Analysis of Synthetic Catalogs

USING THE QTM CATALOG

3.5 Analysis of Synthetic Catalogs

early 1990s (Ben-Zion and Zaliapin, 2020). If more events are identified as clustered for the later years in the catalog, then this could artificially result in fewer mainshocks per year, and hence a lower apparent seismicity rate, as seen in the later half of the HYS catalog. If fewer events are identified as aftershocks in the period before 1991 due to location error, then the likelihood that some of those aftershocks will remain in the catalog after declustering is higher. This means a higher proportion of events with shorter interevent times than there would have been had they been identified as aftershocks, which skews the mean interevent time, and equivalently increases the seismicity rate of the catalog, leading to a fat tail. This could explain why the tail of the catalog subset for the period prior to 1990 is substantially larger than that of the later subset of the catalog (Figure 3.7d). As the location error decreases and the number of events in the catalog increases, as in the case of the QTM catalog, the earthquake IET distribution should approach the exponential distribution.

for six different minimum magnitude cutoffs. The tail of the QTM data, for the most part, clearly does not lie outside the tails of the synthetic data sets.

To verify this, we calculate the p-value using a standard chi-squared goodness of fit test for the QTM data with the null hypothesis that the interevent times of the catalog data are from an exponential distribution. The chi-squared test yields high p-values for all the M^𝑚𝑖𝑛tested except for M^𝑚𝑖𝑛 =1,2 (Figure 3.5), such that for most magnitude windows, there is at least a 22% chance that the count of the interevent times in the QTM data exceeds that predicted by the Poisson model and still comes from an exponential distribution. Thus, in most cases, there is not enough evidence to reject the Poisson model for the QTM earthquake IET distribution at the 5%

significance level. In the cases of M^𝑚𝑖𝑛 = 1,2, however, the p-value is very low (2e-4 and 1.3e-7, respectively), meaning the Poisson hypothesis can be rejected for these particular magnitude cutoffs, even though the deviation is not visually extreme.

The reason why the Poisson model can be rejected for some magnitude cutoffs may possibly be due to random error, but is most likely a result of incomplete declustering and/or non-stationarity of the earthquake occurrences, as examined above with the Schuster spectrum and the annualized IET distributions. The seasonal variation of seismicity detected with the Schuster test seems to have no detectable effect on the IET distribution, and this source of non-stationarity cannot be detected easily based on the departure from the exponential distribution expected for a stationary Poisson process.

3.5.2 Testing the Effect of Non-Stationarity and Aftershocks with Full Syn- thetic Catalogs

With the previous analyses (Figures 3.6 – 3.9), there is already strong indication that it is merely the presence of aftershocks that has the strongest effect on the Poisso- nian (or non-Poissonian) nature of the catalog, but deviations from stationarity in the background rate, as shown in Figures 3.8 and 3.9, may play a role as well. To determine whether non-stationarity alone or incomplete declustering is responsible for the observed fat tail, we generate full synthetic catalogs using a simple-type aftershock sequence model (STAS). We generate non-stationary Poissonian dis- tributed mainshocks using the same annualized𝑡

0as in the QTM and HYS catalogs.

Magnitudes are distributed according to the Gutenberg-Richter law, and aftershock sequences for each earthquake are generated according to the modified Omori’s Law (see supplementary material for details on the generation of the synthetic catalogs).

We create a total of 50 synthetic catalogs for each of the HYS and QTM based mainshock sequences.

We determine the cumulative distributions of the IETs in the same way as before, for both the mainshocks alone (Figure 3.10 a-b) and for the full catalog separately (Figure 3.10 c-d). The IET distributions of the mainshocks alone are plotted against the Poisson model obtained using the mean 𝑡

0 of the HYS or QTM catalogs, re- spectively, and the IET distributions of the full catalog simulations are plotted with the Poisson model calculated using the mean 𝑡

0 of that simulation, including the aftershocks. We compare the two distributions to see if the trends previously seen in the IET CDF of the QTM and HYS data are characteristic of the presence of aftershocks.

Comparing the CDF of the IETs of the synthetic mainshock catalogs, where events are known to come from a Poisson distribution, albeit a non-stationary one, reveals the effect that the magnitude of the non-stationarity has on the IET distribution. The mainshock IET distributions (Figure 3.10 a-b) mostly follow the Poisson model as expected, even with the non-stationarity. Most of the deviation is well within the expected error for synthetic data, as discussed above based on Figure 3.5. Only for the non-stationary case using the𝑡

0of the HYS data is there a more significant tail on the distribution, where the lower standard deviation bound of the synthetic data is actually just above the Poisson model (Figure 3.10a). This suggests that non-stationarity may indeed contribute to the observed tail on the mainshock data.

