Renormalization-Group Transformations and Correlations of Seismicity

The effect of transformations analogous to those of the real-space renormalization group are analyzed for the temporal occurrence of earthquakes. A recently reported scaling law for the distribution of recurrence times implies that these distributions must be invariant under such transformations, for which the role of the correlations between the magnitudes and the recurrence times are fundamental. This approach puts the study of the temporal structure of seismicity in the context of critical phenomena.

Figure 1

Analysis of worldwide earthquakes from 1973 to 2002. Main: recurrence-time probability densities conditioned to $M_{\mathrm{pre}}\ge M_c^{\prime }$, $M_c^{\prime }$ varying from 6 to 7 and with $M\ge 6$. The data collapse indicates that $D_{\uparrow }(\tau )$ may verify the same scaling relation as $D(\tau )$. The solid line is a fit to $D(\tau )$, turning out to be $\propto e^{-\tau /1.4}/{\tau }^{0.29}$. Inset: inverse of the mean recurrence time, $R_{\uparrow }$, scaled by $R$ for recurrence periods started by events with $M_{\mathrm{pre}}\ge M_c^{\prime }$ and ending with $M\ge 5$. The error bars mark 2 standard deviations of the mean value, and the two curves are the linear and exponential fits explained in the text, which do not take into account the last two points.