This study addresses the temporal variations in rockfall activity in the 5.2 km2 calcareous cliffs of the deglaciated Lauterbrunnen Valley, Switzerland. We did this using 19 campaigns of repeated terrestrial laser scans (TLS) over 5.2 yr, power-law predicted behavior from extrapolation of the TLS-derived frequency-magnitude relationship, and estimates of long-time-scale (∼11 k.y.) activity based on the volume of preserved postglacial rockfall talus. Results from the short-time-scale observations indicate no statistically significant difference between TLS observations averaging over 1.5 versus 5.2 yr. Rock-wall retreat rates in both cases are 0.03–0.08 mm/yr. In contrast, the power-law predicted rock-wall retreat rates are 0.14–0.22 mm/yr, and long-term rates from talus volumes are 0.27–0.38 mm/yr. These results suggest (1) short (1.5 yr) TLS inventories of rockfalls provide (within uncertainties) similar frequency-magnitude relationships as longer (5.2 yr) inventories, thereby suggesting short observation periods may be sufficient for hazard characterization from TLS, and (2) higher rock-wall retreat rates over long time scales (Holocene averaged) may reflect debuttressing and stress relaxation effects after glacial retreat, and/or enhanced rockfall activity under periglacial (climatic) conditions.
Rockfalls are efficient agents of erosion, controlling the development of rock slopes, and they can pose a threat to settlements and infrastructure. Rockfalls occur frequently in deglaciated alpine valleys where rock walls are oversteepened, exposed, and more susceptible to failure after glacial retreat. Rock-wall retreat rates under present-day conditions and their temporal change since deglaciation remain less understood. Here, we investigated rock-wall retreat rates over different time scales (∼5 yr to ∼11 k.y.) in the deglaciated Lauterbrunnen Valley of the Bernese Alps, Switzerland (Fig. 1). An improved understanding of these rates is motivated by the need to understand postglacial erosion and the role of rockfalls in the evolution of alpine landscapes.
Rockfall data sets derived from direct measurements (e.g., terrestrial laser scans [TLS] and photogrammetry) cover time scales from hours to years and are often used in modeling rock-wall retreat rates based on rockfall frequency-magnitude distributions over decadal to centennial time scales (Dussauge et al., 2003; Rosser et al., 2005; Barlow et al., 2012; Santana et al., 2012). In contrast, indirect measurements based on volumetric calculation of talus deposits have been used to estimate rock-wall retreat rates over millennial time scales (Curry and Morris, 2004; Sass and Krautblatter, 2007; Siewert et al., 2012). Alpine rock-wall retreat rates vary between both methods. Present-day retreat rates for alpine environments range from 0.01 mm/yr to 0.84 mm/yr, while the Holocene retreat rates range from 0.2 mm/yr to 2.5 mm/yr (Curry and Morris, 2004, and references within). Various factors contribute to this discrepancy, including joint spacing and orientation, and rockfall triggering processes such as paraglacial unloading after deglaciation (Hinchcliffe and Ballantyne, 1999; Arsenault and Meigs, 2005) and periglacial conditions (e.g., enhanced freeze-thaw activity and permafrost degradation).
Although previous studies have reported rock-wall retreat rates, few have provided a complete and continuous coverage of large rock walls with uniform lithology (e.g., Guzzetti et al., 2003). Many studies compare rock-wall retreat rates for localities in different environments (Siewert et al., 2012; Curry and Morris, 2004; Hinchliffe and Ballantyne, 1999). This comparison may identify factors for differing rates, but it provides limited information on the long-term behavior of a given rock mass in one location. This study complements previous work by calculating rock-wall retreat rates over time scales from years to tens of thousands of years for the 5.2 km2 limestone rock walls of a deglaciated valley.
