Bedrock Erosion by Debris Flows at Chalk Cliffs, Colorado, USA: Implications for Bedrock Channel Evolution

. Debris flow erosion into bedrock helps to set the pace of mountain denudation, but there are few empirical observations of this process. We studied the effects of debris flows on bedrock erosion using Structure-From-Motion photogrammetry and multiple real-time monitoring measurements. We found that the distribution of bedrock erosion across the channel cross-section could be generalized as an exponentially decreasing function of height above the channel thalweg. Using this empirical function, we simulated the erosion at a cross-section after the theoretical passage of a migrating knickpoint effectively matching the upstream pre-knickpoint cross-sectional shape to the downstream post-knickpoint cross-sectional shape via debris-flow bedrock erosion.


Introduction
Fluvial bedrock erosion is well described by shear stress and stream power laws that can form a linear power law relationship where bedrock channel slopes are a function of rock erodibility and drainage area [1,2]. However, no such relationship exists for debris flow carved channels. Rather, bedrock channels sculpted by debris flows show a non-linear relationship between slope and drainage area [2,3]. Because of the difficulty in directly measuring bedrock erosion from debris flows, only a few studies have been done to quantify the rate, mechanism, and spatial distribution of bedrock erosion in debris flow channels [4]. Understanding how bedrock erosion will occur both vertically and laterally in debris flow channels has practical implications for infrastructure (such as bridge pier scour) and landscape evolution modeling.
A variety of erosional processes have been identified to erode fluvial bedrock channels: plucking, abrasion, cavitation, and solution weathering [1]. By contrast, debris flow erosion of bedrock channels has been attributed to granular impacts on bare bedrock [5][6][7] via plucking and bedrock abrasion. Plucking of fractured bedrock occurs as large saltating bedload clasts impact the bedrock channel. Abrasion wears away the surface through comminution. However, even a small layer of sediment (~20x the median bed sediment grain size) can prevent erosion [6]. Despite suggestions that granular impacts at the debris flow snout are the primary source of debris flow erosion [8], field observations show that particle impacts can produce erosion during the water-dominated tail of a debris flow as well as the grain-dominated debris flow snout [6].
Modeling of bedrock cross-sectional erosion for fluvial bedrock rivers [9] has assumed that erosion was proportional to the boundary shear stress. For bedrock erosion in debris flow channels, experiments indicate that erosion is proportional to inertial stress [5], and * Corresponding author: frengers@usgs.gov direct experimental measurements show a non-linear velocity gradient [10]. Thus, we expect a non-linear diminution of bedrock erosion with increasing vertical distance above the channel bottom.
In this study we examined a channel reach with frequent debris flows and exposed bedrock at the Chalk Cliffs in central Colorado, USA [11]. The bedrock at Chalk Cliffs is composed of hydrothermally altered quartz monzonite [11] and weathers into small blocks, often along fracture and vein surfaces. Multiple debris flows occur in the catchment each year, and erosion typically occurs by plucking of centimeter-scale blocks from the bedrock [12].
Prior debris flow erosion studies have primarily focused on bedrock erosion at the channel bottom [5,6]. In this study, we explored how bedrock erosion was distributed along the full channel perimeter during debris flows and developed an empirical equation for cross-sectional erosion. We tested the ability of the empirical equation to evolve a cross-section due to debris flow erosion at a knickpoint.

Structure-from-Motion
Photogrammetric surveys of the study channel reach were conducted during bare-bedrock conditions on three dates: 15 Sep. 2015, 17 Sep. 2017, and 26 Sep. 2018. For each of these bedrock channel surveys, the channel bottom was cleaned with a broom to remove any loose sediment so that the photogrammetric surface was limited entirely to bare bedrock. Each of the photogrammetric surveys were conducted with a Nikon AW1 mirrorless camera. Debris flow activity during the summer of 2016 resulted in deposition of sediment in the study reach, such that it was impractical to manually excavate the sediment in order to conduct a bedrock channel survey during the fall of 2016.
We used fixed points as survey control for the photos. Prior to the first survey, 25 rock bolts were drilled into the sides and bottom of the channel to establish the survey control points. The total bolt length was 12.8 cm with 2.2 cm protruding above the bedrock surface. Each bolt was surveyed, using a local coordinate system, with a Topcon total station (accuracy ±1.5 mm). The average uncertainty for each of the three surveys reported by Agisoft [13] was 0.023 m. We estimated the total Structure-From-Motion (SfM) uncertainty using the equation: where U is the total uncertainty (0.025 m), Ereg is the point cloud registration uncertainty from Agisoft (0.023 m), and ETS is the total station measurement uncertainty (0.0015 m). Point clouds from the three survey epochs were compared using the M3C2 method in CloudCompare [14,15] (Fig. 1).

Monitoring
In the study channel reach, a look-down laser with 10 Hz temporal resolution was mounted to a bridge spanning the channel. This laser was used to measure changes in flow depth so that the number and duration of debris flow surges could be identified. In the resulting stage data, we recognized the initial debris flow surges by looking for abrupt changes in the variance of the laser height, and a height change was classified as a debris flow if it had a flow-front height > 15 cm, otherwise it was considered shallow water flow. Debris flow activity was also distinguished from water-dominated flow by visually observing available video footage. All flow peaks following the initial debris flow stage change were labeled as surges for the same event unless the time between surges exceeded 30 minutes.

