A Method to Approximate and Statistically Model the Shape of Triggered Landslides

Abstract:

The planimetric shape of an individual landslide area is controlled by factors such as terrain morphology, material involved and speed, with landslide shapes varying in total area (A_L), type of shape, and their length-to-width (L/W) ratios. Here, we abstract landslide shapes to ellipses, and examine how the corresponding L/W ratios vary as a function of A_L in two substantially complete triggered landslide inventories: (i) 11,111 landslides triggered by the 1994 (M = 6.7) Northridge Earthquake, USA (ii) 9,594 landslides triggered by heavy rain during the 1998 Hurricane Mitch in Guatemala. For each landslide, an ellipse with equivalent area (A_L) and perimeter (P_L) of the original shape was created and a non-dimensional value of the ratio of the ellipse length-to-width (L/W) then calculated. Using Maximum Likelihood Estimation, the statistical distribution of landslide L/W ratio values were then considered for ten landslide area (A_L in m²) categories: 0–99, 100–199, 200–399, 400–799, 800–1599, 1600–3199, 3200–6399, 6400–12,799, 12,800–25,600, and ≥25,600 m². We find that for each of the landslide area categories considered separately, the probability density function p(L/W) as a function of (L/W) approximately follows a three-parameter inverse gamma distribution, which has a power-law decay for medium and large L/W values and exponential rollover for small L/W values. The ‘rollover’ value where p(L/W) is at its maximum, tends to increase with increasing A_L category, from approximately L/W = 1.7 for landslides in the smallest A_L category (0 < A_L < 99 m²), to L/W = 7.5 for landslides in the largest A_L category (A_L ≥25,600 m²). Broadly, this suggests that as A_L increases, L/W increases, i.e. as landslide areas increase, the probability of observing a more elongated shape increases. There is generally good agreement between the two inventories’ statistical distributions in spite of differences in location, triggering mechanism and geology. This work will aid in stochastic modelling of triggered landslide event inventories, where it may not be feasible to deterministically define each landside shape. Using these trends, landslide shape can be approximated as an ellipse, and the length to width ratio of that ellipse selected from a general statistical distribution.