D24 Preliminary models; calibrated magnitude-frequency distributions: landslides & triggers – UCAM-DES

We have obtained digital landslide maps for selected areas in New Zealand, Papua New, Taiwan, and Italy. Each map contains the outlines of thousands of landslides that occurred in a known time interval, due to known trigger populations or events. Although these data sets have different resolutions and underlying methodologies some basic comparisons can be made. For example, we have fit magnitude-frequency distributions of landslides in all study areas with a double Pareto, or power law model and found that the best fit parameters are shared between areas. Specifically, we have found that above a cutoff size, related to the resolution of the mapping, the size-frequency scaling exponent is approximately -1.5, implying that the total surface area disturbed by landslides is dominated by small events, and that the total volume of rock mass displaced by landslides may depend in equal measures on the occurrence of small and large events.

Examples of landslide size distributions, from the western Southern Alps, of New Zealand, plotted as a probability density function p(x) plotted in log [p(x)] versus log(x) form. Solid squares show the probability density of landslides in the Whataroa catchment, mapped at 1:25,000, N = 3986; open circles show the probability-density of landslides in a larger part of the western Southern Alps, mapped at 1:50,000, N = 5086. The data sets show similar scaling of landslide magnitude and frequency. Above a cut-off size, related to the resolution of the mapping and/or a break in the failure mechanism, the data scale as a power-law. This portion of the data is the tail end of the distribution and represents about a quarter of the observed landslides. a is the slope of the best fit power-law, and values are almost identical at a = 1.45 for both data sets.

The following is a brief review of the state of the art :

The magnitude-frequency distribution of landslides is characterized by a maximum at small to intermediate size events (10³ m²) and a broad, negative power law tail for larger landslides. This power law scaling holds true whether the landslide size is defined as the scar area, or the total area disturbed, and whether landslides are triggered over a long period of time, or almost instantaneously; it also holds true if landslide volume is considered instead of area, although volume is typically much more difficult to measure (both in the field and in air photographs). For an idealized landslide size distribution to be power-law distributed across the size range , the size probability density is defined as

, c>0, a>0

where a is the power law scaling exponent, and x is usually defined as planform area. The scaling exponent explicitly determines the impact of large versus small landslides on integrated measures such as the total area disturbed, or the volume of material yielded. Power law scaling is typically observed for areas greater than 1000-5000 m² up to the largest landslide areas for which a distribution can be reliably estimated (of the order of 10⁵ m²).

The power law property of the landslide size distribution introduces several complications. First, the disturbance area and eroded volume of a landslide are highly variable. Second, there is no characteristic landslide scale that dominates the erosion budget: a power-law distribution indicates that events at many scales play an important role. This makes it hard to quantify the pattern and rate at which a mountain landscape evolves by landsliding. At present there exists no mathematical means of assessing the flux of sediment from a zone dominated by landslide mass-wasting. In other words, no differential operator or partial differential equation (analogous to the diffusion equation) yet exists to formalize the relationship between mountain relief and landslide sediment flux (quite apart from the difficulty in calibrating such an equation were it to exist).

Power laws pose further technical problems that have impeded conceptual progress in several respects. It is well known in the statistics community that heavy-tailed distributions are difficult to characterize reliably. The steepness of a negative power-law tail, which represents the relative frequency of small versus large events, cannot be estimated with confidence unless the sample size (the number of landslides) is very large. If the underlying distribution is only asymptotically a power law, as is probably the case with landslides, then the frequency of small to medium events can strongly distort any estimate of the power-law scaling. The practical consequence of erroneous inference is a faulty emphasis on either small or large events. In several recent studies, the steepness of the power-law scaling was underestimated, largely as a result of unsophisticated statistical analysis. This has resulted in the inference that large landslides dominate the erosion budget, since integration of the power-law magnitude-frequency distribution indicated a strong dependence on the largest events. Recent work has shown that the power-law distribution of landslide size-frequency is steep, and reasonably consistently so for a variety of data sets (with some exceptions). The scaling exponent a expresses this steepness and generally varies between 1.3 and 1.5.

