The Weibull – Log-Weibull Transition of Interoccurrence Time of Earthquakes

is a Poisson process. However, we have demonstrated that the Weibull distribution is more appropriate than the gamma distribution on the basis of two goodness-ﬁt-tests. In addition, for large m c , the distribution in a long time domain is similar The study of earthquakes plays a key role in order to minimize human and material losses when they inevitably occur. Chapters in this book will be devoted to various aspects of earthquake research and analysis. The different sections present in the book span from statistical seismology studies, the latest techniques and advances on earthquake precursors and forecasting, as well as, new methods for early detection, data acquisition and interpretation. The topics are tackled from theoretical advances to practical applications.

be noted that the interoccurrence times were analyzed for the events with the magnitude m above a certain threshold m c under the following two assumptions: (a) earthquakes can be considered as a point process in space and time; (b) there is no distinction between foreshocks, mainshocks, and aftershocks. It has been shown that the distribution of the interoccurrence time is also obtained by analyzing aftershock data (7) and is derived approximately from a theoretical framework proposed by Saichev and Sornette (8;9). Abe and Suzuki showed that the distribution of the interoccurrence time, P(> τ), can be described by q-exponential distribution with q > 1, corresponding to a power law distribution (10), namely, ( 4 ) where q t , τ 0 , γ,andǫ are positive constants and ([a] + ≡ max[0, a]).
It has been reported that the sequence of aftershocks and successive independent earthquakes is a Poisson process (11; 12). However, recent works show that interoccurrence times are not independent random variables, but have "long-term memory" (13)(14)(15)(16). Since an interoccurrence time depends on the past, it is difficult to determine the distribution of interoccurrence times theoretically. Therefore, the determination of the distribution of interoccurrence times is still an open problem. Moreover, an effect of a threshold of magnitude on the interoccurrence time statistics is unknown. We study the distribution of interoccurrence times by changing the threshold of magnitude. In this chapter, we review our previous studies (17; 18) and clarify the Weibull -log-Weibull transition and its implication by reanalyzing the latest earthquake catalogues, JMA catalogue (19), SCEDC catalogue (20), and TCWB catalogue (21). This study focuses on the interoccurrence time statistics for middle or big mainshocks.

Data and methodology 2.1 Earthquake catalogue
To study the interoccurrence time statistics, we analyzed three natural earthquake catalogues of the Japan Metrological Agency (JMA) (19), the Southern California Earthquake Data Center (SCEDC) (20) and the Taiwan Central Weather Bureau (TCWB) (21). Information on each catalogue is listed in Table 1, where m min corresponds to the minimum magnitude in the catalogue and m 0 c is the magnitude of completeness, that is the lowest magnitude at which the Gutenberg -Richter law holds. We basically consider events with magnitude greater than and equal to m 0 c because events whose magnitudes are smaller than m 0 c are supposedly incomplete for recording.

Japan Metrological Agency (JMA) earthquake catalogue
JMA catalogue is maintained by the Japan Metrological Agency, covering from 25 • to 50 • N for latitude, and from 25 • to 150 • E for longitude [see Figure 1 (a)] during from 1923 to latest. This catalogue consists of an occurrence of times, a hypocenter, a depth, and a magnitude. In this chapter, we use the data from 1st January 2001 to 31st January 2010. As can be seen from Figure 1 (b), the distribution of magnitude obeys the Gutenberg -Richter law and m 0 c is estimated to be 2.0.

Southern California Earthquake Data Center (SCEDC) earthquake catalogue
SCEDC catalogue is maintained by the Southern California Earthquake Data Center, covering from 32 • to 37 • N for latitude, and from 114 • to 122 • W for longitude [see Figure 2 (a)] during from 1932 to latest. The information of an earthquake, such as an occurrence of times, a hypocenter, a depth, and a magnitude, is listed. Here, we analyze the earthquake data from 1st January 2001 to 28th February 2010. In Figure 2 (b), we demonstrate the magnitude distribution, and we obtain b = 0.97.