The Lauterbrunnen Valley is a deglaciated valley with near-vertical walls consisting of Helvetic limestone (Fig. 1; Fig. DR4 in the GSA Data Repository1). Data on historic rockfalls show large slope failures since 1750 CE, including the 1889 landslide that released >104 m3 of debris (Michel, 1979). Using TLS data, Strunden et al. (2015) detected 122 rockfalls in the valley over an 18 month period. These events ranged in volume from 0.06 m3 to 119.34 m3, with those less than 1 m3 being most common and associated with freeze-thaw cycles. Using seismic signals, Dietze et al. (2017) detected 49 rockfalls over a 6 month period, 10% of which were influenced by freeze-thaw cycles. They inferred a systematic lowering of a rock mass release zone driven by a lowering of the water table in the rock wall. Other potential triggers, such as earthquakes and anthropogenic activity (as shown in Mackey and Quigley, 2014), are unlikely to influence rockfalls in this study area.
TLS Data Collection and Processing
A long-range terrestrial light detection and ranging scanner was used to acquire three-dimensional (3-D) point clouds from 22 scan positions during each campaign. Nineteen (19) campaigns were conducted over 5.2 yr (February 2012 to April 2017). Scans collected from similar positions at different times were aligned using an iterative closest point algorithm. To produce a continuous surface, triangle meshes of reference scans were computed. Surface meshes were compared with point clouds to identify rockfalls between successive campaigns. Rockfall volumes were obtained from cut-and-fill calculations and validated using photos. The details of our TLS data collection, processing, and error evaluation are given in Strunden et al. (2015).
Rock-Wall Retreat Rate Calculations
We used three different approaches to determine retreat rates over different time scales. First, the averaged short-term retreat rates were calculated from the total volume of all rockfalls based on TLS observations collected over 5.2 yr. Second, power-law predicted retreat rates were derived from TLS measurements under the assumption of a power-law distribution of rockfall sizes (for justification, see Strunden et al., 2015). This was done by calculating nonlinear least-squares regression fits and maximum likelihood estimates (MLE). This approach provides long-term extrapolated results, but it assumes a power-law distribution of rockfalls that remains constant over time and requires a maximum event size for exponents ≤1, as is the case here. We set this limit to the largest historic event (104 m3), which occurred 130 yr ago (Fig. DR1, Fig. DR2, and Section 1 in the Data Repository). For a thorough discussion on the cutoff limit, we refer readers to Hergarten (2012). Third, the minimum rock-wall retreat rates over the last 11 k.y. were inferred from the talus volumes using TLS and a digital elevation model on a 2 m grid (swissALTI3D; Fig. DR3 and Section DR2). To compensate for the density difference between talus bulk and intact bedrock, we used a density correction factor of 0.77 for limestone (from Krautblatter et al., 2012; Sass and Wollny, 2001). The density-corrected volume was divided by the rock-wall surface area above each talus fan and the talus production time (i.e., time span elapsed since deglaciation, which was estimated to be 11 k.y.). This talus production time is based on radiocarbon (10.39 ± 0.15 ka) and 10Be surface exposure (12.2–10.8 ka) ages of moraines and bedrock samples located in the study area and the neighboring Hasli valley (Wipf  and Wirsig et al. , respectively). The long-term rates represent minimum estimated rates over the last ∼11 k.y. Uncertainties in this estimate stem from potential incomplete exposure of talus, and the timing of deglaciation (see the Data Repository for an extended discussion).
In total, 316 rockfalls were detected in the 5.2 yr period, 122 of which were identified by Strunden et al. (2015) in the first 18 months (Fig. 2). Rockfall volumes ranged from 0.030 ± 0.004 m3 to 267.27 ± 4.39 m3 (Table DR1). Small rockfalls (<1 m3) were most common (63%), 21% had volumes of 1–3 m3, and 2% were >50 m3.
Short-Time-Scale (<5.2 yr) Wall Retreat Rates
Spatially averaged short-term (5.2 yr) wall retreat rates were calculated for the west wall by dividing the total rockfall volume (1610.87 m3) from the west wall in the 5.2 yr period by its area (3.7 km2). Similarly, the retreat rate for the east wall was obtained using 229.65 m3 of rockfall volume and a wall area of 1.5 km2. The short-term wall retreat rates for the west (0.08 mm/yr) and east walls (0.03 mm/yr) agree with those obtained over an 18 month period by Strunden et al. (2015).