Non-stationarity does not have as strong an effect for the QTM-based synthetic catalog as it does for the HYS-based synthetics, but the variation in 𝑡_𝑜 in the HYS catalog is also an order of magnitude larger than in the QTM, suggesting the variability in the earthquake rate must be above a certain level before it can actually have an effect on the IET distribution.

The IET distributions and the associated Poisson models for the entire catalog with aftershocks, on the other hand, are strikingly different and reveal the effect that the presence of aftershocks has on the IET distribution (Figure 3.10 c-d). When the mean interevent time used in the exponential model is calculated from the interevent times of the full catalog, the resulting model is extremely skewed due to the overabundance of lower magnitude events with shorter interevent times, reducing the overall mean 𝑡

0. This generates the appearance of a large fat tail in the data.

Indeed the synthetic data for the complete catalogs always tails-off to the right (thin light gray lines in Figure 3.10 c-d) relative to their respective Poisson models (dark

0 5 10 Interevent Time 10⁰

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t₀ from HYS Catalog (a)

True Syn. Mainsh.

Poisson Model 1 Data St. Dev.

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(c)

Syn. Data with Afsh.

Poisson Model with Afsh.

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(e)

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Recovered Mainsh.

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10² (f)

Figure 3.10: IET distributions of 50 synthetic catalogs for: (a) Mainshocks generated using the non- stationary annualized𝑡

0’s from the HYS catalog (light gray lines), plotted against the corresponding Poisson model using the mean IET of the whole HYS catalog. (b) Same as (a) but using the non- stationary annualized𝑡

0’s from the QTM catalog. (c) Full synthetic catalogs with aftershocks using the HYS-based mainshocks in (a), with the corresponding Poisson models for each simulation. (d) Same as (c) but using the QTM-based mainshocks in (b). (e) Synthetic mainshocks recovered from declustering the HYS-based synthetic catalogs (dark lines) for simulations that have at least one M

>7 earthquake, versus the true synthetic mainshock distributions for those same simulations (light gray lines) and the known Poisson model, as in (a). (f) Same as (e) but for the QTM-based synthetic catalogs.

thick lines) for all simulations, as expected. This same deviation is what we observe in the undeclustered catalogs, as well as to some extent in the declustered ones.

To test whether the observed tail is an artefact of remaining aftershocks after declus- tering, we decluster the synthetic simulations that contain at least one M>7 earth- quake using the Zaliapin and Ben-Zion (2013) algorithm. This yields 6 declustered catalogs from the simulations based off the HYS data and 4 declustered catalogs from the simulations based off the QTM data. (See supplementary material Figures S1- S10 for results of the synthetic catalog generation and declustering.) The interevent time distributions for the recovered mainshocks obtained from the declustering are shown in Figure 3.10 (e-f) against the distributions for the true synthetic mainshocks for those same simulations and the known Poisson model for the mainshocks as in Figure 3.10 (a-b). For most of the simulations, the Zaliapin and Ben-Zion (2013) declustering predicts the correct proportion of mainshocks to within about 3.8% on average. However, the interevent time distributions of the recovered mainshocks always exhibit a slight fat tail. To determine whether this deviation is significant, given the existing deviation in the synthetic mainshocks, we again perform a basic chi-squared goodness of fit test as was done on the real data. For the HYS-based synthetic catalog, only two cases out of the six that were declustered were significant enough to reject the Poisson model, but for the QTM-based synthetic catalog, the deviations of all but one of the four declustered simulations were significant enough to reject the Poisson hypothesis. This suggests that in the case of the HYS catalog, aftershocks remaining in the catalog after declustering are not the primary cause of the deviation, but rather the non-stationarity itself, since, as was mentioned above, the variation in the earthquake rate of the HYS catalog is greater than that of the QTM catalog. In the case of the QTM catalog on the other hand, the presence of remaining aftershocks in the declustered catalog, even if only a small amount, is enough create a significant fat tail. The synthetic catalogs presented here demon- strate that the fat tail on the earthquake IET distribution is in part due to background non-stationarity, but is also largely an artefact of the presence of aftershocks.