As noted above, in the long-term prediction of seismicity, or in seismic hazard research, it is first assumed that the seismicity follows the Poisson process. Therefore, it is necessary to verify this assumption for the declustered earthquake catalog. Since Gardner and Knopoff (1974) attempted the Poisson test for the declustered catalog in the California area, many researchers have conducted such studies in various areas (Luen and Stark 2012; Shearer and Stark 2012; Wyss and Toya 2000). In particular, Noh (2016) performed the Poisson test on earthquakes observed in and around the Korean Peninsula between 1980 and 2014, to find that the earthquakes with magnitudes greater than or equal to 2.7 that occurred on land followed the Poisson process.
In the present study, the following three Poisson tests are applied to the three declustered earthquake catalogs: (i) the multinomial chi-square (MC) test, (ii) the conditional chi-square (CC) test, and (iii) the Kolmogorov-Smirnov (KS) test. The MC and CC tests pay attention to how the earthquakes are distributed within a certain time period, while the KS test evaluates the distribution of time intervals between earthquake occurrence times.
For the MC test, the method of Shearer and Stark (2012) is adopted as follows. First, the entire observation period is divided into several equal intervals (\({N}_{\dot{i}}\)), and the events occurring within each interval are counted. Then, it is checked whether the number of intervals with a certain number of events agrees well with the theoretical Poisson distribution. Assuming that the seismicity follows the Poisson process, the average event rate \(\left(\lambda \right)\)per interval becomes \(N/{N}_{\dot{i}}\), where the total number of events is N. When the expected number of intervals is at least five, then the smallest and largest intervals (\({K}^{-} and {K}^{+}) \text{c}\text{a}\text{n} \text{b}\text{e} \text{e}\text{x}\text{p}\text{r}\text{e}\text{s}\text{s}\text{e}\text{d}\) by Eqs. (8) and (9) (Shearer and Stark 2012):
$${K}^{-}\equiv min [k : {N}_{\dot{i}}{ e}^{-\lambda }\sum _{j=0}^{k}{\lambda }^{j}∕j!\ge 5]$$
8
$${K}^{+}\equiv max [k : {{N}_{\dot{i}} ( 1- e}^{-\lambda }\sum _{j=0}^{k-1}{\lambda }^{j}∕j!\ge 5]$$
9
The test statistic \({{\chi }_{M}}^{2}\) is obtained with (\({K}^{+}- {K}^{-}+1 )-2\)degrees of freedom by using Eq. (10):
where \({X}_{k} \text{denotes}\) the number of intervals including k events, and \({E}_{k}\) is defined as Eqs. (11):
$${E}_{k}\equiv \left\{\begin{array}{c} {N}_{\dot{i}} {e}^{-\lambda }\sum _{j=0}^{{K}^{-}}{\lambda }^{j}/j! , k={K}^{-} ,\\ \begin{array}{c}\\ \begin{array}{c}{N}_{\dot{i}} {e}^{-\lambda }{\lambda }^{k}/k! , k={K}^{-}+1\dots \dots {K}^{+}-1 ,\\ \\ {N}_{\dot{i}} \left(1-{e}^{-\lambda }\sum _{j=0}^{{K}^{+}-1}{\lambda }^{j}/j!\right) , k={K}^{+}.\end{array}\end{array}\end{array}\right.$$
11
Because the MC test does not distinguish between the numbers of intervals with small and large numbers of events, it is not sensitive to variance in the data. For this reason, the MC test is not as sensitive as certain other tests towards apparent fluctuations in the rate of seismicity (Luen and Stark 2012). This problem can be compensated for by the CC test, which is also known as the Poisson dispersion test. This test divides the entire observation period into several equal intervals to find the variance between the number of events within each interval, along with the average number of events. When the number of events during the k-th observation period is \({N}_{k}\), the test statistic \({{\chi }_{C}}^{2}\) can be expressed as Eq. (12):
$${\chi }_{c}^{2}=\sum _{k=1}^{M}\frac{{\left({N}_{k}-\lambda \right)}^{2}}{\lambda }$$
12
Meanwhile, the KS test can be applied to the distribution of intervals between earthquake occurrences. For an earthquake catalog with (n + 1) events, the earthquake occurrence time intervals are placed in ascending order as \(({x}_{1}\), \({{x}_{2} , \cdots x}_{n }\)), and their average is defined as \(\stackrel{-}{x}\). The time interval distribution function \(F\left({x}_{i}\right)\)is then defined by Eq. (13):
F (\({\text{x}}_{\text{i}}\)) = \({z}_{i}=1-{exp}\left({x}_{i}/\stackrel{-}{x}\right) , i=1, 2, \cdots , n\) (13)
The statistic of the KS test is then expressed by Eqs. (14) (Stephens, 1974):
$$D=\text{max} {( D}^{+ }, {D}^{ -})$$;
\({D}^{+ }=\text{max}(i/n - {z}_{i} ) ,\) \({D}^{- }=\text{max} \left( {z}_{i} -(i-1)/n\right) , 1\le i \le n\) (14)
To execute a hypothesis test using the table provided by Stephens (1974), Eq. (14) must be corrected to give Eq. (15):
(D − 0.2/n ) (\(\sqrt{n}+0.28+0.5 /\sqrt{n}) \to D\) (15)
The null hypothesis is then used to test whether the earthquake catalogs declustered by each of the three methods follow the Poisson process. For each catalog, each Poisson test (MC, CC, and KS) is performed with a significance level of 0.05 for increasing completeness magnitude (Mc) values from 1.0 to 3.0 in steps of 0.1. In general, the null hypothesis is rejected at a small Mc, but is not rejected above a certain limiting Mc. Noh (2016) defined the minimum Mc for which the null hypothesis is not rejected as \(Mp\). The term \(Mp\) is used as the same meaning in this study. In the present work, considering the amount of data, the MC and CC tests were performed for two cases in which the numbers of intervals were 72 and 216, as obtained by dividing the observation period from 2016 to 2021 into 10 and 30 days, respectively. The results in Table 3 indicate that the \(Mp\) values obtained in the MC test for the three declustered earthquake catalogs are in the range of 1.6 to 2.2. Thus, there is no significant difference regardless of the number of intervals. Further, the \(Mp\) value obtained by the Poisson test is seen to be larger than the completeness magnitude (Mc in Table 2). For each of the tree Poisson tests, the largest \(Mp\) value is obtained for the catalog that was declustered using the method of Reasenberg (1985), which removes the smallest number of dependent earthquakes. In addition, it is noted that the CC test gives larger\(Mp\) values than do the other two Poisson tests for the catalogs obtained using the methods of Gardner and Knopoff (1974) and Reasenberg (1985). Hence, given that careful consideration must be paid to the selection of the cutoff magnitude during the process of preparing an earthquake catalog for seismic hazard research, the \(Mp\) value will provide a good reference.
Table 3
The from the three types of Poisson hypothesis test (multinomial chi-square, conditional chi-square, and Kolmogorov-Smirnov) for the declustered catalogs using the methods of Gardner and Knopoff (1974), Reasenberg (1985), and Zhuang et al. (2002)
Catalog
|
Method
|
multinomial chi-square (MC)
|
conditional chi-square (CC)
|
Kolmogorov-Smirnov (KS)
|
Gardner and Knopoff (1974)
|
1.7
|
1.9
|
1.6
|
Reasenberg (1985)
|
1.8
|
2.2
|
1.9
|
Zhuang et al. (2002)
|
1.6
|
1.6
|
1.6
|