An inverse cascade explanation for the power-law frequency-area statistics of earthquakes, landslides and wildfires

Frequency-magnitude statistics for natural hazards can greatly help in probabilistic hazard assessments. An example is the case of earthquakes, where the generality of a power-law (fractal) frequency-rupture area correlation is a major feature in seismic risk mapping. Other examples of this power-law frequency-size behaviour are landslides and wildfires. In previous studies, authors have made the potential association of the hazard statistics with a simple cellular-automata model that also has robust power-law statistics: earthquakes with slider-block models, landslides with sandpile models, and wildfires with forest-fire models. A potential explanation for the robust power-law behaviour of both the models and natural hazards can be made in terms of an inverse-cascade of metastable regions. A metastable region is the region over which an 'avalanche' spreads once triggered. Clusters grow primarily by coalescence. Growth dominates over losses except for the very largest clusters. The cascade of cluster growth is self-similar and the frequency of cluster areas exhibits power-law scaling. We show how the power-law exponent of the frequency-area distribution of clusters is related to the fractal dimension of cluster shapes. The f requencysize statistics of a number of natural hazards appear to satisfy power-law (fractal) scaling to a good approximation under a wide variety of conditions (for a review, see Malamud 2004). These include earthquakes (Aki 1981; Turcotte & Malamud 2002), volcanic eruptions (Pyle 2000), wildfires (Malamud et al. 1998, 2005; Ricotta et al. 1999), landslides (Guzzetti et al. 2002; Malamud et al. 2004), asteroid impacts (Chapman & Morrison 1994; Chapman 2004) and potentially floods (Turcotte & Greene 1993; Turcotte 1994; Malamud et al. 1996; Malamud & Turcotte 2006). In this paper, we will consider the f requency-area statistics of three of these natural hazards: earthquakes, landslides, and wildfires. In each case the 'noncumulative' number of events N with area A satisfies the power-law relation