Fig. 4C). This correspondence is mum dip of bedding along the fl anks of folds is abrupt and occurs at low gradients. Morpho- intuitively appealing given that for high values rarely exceeds 15°–20°) oriented north-north- logically, this gradient-curvature relationship of β relative to μ, fl uxes depend weakly on soil east (Baldwin, 1956). Uplift of the Oregon Coast corresponds to a hillslope with a sharply curved depth. Finally, our analyses (Figs. 4B and 4C) Range commenced in the Miocene (McNeill et * * ε divide and effectively planar sideslopes. This indicate that C4 approaches C3 when E/ 0 <0.5 al., 2000) and continues today as evidenced by pattern results from near-zero soil depths at the and β/μ >2. abandoned wave-cut platforms along the Oregon hilltop (Fig. 2), causing the depth component These results suggest that the transport mod- coast (Kelsey et al., 1996). Rates of rock uplift of transport to be negligible (see Equation 7) els outlined above predict diagnostic gradient- derived via dating of marine terraces adjacent to such that erosion must be accommodated by curvature relationships that should be useful for our study area (latitude ranging from 43° to 45°) high convexity (i.e., rapid change in gradient at assessing model applicability to real landscapes. vary from <0.1 to 0.3 mm yr−1 (Kelsey et al., the hilltop). Moving off the divide to locations Interestingly, as erosion rates approach the 1996) and are generally an order of magnitude ε where gradients exceed 0.3, the downslope maximum soil production rate (E/ 0 >0.9), the lower than geodetic uplift rates derived from increase in fl ux necessary to maintain a steady- two fl ux equations that incorporate soil depth highway leveling and tide gauge data (Mitchell state condition can be accommodated by the (Equations 7 and 15) predict similar steady- et al., 1994). Both short- and long-term uplift incremental increase in soil depth (see Fig. 2) state, gradient-curvature relationships refl ecting rates measured along the coast vary locally due combined with small changes in gradient. With the profound infl uence of soil depth-dependent to vertical movement along faults, although it is ε decreasing E/ 0, the downslope decline in con- processes on hillslope form and evolution (see unclear whether these local variations extend a * * vexity with steepness is more gradual and for E/ C2 and C4 curves in Figs. 4A and 4B). Below, signifi cant distance inland. ε 0 = 0.1, hillslopes with gradient equal to 1.0 are we apply the two-dimensional versions of these The topography of the Oregon Coast Range only slightly less convex than hilltops. models to the Oregon Coast Range to further has been characterized as steep and highly dis- The nonlinear, slope-dependent model (Equa- explore the implications of different fl ux equa- sected with broad regions of relatively uniform tion 8) predicts a sigmoidal relationship between tions on slope morphology. ridge and valley terrain (Dietrich and Dunne, gradient and curvature and a gradual transition 1978; Montgomery, 2001; Reneau and Dietrich, between convex hilltops and steep, planar side- STUDY AREA 1991). Typically, soil is relatively thin (~0.4 m) * slopes (see C3 curve in Fig. 4B). Owing to the on hilltops and sideslopes and thicker (~1–2 m) condition of infi nite fl ux when |dz/dx| = Sc, this The Oregon Coast Range is a humid, soil- in unchanneled valleys that act as preferential model predicts that slopes become effectively mantled, mountainous landscape largely com- source areas for shallow landslides that often planar as angles approach Sc. Once Sc is specifi ed posed of Eocene sedimentary rocks that overlie initiate debris fl ows (Dietrich and Dunne, 1978;

(here we use Sc = 1.0 for simplicity), Equation 19c volcanic basement accreted to the North Ameri- Heimsath et al., 2001a). Soil transport and pro- indicates that the form of the gradient-curvature can plate in the early Tertiary (Orr et al., 1992). duction appear to be driven primarily by bio- curve is ﬁ xed. In contrast, the nonlinear slope- The area is almost exclusively underlain by the genic processes, and overland ﬂ ow on slopes is and depth-dependent model (Equation 15) indi- Tyee Formation, which has been studied in detail absent owing to highly permeable soils. Most cates that the gradient-curvature relationship because of its distinct assemblage of sedimentary studies of decadal- to millennial-scale patterns

(Equation 19d) depends on the value of Sc as well facies (Chan and Dott, 1983; Heller and Dickin- of sediment production and delivery in the β μ ε * as / and E/ 0. Figure 4B shows that C4 values son, 1985; Lovell, 1969). Heller and Dickinson Oregon Coast Range have focused on the cyclic ε approach zero gradually for E/ 0 <0.9, whereas (1985) suggested that the Tyee Formation is a inﬁ lling and evacuation of soil in steep, conver- ε when E/ 0 >0.9, the convex-planar transition sand-rich sequence of turbidite deposits origi- gent areas (Benda and Dunne, 1997; Dietrich is rapid and occurs at low gradients, similar to nated from a delta-fed submarine ramp deposi- and Dunne, 1978; Reneau and Dietrich, 1990). predictions of the depth-slope product model tional system. Since the late Eocene, the Tyee Erosion rates generated by short (~10 yr) and

