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Topography-based modeling of large rockfalls and application to hazard assessment

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Abstract

[1] Rockfalls are among the most important natural hazards in mountainous regions. Similarly to earthquakes and wildfires, their sizes follow a power-law distribution covering an enormous range of sizes. In this paper, the presumably first modeling approach that explains this power-law distribution quantitatively is presented. Applied to the European Alps, the Himalayas and the Rocky Mountains, the model suggests that a power-law exponent of 1.35 with respect to the detached volume is a universal property of rockfalls. Beyond reproducing and explaining existing statistical data, the model allows an estimate on size and frequency of the largest possible rockfalls in a region, which cannot be derived from available rockfall inventories so far.

1. Introduction

[2] Landslides cover an enormous range of sizes from small rockfalls of less than one cubic meter to several cubic kilometers, such as the Flims rockslide in the Alps with a volume of about 8 cubic kilometers [e.g., von Poschinger, 2011]. They can be classified by the mechanism of movement (e.g., falling or sliding) and by the involved material (rock or an unconsolidated regolith layer above the bedrock). Following the majority of the references cited in this paper, the term rockfalls is used for all types of rapid rock mass movements, in particular rockfalls and rockslides, in the following.

[3] While a variety of approaches to predict the runout of rockfalls is available (for a review see, e.g., Volkwein et al. [2011]), very little is known about the probability that a rockfall of a given size occurs in a certain region. Available statistical data [Wieczorek et al., 1992; Noever, 1993; Dussauge et al., 2003; Guzzetti et al., 2003; Malamud et al., 2004; Brunetti et al., 2009; Bennett et al., 2011] indicate that rockfall sizes are power-law distributed,

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where V is the involved volume and p(V) is the probability density of the size distribution. In the studies mentioned above, values of the scaling exponent α in the range between 1.07 and 1.75 were found for the non-cumulative probability density function.

[4] Similar power-law distributions occur in several natural hazards such as earthquakes, wildfires and regolith landslides, so that all these processes have been viewed in the context of self-organized criticality (SOC). The relationship to SOC is well established, although still in discussion for earthquakes [Olami et al., 1992] and for wildfires [Drossel and Schwabl, 1992; Malamud et al., 1998], and there have been some modeling approaches to link regolith landslides to SOC [Densmore et al., 1998; Hergarten and Neugebauer, 1998, 2000; Piegari et al., 2006] as well as laboratory experiments [e.g., Katz and Aharonov, 2006; Juanico et al., 2008]. However, the scaling exponents found for regolith landslides are significantly larger than those of rockfalls [Hovius et al., 1997; Hergarten, 2003; Malamud et al., 2004], namely α ≈ 2.4 with respect to area and α ≈ 2 with respect to volume.

[5] As a consequence of the strongly different scaling exponents found for rockfalls and regolith landslides, the modeling approaches mentioned above cannot be directly transferred to explain or predict the statistics of rockfalls. There was some discussion whether the simplest model of SOC, the so-called “sandpile model” [Bak et al., 1987], could be applied to rockfalls. But apart from the exponent being too low, the physical relationship between this model and gravity-driven mass movements seems to be rather weak [Hergarten, 2002, 2003; Dussauge et al., 2003; Malamud et al., 2004]. In their laboratory experiments originally focusing on regolith landslides, Katz and Aharonov [2006] found block sliding with a rather low scaling exponent α = 1.13 under certain conditions. Although their results suffer from a small range of scales and limited statistics, this seems to be the greatest progress towards understanding the power-law distribution of rockfalls so far. Beyond this, there seems to be no consistent model to reproduce the power-law distribution of rockfalls (and rockslides) quantitatively so far.

