Fractal distributions of stress and strength and variations of b-value

Abstract

Fractal statistics with a fractal dimension near 2.2 are a good approximation for many geological processes. Seismically active fault zones are complex systems with distributed heterogeneities in both stress and strength. The Gutenberg-Richter frequency-magnitude relation for earthquakes obeys fractal statistics between the number of events and the characteristic dimension of the rupture zone. We simulate a 2-D planar fault zone on which the difference between stress and strength follows a fractal distribution to investigate the variation of b-value under different distributions of heterogeneities and ambient stress levels. We hypothesize that earthquakes occur in regions where this difference exceeds a specified value and that the size of each region is a measure of the magnitude of the associated earthquake. Cumulative frequency-magnitude statistics derived from the simulation show a systematic variation in b-value. The b-value has a positive correlation with the fractal dimension of the distribution and is inversely related to the ambient stress level. Realistic b-values ranging from 0.77 to 1.11 are derived by choosing fractal dimensions between 2.2 and 2.4, which is in good agreement with the fractal dimension estimated for the earth's topography. A comparison between the observational data and the simulation suggests that the observed b-value variation before and during earthquake sequences result not only from changes in ambient stress level but also from changes in the fractal dimension of the stress-strength distribution.