Modeling of knickpoint retreat on the Roan Plateau, western Colorado



[1] The Roan Plateau in western Colorado constitutes a natural experiment for studying landscape response to a drop in base level. Late Cenozoic incision of the upper Colorado River led to elevational isolation of the Plateau and initiation of a wave of incision into its southern edge. Knickpoints (oversteepened reaches that contain waterfalls 60–110 m in height) mark the upstream extent of this headward propagating wave. That this incision has occurred in a laterally extensive, well-stratified, and essentially flat-lying bedrock and in an area with relatively uniform climate, implies that it should serve as a good test of existing knickpoint propagation models. We predict the locations of knickpoints by using a stream power-based celerity model, in which knickpoint recession rate is a power function of drainage area and is proportional to rock susceptibility to erosion. Models of the Parachute and Roan drainages (17 and 16 knickpoints, respectively) show expected rapid initial knickpoint propagation rates, which decline as drainage area decreases stepwise at tributary junctions. The modeled positions of knickpoints match well with observed features, using a single combination of parameters to model retreat in both drainage basins. We compare our celerity model results with past studies and explore how longitudinal profile analysis may be used to derive independently the exponent on drainage area in the celerity model.

1. Introduction

[2] Fluvial longitudinal profiles tend to be smooth and concave-up, as river slope decreases with increasing drainage area downstream. Exceptions to this form can provide a sensitive measure of landscape disequilibrium. By studying a landscape in transience, we can place constraints on the rates of geomorphic processes by which a landscape adjusts to a change in boundary conditions. The upstream migration of knickpoints is one mechanism by which a channel network responds to a decrease in base level because erosion by flowing water and the available sediment is enhanced in steep reaches. A knickpoint is defined generally as an abrupt change in slope in a stream longitudinal profile, such as a convex reach in an otherwise concave-up longitudinal profile [e.g., Gardner, 1983; Crosby and Whipple, 2006]. Knickpoints may be obvious waterfalls [e.g., Wohl et al., 1994; Hayakawa and Matsukura, 2003] or more subtle features that are identifiable only after analysis of river slope data [e.g., Bishop et al., 2005; Hayakawa and Oguchi, 2006]. In this study we use the term knickpoint to refer to an anomalously steep reach that is located between reaches of lower slope. As a mechanism of bedrock river incision, knickpoint migration can set the lower boundary condition for adjacent hillslopes [Whipple and Tucker, 1999; Whipple, 2004] and thus dictate landscape response to tectonic and climatic forcing.

[3] A drainage basin with multiple active knickpoints can be used to test bedrock river incision models. Because all the streams within a drainage basin will have experienced the same base level history at the basin outlet, their longitudinal profiles provide independent records of fluvial response to their lower boundary condition. Drainage basin response to base level change has been modeled numerically [e.g., Howard, 1994; Carretier and Lucazeau, 2005], simulated in the laboratory [e.g., Parker and Schumm, 1982; Hasbargen and Paola, 2000], and studied using field and topographic data [e.g., Weissel and Seidl, 1997; Bishop et al., 2005; Crosby and Whipple, 2006].

[4] Crosby and Whipple [2006] studied 236 active knickpoints in the Waipaoa drainage basin on the North Island of New Zealand. These knickpoints mark the headward advance of a wave of incision of the Waipaoa River main stem that began ca. 18 ka. The bedrock of the Waipaoa drainage basin is sedimentary, with varying susceptibilities to erosion based on different structural histories and chemical compositions [Crosby and Whipple, 2006]. The magnitude of incision (50–120 m) is large enough to allow easy discrimination between the eroded portions of the landscape and the relict, preincision terrain upstream of knickpoints and above incised gorges [Crosby and Whipple, 2006]. Crosby and Whipple [2006] tested two models for predicting the position of knickpoints throughout the fluvial network: a model of knickpoint propagation that assumes knickpoint retreat rate is a power function of drainage area, and a model that predicts knickpoint formation at a threshold drainage area.

[5] The Roan Plateau in western Colorado provides a similar natural experiment in bedrock river incision, albeit with an order of magnitude fewer knickpoints than the Waipaoa basin and with a much longer timescale of incision. The purpose of our investigation is to test further the validity of a stream power-based model to predict knickpoint positions, as a prerequisite for understanding specific processes of knickpoint retreat. We explore bedrock river incision into the southern edge of the Roan Plateau by modeling the upstream migration of 33 knickpoints in two catchments directly tributary to the Colorado River. We use a simple detachment-limited stream power model to predict knickpoint migration histories. We use multiple lines of geologic evidence to infer that knickpoint retreat began at approximately 8 Ma (see section 4.3) [Donnell, 1961; Marvin et al., 1966; Johnson, 1975; Donnell et al., 1992; Streufert et al., 1997; Kunk and Snee, 1998; Kirkham et al., 2001; Kunk et al., 2002]. The Roan Plateau bedrock is sedimentary, and lithologic variation appears to exert control on knickpoint form. Knickpoint heights are comparable to those in the Waipaoa (Roan waterfalls are 60 m to at least 110 m tall), yet the magnitude of base level fall in the Roan is much larger (at least 250 m and we suspect up to ∼900 m (see section 5.3)) than in the Waipaoa (∼70 m of base level fall). Our model is identical in form to the first model described by Crosby and Whipple [2006], which assumes knickpoint retreat rate is a power function of drainage area and allows the contributing drainage area to change with stream distance. However, we state that our knickpoint celerity model can be directly derived from the stream power incision model. We use RiverTools software to extract channel profiles from digital elevation model (DEM) data sets and measure for each point along the profile the along-stream distance from the Colorado River confluence, the elevation, and the upstream contributing drainage area.

[6] We find that a stream power-based celerity model works well to predict the positions of knickpoints on Roan Plateau streams. We show that the longitudinal profiles downstream of knickpoints also provide support for the applicability of a stream power model to river incision. In this paper we describe the Roan Plateau field area, present the results of the stream power-based knickpoint celerity model, and discuss the boundary conditions and model assumptions. We compare our model results with several similar knickpoint migration studies, including Bishop et al. [2005], Crosby and Whipple [2006], and Hayakawa and Matsukura [2003]. We also present results of longitudinal profile analysis and discuss ways in which such an analysis can inform the parameterization of the celerity model.

2. Background

[7] The stream power incision model is widely used for predicting the rate of bedrock river incision (E) [Howard and Kerby, 1983; Howard et al., 1994; Whipple and Tucker, 1999]:

equation image

where K is a dimensional resistance to erosion (whose dimensions are dependent on the values of m and n) that combines the effects of lithology, climate, and sediment load (see expressions in the work of Whipple and Tucker [1999]), A is drainage area, S is channel slope, and m and n are positive nondimensional constants that reflect basin hydrology, hydraulic geometry, and erosion process. This model, which was developed following field experiments carried out by Howard and Kerby [1983], assumes steady, uniform flow, detachment-limited conditions (in which sediment transport capacity exceeds sediment supply), and that erosion rate is a power law function of either shear stress or stream power per unit area of channel bed. Stream power per unit area (ω) is defined as ω = ρgQSW−1, where ρ is the density of water, g is gravitational acceleration, Q is water discharge, and W is channel width.

