Power laws statistics of cliff failures, scaling and percolation

Large cliff failures have often been characterized by their size in terms of the horizontal eroded area at the cliff top, and the maximum local retreat of the coastline. Field studies suggest that the frequencies of these two quantities decrease as power laws of their respective magnitudes, defining two different decay exponents. Moreover, the horizontal area increases as a power law of the maximum local retreat, identifying a third exponent. This observation indicates that the geometry of cliff failures are similar for different magnitudes. It is shown that, under a 'scaling hypothesis', the exponents satisfy a precise mathematical relation that relates the exponent of the distributions of magnitudes of events and their geometry - a relation that could be useful to validate the analysis of field catalogs. In order to understand the observed scenario, a numerical model of rocky coast erosion is developed. Despite its minimal character, it reproduces the observed cliff failure statistics. Strikingly, the model precisely reproduces the geometric similarity of cliff failures observed in the field. These results suggest that the theory of percolation, which underlies the erosion model, can possibly be used as a guide to decipher the physics of rocky coast erosion and could provide precise predictions of the statistics of cliff collapses.