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 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 log10M0 = cMs + d with a typical value of c = 1.5; and (3) the static self-similarity scaling law is satisfied, that is, M0L3, 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., M0L3). Nevertheless, for large events, this hypothesis is not satisfied because the shape of large events is a rectangle and not a square (i.e., M0L2). 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, intermediate, and large events, respectively, therefore resulting in D = 3b, D = 2b, 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.

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