1252 Geological Society of America Bulletin, September/October 2008 How well can hillslope evolution models “explain” topography? long-term (~5000 yr) analyses of sediment METHODS Nonetheless, subtle, yet important characteris- yield are commonly 0.05–0.25 mm yr−1 (Bes- tics of our site may infl uence the generality of chta, 1978; Bierman et al., 2001; Heimsath To test how well the transport models our results. We chose the size of the study area et al., 2001b; Reneau and Dietrich, 1991), described above represent processes that shape based on the need to: (1) incorporate several consistent with rates of coastal uplift (Kelsey hillslopes in the Oregon Coast Range, we cali- ridge and valley sequences, and (2) retain com- et al., 1996) and Holocene bedrock chan- brated each model and simulated 500,000 yr of putational effi ciency. nel incision (Personius, 1995). These studies slope evolution to compare the resulting mor- High-resolution, gridded topographic data have been used to argue that an approximate phology with current landscape form. Thus, (with a 2 m grid spacing) for our site along Sul- balance exists between rock uplift and ero- the assumption of approximate steady-state livan Creek, Oregon, reveal a pervasively pock- sion in the Oregon Coast Range such that the conditions (with a uniform erosion rate equal marked surface (Fig. 5A), which refl ects the topographic form may be relatively uniform to 0.1 mm yr −1) serves as a crucial cornerstone stochastic nature of bioturbation (such as pit and with time (Montgomery, 2001; Reneau and of this analysis. Given recent discoveries high- mound features created via tree turnover) as well Dietrich, 1991). This approximate balance is lighting mechanisms by which portions of the as errors associated with the classifi cation of veg- best refl ected in the ubiquity of consistently Oregon Coast Range decidedly do not approxi- etation and bare earth topographic points during spaced ridge and valley terrain in local areas. In mate steady-state erosion (Almond et al., 2007; airborne LiDAR data processing. Although the contrast, deep-seated landslides (Roering et al., Roering et al., 2005), we carefully chose the observed meter-scale roughness serves as the 2005), fl uvial terrace remnants, and debris fl ow section of ridgeline and sideslopes used in this signature of interesting and relevant transport fans produce topographic features that persist study (Fig. 5). Our intention is that the 200 × processes, here we recognize the transient nature for 10–100 k.y., and temporal patterns of sedi- 360 m2 patch of terrain shown in Figure 5B of those features and instead focus our simula- ment production associated with these transient adequately represents the ubiquitous ridge and tion results on the coarse-scale (>5 m) form that features are unconstrained. valley morphology of the Oregon Coast Range. defi nes hillslopes. Consistent with previous studies, we represent episodic transport and production processes with a continuum approach. Because the models discussed here do not Sullivan Crk, Oregon Coast Range apply to fl uvial or debris fl ow processes that Relief=320m, Area = 1km x 1km sculpt valleys and highly convergent, low-order portions of channel networks in the Oregon Coast Range (Stock and Dietrich, 2003), we A only applied the mass conversation and transport equations to hillslope nodes of the topographic grid (Fig. 5B). We identifi ed hillslope nodes by calculating the spatial distribution of drainage area (estimated with a fractional distribution method, e.g., Quinn et al., 1991) and specifying non-hillslope (or valley) points as having area greater than 250 m2. This approach is consistent with geomorphic studies that use a drainage area- slope threshold to map channelization (Mont- gomery and Dietrich, 1992). We chose the 250 m2 area threshold because this value causes the valley network to extend to the approximate loca- B tion of mapped shallow landslides in topographic hollows within our study area (Montgomery et al., 2000). These landslides translated into debris fl ows and thus may refl ect the spatial extent of processes that carve the upper tips of the drainage network. In the subsequent simulations, the background rate of base-level lowering (or rock uplift) is imposed on “valley” (or non-hillslope) cells, and the interior “hillslope” cells evolve as prescribed via the mass conservation, soil pro- Valley cells 180m duction and transport equations. We substituted contour interval: 5m (drainage area > 250m) each of the four transport models into Equation 2 Figure 5. Location map (Lat = 43.464°, Long = −124.119°) and solved Equation 1 using an explicit, fi nite of study site along Sullivan Creek, Oregon Coast Range. difference numerical model. Valley cells span (A) Representative section of topography generated via air- the eastern and western margins of our study borne laser swath mapping (ALSM). (B) Perspective view of area, but the northern and southern margins are terrain used for the simulations in this study. Black cells rep- largely composed of hillslope cells (Fig. 5B). We resent valley nodes identifi ed via a drainage area threshold thus specifi ed boundary conditions of constant and upon which boundary conditions are imposed. baselevel lowering at those locations.

Geological Society of America Bulletin, September/October 2008 1253 Roering

We calibrated the four transport models by 0.008 0.005 evolving the study area for 15,000 yr and deter- A 0.004 0.006

mining the parameter values that generated a )

–1 0.003

distribution of average erosion rates closest yr 0.004 2 −1 eqn. 12 to the steady-state value of 0.1 mm yr . We (m 4 0.002 K chose 15,000 yr for this calibration procedure 0.002 because it was: (1) long enough to eradicate the 0.001

, root volume per soil volume , root volume B meter-scale roughness associated with biotur- 0 0 bation and topographic data misclassifi cation, Dz 0 0.3 0.6 0.9 1.2 1.5 0 0.3 0.6 0.9 1.2 1.5 and (2) short enough that identifi able coarser slope-normal depth, hcosθ (m) slope-normal soil depth, hcosθ (m) scale hillslope modifi cation did not occur (Fer- Figure 6. (A) Decline of root volume per unit soil volume with depth. Data points represent nandes and Dietrich, 1997; Mudd and Furbish, binned averages from 20+ observations from soil pits and landslide scarps in the Oregon 2007; Roering et al., 2001a). Using Equations Coast Range (Gerber, 2004; Schmidt et al., 2001). Gray curve represents Equation 12 fi t to K K K −1 2 6, 7, 8, and 15, we varied values of 1, 2, 3, data where α = 0.0081 and β = 2.6 m and r = 0.63. (B) Variation of K with normal soil depth η 4 and (which is the coeffi cient of the depth- (Equation 14) for η = 0.0045 m2 yr−1 and β = 2.6 m−1. dependent expression for K4), respectively, and calculated the root mean square error (RMSE) of the temporally averaged distribution of erosion rates (Equation 10 in Roering et al., 1999). 8×10 –5 For Equations 8 and 15, we used the previously AB calibrated Sc value of 1.25 (Roering et al., 1999). ) β –1 For Equation 15, we estimated the value of 6×10 –5 by measuring the vertical variation of root density from more than 20 soil pits and landslide scars in the Oregon Coast Range (Gerber, 2004; × –5 Schmidt et al., 2001). Soil production (Equa- 4 10

