Fractal Dimensions of Small, Intermediate, and Large Earthquakes

Abstract

The spatial fractal dimension D of earthquakes (or faults) is often correlated with the slope b of the Gutenberg-Richter law, independently of earthquake size. An already classical formula is Aki9s D = 3 b / c = 2 b . This formula implies the three following hypothesis: (1) the Gutenberg-Richter law log 10 N = a - bM is satisfied; (2) the seismic moment M 0 is related to the surface magnitude M s as log 10 M 0 = cM s + d with a typical value of c = 1.5; and (3) the static self-similarity scaling law is satisfied, that is, M 0 ∝ L 3 , 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., M 0 ∝ L 3 ). Nevertheless, for large events, this hypothesis is not satisfied because the shape of large events is a rectangle and not a square (i.e., M 0 ∝ L 2 ). Therefore, for large events the formula D = 3 b / c should not be used; the formula D = 2 b / c should be used instead. In hypothesis (2), c depends upon event sizes: c = 1, 1.5, and 2 for small, intermediate, and large events, respectively, therefore resulting in D = 3 b, D = 2 b , and D = b , respectively. As a consequence, small earthquakes (or small faults) are distributed within volumes, whereas large earthquakes (or large faults) are distributed along lines.