Scaling laws in earthquake memory for interevent times and distances

Earthquakes involve complex processes that span a wide range of spatial and temporal scales. The limited earthquake predictability is partly due to the erratic nature of earthquakes and partly due to the lack of understanding of the underlying mechanisms of earthquakes. To improve our understanding and possibly the predictability of earthquakes, we develop here a lagged conditional probability method to study the spatial and temporal long-term memory of interevent earthquakes above a certain magnitude. We find, in real data from different locations, that the lagged conditional probabilities show long-term memory for both the interevent times and interevent distances and that the memory functions obey scaling and decay slowly with time, while, at a characteristic time (crossover), the decay rate becomes faster. We also show that the epidemic-type aftershock sequence model, which is often used to forecast earthquake events, fails in reproducing the scaling function of real catalogs as well as the crossover in the scaling function. Our results suggest that aftershock rate is a critical factor to control the long-term memory.

Figure 1

Time series (1981–2017) of (a) interevent times and (b) their corresponding distances for the whole Italian catalog for a magnitude threshold of 3.0. The vertical dashed lines indicate earthquake events with magnitudes above 5.8.

Figure 2

Conditional PDF of (a) the interevent times ${\tau }_1$ and (b) the interevent distances $r_1$ (for magnitude threshold $M_0=3.0$) for all of Italy; the common area between the conditional PDF of the first and third quantiles is $s_{13}=0.43$ and $s_{13}=0.42$ for times and distances, respectively. Note that no memory corresponds to $s_{13}=1$ and full memory to $s_{13}=0$. (c) and (d) Same as (a) and (b) but for lag $k=10$ where the common area is now $s_{13}=0.57$ and $s_{13}=0.67$ for times and distances, respectively. A larger common area indicates less correlation. The black dashed curves indicate the PDFs for all $\tau$ ($r$). The PDFs are normalized in a logarithmic scale.

Figure 3

Memory measure (a) $S\left({\tau }_k\right|{\tau }_0)$ and (b) $S\left(r_k\right|r_0)$ as a function of the lag index $k$ for interevent times and distances, respectively. Colors represent different grid sizes $L$. The shapes of the symbols represent different magnitude thresholds $M_0$. The grid size of $L={14}^{\circ }$ covers the whole region of the Italian catalog. Rescaled memory measure for (c) interevent times and (d) distances. Black dashed lines are fitted power-law curves. Note that the narrow large-$k$ regimes could be fitted also by exponential decay. The vertical red dotted lines indicate the location of the crossover.

Figure 4

Memory measure of the ETAS model catalogs for the area (${34}^{\circ }\mathrm{N}-{48}^{\circ }\mathrm{N}, 6^{\circ }\mathrm{E}-{20}^{\circ }\mathrm{E}$) as a function of the lag index $k$: (a) $S\left({\tau }_k\right|{\tau }_0)$ (interevent times) and (b) $S\left(r_k\right|r_0)$ (interevent distances). Colors and shapes are the same as in Fig. 3. Also shown is the rescaled memory measure for (c) interevent times and (d) distances. The model's parameters are estimated for the Italian catalog [17, 29] (see also [22]). The memory measure $S$ is averaged over 50 independent realizations and each realization includes ${10}^6$ events. The error bars are the standard deviations.