[8] Alternatives to the stream power model [see van der Beek and Bishop, 2003] include (1) the excess stream power model, in which channel incision into bedrock begins once grains on the bed have been mobilized [e.g., Tucker and Slingerland, 1997]; (2) the transport-limited model (erosion is proportional to the divergence of sediment transport) [e.g., Willgoose et al., 1991]; (3) the undercapacity sediment flux model (fluvial entrainment is modeled as a first-order kinetic reaction) [e.g., Kooi and Beaumont, 1994]; and (4) the tools sediment flux model, in which bed load acts both as an abrasive tool and protective cover [Sklar and Dietrich, 1998, 2004]. Recent research has debated the applicability of stream power erosion models to longitudinal profile evolution [e.g., van der Beek and Bishop, 2003; Tomkin et al., 2003]. One weakness of the stream power approach is its insensitivity to the stochastic nature of streamflows that give rise to bedrock river incision, as well as the importance of erosion thresholds [e.g., Snyder et al., 2003; Tucker, 2004; Crosby et al., 2007; Gasparini et al., 2007]. However, because of the simplicity and widespread use of the stream power model, we use it as a starting point for our investigation of Roan Plateau knickpoint migration.

[9] The stream power model can be rewritten to derive the celerity or wave speed of the knickpoint such that knickpoint migration rates are proportional to drainage area (as a proxy for stream discharge), with the susceptibility of bedrock to erosion serving as the proportionality constant [e.g., Rosenbloom and Anderson, 1994; Seidl et al., 1994; Weissel and Seidl, 1998; Whipple and Tucker, 1999; Bishop et al., 2005]. If the slope exponent (n) in equation (1) is unity, then the knickpoint celerity approach describes upstream migration of a knickpoint without significant diffusion in knickpoint form (i.e., parallel retreat). Although the assumptions of steady and uniform flow in the stream power model derivation are not met at Roan Plateau waterfalls, we proceed with the stream power approach as a way to explore the dependence of knickpoint migration rate on drainage area. In this sense, we treat Roan Plateau knickpoints as generic breaks in slope, and choose to ignore momentarily their complicated erosion processes. Niemann et al. [2001] used the stream power law to derive horizontal and vertical velocities of an upstream-propagating knickpoint following a step change in uplift rate. They predicted that the vertical velocity of a knickpoint should be spatially uniform so that knickpoints resulting from a given disturbance should be found at the same elevation within a basin [Niemann et al., 2001]. Deviations from this prediction may allow us to learn more about the details of Roan Plateau landscape evolution, even though the base level fall history is somewhat unconstrained (such that we cannot describe it by a single step change in relative uplift rate).

[10] Prediction of knickpoint positions with a stream power-based celerity model does not, however, provide insight into the relationship between retreat process (e.g., plunge pool scour, groundwater seepage erosion) and stream power. Field and experimental studies, combined with a theoretical foundation for modeling retreat processes, are needed to further explore this relationship. Perhaps the first knickpoints to gain the attention of geomorphologists were waterfalls, in which the stream loses contact with the bed at a free overfall [e.g., Gilbert, 1896]. Knickpoint retreat has typically been attributed to the presence of an erosion-resistant caprock above weaker rocks [e.g., Gilbert, 1896], although von Engeln [1940] and Young [1985] discussed retreat of waterfalls without undermining of a caprock. Knickpoints with vertical faces in either layered rocks with differing erosional resistance [e.g., Wohl et al., 1994] or in dipping strata with extensive jointing [e.g., Miller, 1991] have been assumed to migrate upstream, although Wohl et al. [1994] discussed the importance of base level changes over substrate variability in determining the distribution of active knickpoints. Because our modeling approach here is process-blind (that is, the analysis does not extract information about how knickpoints migrate upstream), we refer the reader to numerous previous discussions of waterfall and headcut retreat processes [e.g., Brush and Wolman, 1960; Young, 1985; Stein et al., 1993; Bennett, 1999; Crosby and Whipple, 2006; Flores-Cervantes et al., 2006; Lamb et al., 2006].

[11] Not every longitudinal profile convexity can be interpreted as a knickpoint that is actively migrating upstream. For an equilibrium longitudinal profile (in which erosion rates are everywhere balanced by uplift rates), equation (1) predicts a tradeoff between the coefficient of erosion (K) and channel slope (S). If climate and contributing drainage area are held constant, as might be expected within a given stream reach or small drainage basin, then anomalies in slope can reflect changes in lithology [e.g., Miller, 1991; Hancock et al., 1999, Figure 2] or sediment load [e.g., Howard and Dolan, 1981; Hanks and Webb, 2006], rather than a change in base level.

[12] River longitudinal profiles often exhibit a power law relationship between drainage area and slope:

equation image

where ks is the steepness index and θ is the concavity index. When a river profile is in steady state with respect to both climatic and rock uplift conditions, the stream power-based model for bedrock erosion (1) can be rewritten in the form of (2), to relate drainage area and local slope:

equation image

where U is the steady state rock uplift rate that must be exactly balanced by bedrock channel erosion for a steady profile, and (U/K)1/n is the steepness index ks [e.g., Whipple and Tucker, 1999]. Regression of channel slope against drainage area can provide estimates of the ratio m/n (or θ, the concavity index) and of the steepness index ks [e.g., Snyder et al., 2000; Kirby and Whipple, 2001]. However, equating the concavity index (obtained by regressing slope against drainage area) to the m/n ratio is only appropriate when the river profile is at steady state, and U, K, m, and n are uniform (see discussion in the work of Tucker and Whipple [2002]). The concavity index for bedrock rivers typically varies from 0.3 to 1.2 [Whipple, 2004], and efforts to constrain the exact numeric values of m and n in the stream power law from field data on erosion rates or numerical modeling have yielded a wide range of results [e.g., Howard et al., 1994; Weissel and Seidl, 1998; Stock and Montgomery, 1999; Tucker and Whipple, 2002]. We return to the importance of slope/area regression as a tool for extracting tectonic information in section 6.

3. Setting

[13] The Roan Plateau (Figures 1 and 2) is in the northeast corner of the Colorado Plateau physiographic province, bounded by the Colorado River on the south and Piceance Creek, a tributary to the White River, on the north. Structurally, the Roan Plateau is in the southern part of the Piceance Basin, which lies between the Douglas Creek arch on the west and the Grand Hogback on the east. The Piceance Basin formed and filled with fluvial and lacustrine sediments during and following the Laramide orogeny, from Late Cretaceous into Eocene time. Carroll et al. [2006] attributed the development of large, long-lived, carbonate-producing Eocene lakes in these basins to sediment starvation caused by a decrease in upstream erosion rate. These decreased erosion rates were caused by exposure of erosion-resistant lithologies within cores of Laramide uplifts (here the White River Uplift, Axial Basin Arch, Douglas Creek Arch, Uncompahgre Uplift, and the Gunnison Uplift [Maclachlan, 1987]) [Carroll et al., 2006]. The Piceance Basin was once occupied by a part of Lake Uinta, in which large quantities of organic-rich marls accumulated; these units have been the target of oil-shale development on the plateau.

Figure 1.

Shaded relief map of Upper Colorado River drainage, showing major physiographic features. Box shows extent of Figure 2. Inset shows approximate location within the state of Colorado. 1–Clear Creek; 2–East Fork Parachute Creek; 3–Government Creek; 4–Roaring Fork River; 5–Eagle River; 6–Piney Creek; 7–Blue River; 8–Troublesome Creek; 9–Williams Fork River; 10–Piceance Creek. A–Aspen; E–Eagle; GC–Glenwood Canyon; GJ–Grand Junction; GS–Glenwood Springs.

Figure 2.