In light of these recent studies it is clear that the area disturbed by landslides, over the long term, is dominated by small to medium scale failures (up to an area of around 10³-10⁴ m²). Most of the landslide data sets assembled over the years are unreliable in their representation of the magnitude-frequency distribution of small to medium scale failures. Only those data sets acquired with great care, high quality air photography, and detailed field verification can be regarded as having counted accurately the smaller landslides. For these very rare data sets, there is convincing evidence for a rollover, or break in scaling, typically at around 1000-5000 m². The mean and most common (modal) size landslides are approximately of this scale, but the strong asymmetry of the landslide size-frequency distribution means that these averages are not equal.

Some major challenges remain. First, most data sets undercount the smaller failures and misrepresent the frequency of the dominant events. The rollover in the magnitude-frequency distribution in these cases is unreliable, and the estimate of the power-law component of the distribution is distorted. Fortunately, this distortion is quantifiable, and a more reliable value of a can be elicited if a censoring model is applied. However, no reliable estimate can be made of the area disturbed for such data sets. Given the importance of such estimates, for example in the evaluation of soil loss and mobilization of particulate organic matter, there is a clear need for high fidelity, regional landslide maps. Second, the volume eroded by landsliding also remains difficult to quantify, particularly where the power-law scaling is steeper than previously thought, since the smaller, poorly enumerated failures are now seen to play a stronger role. This is due, in part, to the fact that the scaling relationship between landslide thickness and landslide planform area remains unclear. This scaling is important because it sets the transformation between the area-frequency and the volume-frequency distributions. For strictly soil/regolith failures, it could be argued that the depth of landslide failure is approximately constant. For failures that involve bedrock, however, it has been argued that the depth of failure likely correlates with landslide length scale, giving a volume to area relation of V ~ A^3/2. At present, neither model has been vindicated with field data, but must be so before any reliable estimate of total landslide sediment flux can be made. If the constant thickness model applies, then the volume eroded by landslides is set by the frequency of the average area landslide and is weakly dependent on the power-law scaling. In contrast, if the scaling thickness model applies, then the total erosion volume is a more equal function of small and large landslides, with a weighting that is a sensitive function of the power-law scaling exponent a.

In a related effort we have looked at the hillslope position of earthquake and rainfall triggered landslides in the Chenyoulang catchment draining the epicentral area of the ChiChi earthquake in Taiwan. Specifically, we have documented landslides triggered by a typhoon in 1996 (Herb), an earthquake in 1999, and a typhoon in 2001 (Toraji). The 1996 data set is thought to capture ‘pure’ storm-driven landsliding, the 1999 data set captures ‘pure’ earthquake-driven landsliding, and the 2001 data set captures a combination of both. The detailed location of landslides was mapped from 20 m resolution SPOT images, with air photos where available, within the Chenyoulang river basin. To quantify the location of landslides relative to the river channel network, we calculated the drainage area, A, of the most downslope point reached by each landslide. The drainage area is used here as an index to distinguish between hillslopes and channels to determine whether landslides delivered sediment to channels. A slope-area plot produced from a 40 m digital elevation model (DEM) indicates that, in the epicentral Chenyoulang catchment, channelization occurs when A > 1 km². Landslides triggered by typhoon Herb had a relatively high probability of delivering sediment to the channel network (they reached a drainage area greater than 1 km² with probability 0.24). In contrast, landslides triggered by the Chi-Chi earthquake originated closer to hillcrests, owing perhaps to the topographic amplification of seismic waves, and had a much lower probability of delivering material to the channel network (0.08). Typhoon Toraji, which occurred after Chi-Chi earthquake, triggered landslides that reached intermediate locations, with 0.13 probability of entering a channel. The marked differences in proximity of landslides to channels during and after the earthquake support the view that substrate material was preconditioned to fail by coseismic damage. The typhoons that followed the Chi-Chi earthquake probably triggered landslides in this weakened material, in addition to remobilizing other coseismically produced debris.

Probability density of the upslope area of the lowermost point reached by landslides triggered by Herb typhoon (blue), Toraji typhoon (green), and Chi-Chi earthquake (red). Upslope area was calculated in ARC/INFO using the D8 flow routing algorithm with a 40 m DEM. The value reported is the maximum found within the perimeter of each landslide scar and its associated deposit.