RMSE (m yr 2 –1 K =0.026 m yr–1 tion 3) was simulated with parameters estimated K1=0.005 m yr 2 ε RMSE=0.0466 mm yr –1 RMSE=0.041 mm yr –1 at this study area by Heimsath et al. (2001b) ( 0 −1 μ −1 ρ −3 2×10 –5 = 0.268 mm yr , = 3 m , r = 2.0 g cm , and ρ −3 0 0.005 0.01 0.015 0.02 0 0.02 0.04 0.06 s = 1.0 g cm ). Because soil depth is difﬁ cult to characterize across spatially extensive areas, we K (m2 yr –1) K (m yr –1) uniformly applied the mean soil depth (0.4 m) as 1 2 the initial vertical soil depth in our simulations. 8×10 –5 For the long-term simulations, we used the CD

)

calibrated parameters and evolved the initial –1 surface for 500,000 yr using the same bound- 6×10 –5 ary conditions and soil production parameters discussed above. This length of simulation time enables the slopes to completely adjust their 4×10 –5 coarse-scale morphology in accord with the

RMSE (m yr K =0.003 m2 yr–1 η 2 –1 speciﬁ ed transport models. We used morpho- 3 =0.0045 m yr –1 logic criteria, such as gradient and curvature RMSE=0.0361 mm yr–1 RMSE=0.032 mm yr 2×10 –5 distributions and gradient-curvature plots (e.g., Fig. 4), to compare the results with the current 0 0.0025 0.005 0.0075 0.01 0 0.005 0.01 0.015 0.02 topography and assess model performance. 2 –1 η 2 –1 K3 (m yr ) (m yr ) Comparing the visual appearance of the simulated landscapes with current topography also Figure 7. Root mean square error (RMSE) for modeled and steady-state erosion rates −1 η provides a useful and straightforward barometer (0.1 mm yr ) as a function of coefﬁ cient values for the K1, K2, K3, and K4 ( ) models (shown in of model predictions. A–D, respectively). Note that the K4 model exhibits the lowest RMSE and thus best approxi- mates the current landscape surface. RESULTS

Model Calibration density profi le data in Figure 6A, obtaining the be the dominant process driving soil production Root data summarized from pits and land- parameter estimates α = 0.0081 and β = 2.6 m−1. in the Oregon Coast Range. slide scars throughout the Oregon Coast Range Interestingly, this β value is similar to Heimsath For each of the four transport models, we indicate that the ratio of root volume to soil vol- et al.’s (2001b) estimate of μ equal to 3 m−1, simulated 15,000 yr of evolution for a range of η ume decreases exponentially with depth, consis- which characterizes the exponential decay of parameter values (K1, K2, K3, and ) and iden- tent with previous studies (e.g., Jackson et al., soil production with depth. This rough corre- tifi ed the parameter values associated with the 1996) (Fig. 6A). We fi t Equation 12 to the root spondence suggests that tree root activity may lowest RMSE (Fig. 7). The nonlinear slope- and

1254 Geological Society of America Bulletin, September/October 2008 How well can hillslope evolution models “explain” topography?

depth-dependent model (K4, which varies with form curvature (Fig. 8G). The depth-slope prod- three models (Figs. 8K–8N). All of the models η , Sc, and h) generated the lowest RMSE of uct model (K2) also produced a broadly curved predicted relatively shallow soils along the rid- −1 0.032 mm yr , whereas the linear model (K1) surface that poorly corresponds with the current geline that became thicker downslope. exhibited the largest RMSE of 0.047 mm yr−1. morphology, although the hillslope-valley tran- η Best-ﬁ t values for K1, K2, K3, and were deter- sitions were more gradual (Fig. 8H) than those Long-Term Simulations: Morphologic 2 −1 −1 mined to be 0.005 m yr , 0.026 m yr , 0.003 for the K1 model. Because ﬂ uxes can be accom- Signatures m2 yr−1, and 0.0045 m2 yr−1, respectively. For the modated by variations in soil depth as well as

nonlinear slope- and depth-dependent model, slope angle, the K2 model produced a somewhat Erosion rates averaged over each 500,000 yr η the calibrated value of enables us to estimate less convex but much gentler surface than the K1 simulation were variable for the different trans-

the variation of K4 with soil depth. Figure 6B model. This outcome arises because downslope port models (Figs. 9K–9N), but the distribution indicates that K4 increases rapidly for thin soils increases in steady-state soil depth (Equation 3) of erosion rates for the last 10,000 yr of each sim- and becomes effectively constant for soil depths served to lessen downslope increases in gradi- ulation was indistinguishable from 0.1 mm yr−1, greater than 1.2 m. In addition, this calibration ent required to generate steady-state erosion. reﬂ ecting the attainment of steady-state condi- allows us to estimate the sediment transport Importantly, neither of these models (K1 or K2) tions. At the conclusion of each simulation, power per unit volume of disturbance agents succeeds in preserving the narrow ridges and we used quantitative representations of surface