Shaded relief map of Roan Plateau. Open circles indicate locations of knickpoints used in modeling study. Diamonds indicate channel heads in the Roan Creek drainage that were not used. MC indicates Mount Callahan (location of basalt flow mapped by Donnell [1961]). Lines and color spectrum indicate structural contours for Mahogany oil-shale zone [from Pitman and Johnson, 1978]. Structural elevations generally decrease to the north. BC—Brush Creek; CAC—Carr Creek; CC—Clear Creek; CNC—Conn Creek; DB—DeBeque; EFF—East Fork Falls on East Fork Parachute Creek; GJ—Grand Junction; KC—Kimball Creek; P—Parachute; R—Rifle; RC—Roan Creek; WFF—West Fork Falls, on West Fork Parachute Creek.

[14] Tertiary sedimentary rocks exposed in the Roan Plateau include the Paleocene and Eocene Wasatch Formation and the overlying Eocene Green River and Uinta Formations. The Wasatch Formation is a fluvial claystone with sandstone beds; the exposed thickness in the Roan Plateau area is ∼76 m [Hail, 1992]. The Green River Formation consists of lacustrine members, including dolomitic marlstone (oil shale), dolomitic clay shale, siltstone, and sandstone. Hail [1992] divided the Green River Formation into four lithologic units, named (in ascending order) the Anvil Points, Garden Gulch, Douglas Creek, and Parachute Creek members. The total thickness of the Green River Formation is 630–990 m. The Parachute Creek member (290–430 m in thickness) contains the richest and thickest oil-shale beds in the Green River Formation [Bradley, 1931], including the Mahogany oil-shale zone (referred to as a ledge at the outcrop). The Mahogany zone varies in thickness from 18 to 50 m, as described on geologic maps [e.g., Johnson, 1975], and is used as a key bed for stratigraphic reference and for structural mapping [Donnell, 1961]. We do not consider the Mahogany ledge to be a true caprock because it appears to dictate strictly neither the location of all knickpoints, nor the upper boundary of the canyon walls; rather, the entire Parachute Creek member of the Green River Formation is a resistant unit that forms cliffs. However, the correlation of Mahogany outcrop elevation with knickpoint elevation (see discussion on Plateau structure below) suggests that the Mahogany ledge does exert some lithologic control on knickpoint form and elevation. Because the Mahogany ledge is laterally extensive, it should exert the same control on all of the tributaries studied. Overlying the Green River Formation is the Uinta Formation, which consists of deltaic-lacustrine clastic rocks, including tuffaceous sandstone, siltstone, and marlstone [Hail, 1992]. The Uinta Formation (∼320 m thick) is interfingered with the Green River Formation and is exposed on the plateau surface, above the canyons and knickpoints.

[15] We are interested in the events that have caused the dissection of the Roan Plateau landscape; in particular, we need to understand the incision history of the Upper Colorado River. Acceleration of Colorado River incision has been attributed to drainage integration within the Colorado Plateau and the carving of the Grand Canyon 6–5 million years ago [Lucchitta, 1972; Young and McKee, 1978]. The degree to which the upper half of the Colorado River has adjusted to this incision is debatable [e.g., Coblentz and Karlstrom, 2003]. The present-day course of the Colorado River ∼80 km upstream of the Roan Plateau can be dated back to approximately 10 million years ago; Larson et al. [1975] dated a basalt flow on Lookout Mountain, a few kilometers east of Glenwood Springs at the west end of Glenwood Canyon (Figure 1), to 10.1 ± 0.5 Ma. Stream gravels that include Precambrian crystalline rocks disconformably overly that basalt flow. Although the elevation of these gravels above the present-day Colorado River is not given, the gravels do indicate an established Colorado River system accessing the cores of Laramide ranges in the headwaters at that time [Larson et al. 1975]. Apatite fission track analysis from drill holes in the Piceance Basin just south of the Colorado River [Kelley and Blackwell, 1990] indicates cooling of the Paleocene Wasatch Group and Upper Cretaceous Mesaverde sandstone after 10 Ma, at a rapid rate of 10–15°C/Myr. This cooling provides additional support for the onset of erosion related to the downcutting of the Colorado River at ∼10 Ma. Geologic mapping [Streufert et al., 1997] and 40Ar/39Ar dating [Kunk and Snee, 1998] of basalt flows near Glenwood Canyon (Figure 1) place the Colorado River at ∼850 m higher than its present-day elevation at 7.8 ± 0.04 Ma, and ∼730 m higher at 3.03 ± 0.02 Ma. These dates suggest incision rates of 0.025 mm/yr between 7.8 and 3.03 Ma, and 0.24 mm/yr during the past 3.03 Ma [Kirkham et al., 2001; Kunk et al., 2002].

[16] The study area consists of the Parachute and Roan Creek drainages. Parachute Creek (drainage area: 500 km2) and Roan Creek (drainage area: 1300 km2) drain south to the Colorado River, and Piceance Creek (drainage area: 1630 km2) drains to the north off the Roan Plateau and into the White River (Figure 2). Maximum plateau elevations are approximately 2600 m, and relief between the plateau surface and the tributary canyon floors typically ranges from 500 to 1000 m. Knickpoints interrupt the longitudinal profiles of tributary streams on the plateau (Figure 3), with dramatic waterfalls at the upstream end of the knickpoint reach (Figure 4). The smooth, rolling top of the plateau surface stands in stark contrast to the steep cliffs and incised canyons etched into the plateau below (Figure 4). Above the knickpoints, stream channels with low slope (which may in part be controlled by structural dip) bound low-amplitude convex hills. Elongate canyons downstream of the knickpoints are fed by small tributary streams hung at the canyon lip. Limited field observations indicate that upstream of the waterfalls, streams flow over smooth bedrock with little sediment cover. Field surveys of East Fork Falls and West Fork Falls reveal waterfall heights of 60 m and 110 m, respectively, and an average slope of the waterfall face of 81° and 61°, respectively. These waterfalls have small plunge pools at their base, and the channels downstream are filled with coarse boulders derived from retreat of the canyon walls, with no visible bedrock exposure. We hypothesize that knickpoints formed as a result of late Cenozoic incision of the Colorado River through the Mahogany oil-shale zone, although the details of that incision history are somewhat unconstrained (see section 4.3). Assuming that incision is focused at the knickpoint zone, the landscape upstream of the knickpoints should be effectively unaware of the base level fall of the Colorado River downstream.

Figure 3.

Longitudinal profiles (distance measured with respect to the Colorado River confluence) for (a) Roan Creek and (b) Parachute Creek basins. Knickpoint profiles for (c) Roan and (d) Parachute indicate knickpoint locations (white circles), and elevation of the Mahogany oil-shale zone outcrop (filled squares).

Figure 4.

(a) View of southeastern Roan Plateau, looking northwest. Thin gray line marks approximate location of Mahogany oil-shale zone. (b) East Fork Falls, an example of a knickpoint (60 m waterfall) on the Roan Plateau. Photo taken in June 2005 during falling limb of hydrograph.

[17] The structure of the Roan Plateau is simple; the axes of gentle folds and related faults generally trend northwesterly. In the eastern portion of the Roan Creek drainage (Figure 2), the Crystal Creek anticlinal nose and the Clear Creek syncline form a gentle fold system with an amplitude of approximately 120 m and a wavelength of approximately 11 km [Donnell, 1961]. Clear Creek runs down the Clear Creek syncline. In the Parachute Creek drainage, rocks generally dip toward the central axis of the stream network (i.e., toward the west in the eastern part of the Parachute Creek drainage, and toward the north in the southern part of the drainage). The horizontal stratification of geologic units and the subtle structure (dips vary on the plateau between 0.01 and 0.04 [Hail, 1992]) appear to control the network geometry and the slope of streams above knickpoints in the Parachute and Roan Creek drainages, although this has not been tested with full numerical modeling of longitudinal profile evolution. Elevation and outcrop data from available geologic maps [e.g., Donnell et al., 1986; O'Sullivan, 1986; O'Sullivan and Hail, 1987] indicate that knickpoint elevations are correlated with along-stream elevations of the Mahogany oil-shale ledge (correlation coefficient ∼0.5) (Figure 3). From this association we assume that at the present time, the Mahogany oil-shale zone significantly affects knickpoint propagation by dictating the vertical position of the knickpoint.

