A correlation between the b value and the fractal dimension of earthquakes

Seismicity has fractal structures in space, time, and magnitude distributions, as expressed by the fractal dimension D, Omori's exponent p, and the b value, respectively. We expect that there is correlation among these scaling parameters. Aki (1981) speculated that there is a relation D = 3b/c (c = 1.5) between the b value and the fractal dimension D of fault planes. We point out that Aki's fractal dimension corresponds to the capacity dimension D0 and may be compared with the correlation dimension D2, obtained from the spatial distribution of earthquakes. By analyzing the actual earthquake catalogue, we calculated the fractal dimension D2 and the b value. Our result does not support Aki's speculation that D0 = 3b/c, but shows, on the contrary, that there is a negative correlation (D = 2.3 − 0.73b) between the b value and the fractal dimension of the spatial distribution of earthquakes in the Tohoku region.

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