(e.g., tree roots), Pv, by rearranging Equation 14 steep, nearly planar sideslopes characteristic of morphology to determine the extent to which ηβ 2ρσ α −2 −3 −1 as PV = ( gSC )/(2 ) = 1.1 J m m yr Oregon Coast Range slopes. the current landscape form was maintained. The (input parameters are summarized in Table 1). The nonlinear models (Equations 8 and 15) current surface exhibits curvature values that generated surfaces more consistent with the cur- become increasingly planar with steepness (see Long-Term Simulations: Spatial Patterns rent form in that the primary ridgeline remained fi lled black circles in Fig. 9A) consistent with relatively sharp, and roughly planar slopes our one-dimensional, steady-state gradient-cur- To evaluate our long-term (500,000 yr) extended above the tips of the drainage net- vature analysis for the nonlinear slope-depen- * * model predictions, we constructed three-dimen- work (Figs. 8I–8J). In addition, the distribution dent transport models (see C3 and C4 curves in sional shaded relief images and gradient maps of hillslope gradients for these two models is Fig. 4C). In contrast to the current morphology, for comparison with the initial topography roughly coincident with that for the current land the linear model (K1) predicts that curvature val- (Figs. 8A–8J). The current topography is dis- surface because a large proportion of the study ues are independent of gradient (Fig. 9B). The tinguished by a relatively narrow north-south– area and model surfaces exhibit slope gradients depth-slope product model (K2) predicts a slight trending ridge line, steep sideslopes (gradients between 0.8 and 1.0 (Figs. 8F, 8I, and 8J) as rep- decrease in convexity with gradient, which is between 0.8 and 1.0), and meter-scale noise resented by warm (yellow and orange) colors. consistent with our one-dimensional predictions associated with bioturbation and airborne laser The K4 ridgeline is narrower than that for the K3 (Fig. 4A), but inconsistent with our study site swath mapping (ALSM) classifi cation errors model and better represents the initial morphol- form (Fig. 9A). On gentle slopes, the K1 and K2 (Figs. 8A and 8F). The simulated landscapes ogy of our study site (Fig. 9A). models predict curvature values signifi cantly (Figs. 8B–8E) exhibit much smoother surfaces, Interestingly, none of the models maintained closer to zero than our observations for the cur- although each has a distinctive morphologic the local sharpness (high convexity) of at least rent landscape (Figs. 9B and 9C). pattern driven by topography-transport feed- two small subsidiary ridges that trend west (to The nonlinear slope-dependent models (K3 backs. In contrast to the current land surface, the right) in the central portion of our study and K4) produced surfaces with gradient-curva- the linear model (K1, Equation 6) generated area. The along-axis steepness of these iso- ture patterns similar to the current topography a broadly curved ridgeline and locally steep lated features elicited relatively rapid erosion (Figs. 8D and 8E). The modeled surfaces exhib- hillslope-valley transitions (Figs. 8B and 8G). and “rounding” during the simulations. These ited negligible variability in curvature for low- For this model, erosion scales with total curva- locally sharp features are not an areally exten- gradient slopes (hilltops), whereas the current ture (equal to the sum of planform and profi le sive component of the study landscape, but their slopes have signifi cant variation in curvature at curvature), which caused steep, subtly concave distinctive appearance warrants attention. Also, these locations. Nonetheless, the trend expressed slopes above the tips of the channel network we observed a signifi cantly broader distribution by the binned average gradient-curvature data

to develop high-proﬁ le convexity (and become of soil depths for the K1 model compared with (larger black circles) shows a “sigmoidal” pat- extremely steep) to balance the absence of plan- more subdued soil depth variations for the other tern similar to the model predictions (Figs. 9D and 9E). Consistent with the initial morphology, both nonlinear models predicted a broad range of curvature values for slopes near the critical value, TABLE 1. PARAMETER DEFINITIONS AND VALUES S K ρ ρ •3 c. Notably, the 4 (nonlinear slope- and depth- r, s = Rock and soil bulk density (gm cm ) 2.0, 1.0** ε •1 dependent) model better represents the current 0 = Maximum soil production rate (mm yr ) 0.268* μ = Soil production exponential coefficient (m•1) 3.0* gradient-curvature pattern shown by the ﬁ lled E = Steady-state erosion rate (mm yr•1) 0.1* ε black circles in Figure 9A because it predicts E/ 0 = Ratio of erosion rate to maximum soil production rate 0.37 2 •1 higher convexity ridges (∇2z = –0.078 for |∇z| = K1 = Transport coefficient (m yr ), Equation 6 0.005 K = Transport coefficient (m yr•1), Equation 7 0.02 2 2 0) than does the K model (∇ z = –0.066 for |∇z| = K = Transport coefficient (m2 yr•1), Equation 8 0.003 3 3 0). As shown in Figures 8D and 8E, both nonlin- Sc = Critical gradient, Equations 9 and 15 1.25** α = Root density at the surface (m3/m3) 0.0081 ear models produced ~300 points (~1% of the Β = Root density exponential decay term (m•1) 2.6 η 2 •1 study area) with high convexity and steep slope = Transport coefficient (m yr ), Equation 14 0.0045 2 P = Power per unit volume roots (J m•2 m•3 yr•1) 1.1 angles (|∇z|>0.7 and ∇ z<–0.05). These points v *(Heimsath et al., 2001b) fall outside of the gradient-curvature pattern **(Roering et al., 1999) illustrated in our one-dimensional formulation

Geological Society of America Bulletin, September/October 2008 1255 Roering

Gradient 1.5 1.25 1 0.75 0.5 0.25 0 F A

Current Topography (t=0) Soil depth (m) 3 2.5 2 1.5 1 0.5 0 G K t=500ky B

qKzs =−1 ∇

H L t=500ky C

qKhzs =−2 cosθ ∇

I M t=500ky D

Kz∇ q =− 3 s 2 1−∇()zSc J N t=500ky E

Kh()∇ z q =− 4 s 2 1−∇()zSc 180 m 180 m contour interval=5m Figure 8. Comparison of simulation surfaces with current topography. (A–E) Perspective-view, shaded relief images of current and modeled topography. Modeled surfaces reﬂ ect 500,000 yr of evolution via the calibrated parameters given in Figure 7. (F–J) Spatial variation of hillslope gradient for current and modeled surfaces. The current surface is pockmarked due to bioturbation and data errors, whereas the modeled surfaces are uniformly smooth because of the continuum assumption used here. The nonlinear slope-dependent models (I and J) best represent the sharp, steep-sided slope morphology of the ﬁ eld site. (K–N) Spatial variation of simulated soil depth for the four transport models. Each model predicts thin soils near the ridge top and thicker soils along sideslopes.