[18] Numerous landslides caused by rockslides and slump movement (both recent and Pleistocene in age) have been mapped in the Roan Creek basin [e.g., Umstot, 1989], while canyon walls in the Parachute Creek basin are more stable [Schumm and Olson, 1974]. The landslide distribution has been explained by slight differences in the lithology and mineralogy of rock units [Schumm and Olson, 1974]. Rocks of the Roan Creek area have a lower organic content, higher content of clastics, and a greater percentage of unstable clay minerals, making them inherently weaker and more susceptible to erosion than rocks in the Parachute Creek basin [Schumm and Olson, 1974].

[19] Precipitation in the Roan Plateau area is relatively uniform throughout the year, averaging 2.5 cm per month at low elevations near the town of Rifle (Western Regional Climate Center Station 057031,, accessed March 2006) for an average annual precipitation of 30 cm. Precipitation is higher on the plateau surface, where average annual precipitation can reach 50–65 cm [Taylor, 1987; Bureau of Land Management, 2004]. Between October and April, most of the precipitation falls as snow, whereas thunderstorms provide precipitation during the summer months. Peak stream discharges are associated with snowmelt runoff and occur between late April and late May. Records from 13 USGS gages on streams in the Parachute and Roan Creek basins (periods of record between 6 and 25 years) indicate that mean annual discharges range from 0.02 to 1.25 m3/s. Maximum daily flows of up to 60 m3/s have been measured at gages with larger drainage areas. Groundwater flow from seeps and springs does contribute to surface runoff, although discharge measurements between 1981 and 1983 from springs in the Parachute and Roan Creek basins indicate a mean annual discharge of only ∼3 × 10−4 m3/s (based on measurements at 129 springs) [Butler, 1985].

4. Methods

[20] Our methods can be divided into three parts: (1) extracting elevation, river slope, and drainage area data from digital elevation models to identify knickpoints and their associated oversteepened zones; (2) solving for the best fit parameters in the celerity model (erosion coefficient K and area exponent m) that best predict knickpoint locations in the Parachute and Roan Creek basins; and (3) performing power law regressions of channel slope versus drainage area.

4.1. Topographic Analysis and Knickpoint Identification

[21] We used 10-m and 30-m Digital Elevation Model (DEM) data from the USGS Seamless site (, accessed 27 October 2004 and 10 February 2006) for the Roan Plateau. We built 8-directional flow and upstream area grids for each DEM using RiverTools software. The flow and area grids allowed us to extract longitudinal profiles as well as the distribution of drainage area along each stream of interest (17 streams in the Parachute basin and 16 in the Roan) (Figure 3). Longitudinal profiles from DEMs can often have a jagged appearance with sharp steps where streams cross each contour line, as a result of the way in which DEMs are generated from hard copy topographic maps. To obtain the more realistic slope values needed to identify knickpoints and to extract river profile concavity and steepness indices, we calculated river slope using a fixed vertical interval [see Wobus et al., 2006] of 12.192 m (40 ft) because this is the original contour interval from USGS 7.5-min topographic maps used to construct the Roan Plateau DEM data. We identified knickpoints on Roan Plateau tributaries as local maxima in slope in log-log slope-area space (Figure 5).

Figure 5.

Parachute Creek tributary (a) river slope versus drainage area data and (b) distribution of upstream drainage area at knickpoints. Roan Creek tributary (c) river slope versus drainage area data and (d) distribution of upstream drainage area at knickpoints. For Figures 5a and 5c, all channels have different symbols and a solid line connects symbols for one channel.

4.2. Knickpoint Celerity Model

[22] We used a stream power-based knickpoint celerity model to predict a time series of knickpoint positions on the Roan Plateau [e.g., Rosenbloom and Anderson, 1994; Crosby and Whipple, 2006]. We begin by assuming that plucking is the dominant erosion process, which allows n = 1 [Whipple et al., 2000]. Limited observations of channel beds upstream of Roan Plateau knickpoints revealed blocky morphology that was defined by joint and bedding planes, which is consistent with quarrying or plucking as the dominant erosion process [Hancock et al., 1998]. This allows us to rewrite (1) as:

equation image

We rearrange the dz terms to arrive at:

equation image

where dx/dt is the rate of upstream knickpoint migration in m/yr, K is the detachment limited erosion coefficient, here with units of m(1−2 m)/yr, A is upstream drainage area in m2, and m is a nondimensional constant that depends on basin hydrology, channel geometry, and erosion process [Whipple and Tucker, 1999]. This expression describes the upstream propagation speed (or celerity) of a knickpoint and is identical in form to an expression used by Crosby and Whipple [2006], although we specify that K is indeed the same K as the detachment limited erosion coefficient, instead of an arbitrary constant. Our celerity model follows the work of Rosenbloom and Anderson [1994] and Bishop et al. [2005], although we have no channel diffusion term, we allow drainage area to change realistically with distance, and we solve for retreat rate, not total retreat distance. We note that our celerity model does not specify changes in channel width, sediment flux, the effects of stochastic climate events, or the presence of an erosion threshold. A critical drainage area term could be included in equation (5) to note the presence of a threshold below which fluvially driven knickpoint retreat processes would stall or change to mass wasting or block failure processes [e.g., Weissel and Seidl, 1997; Crosby and Whipple, 2006; Wobus et al., 2006]. At this point we lack information about how the specific retreat processes vary with stream power; we therefore cannot either firmly support or refute the importance of a threshold in Roan Plateau knickpoint retreat.

[23] We used (5) to model numerically the upstream migration of a knickpoint that begins at the Colorado River confluence with each drainage basin, using a dynamic time step approach that was described by Crosby and Whipple [2006]. At each cell in a longitudinal profile, we calculated a knickpoint speed based upon the contributing drainage area at the downstream end of the cell [e.g., Crosby and Whipple, 2006]. That speed, along with the cell size, was used to calculate the travel time within that cell. The knickpoint moves upstream, one cell at a time, until the sum of all travel times is greater than or equal to the assumed knickpoint travel time (or time since initiation of base level drop, in our case 8 Ma) [e.g., Crosby and Whipple, 2006].

[24] To constrain the parameters K (detachment-limited erosion coefficient) and m (exponent on drainage area) in the celerity model, we used a brute force two-parameter search [e.g., Stock and Montgomery, 1999; Crosby and Whipple, 2006] to evaluate the combination of K and m that best predicts the knickpoint locations, assuming a knickpoint initiation time of 8 Ma (discussed below). We evaluated model misfit by the streamwise distance between the observed knickpoint location and our model prediction [e.g., Crosby and Whipple, 2006], with a negative value meaning that the modeled knickpoint is upstream of the observed knickpoint. For each parameter combination, we calculated the sum of the squares of the streamwise differences in all the individual knickpoints as an indicator of model performance [e.g., Crosby and Whipple, 2006] (excluding Brush Creek because it is an obvious outlier). We allowed m to vary linearly between 0.2 and 1.3 over 30 values and K to vary logarithmically between 10−12 and 10−4 over 300 values. The positions of 33 knickpoints from both the Parachute and Roan Creek basins were modeled together because we assume (1) simultaneous knickpoint initiation in both drainages and (2) similar values of K and m for each drainage.

