Catalog and Local Declustering Method

We use moment magnitudes (Mw) and times from an earthquake catalog compiled for use by the US Geological Survey's Prompt Assessment of Global Earthquakes for Response system (PAGER-CAT) (12) for 1900 to 30 June 2008 seismicity, and the Preliminary Determination of Epicenters monthly and weekly (PDE and PDE-W) catalogs (available from the US Geological Survey National Earthquake Information Center web site, http://earthquake.usgs.gov/earthquakes) from 1 July 2008 to 13 August 2011. We consider only M≥7 events to reduce catalog completeness issues that may dominate results at smaller magnitudes. Nonuniformity in earthquake magnitude assignments is a serious issue in estimating earthquake rate changes (13, 14). PAGER-CAT attempts to use consistent moment magnitude estimates (see http://earthquake.usgs.gov/research/data/pager/PAGER_CAT_Sup.pdf), but the catalog might still have artificial changes in rate related to magnitude estimation.

Separating triggered aftershock seismicity from “background” seismicity is not trivial, and a variety of methods have been proposed. Because our focus here is on the global scale, we adopt the conservative and simple approach of removing events for which preceding larger earthquakes occur within 3 yr and 1,000 km. Declustering in this manner removes many events that might not traditionally be classified as aftershocks. For example, we remove both the March 2005 M 8.6 and September 2007 M 8.5 Sumatra earthquakes, retaining only the December 2004 M 9.0 Sumatra-Andaman earthquake. We do this because we want to consider only whether distant events are correlated, such as the 2010 M 8.8 Chile and 2011 M 9.0 Japan earthquakes, not whether regional-scale clustering may exist. Thus, it is important to decluster events at distances of less than about 1,000 km because they are not independent from a global perspective. We explore the effects of changing these declustering criteria below.

Magnitude Versus Time

Fig. 1 shows earthquake magnitudes versus time and smoothed yearly rates of M≥8, M≥7.5, and M≥7 activity. As expected, there are many more small earthquakes than large earthquakes, consistent with a Gutenberg-Richter (GR) power law relationship. The GR b value for the declustered catalog is approximately 1.3 for M≥7.5 earthquakes. There are only 16 earthquakes of M≥8.5, which limits the power of statistical tests of whether these giant events cluster in time. The eye is a poor judge of randomness and tends to find patterns in random sequences (e.g., ref. 15). Thus, the simple appearance of clustering should not be considered convincing evidence for nonrandomness.

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Fig. 1.

(A) Global earthquake magnitudes since 1900 after regional declustering of events. (B)–(D) Yearly rates of M≥8, M≥7.5, and M≥7 earthquakes. Rates are five-year running averages.

Nonetheless, past researchers have pointed to several possibly anomalous features that are visible in this plot. First, there were a disproportionate number of very large M≥8.5 earthquakes between 1950 and 1965. Second, there was a dearth of such large earthquakes in the 38 yr from 1966 to 2003. Finally, since 2004 there has been an elevated rate of M≥8 earthquakes: The five-year running average is at a record high, although there have been rates nearly as high in the past. These anomalies are evident only for the largest earthquakes and are much weaker or absent for smaller earthquakes. This observation implies that if the large earthquake clustering is caused by a physical mechanism, the mechanism must affect M≥8 earthquakes without changing the rate of smaller events. This property is inconsistent with the triggering behavior implied by aftershock sequences, which are observed to have Gutenberg-Richter magnitude-frequency relationships reflecting a preponderance of smaller events (e.g., ref. 16).