1256 Geological Society of America Bulletin, September/October 2008 How well can hillslope evolution models “explain” topography?

gradient, |∇z| gradient, |∇z| 0 0.3 0.6 0.9 1.2 1.5 00.40.81.21.62 0 A F median=0.81 )

-1 IQR=0.71-0.88 -0.05

-0.1 count t=0 curvature (m curvature current morphology t=0 erosion rate (mm yr-1) soil depth (m) -0.15 0.05 0.075 0.1 0.125 0 0.3 0.6 0.9 1.2 1.5 0 B G K O

) median=0.91 median=0.092 -1 IQR=0.73-1.1 IQR=0.078-0.098 -0.05 broad convex median=0.59

hilltops count -0.1 IQR=0.51-0.70 qKzs =−1 ∇

curvature (m t=500ky -0.15 0 C H L P

) median=0.76 median=0.1 -1 IQR=0.64-0.87 IQR=0.096-0.104 -0.05 broad convex hilltops count -0.1 qKhz=−cosθ ∇ median=0.51 s 2 IQR=0.45-0.57 curvature (m t=500ky -0.15 0 D I M Q ) median=0.85 median=0.099 -1 IQR=0.74-0.94 IQR=0.095-0.101 -0.05

−∇K z q = 3 count -0.1 s 2 median=0.55 1−∇()zSc curvature (m t=500ky IQR=0.50-0.59 -0.15 0 E J median=0.84 median=0.1 N R )

-1 IQR=0.75-0.93 IQR=0.096-0.102 -0.05

−∇Kh() z q = 4 count -0.1 s 2 1−∇()z Sc median=0.54 curvature (m t=500ky IQR=0.50-0.59 -0.15 0 0.3 0.6 0.9 1.2 1.5 00.40.81.21.620.05 0.075 0.1 0.125 0 0.3 0.6 0.9 1.2 1.5 gradient, |∇z| gradient, |∇z| erosion rate (mm yr-1) soil depth (m) Figure 9. Comparison of gradient-curvature plots and distributions of gradient, erosion rate, and soil depth for current and modeled surfaces shown in Figure 8. (A–E) Variation of curvature with gradient for current and modeled surfaces. Only the nonlinear slope-dependent models (D and E) predict a substantial decrease in convexity with gradient, consistent with the current landscape. (F–J) Distributions of gradient for current and modeled surfaces. (K–N) Temporally averaged (500,000 yr) distributions of modeled erosion rates. (O–R) Distributions of soil depth for the modeled landscapes. The ﬁ lled black circles in (A) represent binned average curvature values for our study site. While erosion rates averaged over the entire simulation (K–N) were variable, the distribution of erosion rates for the last 10,000 yr of each simulation was indistinguishable from 0.1 mm yr−1.

Geological Society of America Bulletin, September/October 2008 1257 Roering

(Fig. 4B) and correspond to cells immediately model (Fig. 8O) in comparison with the other The K3 model, however, predicts rapid erosion adjacent to the valley cells, such that they may models (Figs. 8P–8R). The median soil depth rates (>0.15 mm yr−1, shown by warm colors) be subject to boundary effects associated with for all of the models was 0.5–0.6 m, consistent for much of the high convexity terrain (e.g., our numerical analysis. with ﬁ eld measurements, although the ﬁ eld- ridges) in the study site. In contrast, hilltop

The distribution of gradient values and mod- derived values exhibit greater variability driven erosion rates predicted with the K4 model are eled erosion rates further distinguishes the trans- by the stochastic nature of soil production pro- more consistent with the background rate of port model predictions (Figs. 9F–9J). In contrast cesses (Heimsath et al., 2001b). 0.1 mm yr−1. Distributions of average erosion ∇ to the current gradient distribution, the K1 and K2 rate for low-gradient (| z|<0.4) terrain dur- models generated distributions with substantial Morphologic Predictions of the Nonlinear ing the fi rst 50,000 yr reveals an abundance of concentrations of low-gradient values. These Slope-Dependent Models rapidly eroding terrain for the K3 model com- values correspond to the broadly curved, gentle pared to the K4 model (Figs. 10A and 10B). ridgelines illustrated in Figures 8B and 8C. In Although the two nonlinear slope-dependent Because fl ux rates calculated with the K4 model particular, the K1 model gradient distribution models (K3 and K4) predict similar morphologic increase with soil depth and steady-state soil is characterized by a large interquartile range, trends via the long-term simulation, the differ- depth increases with slope angle (Equation 3), whereas the other model distributions more ence in predicted ridgetop convexity reveals K4-modeled slopes attain sharper hilltop con- closely refl ect the magnitude of dispersion in the an important distinction. The K4 model better vexity than when fl ux does not depend on soil current gradient distribution. Slope angles for preserves the sharp ridgeline convexity that depth (e.g., K3 model). the K2 model were somewhat gentler than the is characteristic of our study area. To clarify study site slopes, whereas distributions for the this difference in model behavior, we exam- DISCUSSION

K3 and K4 models were diffi cult to distinguish ined the fi rst 50,000 yr of evolution for the from the current slope distribution. Temporally two models and plotted the spatial distribution The slope- and depth-dependent nonlinear averaged erosion rates were also broadly distrib- of average erosion rate (Figs. 10C and 10D). transport (K4) model proposed here (Equa- uted for the K1 model (Fig. 9K) refl ecting the We chose 50,000 yr for this analysis because tion 15) best preserved the topographic charac- long period of time some portions of the land- this length of time facilitates an initial phase teristics of our study site. In contrast to previous scape required to attain the spatially constant, of landform modifi cation substantial enough to studies (Fernandes and Dietrich, 1997; Roer- steady-state curvature value. In contrast, the reveal differences between the two models. For ing et al., 1999), our new formulation indicates other three models exhibited erosion rate distri- steep and subtly concave slopes that occur just that hilltop curvature for soil-mantled slopes butions clustered around 0.1 mm yr−1 indicating above the upper tips of the valley network, both varies nonlinearly with the steady-state erosion their relatively rapid attainment of steady-state models predict a similar erosion rate pattern rate. According to our one-dimensional analy- conditions (Figs. 8L–8N). The distribution of with values generally lower than 0.1 mm yr−1 sis of nonlinear depth- and slope-dependent soil depth was also highly variable for the linear (shown by cool colors in Figs. 10C and 10D). transport (Equation 18d), steady-state hilltop