4.3. Initiation of Incision

[25] The boundary condition that dictates knickpoint initiation was based upon two assumptions. First, we assumed that channel incision rates in Glenwood Canyon discussed above in section 3 can be extrapolated ∼80 km downstream to the Roan Plateau, such that ∼850 m of incision has occurred there in the past 7.8 Ma. This implies that the Colorado River profile between Glenwood Canyon and the Roan Plateau had the same longitudinal profile form 7.8 Ma as observed today. Second, we assumed that the Mahogany oil-shale zone must have been cut through at the Colorado River-Parachute Creek confluence before knickpoints began to migrate upstream. Even though the Mahogany is not a true caprock (as discussed above), the spatial and elevational correlation with knickpoint locations suggests that the Mahogany plays a significant role in controlling knickpoint form. The Mahogany oil-shale zone structural contours suggest that it would crop out ∼890–895 m above the present-day Colorado River confluences with Parachute Creek and Roan Creek [Johnson, 1975; Donnell et al., 1992]. Because this is just slightly higher than the ∼850 m of incision in Glenwood Canyon since 7.8 Ma, we assume that the Colorado River was at this elevation ∼8 Ma (Figure 6). Additional support for this paleoelevation comes from the presence of a small basaltic lava flow on the summit of Mt. Callahan, on the very southern end of the Roan Plateau ∼166 m higher than the expected Mahogany oil shale zone outcrop and ∼1060 m higher than the present-day Colorado River (Figure 2). This flow, which overlies rounded, quartzite gravels (which were presumably transported by the Colorado River), has been described as the northernmost scrap of a once continuous feature that was derived from the same source as basalt flows on Grand Mesa to the south [Donnell, 1961]. One Grand Mesa basalt flow has a whole-rock K-Ar age of 9.7 ± 0.5 Ma [Marvin et al., 1966], and more recent 40Ar/39Ar dating has yielded basalt flow ages ranging from 10.76 ± 0.24 Ma to 9.49 ± 0.16 Ma [Kunk et al., 2002]. Incision of the Colorado River through the basaltic lava flow surface on Mt. Callahan therefore likely postdates the age of Grand Mesa basalts.

Figure 6.

Diagrammatic summary of assumptions related to knickpoint initiation timing. Upper Colorado River longitudinal profile upstream of the Colorado-Utah border (thick line), with two Roan Plateau tributary streams: Conn Creek (long-dashed line) and East Fork Parachute Creek (short-dashed line). Also shown are projected elevations of the Mahogany oil-shale zone at the Parachute Creek confluence (open circle at 2434 m) and ∼7 km upstream of the Roan Creek confluence (filled circle at 2439 m). We assume that knickpoints on the Roan Plateau were initiated when the paleo-Colorado River incised through these points. Thin line indicates paleolongitudinal profile of the Colorado River at time of this incision event, assuming an identical profile form to the present-day longitudinal profile. Timing of inititation of incision is constrained by the approximate elevation of incision markers in Glenwood Canyon [Streufert et al., 1997; Kunk and Snee, 1998; Kirkham et al., 2001; Kunk et al., 2002] (diamonds labeled with ages).

[26] The celerity model for upstream-propagating knickpoints on the Roan Plateau requires that we specify a start time. Our conceptual model is that once the Colorado River incised through the Mahogany oil-shale zone, the base level for Parachute and Roan Creeks rapidly fell, initiating upstream-propagating knickpoints. Although the details of the base level fall history beyond knickpoint initiation are unimportant for the celerity model, we eventually wish to constrain that history (see section 6). We assume that the Parachute and Roan Creek knickpoints were initiated simultaneously because the Mahogany oil-shale marker is projected to occur at a similar elevation at each confluence (Figure 6). We return to the assumption of simultaneous knickpoint initiation in section 6.

4.4. Longitudinal Profile Analysis

[27] We performed linear regressions in log-log slope-area space for all Roan Plateau streams [e.g., Snyder et al., 2000; Kirby and Whipple, 2001]. Although this analysis is somewhat peripheral to our central objective of predicting knickpoint location, it is still important for understanding the response of the Roan Plateau streams to base level fall. First, we evaluated the concavity index and normalized steepness index for channel segments upstream of knickpoints, as a basis for performing a total incision calculation, following the methods described by Schoenbohm et al. [2004]. Second, we evaluated the concavity index and normalized steepness index for channel segments downstream of knickpoints. If these channels have reached a new equilibrium state following upstream passage of a knickpoint, their concavity indices might be comparable to the m/n ratio in the stream power model [e.g., Niemann et al., 2001; Whipple and Tucker, 1999].

[28] Linear regressions in log-log slope-area space were performed on segments upstream of the knickpoints, and segments downstream of the knickpoints. For channel segment regressions upstream of knickpoints, we included only those points for which slope increased with distance upstream. This allowed us to exclude any oversteepened portions (e.g., due to flow acceleration upstream of a waterfall [Rouse, 1936; Gardner, 1983; Haviv et al., 2006]). We did not use a threshold drainage area to limit the upstream extent of this regression (i.e., to exclude steep, debris flow dominated headwaters [e.g., Snyder et al., 2000]) because we do not have field data on channel processes to justify such an upper limit, and we wish to maximize the amount of data available for regression upstream of knickpoints (particularly for those knickpoints at small drainage areas).

[29] For lower channel segment regressions, we included all points downstream of the knickpoint and extending to the Colorado River confluence. The upstream limit of this regression was difficult to define because Roan Plateau streams exhibit considerable oversteepening, even several kilometers downstream of the obvious waterfalls. To address this ambiguity, and in order to avoid confusing oversteepened reaches with “adjusted” high steepness reaches (see discussion in the work of Wobus et al. [2006]), we used three different slope values (0.1, 0.2, and 0.5) to define the upstream limit of lower channel segment regressions. These slope limits are arbitrary because we currently lack field data about channel process transitions that might be helpful in setting regression limits [e.g., Snyder et al., 2000], and the discontinuous distribution (clustering) of drainage area makes selecting individual regression limits unreliable. For both upper and lower channel segments, we calculated normalized steepness indices using an appropriate average concavity value as a reference concavity (Table 1) [e.g., Snyder et al., 2000].

Table 1. Longitudinal Profile Parameters for Upper Channel Segmentsa
 Concavity Index (Mean ± Std. Dev.)Normalized Steepness Index, θref = 0.21 (Mean ± Std. Dev.)Normalized Steepness Index, θref = 0.50 (Mean ± Std. Dev.)
  • a

    Regression performed on stream channel segments upstream of any oversteepening associated with the knickpoint (e.g., drawdown reach above the waterfall); no upstream regression limit was used (for example, based on drainage area [e.g., Snyder et al., 2000]). In each basin, one upper channel segment was convex-upward and was excluded from the average.

Parachute (n = 16)0.23 ± 0.081.36 ± 0.4033 ± 31
Roan (n = 15)0.19 ± 0.081.04 ± 0.2716 ± 20
Mean0.21 ± 0.081.20 ± 0.3825 ± 27

5. Results

5.1. Topographic Analysis and Knickpoint Identification

[30] We found 33 knickpoints on tributaries suitable for our analysis: 17 in the Parachute Creek drainage, and 16 in the Roan Creek drainage (Figure 2). Knickpoint elevations range between 2056 and 2425 m in the Parachute drainage and between 2222 and 2389 m in the Roan drainage. The drainage area upstream of each knickpoint ranges between 0.75 and 73 km2 (Parachute) and between 1.4 and 40 km2 (Roan). This range in drainage area (Figure 5) strongly suggests that the knickpoints have not all stalled at a critical drainage area and are indeed actively migrating; this is a transient landscape still in the midst of its response to Colorado River incision. We excluded three streams on the western side of the Roan Creek drainage basin (Carr Creek, Roan Creek, and Kimball Creek; Figure 2) from our analysis because their longitudinal profiles display odd convexities not seen in the other 33 streams. These three streams do not contain obvious, well-defined knickpoints; their channel slope generally increases with distance upstream. It is difficult to tell for these streams if the knickpoints have propagated all the way to the headwaters or if the knickpoints diffused away on their journey upstream. We attribute the different longitudinal profile pattern of these streams to lithologic variation (see section 3), which may have resulted in higher erosion rates and a more complete adjustment of these channels to base level fall.