Monte Carlo Tests

How statistically significant are these anomalies for the large earthquakes? Addressing this question is complicated by the fact that virtually every realization of a random process will have features that appear anomalous. If the statistical test is chosen after looking at the data, the true significance level or p value can be substantially larger than the nominal value computed as if the test had been chosen before collecting the data (more about this topic later). Nonetheless, it is interesting to find the apparent probability of observed anomalies under various assumptions about the underlying process. For instance, one might assume that seismicity follows a homogeneous Poisson process with the expected rate equal to the observed rate in the catalog and generate a series of random catalogs. The fraction of these catalogs that have anomalies like those observed—or even more “extreme”—is an estimate of the p value of the hypothesis that seismicity satisfies the assumptions in the simulation, which include conditioning on the estimated rate.

This approach was used by Bufe and Perkins (2) to assess the 1950–1965 peak in great earthquake activity. They estimated that there was only a 4% chance that the three M≥9 earthquakes observed through 2001 would occur within an 11.4-yr period for a 100-yr catalog and that there was only a 0.2% chance that seven of the nine M≥8.6 earthquakes observed through 2001 would occur within any 14.5-yr period. We have not recalculated these probabilities, but note that they are likely underestimates for at least two reasons. First, the rate of M≥9 and M≥8.6 earthquakes that Bufe and Perkins used in their calculations is almost certainly lower than the true long-term rate, as activity since 2001 has shown. Second, and more importantly, they appear to have selected details of their statistical tests, such as the magnitude thresholds, to maximize the apparent anomaly. As mentioned above, this approach causes the p value or significance level to appear to be smaller than it really is, taking into account the post hoc selection.

Bufe and Perkins (2) also considered the gap in M≥8.4 earthquakes between 1966 and 2001 and estimated that a 36-yr gap would occur in only 0.5% of random catalogs of 18 M≥8.4 earthquakes. We show below that this gap is perhaps the most anomalous feature in the global catalog. However, the 0.5% value is misleadingly low because (i) Bufe and Perkins included among the 18 events a 23 July 1905 M 8.4 earthquake that is likely an aftershock of a nearby M 8.5 earthquake occurring 14 d earlier, and (ii) Bufe and Perkins apparently selected the M≥8.4 cutoff to maximize the apparent anomaly (there were three M 8.3 earthquakes in their catalog between 1966 and 2001).

We use Monte Carlo simulations to estimate the probability both of the recent elevated rate of large earthquake activity and of the gap that preceded it, under the null hypothesis that seismicity follows a Poisson process that generates exactly as many events as were in fact observed. That is, under the null hypothesis, the number of events is given and the times of these events are independent, identically distributed (iid) random variables, all with a uniform distribution on the interval in days [0, 40,767]. Our estimates are based on 100,000 random catalogs simulated from that joint distribution. The estimated probabilities are the fractions of those 100,000 catalogs that have the apparent anomaly at issue, for instance, the fraction that have at least a given number of events within an interval of a specified length.

Nine of the 75 (after local declustering) M≥8 earthquakes occurred in the 2,269-d period between the 23 December 2004 M 8.1 Macquarie earthquake and the 11 March 2011 M 9.0 Tohoku earthquake. Under the null hypothesis, there is about an 85% chance that at least nine of 75 events would occur within 2,269 d of each other: The recent elevated rate of large earthquakes is hardly surprising even if regionally declustered seismicity follows a homogeneous Poisson process.

Three of 16 M≥8.5 declustered earthquakes occur during the 2,266 d between the 26 December 2004 M 9.0 Sumatra earthquake and the Tohoku earthquake. Under the null hypothesis, there is about a 97% chance that at least three of 16 events will occur within 2,266 d of each other. Even if we cherry-pick the lower magnitude threshold to be 8.8 (the size of the 27 February 2010 Maule earthquake), so that three of six M≥8.8 events occur in a 2,266-d interval, this event concentration has a 14% chance under the null hypothesis that regionally declustered seismicity is Poisson.

The lack of M≥8.5 events in the approximately 40 yr between 4 February 1965 and 26 December 2004 is more anomalous than the recent elevated rate. Under the null hypothesis, the probability that 16 events in a 111-yr interval would contain such a long gap is only about 1.3%. However, this feature was selected in retrospect, and it is essentially always possible to find a feature of any specific realization of a random process that appears improbable—that only a small fraction of random realizations would have. Hence, this gap is hardly evidence that the underlying process is nonuniform.