erosion rate (mm yr -1) erosion rate (mm/yr) 0 0.05 0.1 0.15 0.2 C mean=0.127 Figure 10. Comparison of spa- std. dev.=0.037 A tial variations in erosion for the 0.2 ﬁ rst 50,000 yr using the K3 and K3 K4 models. (A–B) Distribution of

count model erosion rates on terrain gentler

than 0.4 predicted using the K3 t=50ky 0.15 and K models, respectively. The gradient<0.4 4 K4 model predicts a narrower distribution that better corresponds with the steady-state 0.1 −1 mean=0.101 D erosion rate of 0.1 mm yr . std. dev.=0.029 B (C–D) Spatial variation of erosion rates after 50,000 yr of

0.05 simulation time using the K3 K4 and K4 models, respectively. The

count model K3 model predicts rapid erosion rates along the ridgelines, t=50ky 0 exceeding the steady-state value gradient<0.4 and reducing hilltop convexity. 0 0.05 0.1 0.15 0.2 -1 erosion rate (mm yr ) 180 m

1258 Geological Society of America Bulletin, September/October 2008 How well can hillslope evolution models “explain” topography?

2 2 ε curvature, (d z/dx )HT, defi ned as the curvature refl ected in 0), including variations in lithology, hillslope transport models. In contrast, our mor- where dz/dx = 0, is given by: weathering, and biogenic activity. For our study phologic observations of natural slopes incor- site, values of β and μ are of the same order (β/μ porate >19,000 data points and show consistent ⎛ 2 ⎞ −()ρρ()E η = 0.87), suggesting that the depth-dependencies patterns in gradient-curvature space as defi ned dz = rs ⎜ 2 ⎟ ()βμ() ε . (20a) of soil production and soil transport processes via spatially extensive data. Our approach aver- ⎝ dx ⎠ 1− e ln E 0 HT may refl ect the same suites of processes (e.g., ages the morphologic characteristics of numer- tree turnover and root growth). As a result, we ous hillslopes, reinforcing the systematic nature Equation 20a indicates that mapping erosion might expect that β/μ ≈1 in many fi eld settings. of gradient-curvature trends. rate via hilltop convexity requires estimates of To explore the morphologic implications of Our assessment of model performance depends soil transport and production parameters. Using landscapes wherein erosion rates approach the on the assumption of steady-state hillslope ero- our calibrated parameters (Table 1), Equa- maximum soil production rate, we conducted a sion for our Oregon Coast Range study site. Cer- 2 2 tion 20a indicates that (d z/dx )HT = −0.078, con- two-dimensional hillslope evolution simulation tainly, large landsliding, drainage capture, dif- sistent with the results of our two-dimensional of our study site and imposed a baselevel lower- ferential incision driven by resistant mafi c dikes, −1 ε simulations (Fig. 9E). Dimensionless hilltop ing rate of 0.24 mm yr such that E/ 0 = 0.9. and lateral bedrock channel migration, conspire curvature (calculated as positive here to allow After 500,000 yr, the model hillslopes, which in various locales of the Oregon Coast Range for plotting on a log scale) varies with erosion had attained a temporally invariant form, were to impose erosion rates signifi cantly higher or rate according to: steep with extremely sharp hilltops (Figs. 12A lower than the assumed 0.1 mm yr −1 used here and 12B). Hillslope gradients (Fig. 12d) had a (Almond et al., 2007; Baldwin and Howell, median value of 1.08 and more than 90% were 1949; Kobor and Roering, 2004; Moeller, 1990; −=* 1 C4()HT ()βμ() ε . (20b) steeper than 0.9 as low-gradient terrain became Personius, 1995; Roering et al., 2005). Fortu- − ln E 0 1 e exceedingly sparse. Soil depths for the simu- nately, the topographic manifestation of these lated landscape averaged 0.2 m and 90% of the processes is signifi cant and readily identifi able ε → As E/ 0 1, the denominator of Equation 20b landscape had soils thinner than 0.3 m. The via visual inspection or digital elevation model decreases rapidly and hilltops become exceed- gradient-curvature relationship for the modeled (DEM) analysis. In the late 1980s, the study site ingly sharp (i.e., exhibit a narrow convexity) slopes (Fig. 12C) exhibited a convex-upward used here was chosen for extensive hydrologic (Fig. 11). In essence, Equation 20b describes form, consistent with the results of our one- study (Anderson and Dietrich, 2001; Montgom- ε β μ ε how hilltop convexity varies with E/ 0 and / dimensional analysis for hillslopes with E/ 0 = ery et al., 1997; Torres et al., 1998) because of in addition to proportional changes associated 0.9 (Fig. 4B). Hilltop curvature for this scenario its collection of characteristic ridge and valley η ε ∇2 ≈− with E and . Small changes in E/ 0 can invoke is highly convex ( z 1.1), refl ecting the non- sequences (Montgomery and Dietrich, 1989). * ε large changes in –C4(HT) such that relative mag- linear dependency of E/ 0 on hilltop morphology Although at least two of the models presented nitudes of erosion rate (E) and the maximum as predicted by Equation 20a, which predicts above generate morphologic patterns similar to ε 2 2 ε soil production rate ( 0) exert a strong control that d z/dx HT = −1.17 when E/ 0 = 0.9. This the current land surface, we cannot rule out the on slope morphology. Interestingly, this formu- sharp-crested model morphology arises because possibility that transient adjustments are mani- lation indicates that the observed variability of soil depths are shallow, such that hilltops must fest. Given the ubiquity of erosion rates between hilltop convexity for natural slopes, which is develop high convexity to accommodate rapid 0.05 and 0.2 mm yr−1 observed in characteristic substantial as shown in Figure 9A, may partly erosion. On sideslopes, by contrast, the nonlin- ridge and valley terrain of the Oregon Coast derive from small variations in processes and ear slope-dependency on transport rates enables Range (Bierman et al., 2001; Heimsath et al., properties that infl uence soil production (as the base-level lowering rate to be accommo- 2001b; Reneau and Dietrich, 1991), the mag- dated with modest increases in slope angle. nitude of associated deviations in morphologic This simulation may be relevant to many natural trends should not signifi cantly hamper our abil- settings, including the steep, sharp-crested, and ity to compare and differentiate model behavior. 100 thin soil-mantled Verdugo Hills described by Nonetheless, a more spatially extensive testing Strahler (1950). In addition, other portions of of our calibrated equations should be conducted, the San Gabriel Mountains experiencing rapid including areas that appear to show decidedly uplift exhibit extremely sharp hilltops and thin non-steady morphologic features. * –C 4(HT)10 (and likely tenuous) soil mantles (Whipple et Perhaps the most readily observed result of al., 2005), consistent with our conceptualization. our simulations is that meter-scale topographic Similarly, Young (1961) describes sensitive feed- “noise” ubiquitous across the current landscape β/μ=1 backs between soil thickness and slope angle for gives way to smooth modeled land surfaces 1 catchments in Wales such that soils become thin (Figs. 8A–8E). Using a wavelet-based analysis 0 0.2 0.4 0.6 0.8 1 and patchy as slope angles approach 45°–50°. of variations in the dispersion of local curva- E/ε 0 Signifi cant uncertainty accompanies the esti- ture, Lashermes et al. (2007) demonstrated that Figure 11. Variation of dimensionless hilltop mation of topographic derivatives used to dis- the length scale at which topography becomes curvature (denoted here as positive to facili- tinguish our model predictions (Figs. 9B–9C). “noisy” in the Oregon Coast Range is ~10 m. ε tate plotting on a log scale) with E/ 0 for a For example, fractional standard error estimates Below this length scale, pit and mound features one-dimensional, steady-state solution of the for curvature values (error bars are not shown in associated with tree turnover and other forms

