Earthquakes Descaled: On Waiting Time Distributions and Scaling Laws

Recently, several authors have used waiting time distributions for large earthquake data sets to draw conclusions regarding the physics of earthquake processes. We show, theoretically and by simulation, that a characteristic kink in observed waiting time distributions does not have the physical significance of separating correlated and uncorrelated earthquakes. It also follows from our discussion that the Omori law is not trivially related to a proposed scaling law and that caution must be taken before the spatial scaling exponent of the law is interpreted as a fractal dimension of seismicity.

Figure 1

Characteristics of waiting time distributions for a single aftershock sequence. Solid lines—theoretical distributions [Eq. (2)], symbols—empirical distributions from simulated aftershock sequences. In (a), (b), and (c) the histogram count is normalized by the size of each histogram bin. (a) Waiting time distributions for Omori sequences with ${10}^6$ ($C_1=49902$), ${10}^5$ ($C_1=4990$), and ${10}^4$ ($C_1=499$) events collected over 16 years ($C_2=0$, $p=1$). (b) Waiting time distributions for different values of $p$: 0.7, 1, 2, and 2.5. $C_1=40$, $C_2=0$ and observation period $T=1000\mathrm{s}$. Dashed lines—asymptotes to the power law regimes. The distributions are shifted for clarity. (c) Waiting time distribution for the same aftershock sequence observed for 1 month, 6 months, 4 years, and 64 years. $C_1=50$, $C_2=0$, $p=1$. (d) Waiting time distributions from (a) with waiting time axis rescaled by ${10}^{-m}$ where $m=4$, 5, and 6, respectively.

Figure 2

Waiting time distribution for realistic earthquake data sets covering a period of 16 years. The distributions are not normalized with the bin size, i.e., they correspond to Fig. 4 in BCDS [1]. (a) Waiting time distribution for an aftershock sequence (${10}^4$ events, $C_1=499$, $C_2=0$, $p=1$) with Poisson-distributed background seismicity (2000 events). Solid line—theoretical distribution for the single aftershock sequence [Eq. (2)]. (b) Waiting time distributions for 25, 50, and 100 main shock-aftershock sequences. For details see the text. (c) Proportion of correlated and uncorrelated successive events for each bin of the distribution in (a). (d) Proportion of correlated and uncorrelated successive events for each bin of the distribution in (b) with 25 aftershock sequences. Also shown is the proportion of correlated waiting times for 400 aftershock sequences.