Crustal Deformation and Fractals, a Review

The frequency—magnitude statistics of earthquakes have long been known to satisfy the Gutenberg—Richter relation; it is easy to show that this relation is fractal with D=1.8–2. Since it is generally accepted that individual faults have characteristic earthquakes, it follows that the number—size statistics of faults are also fractal. Limited field studies indicate that the surface exposures have a two—dimensional fractal dimension of D ~ 1.6. (The seismic and fault fractal dimensions need not be equal since the repeat time of earthquakes can also have a power—law dependence on scale.) Fragments have a fractal number—size relation with D ~ 2.5 under a wide range of conditions. One model for tectonic deformation is comminution; such a scale—invariant model appears to be consistent with the fractal correlations discussed above. Seismicity has many of the characteristic features of “self—organized criticality”. Energy (strain) is continuously added to the crust and is lost in discrete events (earthquakes) that have a fractal frequency—size distribution. It appears reasonable to hypothesize that the continental crust is everywhere in this critical state (similar to the critical stress associated with perfect plasticity). Evidence for this comes from the distribution of intraplate earthquake and from the occurrence of induced seismicity almost anywhere an artificial reservoir is filled.

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