To illustrate the effect of post hoc selection on nominal p values, we performed the following experiment. We generated 1,000 different catalogs of 330 events with random times uniformly distributed between 0 and 40,767 d and random magnitudes of 7.5 ≤ M ≤ 9.6 at 0.1 increments, assuming a b value of 1.3 (close to that observed for the declustered catalog). For each catalog we searched for event clusters, defined as the greatest concentration in time of n events of M≥M_min, where the minimum magnitude M_min varied from 8.0 to 9.0 in steps of 0.1, and the number of events in the cluster n ranges from 2 to 15. Using the Monte Carlo approach described above, for every (M_min, n) pair, we estimated the probability of its cluster, given the total number of events of M≥M_min in the random catalog, and found the “least likely” or “most surprising” cluster. We found that 91% of the 1,000 random catalogs had a cluster that should occur less than 10% of the time, 74% of the catalogs had a cluster that should occur less than 5% of the time, and 30% of the catalogs had a cluster that should occur less than 1% of the time. The (M_min, n) values corresponding to the most surprising cluster differ for different catalogs. If the analyst selects the most anomalous feature in a specific dataset, the nominal p value, which ignores the fact that the feature was selected after looking at the data, is generally much smaller than the true p value, which accounts for that selection. This simulation considered only a single cluster of events, but searching for gaps or for more than one cluster would also result in a nominal p value much smaller than the true p value, because the surprising feature was chosen after looking at the data.

Tests of the Poisson Hypothesis

A more general question is whether the earthquake catalog, after removing regional clustering, is statistically distinguishable from a realization of a homogenous Poisson process. We consider three tests, all of which condition on the number of events after declustering, so that the times of those events are iid uniform random variables under the null hypothesis. For additional tests, see ref. 17.

The first test compares the empirical distribution of the times with the uniform distribution. For each magnitude threshold, we determine the times of (locally declustered) events of that magnitude or above. We then perform a Kolmogorov-Smirnov (KS) test (e.g., ref. 18) of the hypothesis that those times are a sample of iid uniform random variables, estimating the p value by simulation. The second and third tests use the chi-square statistic but in different ways. Both partition the observation period into equal-length windows (we used N_w = 100 windows). These tests are more complicated than the KS test and require ad hoc choices, such as the lengths of the windows; the p values depend on those choices.

The first chi-square test (the Poisson dispersion test) uses the fact that the conditional joint distribution of the number of events in different windows, given the total number of events, is multinomial with equal category probabilities. The test statistic for this test is proportional to the variance of the counts across windows. We estimated the p value by simulating 100,000 catalogs with the same number of events that the declustered catalogs contained and iid uniformly distributed event times.

The second chi-square test (the multinomial chi-square test) assesses how well the numbers of windows with various numbers of events agree with the numbers expected for homogeneous Poisson seismicity. To perform the multinomial chi-square test, we calculated the observed average rate λ of events per period. On the assumption that seismicity follows a Poisson distribution with the expected rate per period equal to the average rate λ, let K^- denote the smallest integer such that the expected number of periods with no more than K^- events is at least five, and let K⁺ denote the largest integer such that the expected number of periods with at least K events is at least five:* [1][2]Define [3]Let X_K^- denote the number of periods that contain K^- or fewer events. For k = K^- + 1,…,K⁺ - 1, let X_k denote the number of periods that contain k events. And let X_K⁺ denote the number of periods that contain K⁺ or more events. The test statistic is [4]The values of K^- and K⁺ were 3 and 12 for the 759 M≥7.0 events, 1 and 7 for the 330 M≥7.5 events, and 0 and 2 for the 75 M≥8.0 events, respectively.