Taiwan Central Weather Bureau (TCWB) earthquake catalogue
TCWB catalogue is maintained by the Central Weather Bureau, covering from 21 • to 26 • Nfor latitude, and from 119 • to 123 • E for longitude [see Figure 3 (a)]. This catalogue consists of an occurrence of times, a hypocenter, a depth, and a magnitude. We use the data from 1st January 2001 to 28th February 2010. As shown in Figure 3    magnitude range, 1.9 ≤ m ≤ 6.7. b-value is calculated from the slope of the distribution, and is estimated to be 0.90.

Methodology (How to detect the appropriate distributions)
Our method is similar to the that of previous works (17; 18) (see Figure 4).
1. We divided the spatial areas into a window of L degrees in longitude and L degrees in latitude. 2. For each bin, earthquakes with magnitude m above a certain cutoff magnitude m c were considered. 3. We analyzed interoccurrence times greater than and equals to h day.
For each bin, we analyzed interoccurrence times using at least 100 events to avoid statistical errors. h and L are taken to be 0.5 and 5, respectively. It is noted that for SCEDC and T C W B ,w ea n a l y z ee a r t h q u a k ec o v e r i n gt h ew h o l er e g i o n . A ss h o w ni nF i g u r e4 ,w e investigated the interoccurrence time statistics for different 16 regions (14 regions in Japan, Southern California, and Taiwan). Aftershocks might be excluded from the study based on the information from previous studies (6; 12).
One of our main goals in this chapter is to determine the distribution function of the interoccurrence time. Here, we will focus our attention on the applicability of the Weibull distribution P w , the log-Weibull distribution P lw (22), the power law distribution P pow (10), the gamma distribution P gam (in the case of δ = 1 in the paper (6)), and the log normal distribution P ln (23), which are defined as where α i , β i ,andh are constants and characterize the distribution. Γ(x) is the gamma function.
i stands for an index number; i = 1, 2, 3, 4, and 5 correspond to the Weibull distribution, the log-Weibull distribution, the power law distribution, the gamma distribution, and the log normal distribution, respectively.
The Weibull distribution is well known as a description of the distribution of failure-occurrence times (24). In seismology, the distribution of ultimate strain (25), the recurrence time distribution (26; 27), and the damage mechanics of rocks (28) show the Weibull distribution. In numerical studies, the recurrence time distribution in the 1D (4) and 2D (29) spring-block model, and in the "Virtual California model" (30) also exhibit the Weibull distribution. For α 1 = 1a n dα 1 < 1, the tail of the Weibull distribution is equivalent to the exponential distribution and the stretched exponential distribution, respectively. The log-Weibull distribution is constructed by a logarithmic modification of the cumulative distribution of the Weibull distribution. In general, the tail of the log-Weibull distribution is much longer than that of the Weibull distribution. As for α 2 = 1, the log-Weibull distribution is equal to a power law distribution. It has been shown that the log-Weibull distribution can be derived from the chain-reaction model proposed by Huillet and Raynaud (22).
To determine the best fitting for the distribution of the interoccurrence time data, we used the root mean square (rms) and Kolomogorov-Smirnov (KS) tests as the measure of goodness-of-fit. The definition of the rms value is where x i is actual data and x ′ i is estimated data obtained from P(τ). n and k indicate the numbers of the data points and of the fitting parameters, respectively. In this study, the rms value is calculated using the cumulative distribution for decreasing the fluctuation of the data. The most appropriate distribution is, by definition, the smallest rms value. Also, in order to use the KS test, we define the maximum deviation of static DKS, which is so-called Kolomogorov-Smirnov statistic, as where y i and y ′ i mean the actual data of the cumulative distribution and the data estimated from the fitting distribution, respectively. Then, the significance level of probability of the goodness-of-fit, Q,isdefinedas  where where n ′ stands for the number of data points.
It is known that the preferred distribution shows the smallest value of DKS and the largest value of Q (31).