K4 model (Equation 20b). Dimensionless cur- our plots to improve clarity) range from 20% to of bioturbation dominate the topography. Mis- vature increases rapidly as the steady-state 50%. As a result, data sets based on small sam- classiﬁ cation of the ALSM data may also con- erosion rate approaches the maximum soil ple sizes may compromise the diagnostic ability tribute to this meter-scale topographic rough- production rate. of gradient-curvature plots for distinguishing ness. At length scales greater than 10 m, the

Geological Society of America Bulletin, September/October 2008 1259 Roering

Gradient Base-level lowering =0.24 mm/yr 1.25 1 0.75 0.5 0.25 ε E/ 0 = 0.9

t=500ky B

A Figure 12. Summary of simula-

tion results of the K4 model after 500,000 yr with a base-level lowering rate of 0.24 mm yr−1, such that E/ε = 0.9. (A) Perspective shaded Kh()∇ z 0 q =− 4 relief image (compare to current s 2 surface shown in Fig. 8A); (B) spa- 1−∇()zSc 180 m tial variation of modeled gradient, contour interval=5m and (C) variation of curvature 0 with gradient. Note that the convex C D upward gradient-curvature pattern ) is consistent with our analytical, -1 -0.3 one-dimensional solution shown median=1.08 in Figure 4B. (D) Distribution of -0.6 IQR=0.99-1.13 hillslope gradient for the modeled surface.

-0.9 count

curvature (m -1.2 t=500ky -1.5 00.30.60.91.21.50 0.30.60.91.21.5 gradient gradient

morphologic signature of hillslopes and valleys our gradient-curvature plots to only show planar nonlinear model (Equation 15) has three param- predominates. Because our simulations do not or convex terrain, although the simulations for eters (whereas the previous nonlinear model has β account for the stochasticity of biogenic-driven the K3 and K4 models did generate slopes with only two), can be estimated via fi eld-based transport processes or episodic fi re-driven trans- slight concavity that accounted for <15% of the data and thus is not a “fi t” parameter. port (Roering and Gerber, 2005), our simulated study area. The linear and depth-slope product Our delineation of the valley network does slopes do not exhibit meter-scale roughness but models (Equations 6 and 7) are unable to gener- not correspond to the location of channels with instead average across these temporal variations ate concave terrain that experiences erosion. defi nable banks but instead accounts for the in fl ux. Future studies should explicitly account Analogous to the analysis by Yoo et al. (2005), upstream extent of valley sculpting via shallow for statistical variations in fl ux associated with we use our analysis of erosion and transport rates landslides that trigger debris fl ows (Stock and natural processes. to estimate power per unit volume of biota, PV. Dietrich, 2006). The location of shallow land- Traditionally, the erosion of concave terrain By assuming that gopher-driven soil transport at slide initiation likely varies temporally accord- has been associated with advective processes their California fi eld site and documenting how ing to climatic and vegetation changes (Reneau such as overland fl ow erosion (e.g., Smith and gopher density varies with soil depth, Yoo et al. and Dietrich, 1990), however, and as a result we Bretherton, 1972). As noted by Dietrich et al. (2005) calculated how fl ux rates adjust to cli- acknowledge that other drainage area thresh- (2003), nonlinear slope-dependent models mate-driven perturbations in gopher population old values are defensible. Our area threshold (e.g., Equations 8 and 15) can accommodate density, enabling an explicit linkage between (250 m2), however, is consistent with the value landscape lowering for steep, slightly concave geomorphic processes and ecology. Our calibra- of drainage area per unit contour (a/b = 57 m, slopes. In our simulations, steep slopes perched tion of the K4 model (Equation 14) assumes that where b = 4 m) that defi nes the hillslope-channel above the upper tips of the valley networks bioturbation associated with root action (e.g., transition at this site (Roering et al., 1999) and exhibit concavity yet experience erosion, con- growth, dilation, or tree turnover) dominates thus may refl ect the effective long-term average sistent with that assertion. The nonlinearity of transport in the Oregon Coast Range, which is a extent of valley-sculpting processes. the transport relationships facilitate this behav- reasonable assumption given the high frictional Our analysis highlights the importance of soil ior as erosion is a complex combination of plan- strength of the coarse colluvial soils (Schmidt et depth in modulating process-scale morphology form and profi le gradient and curvature values al., 2001). Nonetheless, additional disturbance in soil-mantled landscapes. Anderson (2002) scaled by the degree to which the nonlinearity mechanisms, such as mammal burrowing, may reasoned that transport rates should depend affects transport rates (Equation 9 in Roering et contribute to soil mobilization and displacement on the depth of available soil and developed a al., 1999). To avoid confusion, we constructed but are not incorporated here. Although our new freeze-thaw model whereby fl ux rates attained a

