Short Notes Fractal Dimensions of Small, Intermediate, and Large Earthquakes

Abstract The spatial fractal dimension D of earthquakes (or faults) is often cor-related with the slope b of the Gutenberg–Richter law, independently of earthquake size. An already classical formula is Aki’s D 3b/c 2b. This formula implies the three following hypothesis: (1) the Gutenberg–Richter law log10N a bM is satisfied; (2) the seismic moment M0 is related to the surface magnitude Ms as log10 M0 cMs d with a typical value of c 1.5; and (3) the static self-similarity scaling law is satisfied, that is, M0 L3, where L is the characteristic dimension of the fault. Hypothesis (3) implies that events are small or intermediate and break on a square plane (i.e., M0 L3). Nevertheless, for large events, this hypothesis is not satisfied because the shape of large events is a rectangle and not a square (i.e., M0 L2). Therefore, for large events the formula D 3b/c should not be used; the formula D 2b/c should be used instead. In hypothesis (2), c depends upon event sizes: c 1, 1.5, and 2 for small, inter-mediate, and large events, respectively, therefore resulting in D 3b, D 2b, and D b, respectively. As a consequence, small earthquakes (or small faults) are dis-tributed within volumes, whereas large earthquakes (or large faults) are distributed along lines