Fractal analysis of three‐dimensional spatial distributions of earthquakes with a percolation interpretation

Although many studies have shown that faults and fractures are self‐similar over a large range of scales, none have tested the fault structure for self‐similarity in three dimensions. In this study, earthquake hypocentral locations in central and southern California were used to illuminate three‐dimensional (3‐D) fault structures, for which we measured the fractal capacity dimension, D0(3‐D). Hypocentral distributions from the Joshua Tree, Big Bear, and Upland aftershock sequences, as well as background seismicity at Parkfield were found to be fractal, where D0(3‐D) increased with increasing event density, asymptotically approaching a stable value. The Joshua Tree data set stabilized at D0(3‐D) = 1.92 ± 0.02, the Parkfield data set asymptotically approached D0(3‐D) = 1.82, and the Big Bear data set approached D0(3‐D) = 2.01. As a test of the effects of location errors upon the measured value of D0(3‐D), the Upland aftershock data were located with both the southern California Hadley and Kanamori (1977) (H‐K) velocity model, and the more accurate Hauksson and Jones (1991) (H‐J) velocity model. Events located with the H‐K model asymptotically approached D0(3‐D) = 2.07, and events located with the H‐J model approached D0(3‐D) = 1.79, suggesting that improved hypocentral locations may decrease the measured fractal dimension. One interpretation of our results of D0(3‐D) ≤ 2.0 for all of the hypocentral data is that earthquakes only occur on the “percolation backbone” of a fault network, i.e., the active part of the network that accommodates finite strain deformation (Sahimi et al., 1993). We show that a percolation model that allows for healing of previously broken bonds is consistent with this interpretation.