The Kolmogorov-Smirnov test and the Poisson dispersion test are sensitive to whether the rate varies with time, that is, to clustering. In contrast, the multinomial chi-square test is more sensitive to whether the distribution of the number of events in each window differs from the distribution expected if declustered seismicity were Poisson. For instance, if event times were equispaced, neither the KS test nor the Poisson dispersion test would reject the Poisson hypothesis, but the multinomial chi-square test would, given enough data.

Table 1 gives the results of all three tests, computed separately for minimum magnitudes of 7.0, 7.5, and 8.0. For the catalogs with aftershocks removed (i.e., Fig. 1), the p values range from 7.5% to 94%. If we remove all smaller events within 3 yr and 1,000 km, regardless of whether they are foreshocks or aftershocks, the p values range from 16.7% to 95.1%. For both magnitude thresholds and all three tests, the observed distribution of declustered event times is consistent with the hypothesis that declustered event times follow a homogeneous Poisson process. These results agree with those of Michael (19), who concluded that the global M≥7.5 and M≥8 catalogs (1900–2011), after aftershock removal, are well described by a Poisson process. Our smallest estimated p values, after declustering, occur for the M≥7 catalog, which has the largest number of events and is thus most sensitive to small rate changes, but even these values are not less than 5%.

One might ask what it would take to reject the hypothesis that regionally declustered seismicity follows a homogeneous Poisson process. Suppose the declustered catalog were extended by a year. How many more globally dispersed events of a given magnitude would have to occur in that year for the tests to reject the null hypothesis? For M≥8.5 events, the hypothesis would be rejected by the Poisson dispersion and multinomial chi-square tests if three more such events were to occur in the year following the end of the catalog, increasing the total from 16 to 19 events. The p values for the KS, Poisson dispersion (PD), and multinomial chi-square (MC) tests then would be about 16.2%, 0.9%, and 1.6%, respectively. For M≥8.0 events, the hypothesis would be rejected by the multinomial chi-square test if seven more such events were to occur in the year; the p values for the KS, PD, and MC tests then would be 6.0%, 5.3%, and 2.1%, respectively.

As mentioned above, it is essentially always possible in retrospect to find some feature of a dataset that would be prospectively unlikely under the null hypothesis, so it is essentially always possible—after looking at the data—to find or contrive some test that formally rejects the null hypothesis (in this case, for instance, a test on the basis of the longest gap between M≥8.5 events). But, as the simulations in Monte Carlo Tests show, the formal significance level or p value would not be meaningful, because it does not take into account the “data snooping” involved in selecting the test.

Sensitivity to Declustering Parameters

We have shown that the global catalog of large earthquakes, with aftershocks or foreshocks and aftershocks removed as described above, is statistically indistinguishable from a homogeneous Poisson process. However, it is well known that earthquakes cluster in local and regional catalogs: There are swarms, foreshocks, and aftershocks. Thus, we should expect the p value of the Poisson hypothesis to be lower for the original catalogs than for the declustered catalogs, as Table 1 confirms: The p values are generally rather smaller for the original catalogs than for the declustered catalogs (except for the multinomial chi-square test, which, as mentioned, measures something other than clustering). The original (undeclustered) catalog for M≥7.0 earthquakes is clearly inconsistent with the Poisson hypothesis. But the p values are not small for the original M≥7.5 and M≥8.0 catalogs. Even without declustering, the null hypothesis that times of large earthquakes follow a homogeneous Poisson process would not be rejected by any of these tests.

Because the original catalog for M 7.5 and larger events is consistent with the Poisson hypothesis, our conclusions for large earthquakes clearly do not depend strongly on details of the declustering method. For example, milder cutoffs of 400 d and 333 km give KS p values of 56.5% and 35.9% for the M≥8 and M≥7.5 declustered catalogs (aftershocks removed), respectively. The estimated p values depend on the choice of declustering parameters, but our main conclusions do not depend upon these details.