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 = 3 b / 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 D 0 and may be compared with the correlation dimension D 2 , obtained from the spatial distribution of earthquakes. By analyzing the actual earthquake catalogue, we calculated the fractal dimension D 2 and the b value. Our result does not support Aki's speculation that D 0 = 3 b / c , but shows, on the contrary, that there is a negative correlation ( D = 2.3 − 0.73 b ) between the b value and the fractal dimension of the spatial distribution of earthquakes in the Tohoku region.