5.2. Knickpoint Celerity Model

[31] The misfit between the modeled and the observed knickpoint locations on the Roan Plateau was minimized with m = 0.54 and K = 1.37 × 10−7 (m0.08 yr)−1 in equation (5) (Figure 7). The following ranges of model parameter combinations yielded model misfit within 4% of the minimum misfit: K ranging from 3.33 × 10−8 to 2.87 × 10−7, and m ranging from 0.50 to 0.62. The spatial pattern of knickpoint locations is matched well by our model predictions (Figure 8). In the Roan Creek drainage, Brush Creek is a noticeable outlier with respect to model performance, and may reflect a transition toward less resistant bedrock in the west (see discussion of landslides in section 3 and also section 6.1). Brush Creek was not used in model calibration, although we do indicate model results on Figure 8. Most (25 of 32) predicted knickpoints are within 2 km of their real counterparts, with 21 knickpoints arriving within 5% of the observed knickpoint distance from the Colorado River confluence (Figure 9). Mean residuals are −0.78 ± 1.37 km for the Parachute Creek basin (a negative residual indicates that the modeled knickpoint is upstream of the observed knickpoint), and 1.25 ± 0.93 km for the Roan Creek basin. Combined residuals for both basins average 0.17 ± 1.56 km. As expected from the upstream decline in drainage area, modeled knickpoint migration rates decrease as the knickpoints propagate upstream. Modeled rates range from a maximum of 7.1–11.9 mm/yr at the confluence of each drainage with the Colorado River, to 0.3–2.3 mm/yr at the best fit positions (Figure 10). Given the large uncertainties in our model start time, we consider the value of the best fit K to be preliminary. Changing the knickpoint initiation time by two orders of magnitude to 80,000 years ago increases K by two orders of magnitude. Large fluctuations in K could be expected as a result of changes in climate over the model duration; our best fit value of K thus represents a spatial and temporal average of all K values.

Figure 7.

Contours of knickpoint celerity model misfit for given combinations of K and m. Reasonable values fall within a narrow trough, with the best fit combination marked by a star. Inset plot shows vertical slices through the contour plot at specific values of m. For each value of m, a different K value is required to produce a minimum model misfit. The best fit combination has the lowest total misfit and is marked by a star.

Figure 8.

Modeled and observed knickpoint locations on the Roan Plateau. Solid circles indicate the modeled knickpoint positions and stars indicate observed knickpoint locations obtained from DEM analysis. BC–Brush Creek.

Figure 9.

(a) Histogram of knickpoint celerity model residuals (n = 32), expressed as a fraction of observed knickpoint distance from the Colorado River confluence. (b) Modeled versus observed knickpoint distances from the Colorado River. Straight line is a 1:1 relationship. Equation in legend is for a linear regression through all data points (R2 = 0.97).

Figure 10.

Modeled knickpoint migration rates in mm/yr for streams in the Parachute Creek drainage (solid lines) and the Roan Creek drainage (dashed lines).

5.3. Longitudinal Profile Analysis

[32] Concavity indices for channel segments upstream of knickpoints were similar for Parachute and Roan Creek streams, with a mean concavity (θ) = 0.21 (Table 1). Including a threshold drainage area to limit the upstream extent of this regression [e.g., Snyder et al., 2000] decreases the confidence of regression estimates, and still results in somewhat low concavity indices. For example, using a regression limit of 105 m2 in the Parachute basin yields an average concavity index of 0.36, which is still considered low [Whipple, 2004]. Although we do not know the erosion history of the upper plateau surface, we attribute the low concavity indices to diffuse steepening above the knickpoints (see section 6.3). Normalized steepness indices, calculated using two different reference concavities (θref = 0.21 and 0.50), were within error of each other for upper segments in the two catchments (Table 1). If we are correct in assuming that the upper channel segments in the Parachute and Roan basins have had a similar base level history, then similar channel concavity and normalized steepness indices imply that the characteristics of the bedrock controlling upper plateau incision are similar in the two catchments, as the geology would suggest.

[33] We projected each upper channel segment profile downstream to the Colorado River confluence [e.g., Schoenbohm et al., 2004], using a mean concavity index (θ = 0.21), drainage area and slope data, the normalized steepness indices in Table 1 and equation (2). The mean and standard deviation of incision estimates were 195 ± 221 m for the Parachute drainage (n = 16) and 136 ± 263 m for the Roan drainage (n = 15) (Table 2). Excluding negative values, the average difference in elevation between the downstream end of the projected profile and the modern Colorado River elevation was ∼250 m for both drainage basins (n = 25) (see Figure 11 for an example of our channel projection results). This incision estimate is much smaller than the anticipated estimate of base level fall of ∼894 m constrained by the Mahogany oil-shale zone outcrop elevation at the Colorado River confluence. For the sake of comparison, we repeated the downstream projection analysis using a larger concavity index (θ = 0.5); this resulted in larger estimates of base level fall that were more consistent with geologic evidence (Table 2).

Figure 11.

Example of channel projection results for Roan Plateau streams, following the methods of Schoenbohm et al. [2004]. (a) Slope and area data for East Fork Parachute Creek, with different symbols for each channel segment and mean concavity values for all 32 streams. Dashed lines with shaded background indicate typical regression lines. Thin dotted line at slope = 0.1 indicates upper limit for lower segment regression (to avoid any transient oversteepening dowstream of knickpoints). (b) Longitudinal profile of East Fork Parachute Creek (thick line), the projected paleolongitudinal profile and uncertainty based upon steepness estimate (thin line bounded by dashed lines), and the estimated elevation of the Mahogany oil-shale zone at the Colorado River confluence (star). Circles on East Fork longitudinal profile indicate the location of slope area regression limits; here, the upstream end of the lower channel segment was set at slope = 0.5; these regression limit circles bound a 60-m waterfall.

Table 2. Base Level Fall Estimates Using Downstream Projection Method
BasinMean ± Std. Dev., θref = 0.21, mMean ± Std. Dev., θref = 0.5, m
Parachute (n = 16)195 ± 221749 ± 126
Roan (n = 15)136 ± 263801 ± 105

[34] Concavity indices for channel segments downstream of knickpoints varied depending on the upstream limit of the slope-area regression (Table 3). As the slope limit for regression increases, the concavity index also increases, because the regression segment includes steeper values that may be part of the “knickpoint zone” (Figure 3 and Table 3). Channels with small drainage area at the knickpoint tended to have lower downstream segment concavities for a given slope-based regression limit, suggesting that these regression limits do not treat tributaries of different sizes equally. However, they do provide an indication of the typical concavity indices and the sensitivity of those values to regression limits (Table 3). Normalized steepness indices (calculated using a reference concavity, θref = 0.50) were within error of each other for the Parachute and Roan catchments (Table 3).

