Asymmetry in earthquake interevent time intervals

Here we focus on a basic statistical measure of earthquake catalogs that has not been studied before, the asymmetry of interevent time series (e.g., reflecting the tendency to have more aftershocks than spontaneous earthquakes). We define the asymmetry metric as the ratio between the number of positive interevent time increments minus negative increments and the total (positive plus negative) number of increments. Such asymmetry commonly exists in time series data for non-linear geophysical systems like river flow which decays slowly and increases rapidly. We find that earthquake interevent time series are significantly asymmetric, where the asymmetry function exhibits a significant crossover to weak asymmetry at large lag-index. We suggest that the Omori law can be associated with the large asymmetry at short time intervals below the crossover whereas overlapping aftershock sequences and the spontaneous events can be associated with a fast decay of asymmetry above the crossover. We show that the asymmetry is better reproduced by a recently modified ETAS model with two triggering processes in comparison to the standard ETAS model which only has one.

Based on earthquake catalogs, we consider seismic events above a certain magnitude threshold (i.e. the magnitude of completeness for the given catalogue). For this sequence, we define the time interval between two successive earthquake events i+1 and i as the interevent time τ i (in days). The lagged interevent time increment is defined as ∆τ (k) i = τ i+k − τ i for a lag k where k is a positive integer lag. Following the above, the asymmetry measure of interevent times is defined as the ratio between the number of positive interevent time increments, N p , minus the number of negative increments, N n , and the total (positive plus negative) increments (Ashkenazy et al., 2008): where Θ(τ ) = 1 when τ > 0 and otherwise it is zero. We exclude the zero increments 115 ∆τ (k) i = 0 from the calculation; the number of zero increments is indeed very small.

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U is bounded between -1 (monotonically decreasing sequence) and 1 (monotonically in-117 creasing sequence). When U is close to zero, the time series is symmetric (for example, 118 the PDF is nearly symmetrical close to ∆τ (k) = 0 for k = 1 but highly asymmetric  2.2 Generalized ETAS model 125 We also study the asymmetry of synthetic catalogs based on the ETAS model in comparison to the asymmetry observed in the time series of real records. We use the ETAS model as a null hypothesis , since it is the most widely used statistical model to simulate the spatiotemporal clustering of seismic events (Ogata, 1988(Ogata, , 1998. The earthquake sequence in the ETAS is defined as a stochastic Hawkes (point) process. We use the Gutenberg-Richter law (where b = 1, truncated at M max ) to independently generate the magnitude of each earthquake (≥ M 0 ). For the ETAS model, the conditional intensity function λ (which is basically the rate of earthquakes) at time t with the seismic history H t prior to t is given by where µ is the background rate to generate spontaneous earthquakes estimated from the real catalogs (Zhuang et al., 2010;Zhuang, 2012 Figure S1 where the distributions of the magni-147 tudes (≥ 3) follow the Gutenberg-Richter law.

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First, we obtain interevent times of the Italian earthquake catalog (using the thresh-150 old of magnitude M 0 = 3.0) and their increments for the lag index k = 50. The re-151 sults are shown in Fig. 1. As can be seen, interevent times decrease abruptly and then 152 increase gradually after the occurrence of a large earthquake (Fig 1(a)), consistent with random process shown at high k. As discussed above, we expect the presence of asym- indicates the crossover lag kc ≈ 50 at which the asymmetry measure U for interevent times is maximal.
SC and JA respectively. We also consider and observed the asymmetry for different re-211 gion sizes as shown in Fig. S3(a). A smaller region size shows a larger asymmetry since  Moreover, the crossover can be scaled with respect to region size in Fig. S3(b). Figure   214 S4 shows the weak asymmetry for the global earthquake catalog. 215 We now aim to explain the mechanism underlying the observed asymmetry mea-  Fig. 4(b)). The indirect triggered events and the 226 spontaneous events can decrease the interevent times as shown in Fig. 4(b) (τ 2 , τ 3 and 227 τ 5 are smaller than τ 1 ). The above considerations implies that the events below the crossover 228 lag k c are mainly triggered by a mainshock. Above the crossover (k > k c ), the sequences 229 for spontaneous and triggered events will overlap with high probabilities resulting in a 230 fast decay of asymmetry.

