Long-Term Clustering, Scaling, and Universality in the Temporal Occurrence of Earthquakes

Analyzing diverse seismic catalogs, we have determined that the probability densities of the earthquake recurrence times for different spatial areas and magnitude ranges can be described by a unique universal distribution if the time is rescaled with the rate of seismic occurrence, which therefore fully governs seismicity. The shape of the distribution shows the existence of clustering beyond the duration of aftershock bursts, and scaling reveals the self-similarity of the clustering structure in the space-time-magnitude domain. This holds from worldwide to local scales, for quite different tectonic environments and for all the magnitude ranges considered.

Figure 1

Recurrence-time distributions without and with rescaling. (a) Probability densities from the NEIC-PDE worldwide catalog for several regions, $L$, and $M_c$. For $L\ge 45°$, all the regions with more than 500 events are shown, whereas for $L\le 22.5°$, only regions with moderate aftershock activity are displayed (for these, the total number of events versus time shows a dominant linear behavior). The vector $(k_x,k_y)$ labels the different regions, for which the coordinates of the center can be obtained as $x=x_{\min }+(k_x+0.5)L$, $y=y_{\min }+(k_y+0.5)L$, with $x_{\min }=-180°$, $y_{\min }=-90°$. (The $360°\times 180°$ region, which covers the whole planet, has been included for completeness.) (b) Previous data, after rescaling, with a fit of the scaling function $f$. (c) Rescaled distributions from local catalogs. SC88, SC95, and SC84 refer to Southern California for the years 1988–1991, 1995–1998, and 1984–2001. To obtain region coordinates, use the previous formula with $(x_{\min },y_{\min })=(-124°,29°)$, $(-123°,30°)$, $(125°,25°)$, $(-20°,30°)$, and $(-10°,45°)$ for SC88-95, SC84, Japan, the Iberian Peninsula, and the British Islands, respectively. The function displayed is the fit obtained from the NEIC-PDE catalog.

Figure 2

Analysis of aftershock sequences. (a) Decay of the rate after a mainshock and illustration of the Omori law for the following earthquakes in Southern California: Chalfant Valley (July 21, 1986, $M=5.9$), Landers (June 28, 1992, $M=7.3$), Northridge (Jan. 17, 1994, $M=6.7$), and Hector Mine (Oct. 16, 1999, $M=7.1$). Regions of diverse size $L$ are considered, all of these including the mainshock. Some curves are shifted for the sake of clarity. (b) Distributions of recurrence times for the previous sequences. (c) Distributions $\psi$ of the dimensionless time $r_{xy}(t)\tau$, in total agreement with the universal scaling function $f$ derived previously for the NEIC-PDE data.