Table 3. Longitudinal Profile Parameters for Lower Channel Segments
 Concavity Index (Mean ± Std. Dev)Normalized Steepness Index (Mean ± Std. Dev)
d/s of S > 0.1ad/s of S > 0.2bd/s of S > 0.5cd/s of S > 0.5c, θref = 0.5
  • a

    Regression performed on stream channel segments downstream of the knickpoints, below slopes >0.1.

  • b

    Same as a, but upper regression limit is for slopes >0.2.

  • c

    Same as a, but upper regression limit is for slopes >0.5.

  • d

    Here, n = 15 for this regression because all slopes for one stream (Brush Creek) were below 0.5.

Parachute (n = 17)0.53 ± 0.170.63 ± 0.150.73 ± 0.13531 ± 137
Roan (n = 16)0.47 ± 0.070.59 ± 0.070.67 ± 0.06d512 ± 78
Mean0.50 ± 0.130.61 ± 0.120.70 ± 0.11522 ± 112

6. Discussion

6.1. Knickpoint Celerity Model

[35] A simple knickpoint migration model in which retreat rate is a power law function of drainage area performs surprisingly well at predicting the spatial distribution of present-day knickpoints on the Roan Plateau. This model makes no prediction about whether knickpoints change form as they migrate upstream and is insensitive to slope (because we specified n = 1 in order to derive the celerity equation). Because this model works well to represent the observed knickpoint distribution, it provides a baseline against which other models can be tested. The difference between model residuals suggests that the Parachute Creek and Roan Creek knickpoints positions might be predicted more accurately with separate parameter combinations. Roan Creek modeled knickpoints generally had positive residuals (on average, modeled knickpoints were downstream of observed knickpoints), implying that a larger K value could result in a better fit to the observed knickpoints. This difference in rock susceptibility to erosion (which could be caused by differences in lithology or mineralogy of the rock units) may also explain the considerable misfit of our model to the Brush Creek knickpoint. Brush Creek may represent a transition between the less resistant, landslide-prone bedrock to the west (which is drained by streams that we excluded from our modeling exercise) and the more resistant bedrock to the east.

[36] Alternatively, the difference in model residuals between the Parachute and Roan drainages could reflect nonsimultaneous knickpoint initiation. If Colorado River incision was nonuniform and progressed upstream, then Roan knickpoints may have initiated earlier than Parachute knickpoints (thus explaining the positive Roan residuals when we require simultaneous initiation). As a simple thought experiment, we can use a knickpoint retreat rate of 1 mm/yr (typical for both drainages at the end of the model run) to calculate the necessary time lag in knickpoint initiation. This retreat rate and the difference in average model residuals (0.78 + 1.25 km = 2.03 km) imply a ∼2 million year time lag in knickpoint initiation. It is plausible that a knickpoint on the Colorado River could have propagated ∼20 km upstream from the Roan Creek confluence to the Parachute Creek confluence at a rate of ∼10 mm/yr. However, we currently lack information to discriminate between progressive knickpoint initiation and a transition in lithology as the best explanation for differences in the knickpoint celerity model performance between the Parachute and Roan drainages.

6.2. Comparison With Other Studies

[37] Knickpoint retreat rates in natural settings are typically too slow for real-time measurement, except in cases where the underlying rock is especially weak (e.g., badland shales studied by Howard and Kerby [1983]), or where discharge is sufficiently large to accomplish rapid erosion (e.g., retreat of Niagara Falls, as first studied by Gilbert [1896, 1907]). Dramatic variations in knickpoint retreat rate can depend on climate, substrate erodability, magnitude and type of incision signal, and changes in drainage area distribution (e.g., stream capture). While average retreat rates can be estimated by dating the age of knickpoint initiation and measuring the distance of retreat, this calculation fails to account for either the decline in discharge in the upstream direction (causing decline in retreat rate through time) or variation in discharge due to changes in climate during the period of retreat. This method has been used by various workers [e.g., Seidl et al., 1994; Hayakawa and Matsukura, 2003; Bishop et al., 2005] to yield knickpoint retreat rates ranging from 0.5 to 2000 mm/yr. Our modeled retreat rates are comparable to cosmogenically derived (36Cl) average short-term retreat rate of 1.75 mm/yr documented on Dead Sea knickpoints [Enzel et al., 2005].

[38] Below we compare several recent knickpoint migration studies that are calibrated by field and topographic data with the results of our study (Table 4). Even though these studies address Quaternary rates of change, they applied similar methods and provide a useful comparison. Hayakawa and Matsukura [2003] used known recession rates of nine waterfalls in Japan to calibrate an empirical model for waterfall retreat. Their model predicts that the retreat rate depends on the drainage area upstream of the waterfall, the mean annual precipitation in the catchment, the area of the waterfall face, and the unconfined compressive strength of the bedrock. Their best fit relationship between modeled and observed retreat rates predicts that retreat rate will scale with drainage area to the power of 0.73, if all other variables are held constant. That their model explicitly includes the effects of precipitation and rock strength allows it to be applicable to other waterfalls in different climates and lithologies. However, the recession rates used to calibrate the model (calculated by dividing the total retreat distance by the waterfall age) are again averages over the entire retreat history. Early in the retreat, rates will likely have been much faster and will have slowed as the contributing drainage area decreased.

Table 4. Comparison of Knickpoint Retreat Model Parameters
Retreat RateConstantVariablePowerMaximum Duration of Incision, yearsStudy
Meters/year7.9 × 10−9Upstream drainage area1.12518,000Crosby and Whipple [2006]
Meters/year0.015Upstream drainage area1.166,150Bishop et al. [2005] (using data from Hayakawa and Matsukura [2003])
Meters/14 ky50.8Total drainage area1.2414,000Bishop et al. [2005]
Meters/yr1.37 × 10−7Upstream drainage area0.54∼8 × 106This study

[39] Bishop et al. [2005] expanded upon Hayakawa and Matsukura's [2003] work by studying knickpoint retreat in Eastern Scotland. They tested for a power law relationship between the drainage area draining to the basin outlet across each knickpoint and the knickpoint retreat distance (assuming a common initiation age). They also tested their model with retreat rate data from Hayakawa and Matsukura's study and found that their simpler model, which uses only drainage area as a variable, is a good first-order predictor of retreat distance, especially because precipitation does not vary greatly for the Japanese streams. Neither Hayakawa and Matsukura [2003] nor Bishop et al. [2005] explicitly considered stepwise changes in retreat rate over time due to changes in contributing drainage area as tributaries are passed. However, these studies do demonstrate that to first order, drainage area exerts an important control on the rates of landscape evolution.

[40] Our work is similar to a basin-wide knickpoint study in the Waipaoa basin in New Zealand, in which Crosby and Whipple [2006] compared two different knickpoint models; one in which knickpoints form at a threshold drainage area, and the other that predicts an upstream migration rate dx/dt = CAp, where C = 7.9 × 10−9 (m1.25 yr)−1 and p = 1.125 (although the difference between p = 1.125 and p = 1 was not statistically significant). This progressive retreat model predicts a much stronger dependence of knickpoint migration rate on drainage area than does the formulation presented here (in which the best fit exponent on A is 0.54). A large exponent on drainage area can lead to spurious knickpoint migration rates (>100 m/yr) when upstream drainage areas are large. Yet, as Crosby and Whipple's modeled knickpoints move upstream, their migration rates decline and converge to reasonable values. Their model yields migration rates of 3–44 mm/yr for knickpoints with upstream drainage areas between 0.1 and 1 km2. However, these apparent knickpoint retreat rates are probably not an accurate portrayal of Waipaoa basin response to base level fall. As discussed by Crosby et al. [2007], knickpoints in the Waipaoa are most frequently found just upstream of tributary junctions. This distribution is consistent with theoretical and numerical predictions that use sediment-flux dependent incision rules to predict the formation of hanging valleys at tributary junctions [Crosby et al., 2005, 2007].