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The ETAS model is widely used to simulate and study the temporal clustering of      catalog, the dotted line (see Fig. 5(a)). Moreover, the crossover point is different for EM1 266 (k c ≈ 60) and EM2 (k c ≈ 50). Thus, the crossover of EM2 is closer to the observed 267 one ( Fig. 5(a)). We thus conclude that the two-alpha (α 1 > α 2 ) ETAS model exhibits the real one (see SI, Figure S5).

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We further study the dependence of the interevent time increments ∆τ k i on the mag- to zero. Thus, the asymmetry measure close to zero could be due to the magnitude sim-280 ilarity for small lag-index k, as the PDF interevent time increments is maximal close to 281 zero (Fig 2(a) 10 0 10 0 10 1 10 1 10 2 10 2 10 0 10 0 10 1 10 1 10 2 10 2 10 0 10 0 10 1 10 1 10 2 10 2 for the real catalog, from values closer to zero for small lag-index k to values of EM1 and 290 EM2 at large k (Fig. 5(c)). These results indicate that the magnitude similarity reported While the asymmetry of EM2 is similar to the asymmetry of the real catalog for k > 300, it is significantly higher for smaller k (Fig. 5(a)). The Italian catalog we used has been reported to be complete above magnitude threshold 3.
where all past events with t i < t are considered. We define Φ (m|M where m i is the magnitude of past event i, and t − t i is the time since the past event.

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The parameter σ = 0.6 is chosen following Petrillo and Lippiello (2021) and Seif et al. 3.5 to generate synthetic catalogs with different degree of incompleteness.

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To generate the synthetic incomplete catalogs based on EM0, EM1 and EM2 (rep-298 resented as EM0I, EM1I and EM2I respectively), we remove an aftershock i from the 299 synthetic complete catalogs with a probability given by i Φ (m|M i (t − t i ) , σ) (Petrillo eters unchanged. With this procedure, the level of incompleteness of each synthetic cat-303 alog was 5%, 10% and 20% for δ 0 = 4.5 , 4.0, and 3.5 respectively for EM1I and EM2I; 304 the percentages indicate the relative number of events that has been removed from the 305 complete catalog. It is apparent from our results (see Figure 6) that the asymmetry weak-306 ens as the degree of incompleteness is higher. Still, for both models, the asymmetry is 307 overestimated for small lag index k in comparison to the real catalog and weakens when 308 the catalogs are more incomplete. Figure S6 shows similar results when using σ = 0.6, 309 1.2, 1.8 to control the degree of incompleteness. We also try to control the parameter 310 A to keep the same number of earthquakes for EM0I, EM1I and EM2I and the results 311 are shown in Figure S7. fied value of k c 10 bM0 . We suggest that the Omori law is associated with the increase of 318 the asymmetry below the crossover and has a decreasing influence above this crossover.

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This is probably due to the overlapping of different triggered aftershocks and the spon-320 taneous events that lead to a fast decay of asymmetry above the crossover. The de-clustering here and its associated crossover may help to resolve this question although this requires 324 further investigation.

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In the standard ETAS model whose parameters are estimated by MLE, the increase 326 of asymmetry and the crossover cannot be reproduced. When the α-value is increased, 327 a large mainshock can trigger more aftershocks such that there exists an increasing trend 328 and a crossover in the standard ETAS model. This demonstrates that the common α-329 value is indeed underestimated by MLE. However, the crossover value of k is larger and 330 the asymmetry above the crossover is significantly higher and decays slower in the stan-331 dard ETAS model with large α than the real one. The generalized ETAS model with two 332 α-values (α 1 > α 2 ) in short and long time scales exhibits similar asymmetry behavior 333 as that of the real catalog for lags larger than the crossover lag k c . Yet, the asymmetry 334 for small lag-index k is overestimated by both models (one and two α-value). We sug-335 gest that the short-term symmetrical behavior can be attributed to the magnitude sim-336 ilarity in real data which is missing in both models. The additional advantage of the gen-337 eralized ETAS model is its ability to reproduce the observe memory in earthquake cat-338 alogs as reported in (Zhang et al., 2021). Thus, generally speaking, the asymmetry find-339 ings reported here may be used to improve earthquake forecasting models as the asym-