2. A Simple Model for Rock Detachment

[6] The model introduced in the following is to some extent inspired by the basic models of self-organized criticality [Bak et al., 1987; Olami et al., 1992] where avalanches propagate on a lattice if local thresholds are exceeded. It is assumed here that slope stability at any location depends on the slope gradient, and all other contributions to rock instability in nature such as fracturing are mimicked by random impacts. Slope gradient is computed using the D8 (deterministic eight-node) algorithm [O'Callaghan and Mark, 1984] where the slope of a site is determined by the steepest descent among its eight (direct and diagonal) neighbors on a rectangular lattice. This algorithm is widely used in hydrological applications. It is further assumed that slopes below a lower threshold slope smin remain stable under all conditions, while slopes above an upper threshold slope smax are destabilized by any impact. For slopes s between smin and smax a linear increase of the probability of instability in case of an impact is assumed:

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If a site becomes unstable, material is removed until its slope decreases to smin. The downslope motion of unstable rock masses and their deposition is not computed, only the volume of detached material is recorded. The effect of the event on its vicinity, i.e., progressive destabilization in the source area of the rockfall, is mimicked by exposing the eight neighbored sites to the same random impact as the unstable site, so that each of them may become unstable with a probability given by equation (2), too. Those sites which received in impact without becoming unstable are assumed to be stable at their present slope and cannot be destabilized by further impacts unless their slope increases as a consequence of further removal of material at neighbored sites. This is realized by replacing smin of these sites by the present value of s.

[7] This model is applied to the data of the recently released version 2 of the ASTER Global Digital Elevation Model (a product of METI and NASA) with a resolution of 1 arc second (about 20–30 m in the regions considered in the next section). Although improved compared to the previous version, the elevation data still contain a considerable amount of errors, and the model is obviously very sensitive to such errors. Points or areas of erroneously low elevations may result in large events ending with a crater-shaped topography. In order to avoid such artifacts, all local depressions in the DEM were first filled to form some kind of lakes as it is usually done in hydrological applications. However, there may still be artificially oversteepened slopes around such areas. Therefore, instability was artificially prohibited in a region around each filled-up site. Choosing the radius of this region as 5 times the amount by which the site has been lifted turned out to be sufficient to get around the majority of the DEM errors.

3. Results and Discussion

3.1. The Event-Size Distribution

[8] Figure 1 shows the probability density obtained for the Alps (for simplicity the rectangle from 43°N to 48°N and 5°E to 16°E) and different values of smin and smax. Statistics are based on 106 events for each parameter set. As the model does not include long-term driving processes, e.g., fluvial incision of valleys, that steepen the relief, slope gradients will decrease through time in the mean. As a consequence, the potential for large events would dramatically decrease during such a long simulation, causing a bias in the statistics. This is avoided by restoring the original surface after each event, so that each of the 106 events starts from the original topography.

Figure 1.

Probability density of the rockfalls in the Alps predicted by the model for different parameter values. The straight line corresponds to a power-law distribution with an exponent α = 1.35.

[9] In addition to the automatic treatment of DEM errors described above, the largest events were checked visually, and those which seemed to be related to DEM errors were removed. It was found that the fraction of potentially erroneous events strongly decreases with decreasing event size, and that they become statistically negligible for V < 0.1 km3.

[10] As shown in Figure 1, the probability density roughly follows a power law (equation (1)) for all considered parameter values. The coincidence with the power law is much better for smax = 5 than for smax = 10 where the propagation of instability is obviously inhibited according to equation (2). The impact of smin on the power-law distribution is very weak. The finding that most rockfalls take place at s ≥ 1 [Bennett et al., 2011] suggests to consider the case smin = 1 (corresponding to 45° slope angle) and smax = 5 in the following.

[11] Visual correlation leads to an exponent α = 1.35. This value is perfectly in the middle of the range found in rockfall inventories in different regions on earth. Using different methods of analysis, exponents α = 1.07 [Malamud et al., 2004], α = 1.1 [Guzzetti et al., 2003], α = 1.19–1.23 [Noever, 1993], α = 1.41–1.51 [Dussauge et al., 2003] and α = 1.75 [Bennett et al., 2011] were obtained. As a variation of more than 0.4 in α was found by applying different methods to the same data set [Dussauge et al., 2003; Malamud et al., 2004], even the entire variation in α may be a spurious effect of limited statistics.

[12] At this point it should be emphasized that the modeled distributions concern the detached volumes, while real rockfall inventories refer to the deposited volumes which are in general larger due to dilatancy. However, it is easily recognized that the effect of the difference on the scaling exponent α is almost negligible.