[41] The assumed duration of knickpoint retreat for Roan Plateau streams (8 Ma) is much longer than in the Waipaoa basin (18 ka), even though knickpoint retreat distances are shorter (between 10 and 55 km) on the Roan Plateau than in the Waipaoa (between 30 and 90 km) [Crosby and Whipple, 2006, Figure 7]. The much longer response time on the Roan Plateau likely reflects differences in both lithology (resistance to erosion) and climate (forcing for erosion), and perhaps in knickpoint retreat processes. Bedrock erosion rates can vary by many orders of magnitude depending on lithology [e.g., Sklar and Dietrich, 2001], but we have no basis yet for quantitative comparison of bedrock susceptibility to erosion between the Roan Plateau and the Waipaoa drainage basin (such as Schmidt hammer measurements). Erosion rates are likely lower on the Roan Plateau because of its arid climate, or the dependence of peak stream discharge on spring snowmelt (which may reduce the number of discharge events capable of bed scour). At this point we do not have sufficient information to compare the dominant knickpoint retreat processes of the Roan Plateau and the Waipaoa basin.

6.3. Longitudinal Profile Analysis

[42] Downstream projection of channel profiles using the properties of upper channel segments [e.g., Schoenbohm et al., 2004] is probably inappropriate for estimating the magnitude of base level fall on the Roan Plateau, given the low average concavity index (θ = 0.21). Projections using a larger concavity index (θ = 0.5) resulted in more geologically reasonable estimates of base level fall, but this concavity index does not accurately describe the upper channel segments. The downstream projection method assumes that the upper plateau is a “relict” landscape that reflects pre-incision conditions. While this is a tempting simplification, it is not strictly true in the Roan Plateau. Recall that the Niemann et al. [2001] model suggested that following a step change in relative uplift rate, upstream-propagating knickpoints should experience steady vertical velocity and should all therefore be found at the same elevation. The Roan knickpoints are found at a variety of elevations and correlate well with the outcrop of the Mahogany oil-shale zone (Figure 3), which in some drainages dips away from the Colorado River (Figure 2). It is therefore unlikely that the Roan Plateau knickpoints obey this uniform vertical velocity, perhaps because they have experienced a relative uplift history that is more complicated than a single step change. It is also possible that early in their migration history (before branching into tributaries), knickpoint elevations declined through time as they followed the elevation of the upstream-dipping Mahogany oil-shale zone. We cannot ignore the effects of structure in understanding how Roan Plateau knickpoints have retreated. A direct consequence of the lowering of the knickpoint elevation with time is that the upper channel segments would experience a dropping base level with time and thus experience profile steepening (which can explain the low observed concavity indices). In turn, this would disallow use of the projection of their longitudinal profiles to the Colorado River as a means of estimating the magnitude of base level lowering that drove the knickpoints.

[43] Longitudinal profile analysis of lower channel segments on the Roan Plateau allows evaluation of channel adjustment following base level fall. Although it is difficult to determine which regression slope limit yields the most meaningful channel concavity index for the lower channel segments, a slope limit of 0.2 resulted in the highest average R2 regression values (0.72 and 0.78 for the Parachute and Roan basins, respectively). The concavity indices determined with a slope limit of 0.2 fall within the error range of the best fit area exponent (m) in the knickpoint celerity model (Table 3). This coincidence suggests that downstream channel segments may have reached a new equilibrium state, such that their concavity indices are comparable to the ratio m/n in the stream power model [e.g., Niemann et al., 2001; Whipple and Tucker, 1999] because we have specified that n = 1. This would further imply that the Roan Plateau knickpoints may be analogous to the knickpoints described by Whipple and Tucker [1999] and Niemann et al. [2001], in which a break in channel steepness migrates upstream. The dramatic waterfalls on the Roan Plateau, which define the upstream end of a large oversteepened zone, can be viewed as a consequence of lithologic variation that is not evident in these other studies.

[44] However, the similarity between channel concavity indices and the celerity model drainage area exponent does not necessarily prove that downstream of knickpoints, channels have reached a new equilibrium state. Equating parameters from slope-area regression to stream power exponents requires the assumptions that (1) the river profile is in steady state with respect to climate and rock uplift (base level fall); and (2) rock uplift rate and the coefficient of erosion (K) are uniform throughout the channel reach [e.g., Snyder et al., 2000]. It seems unlikely that either condition holds for channels downstream of knickpoints on the Roan Plateau. Climate has certainly not been steady over the several million year duration of knickpoint retreat [e.g., Zachos et al., 2001], and we do not have enough information about Colorado River incision to assume a steady base level fall. The effective coefficient of erosion (K) may be nonuniform immediately downstream of Plateau waterfalls, where boulder input from canyon walls likely acts to slow erosion and oversteepen these channel segments. Numerical experiments by Gasparini et al. [2006] indicate that when an increase in sediment flux (relative to carrying capacity) acts to decrease incision rates, then an increase in uplift rate across the network acts to increase channel slopes. Because of these complications, Roan Plateau channels downstream of knickpoints are likely not at steady state; we therefore do not claim to have calibrated the m/n ratio in the stream power model through our longitudinal profile analysis.

6.4. Future Work

[45] Future work must address the base level history for Parachute and Roan Creeks (determined by the rates of Colorado River incision since 8 Ma) to provide a better quantitative constraint on the boundary condition for both the knickpoint celerity model and evolution of longitudinal profiles. The Roan Plateau may also provide an opportunity to assess the role of sediment armoring in channel profile evolution immediately downstream of knickpoints, as boulders derived from the retreating waterfall are both weathered and comminuted. In addition, our ability to accurately predict knickpoint retreat distances as a function of drainage area strongly provides insight into which specific knickpoint retreat processes must dominate. Potential discharge-dependent (and hence drainage area-dependent) knickpoint retreat processes on the Roan Plateau include plunge-pool scour and weathering caused by freeze-thaw and wetting-drying cycles.

7. Conclusions

[46] The simple stream power-based model for knickpoint retreat works remarkably well in explaining the present locations of 32 knickpoints on the Roan Plateau. The best fitting exponent on drainage area (m = 0.54) implies roughly square root dependence of retreat rate on drainage area. If the 8 Ma time of knickpoint initiation caused by Colorado River incision is correct, then our model predicts that knickpoint speeds began at 7.1–11.9 mm/yr and have dropped to 0.3–2.3 mm/yr at their present locations. These values and the corresponding erodability constant (K) will adjust as we constrain timing of Colorado River incision more tightly. That the Roan Plateau knickpoints are incising into bedrock structurally controlled by minor dips of the more resistant formation members dictates that (1) knickpoint elevations vary with the elevation of those resistant members, and (2) the landscape upstream of the knickpoints will respond to downstream incision before the knickpoint arrives. The success of this simple model strongly demands a more process-specific exploration of knickpoint evolution and suggests that over long timescales these processes can be described as being controlled by drainage area.


[47] We thank Ben Crosby, David Harbor, and Kelin Whipple for thorough reviews that significantly improved the original manuscript. We thank David Thul for assistance in the field and Encana Oil & Gas (USA) for property access. This research was conducted with support from the National Science Foundation (EAR-0545537) to R. Anderson, a fellowship from the Colorado Mountain Club to M. Berlin, a Geological Society of America for Graduate Student Research grant (8060-05) to M. Berlin, and a National Science Foundation Graduate Research Fellowship to M. Berlin.