A fractal model of earthquake occurrence: Theory, simulations and comparisons with the aftershock data

Abstract

Abstract. Our understanding of earthquakes is based on the theory of plate tectonics. Earthquake dynamics is the study of the interactions of plates (solid disjoint parts of the lithosphere) which produce seismic activity. Over the last about fifty years many models have come up which try to simulate seismic activity by mimicking plate plate interactions. The validity of a given model is subject to the compliance of the synthetic seismic activity it produces to the well known empirical laws which describe the statistical features of observed seismic activity. Here we present a review of one such, purely geometric, model of earthquake dynamics, namely The Two Fractal Overlap Model. The model tries to emulate the stick-slip dynamics of lithospheric plates with fractal surfaces by evaluating the time-evolution of overlap lengths of two identical Cantor sets sliding over each other. As we show later in the text, some statistical aspects of natural seismicity are naturally captured by this simple model. More importantly, however, this model also reveals a new statistical feature of aftershock sequences which we have verified to be present in nature as well. We show that, both in the model as well as in nature, the cumulative integral of aftershock magnitudes over time is a remarkable straight line with a characteristic slope. This slope is closely related to the fractal geometry of the fault surface that produces most of thee aftershocks. We also go on to discuss the implications that this feature may have in possible predictions of aftershock magnitudes or times of occurrence. IntroductionMany models of natural seismicity have been proposed over the last half-decade and more. These cover a very broad range of approaches ranging from purely physical to purely statistical. The relative successes of these very diverse models is testament to the great complexity involved in the production of natural seismicity and points to the fact that the correct model is probably neither extreme and is some elusive combination of these approaches. However, an approach that has been less pursued to some extent is the investigation of the role of the fractal surface geometry of a fracture surface topography (or self affine for most natural surfaces) in producing the known statistical properties of natural seismicity. The model we propose in this article incorporates this fractal topography of the fracture surface in a simplistic way.

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