Breaks in Fractal Scaling of Real and Synthetic Earthquake Catalogues

Earthquake generation within the crust is the result of a series of complicated spacio-temporal interactions between different tectonic blocks and units. The end product of the process is a function of both long term deterministic-chaotic processes in a regional scale and short-term Self-Organized Critical (SOC) processes of a local nature [e.g. McCloskey and Bean,1994; ]. In the past three to four decades many models of seismicity have been developed [e.g. Burridge and Knopoff, 1967; Huang and Turcotte, 1990; McCloskey, 1993; Ben-Zion, 1996] trying to model the observed patterns of earthquake generation and seismicity. Some of these studies have shown that it is possible to reproduce the main features of the real earthquake populations. In this study the fractal dimension of SCSN, JMA and ISC seismicity catalogues have been studied. the aim was to see whether all the different sizes of earthquakes within a catalogue (i.e. a single spacio-temporal window) belong to the same population and whether any breaks in fractal scaling exists within the catalogue concerned. Furthermore, a selection of synthetic earthquake models were analyzed with the same approach to determine whether they are able to reproduce the same results as empirical ones. Subsequent analysis of the data have revealed several distinct breaks in fractal scaling of earthquakes of different magnitudes. In other words, it emerged that small and large earthquakes in each catalogue are obeying different fractal dimensions hence belonging to different earthquake populations. It is possible to associate one of the breaks, observed in the SCSN catalogue to the average thickness of the seismogenic crust of California ( ∼ 15 km as calculated by Nazareth and Hauksson, 2004). With the same technique used for the empirical catalogues, three different synthetic catalogues [McCloskey, 1993; Ben-Zion, 1996; Khademi and McCloskey, this study ] were analyzed. Results have shown that all the models are able to predict the fractal dimension of the empirical catalogues to some degree and further, two latter models are able to simulate the breaks in the fractal scaling. The fractal dimensions from models F, U, M and A of Ben-Zion [1996] are generally in good agreement with observed dimensions of empirical catalogues, though the observed breaks in the empirical catalogues cannot be seen in these synthetic models. However, a gradual decrease in the fractal dimension with increasing treshold magnitude can be observed. The McCloskey [1993] Chaotic-SOC hierarchical model, with reasonable accuracy, predicts both the fractal dimensions and dimension breaks, observed in the empirical catalogues. The model's success in the prediction of behaviour of empirical data is particularly due to the combination of low-dimensional chaotic behaviour of bigger blocks (i.e. larger events) and high-dimensional SOC behaviour of smaller blocks (i.e. small events resulting from activity of smaller portions of the main fault or adjacent minor discontinuities). And finally the Khademi-McCloskey (KMC) model is able to reproduce both the dimension and one of the breaks of scaling. But, the model is unable to produce more than one break in scaling (i.e. to distinguish more than two earthquake populations within the same dataset). It is concluded that hierarchical earthquake models (though with some modifications) can be used to extend the temporally limited empirical catalogues to much longer time spans and to overcome the temporal limitations of the existing empirical catalogues.