Power-Law Predicted Wall Retreat Rates
The empirical log-frequency and log-magnitude distributions for the 5.2 yr rockfall data set show a rollover and power-law tail (Fig. 3). A nonlinear least-squares regression and MLE were used to determine the frequency-magnitude relationship. The inferred rollover volumes, based on optimization of the R2 value and the Kolmogorov-Smirnov statistic, differed for the two methods (Fig. DR1). We observed deviations from pure power-law behavior and a systematic trend in fit parameters for rollover volumes of 0.29–0.92 m3. For these values, the MLE of the scaling exponent b ranged from 0.61 to 0.72. The power-law fit parameters agree with previous values obtained for a shorter observation duration (1.5 yr; Table DR2, Fig. DR2C). However, the frequency of rockfalls in the 5.2 yr data set is lower compared to the 1.5 yr data set (Fig. 3). This is particularly evident for rockfalls smaller than 2 m3.
Using the above parameters, the power-law predicted eroded volume per year is (0.70–1.13) × 103 m3, and wall retreat rates are 0.14–0.22 mm/yr. Using observations from Strunden et al. (2015), we recalculated the total eroded volume and power-law predicted wall retreat rates (i.e., 0.62 ± 0.13 × 103 m3 and 0.12 ± 0.03 mm/yr, respectively; Table DR2).
Long-Time-Scale (11 k.y.) Wall Retreat Rates
The measured talus volumes ranged from 2.1 × 106 m3 to 4.3 × 106 m3 (Fig. 1). These are considered minimum estimates due to possible loss of material or incomplete exposure due to alluvial infilling in the valley. However, no field evidence supports transport of talus out of the valley. The long-term averaged wall retreat rate of >0.33 mm/yr was calculated by dividing the talus volume estimated for each talus section by the surface area of the rock wall above that section (Table DR3). Due to their complex morphology, talus sections WT1 and WT4 were excluded from retreat rate calculations.
Power-law relationships can be used to estimate the return time of large rockfall events and long-term erosion. For a sample size of >300 rockfalls (this study), many of the issues associated with producing robust power-law fits are negligible (Clauset et al., 2009; Strunden et al., 2015). We found that power-law predicted calculations were sensitive to the choice of rollover values. Larger rollover values yielded larger values of exponent b and a longer return time for a 1000 m3 rockfall and smaller values of long-term erosion (Fig. DR2), demonstrating that a rockfall frequency-magnitude relationship does not follow a pure power law, and caution must be exercised when calculating power-law fit parameters. To account for this, ranges of values are reported here to encompass the variability in the data and range of possible solutions (Table DR2).
Comparison of 1.5 yr and 5.2 yr Rockfall Data Sets
TLS measurements from two different time intervals (1.5 yr and 5.2 yr) were compared to evaluate the effect of observation time on the frequency-magnitude relationships. Power-law exponents and extrapolated wall retreat rates from both data sets agree within error (Fig. 3; Table DR2C, Fig. DR2D). However, there was a systematic decrease in the frequency of smaller events for the 5.2 yr data set (Fig. 3). Additionally, the number of larger events was higher in the 5.2 yr data set (three events over 250 m3), which permitted extrapolation to larger volumes. The differences may be explained by the stochastic nature of rockfalls.
Comparison of Wall Retreat Rates
The long-term wall retreat rates were an order of magnitude higher than the short-term rates (>0.33 vs. 0.03–0.08 mm/yr). This difference can reflect sensitivity to observation time scales and/or different erosional processes at work. For example, in glaciated landscapes, glaciers not only increase erosion rates during glaciation, but they can result in a postglacial increase in rates that decays over time scales of 101–104 yr (Stoffel and Huggel, 2017). Below, we discuss potential sources of this observed difference in short- and long-term retreat rates.
