Power Law Distribution and its inferences in understanding Landslides Empirical Dataset

Landslides count among the most exacting of natural hazards given its potential to cause loss to human life as well as socio-economic disruption. Due to burst in the human population, anthropological activities are bound to move to landslide-prone areas. Understanding when and where this natural hazards will occur next is, therefore, an outstanding challenge in the earth sciences. Either the problem is to find out what triggers the landslides or to observe the laws of chance. Among various statistical distributions which are used to predict the landslides, the power-law distribution has attracted attention for its remarkable properties, which lead to surprising physical consequences, and for its appearance in a landslide data across the globe.

Uttarakhand (dataset 1) and Himachal Pradesh (dataset 2) states of India are taken as case studies as landslide incidences are frequent in the region. The topography of the area and its surroundings is rugged and mountainous with several streams and snow-covered peaks. Maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov-Smirnov statistic methods are used to estimate power-law parameters such as power-law exponent (α), lower cut off (x_min) and goodness of fit (p-value). According to Caluset et al. (2009), if the resulting p-value is more than 0.1, the power law is a reasonable premise for the data. Otherwise, it is rejected. The p-value calculated for dataset 1 and 2 is 0.1 and 0.19. Hence, power-law distribution is a plausible hypothesis for the data. The value of α and x_min are 2.13 and 7191.5 m² for dataset 1, and 2.49 and 923 m² for dataset 2.

There are two inferences from the result. First, the dataset is exhibiting scale invariance in the range [7191.5 , 347184] and [923, 12203] for dataset1 and 2 respectively while data consists of landslides triggered by different forces such as rainfall, earthquake and anthropogenic intervention. Scale-free behaviour of the dataset in the mentioned range is suggesting a common mechanism of failure at different scales, which is independent of the driving force. Second, distribution has a fat tail, infinite variance and unstable means, which suggests that extreme events are likely to occur. Absent stable mean and finite variance implies that the probabilistic assessment of landslides occurrence is much difficult.