Multifractal measures of time series of earthquakes

The generalized fractal dimensions are measured for the time series based on two complete earthquake catalogues: one with M greater than or equal to 6 earthquakes occurring in the north-south seismic belt of mainland (China during the 1900-1990 period published by Ma er al. (1992) and the other with M greater than or equal to 5.5 earthquakes occurring in southern California, USA during the 1915-1994 period compiled by Press and Alien (1995). The log-log plot of C-q versus t, where C-q(t) is the generalized correlation integral and t is the interoccurrence time in years between two events, at positive q shows a linear distribution when t < t(c). D-q is the slope of this linear portion. The value of t(c) decreases from 50.1 to 39.8 years for Chinese earthquakes and fi om 50.1 to 31.6 years for southern California events as q is increased from 0 to 15. For M greater than or equal to 6 Chinese earthquakes, the well-distributed, monotonically decreasing function of D-q with increasing q would imply that such earthquakes have formed a multifractal time series. In contrast, the M greater than or equal to 5.5 southern California earthquakes might have not yet formed a complete multifractal time series or the number of these events is too small to accurately estimate the multifractal dimensions, especially for large qs. Different degrees of complexity of fault distributions in the two seismic regions might also be a factor in causing the difference in the D-q-q relations. In addition, the results also suggest that a D-q-q relation is better than the first three commonly-used values of D-q to completely represent a multifractal time series.