Magnitude distribution of earthquakes: two fractal contact area distribution

Abstract

"Plate tectonics" is an observed fact and most models of earthquake incorporate it through the frictional dynamics (stick-slip) of two surfaces where one surface moves over the other. These models are more or less successful to reproduce the well known Gutenberg-Richter type power law in the (released) energy distribution of earthquakes. During a sticking period, the elastic energy gets stored at the contact area of the surfaces and is released when a slip occurs. Therefore, the extent of the contact area between two surfaces plays an important role in the earthquake dynamics and the power law in energy distribution might imply a similar law for the contact area distribution. Since fractured surfaces are fractals and tectonic plate-earth's crust interfaces can be considered to have fractal nature, we study here the contact area distribution between two fractal surfaces. We consider the overlap set (m) of two self-similar fractals, characterized by the same fractal dimensions (d_f), and look for their distribution P(m). We have studied numerically the specific cases of both regular and random Cantor sets (in the embedding dimension d = 1), gaskets and percolation fractals (in d = 2). We find that in all the cases the distributions show an universal finite size (L) scaling behavior P'(m') = L^α P(m, L ); m' = mL^-α, where a = 2(d_f - d). The P(m), and consequently the scaled distribution P'(m'), have got a power law decay with m (with decay exponent equal to d) for both regular and random Cantor sets and also for gaskets. For percolation clusters, P(m) (and hence P'(m')) have a Gaussian variation with m.