First, we found that over the short-term (5.2 yr), small rockfalls (<1 m3) were the most frequent in the Lauterbrunnen Valley, and those over 100 m3 (total of 4) were uncommon. However, several large rockfall episodes (e.g., several 103 to 105 m3 rockfalls between 1750 and 1947 CE) have occurred in the past and released large volumes of debris, forming talus slopes and depositing boulders beyond the base of talus cones. The short-term wall retreat rates calculated over 5.2 yr exclude such large events, whereas retreat rates calculated over longer time intervals (11 k.y.) include these events. Second, previous work has documented that long-term rates are sensitive to temporal changes in the stability of rock walls. For example, following deglaciation, a debuttressing of rock walls occurs from ice removal and releases of confining stresses, which can lead to increased rockfalls (André, 1997; Ballantyne, 2002; Curry and Morris, 2004; Sass and Krautblatter, 2007; Korup et al., 2012; Messenzehl et al., 2017). Similarly, recent studies in other areas of the Bernese Alps have documented that paraglacial slope adjustment by rockfalls has occurred in response to the reduction of ice volume in recent decades (Keusen et al., 2007; Zumbühl et al., 2008). Since these effects diminish through time, lower short-term compared to long-term rates may reflect progressive relaxation of postglacial stresses release and an increase in rock-wall stability (Hinchliffe and Ballantyne, 1999).
Finally, postglacial climatic changes during the Holocene have been suggested to affect denudation in glaciated valleys (e.g., Stock et al., 2009; Yanites and Ehlers, 2012) and provide favorable conditions for rockfalls (André, 1997; Hales and Roering, 2007; Ravanel and Deline, 2011; Gallach et al., 2018). For example, early Holocene warming and permafrost degradation (ca. 10–9 ka) and mid- to late Holocene humid and cold phases (ca. 5–3 ka and ca. 2–1 ka) are often cited as possible triggers of mass-wasting events in the European Alps (Prager et al., 2008; Borgatti and Soldati, 2010; Ivy-Ochs et al., 2017). This is supported by cosmogenic radionuclide dating of large slope failures elsewhere in the Alps that are correlated to a cold phase at 8.2 ka (Ostermann et al., 2012) and a period of intense rainfall events at 4.2 ka (Zerathe et al., 2014).
In light of the above mechanisms documented in the European Alps, the higher long-term retreat rates identified here may reflect periods of enhanced rockfall activity resulting from the combined effects of paraglacial processes (e.g., bedrock jointing inherited from stress release) and changing periglacial conditions (e.g., permafrost degradation).
This study investigated temporal variations in rock-wall retreat rates in the 5.2 km2 calcareous cliffs of the Lauterbrunnen Valley, Switzerland. Short-term retreat rates, power-law exponents, and derived values do not appear to be sensitive to observation periods of 1.5 yr versus 5.2 yr, within error (Fig. 3). Power-law predicted and long-term (11 k.y.) retreat rates calculated from talus accumulation are an order of magnitude higher than the short-term rates. The lower short-term rates (<5.2 yr) reflect frequent small rockfalls and exclude rare large events that dominate retreat rates over longer time scales. Higher long-term rates likely reflect periods of increased rockfall activity enhanced by stress release due to unloading and debuttressing of the rock wall following deglaciation, and the influence of changing periglacial (climatic) conditions in the past 11 k.y.
We thank Josy Burke (née Strunden), Sarah Falkowski, Lorenz Michel, and Michael Kloos for terrestrial laser scan field assistance. We are grateful to Nick Rosser, Alex Densmore, and Michael Krautblatter for valuable discussions in the early phases of this research, and to the residents of Lauterbrunnen Valley for their support. The manuscript benefited from constructive comments by Stefan Hergarten, Fritz Schlunegger, and one anonymous reviewer. We thank Mark Quigley for editorial handling. The funding for this study was provided by the German Science Foundation grant EH329/18–1 to Ehlers. The point cloud data and MATLAB codes used in this study are available upon request.