Cross-section Evolution
We examined bedrock cross-sectional erosion by looking at a portion of the channel above and below a transient knickpoint. The average channel cross-section upstream from the knickpoint is wider than downstream from the knickpoint, where the channel narrows ( Fig. 1-3). This is the natural consequence of upstream knickpoint propagation driven by debris flow erosion. We hypothesize that as the knickpoint advances upstream, debris flows will erode the bedrock channel bottom and the upstream cross-section will evolve to a shape similar to the downstream cross-section. Here we test that hypothesis by simulating bedrock erosion using observations of channel bedrock erosion from repeat photogrammetric surveys.
We created a characteristic average cross-section for the upstream and downstream reaches separately. We extracted cross-sections every 0.5 m (more than 80 cross-sections in total) and averaged the height above the channel bottom every 1 cm for both upstream and downstream reaches (Fig. 3a-b dashed black line). For simplicity, we show the cross-sections from the initial survey (September 2015) in Fig. 3, but the average shape of the channel is similar for all of the surveys.   (Fig. 3c). The erosion values were normalized by the observed time that debris flows were active on a bedrock channel bed. These data were used to find best-fit parameters for Equation 2 (Fig.  3c).
We applied Equation 2 to the average cross-sectional shape upstream from the channel knickpoint, which represents a channel cross-section before the knickpoint passage (Fig. 2a). We ran the model forward in time with one time step representing a year of debris flow erosion, assuming a debris flow bedrock erosion duration of 20 minutes per year. We calculated the root mean square error (RMSE) offset between the elevation of each point in the model eroded cross-section with each point in the control cross-section (downstream) and found the time that resulted in the simulated crosssection with a minimum RMSE.

Monitoring
Our monitoring data show 21 debris flow events between September 2015 and September 2018 with 184 observed surges. The average maximum depth for each surge was 0.9 m (max = 1.9, min = 0.2), and the average number of surges per debris flow event was 8.7 (max = 42, min = 1). Cumulative debris flow activity lasted over several hours. The debris flow activity over bare bedrock between the survey epochs was 16 min, 47 min, and 16 min.

Bedrock Erosion Patterns and Magnitude
The SFM point cloud difference was used to observe the patterns and magnitude of bedrock erosion between SfM surveys (Figs. 1 and 2). The only major deposition is due to accretion of a debris flow levee (Fig. 1). Minor deposition high on the channel side walls is likely due to dry ravel accumulation (Fig. 1b). The change detection map shows that bedrock erosion is focused around two morphologic features: knickpoints and potholes ( Figs. 1 and 2). The average erosion depth for the channel area (69 m 2 ) was 6.8 cm between 2015 and 2018.

Bedrock Erosion Patterns and Bedforms
Bedrock erosion is concentrated near bedrock bedforms. For example, erosion occurs downstream from a knickpoint ( Figs. 1 and 2b) where debris flow activity steadily eroded the bed during the study period. The bedrock study channel also contains numerous potholes (Fig. 1), and erosion concentrates in these areas, as illustrated by contour lines representing total erosion between 2015 and 2018 that form closed rings around the potholes (Fig. 2a). Pothole erosion in the bedrock substrate can take the form of general deepening and widening (Fig. 2c). Between 2015 and 2017 we see the pothole in Fig. 2c become both deeper and wider. Another pothole shows migration in the downstream direction over time, with most of the erosion occurring on the portion of the pothole surface facing upstream (Fig. 2e).

Cross-section evolution
Equation 2 was applied to the average cross-section upstream from the knickpoint in order to simulate how the cross-section would evolve from bedrock erosion simulated by Equation 2 (Fig. 4). We found that the lowest RMSE between the erosion-simulated crosssection and the real downstream cross-section occurred after 6 model iterations, indicating that it would take approximately 6 years of debris flow erosion (assuming 20 minutes per year of debris flow erosion on exposed bedrock) for the upstream cross section to more closely resemble the downstream cross-sectional shape.

Discussion
Direct measurements of debris flow bedrock erosion are difficult because advanced warning of debris flow activity is rarely available. However, this study captures spatially continuous measurements of debris flow generated bedrock erosion within a channel, using SfM measurements of erosional change. Our erosion measurements show that erosion is concentrated and localized in specific channel forms: potholes and a knickpoint. The measurements indicate that, although bedrock erosion may be controlled by random granular impacts [6], bedrock bedforms that create hydraulic flow disturbances nucleate erosional change. Prior research in fluvial geomorphology shows the importance of bedforms in creating flow instabilities, and therefore increased erosion near bedforms in a debris flow channel indicates that this process likely reflects a pattern that could be expected in other debris flow channels. The measurements available from this site are insufficient to directly link rock erosion to mechanistic flow parameters; however, our empirical observations provide a generalizable framework for predictions of bedrock erosion from debris flows. The exponential equation for debris flow erosion (Equation 2) helps to explain the general U-shaped pattern of the channel cross-sections. In addition, the magnitude of crosssectional erosion measurements illustrates the general lack of strath terraces in the bedrock channel. That is, given a headward-migrating knickpoint and high vertical channel incision rates the conditions are set to preserve strath terraces from the knickpoint passage. However, our erosional measurements indicate sufficient erosion rates exist such that nascent bedrock terraces will be destroyed by subsequent debris flows as well as slope weathering (Fig. 3c). Consequently, this study provides important information of distributed bedrock erosion that helps to define an empirical relationship for debris flow induced bedrock erosion.

Conclusions
This study captures bedrock erosional change in a steep debris flow channel over three years using SFM photogrammetry. We found that erosional change was concentrated around key bedforms (i.e., potholes and knickpoints). More generally, we used erosional measurements from channel cross-sections extracted every 0.5 m in our study reach to define an equation for bedrock erosion, showing that the measured bedrock erosion varied as a function of channel height.