Fractal Analysis of Three-Dimensional Spatial Distributions of Earthquakes with a Percolation Interpretation

Michelle Robertson Haver, Charles G. Sammis, Muhammad Sahimi, & Aaron Martin

Published 1995, SCEC Contribution #117

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, D 0(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 D 0(3-D) increased with increasing event density, asymptotically approaching a stable value. The Joshua Tree data set stabilized at D 0(3-D) = 1.92 ± 0.02, the Parkfield data set asymptotically approached D 0(3-D) = 1.82, and the Big Bear data set approached D 0(3-D) = 2.01. As a test of the effects of location errors upon the measured value of D 0(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 D 0(3-D) = 2.07, and events located with the H-J model approached D 0(3-D) = 1.79, suggesting that improved hypocentral locations may decrease the measured fractal dimension. One interpretation of our results of D 0(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.

Robertson Haver, M., Sammis, C. G., Sahimi, M., & Martin, A. (1995). Fractal Analysis of Three-Dimensional Spatial Distributions of Earthquakes with a Percolation Interpretation. Journal of Geophysical Research, 100(B1), 609-620.