[13] In the context of risk assessment, the breakdown of the power-law distribution at large event sizes may be even more important than the power-law exponent itself [Hergarten, 2004]. As shown in Figure 1, the model suggests that rock failures with V > 0.1 km3 occur less frequently than predicted by the power law in the Alps. The two largest events predicted by the model involve about 0.5 km3, which in return implies that the present relief of the Alps does not carry the potential for huge rockfalls of several cubic kilometers such as the Flims rockslide.

[14] The two largest events are illustrated in Figure 2. Both are located in the Swiss Alps, one in the Lauterbrunnen valley and the other above the Klöntal lake. A visualization of the Lauterbrunnen event is provided as auxiliary material in order to illustrate how the model works. Although similar in size, these two events obviously differ in their topographic characteristics. While the Lauterbrunnen event is related to the extremely steep walls of a glacial valley, the Klöntal event corresponds to the breakdown of a complete mountain top. The slopes at the Klöntal event are in a range where sliding or avalanching shall be dominant, while parts of the walls in the Lauterbrunnen valley are steep enough to allow a significant amount of falling, too, although, this contribution would probably not be recognized in the deposits.

Figure 2.

The two largest events predicted for the Alps (V ≈ 0.5 km3, red). The black lines correspond to smaller events predicted for a 2000 year time span (see section 3.2). Top: Lauterbrunnen valley, view from south. Bottom: Klöntal lake, view from north.

[15] For comparison, the same simulation was performed for the central part of the Himalayan region (26–31°N, 82–92°E) and for the southern part of the Rocky Mountains (35–45°N and from 105°W to the West Coast). Although the topographic characteristics of the three mountain belts strongly differ, the power-law distributions of the rockfall sizes predicted for the three orogens coincide almost perfectly (Figure 3). The only noticeable difference concerns the cutoff at large event sizes. For the Himalayas, the distribution follows a power law up to V ≈ 0.5 km3, and the largest events involve a volume V ≈ 4.5 km3, which is ten times larger than in the Alps. In contrast, all rockfall volumes predicted for the southern part of the Rocky Mountains are smaller than 0.25 km3. Furthermore, almost all events with V > 0.02 km3 are located in the Yosemite and Grand Canyon regions, while large events are widely distributed in the Alps and Himalayas. These results suggest that a power-law distribution with α = 1.35 is universal for rockfalls, while regional differences are reflected in the cutoff behavior at the largest event sizes.

Figure 3.

Probability density of the rockfalls in the Himalayas and the southern Rocky Mountains predicted by the model compared to the Alps. The straight line corresponds to a power-law distribution with an exponent α = 1.35.

3.2. Hazard Assessment

[16] The model introduced in this paper does not involve any absolute timescale and thus does not allow to specify the frequency of rockfalls in a given region. However, it is in principle possible to assign a rough absolute timescale to the model from existing statistical data. The largest events that occurred in the Alps during the last decades took place at Morignone (Val Pola, Italy, 1987, V ≈ 0.04 km3) and Randa (Matter valley, Switzerland, 1991, V ≈ 0.03 km3). This may lead to a crude estimate that one event with V > 0.03 km3 occurs per 10 years in the mean, although this is rather an order of magnitude than an exact value. Transferred to the simulation of 106 events, this results in a probability of the Lauterbrunnen event of one per 500 years, and one per 350 years for the Klöntal event. If the power-law distribution with α = 1.35 held up to events of several cubic kilometers, one event of the size of the Flims rockslide or even larger (V ≥ 8 km3) should occur in Alps per 70 years in the mean, which is obviously not the case.

[17] Figure 4 gives a map of the rockfalls with V ≥ 10−3 km3 predicted for a 2000 year time span in the Alps according to the timescale estimated above. In contrast to the large statistics considered in the previous section, the map is based on a simulation of 1078 subsequent events without restoring the topography after each event. This map can be viewed as a first step towards a large-scale hazard map. It shows a rather inhomogeneous spatial distribution of large rockfalls. The potential for large rockfalls seems to be particularly high along the northern margin of the Alps and in the southern part at latitudes between about 11°E and 12.5°E.