1260 Geological Society of America Bulletin, September/October 2008 How well can hillslope evolution models “explain” topography?

Baldwin, E.M., 1956, Geologic map of the lower Siuslaw maximum value for soils greater than the depth as well as gradient and curvature distributions, River area, Oregon: U.S. Geological Survey Oil and Gas of frost penetration. Our approach here is analo- to assess model performance. The two models Investigations Map OM-186. gous and is also consistent with a rate-process– for which fl ux varies linearly with slope gra- Baldwin, E.M., and Howell, P.W., 1949, The Long Tom, a for- mer tributary of the Siuslaw River: Northwest Science, based approach that also accounts for the depth dient generated broad, convex hilltops incon- v. 23, p. 112–124. of disturbance penetration in estimating net fl ux sistent with the current study site morphology. Benda, L., and Dunne, T., 1997, Stochastic forcing of sediment supply to channel networks from landsliding and debris (Roering, 2004). Feedbacks between ecological The two nonlinear slope-dependent models fl ow: Water Resources Research, v. 33, p. 2849–2863, processes and soil depth are important to con- predicted slopes along which curvature values doi: 10.1029/97WR02388. sider, but are beyond the scope of this contri- became increasingly planar with steepness. The Beschta, R.L., 1978, Long-term patterns of sediment production following road construction and logging in the bution. For example, moisture and nutrient con- proposed nonlinear slope- and depth-dependent Oregon Coast Range: Water Resources Research, v. 14, straints may favor a different assemblage of fl ora model best preserved the sharp, narrow hilltops p. 1011–1016. and fauna, if soils thin below a particular depth, and steep nearly planar sideslopes characteristic Bierman, P., Clapp, E., Nichols, K., Gillespie, A., and Caf- fee, M., 2001, Using cosmogenic nuclide measurements affecting the magnitude and depth-dependency of our study site. Our new formulation predicts in sediments to understand background rates of ero- of transport. In the absence of soil, additional that steady-state hilltop convexity varies nonlin- sion and sediment transport, in Harmon, R.S., and Doe, W.M., eds., Landscape erosion and evolution modeling: processes not addressed in this contribution early with erosion rate such that as erosion rates New York, Kluwer Academic Plenum, p. 89–115. (such as rockfall) likely ensue. In contrast to approach the maximum soil production rate, Braun, J., and Sambridge, M., 1997, Modelling landscape the depth-slope product model (Equation 7), hilltops become exceedingly sharp. evolution on geological time scales: A new method based on irregular spatial discretization: Basin Research, our proposed model (Equation 14) explicitly v. 9, p. 27–52, doi: 10.1046/j.1365-2117.1997.00030.x. accounts for how the distribution of transport ACKNOWLEDGMENTS Braun, J., Heimsath, A.M., and Chappell, J., 2001, Sediment agents varies with soil depth. Our formulation transport mechanisms on soil-mantled hillslopes: Geol- The author was supported by National Science ogy, v. 29, p. 683–686, doi: 10.1130/0091-7613(2001)0 thus avoids the problematic scenario for which 29<0683:STMOSM>2.0.CO;2. Foundation grant EAR-0309975. Discussions with the depth-slope product model predicts propor- Burbank, D.W., Leland, J., Fielding, E., Anderson, R.S., Bill Dietrich and Simon Mudd greatly improved Brozovic, N., Reid, M.R., and Duncan, C., 1996, Bed- tional increases in fl ux as soils become inordi- the manuscript. Reviews by Jon Pelletier and Arjun rock incision, rock uplift and threshold hillslopes in the nately thick (i.e., much thicker than the depth Heimsath sharpened the manuscript and were highly Northwestern Himalayas: Nature, v. 379, p. 505–510, along which transporting agents act). appreciated. doi: 10.1038/379505a0. Canadell, J., Jackson, R.B., Ehleringer, J.R., Mooney, H.A., Sala, O.E., and Schulze, E.D., 1996, Maximum rooting REFERENCES CITED CONCLUSIONS depth of vegetation types at the global scale: Oecologia, v. 108, p. 583–595, doi: 10.1007/BF00329030. 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p. 1041–1069. Roering, J.J., Kirchner, J.W., and Dietrich, W.E., 1999, MANUSCRIPT RECEIVED 30 JUNE 2007 Lashermes, B., Foufoula-Georgiou, E., and Dietrich, W., Evidence for nonlinear, diffusive sediment transport REVISED MANUSCRIPT RECEIVED 28 DECEMBER 2007 Channel network extraction from high resolution topog- on hillslopes and implications for landscape morphol- MANUSCRIPT ACCEPTED 3 JANUARY 2008 raphy using wavelets: Geophysical Research Letters, ogy: Water Resources Research, v. 35, p. 853–870, doi: v. 34, L23S04, doi: 10.1029/2007Gl031140. 10.1029/1998WR900090. Printed in the USA

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