Figure 4.

Rockfalls with V ≥ 10−3 km3 predicted for a 2000 year time span in the Alps. Black: V ∈ [0.001,0.01] km3 (756 events), blue: V ∈ [0.001,0.01] km3 (301 events), red: V ≥ 0.1 km3 (21 events).

[18] However, quantitative assessments based on this model must be treated with some caution. First, the time scale given above is very rough or even just an order of magnitude, and this uncertainty immediately affects the probabilities per time. And second, variations in the parameters smin and smax have a stronger influence on the largest events than on the power-law distribution itself (Figure 1). If, e.g., the thresholds at both locations considered above were 20 % higher than elsewhere (smin = 1.2 and smax = 6), both events would involve volumes of “only” 0.3 km3 and be predicted at probabilities of one per 1100 years (Lauterbrunnen) and one per 700 years (Klöntal). So the model provides a rather simple and efficient tool for a first step of hazard assessment, but for a serious assessment on a regional or even local basis, estimates of the model parameters going beyond the first guess smin = 1 and smax = 5 are required. The question how to combine knowledge on rock type, tectonics and climate to obtain such an estimate is, however, open.

3.3. Relationship to Self-Organized Criticality

[19] The occurrence of power-law distributions with an apparently universal exponent in different regions strongly supports the idea that rockfall dynamics are governed by SOC, as it may be the case for earthquakes, wildfires, and regolith landslides. Long-term driving forces such as fluvial or glacial erosion locally steepen the relief and thus supply the potential for rockfalls (and for other types of mass movements). The two competing processes may approach some kind of dynamic equilibrium which exhibits critical properties in some cases, reflected by a power-law distribution.

[20] However, the approach presented here strongly differs for the basic models of SOC [Bak et al., 1987; Drossel and Schwabl, 1992; Olami et al., 1992] and from the landslide models mentioned above. All these models are composed by two components, long-term driving (e.g., growth of trees, tectonic forces, fluvial incision, weakening of soil) and rapid relaxation (“avalanching”) if a threshold is exceeded. The rockfall model only consists of threshold behavior and avalanching (i.e., progressive detachment), while long-term driving is not included. Instead, the model inherits the topography of the considered region and directly proves that this topography is critical (or close to critical) with respect to the considered progressive slope failure mechanism.

[21] From this point of view, the model describes only half of a SOC system. As a consequence, the model cannot specify the efficiency of different long-term driving forces (e.g. fluvial or glacial erosion) in making the relief critical. On the other hand, it reveals a difference in criticality between the considered regions. All show a cutoff in the distribution which is much below the system size, so that they must be slightly subcritical. The Himalayas seem to be closer to a critical state than the Alps and the Rocky Mountains. One may speculate that both the Alps and the Himalayas may have been critical at the end of the last glacial period. In return, fluvial erosion being responsible for landform evolution of the southern Rocky Mountains may be too weak to bring the surface very closely to its critical state.

4. Conclusion

[22] Application of a simple, topography-based model to three mountain belts suggests that rockfall volumes (considered together with rockslides) follow a power-law distribution (equation (1)) with a universal scaling exponent α = 1.35, in very good agreement with available statistical data. Despite the obvious importance of fracturing on rock destabilization, this result suggests that the size distribution of rock failure events is not inherited from a pre-defined pattern of fragmentation, but arises from a progressive failure process.

[23] In contrast to the scaling exponent α, the cutoff of the distribution at large event sizes varies from region to region and depends on the model parameters. Therefore, geology and climate must be considered for local or regional hazard and risk assessment, but the dependence of the model parameters on rock properties, tectonics and climate is yet unknown.

[24] The results strongly support the idea that rockfalls are governed by SOC. However, the considered mountain belts seem to be slightly subcritical. Quantifying the degree of criticality depending on the driving forces (glacial and fluvial erosion) may be subject of subsequent studies.

Acknowledgments

[25] This work was funded by the Austrian Science Fund (FWF): P19733-N10 and EUROCORES TopoEurope I152.

[26] The Editor thanks one anonymous reviewer for his/her assistance in evaluating this paper.

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