Interoccurrence time statistics in Japan
First, we analyze the JMA data. Here, we consider the two region; Okinawa region The cumulative distributions of the interoccurrence times for different m c in Okinawa region and in Chuetsu region are displayed in Figures 5 and 6, respectively. We carried out two statistical tests, the rms and the KS test so as to determine the distribution function. The results for large magnitude (m c = 4.5) in Okinawa and Chuetsu are shown in Table 2 and 3, respectively. For Okinawa, we found that the most suitable distribution is the Weibull distribution in all tests. In general, there is a possibility that the preferred distribution is not unique but depends on the test we use. However, the results obtained in Table 2 provide evidence that the Weibull distribution is the most appropriate distribution. As for Chuetsu, by two tests, the preferred distribution is suited to be the Weibull distribution as shown in Table 3, where the Weibull distribution is the most prominent distribution in the two tests. It follows that the Weibull distribution is preferred.    Table 3. Results of rms value, DKS, and Q for different distribution functions for Chuetsu area (m c = 4.5). The error bars mean the 95% confidence level of fit.  Table 4. Interoccurrence time statistics of earthquakes in Okinawa region. The error bars mean the 95% confidence level of fit.
However, the fitting accuracy of the Weibull distribution becomes worse with a gradual decrease in m c . We now propose a possible explanation which states that "the interoccurrence time distribution can be described by the superposition of the Weibull distribution and another distribution, hereafter referred to as the distribution P X (τ), where p is a parameter in the range, 0 ≤ p ≤ 1 and stands for the ratio of P w divided by P(τ). The interoccurrence time distribution obeys the Weibull distribution for p = 1. On the other hand, it follows the distribution P X (τ) for p = 0. Here, the log-Weibull distribution, the power law distribution, the gamma distribution, and the log normal distribution are candidates for the distribution P X (τ).
Next we shall explain the parameter estimation procedures; (A); the optimal parameters are estimated so as to minimize the differences between the data and the test function by varying five parameters, α 1 , β 1 , α i , β i and p.  Table 5. Interoccurrence time statistics of earthquakes in Chuetsu area. The error bars mean the 95% confidence level of fit.
If there is a parameter, where C v , the ratio of the standard deviation divided by the mean for a parameter exceeds 0.1, another estimation procedure, (B), is performed.
(B); the Weibull parameters, α 1 and β 1 , and the parameters of P X (τ), α i and β i , are optimized dependently and then p is estimated.
According to those procedures (A) and (B), we obtain the fitting of results of P(τ).Th er esult s for Okinawa and Chuetsu region are listed in Table 4 and 5. We assume that the Weibull distribution is a fundamental distribution, because p becomes unity for large m c ,whichmeans that the effect of the distribution P X (τ) is negligible. As observed in Table 4 and 5, the log-Weibull distribution is the most suitable distribution for the distribution P X (τ) according to the two goodness-of-fit tests. Thus, we find that the interoccurrence times distribution can be described by the superposition of the Weibull distribution and the log-Weibull distribution, namely, P(τ) is controlled by five parameters, α 1 , α 2 , β 1 , β 2 ,andp.

Interoccurrence time statistics in Southern California
Second, we analyze the interoccurrence time statistics using the SCEDC data. The cumulative distributions of interoccurrence time for m c = 4.0 and m c = 2 . 0a r es h o w ni nF i g u r e8( a ) and (b), respectively. By the rms test and KS test, we confirmed that the Weibull distribution is preferred for large m c (m c = 4.0) [see Table 6], which is the same result as that from JMA      Table 6. Results of rms value, DKS, and Q for different distribution functions in the case of m c = 4.0 for Southern California earthquakes. The error bars mean the 95% confidence level of fit.
data. Unfortunately, the fitting accuracy of the Weibull distribution gets worse by decreasing a threshold m c . We propose the following hypothesis, "the interoccurrence time distribution can be described by the superposition of the Weibull distribution and the distribution P X (τ)" which is the same in 3.1. As shown in Table 7, the log-Weibull distribution is the most suitable for the distribution P X (τ), because the smallest rms-value, the smallest DKS value and the largest Q value can be obtained. Therefore, we find that the Weibull -log-Weibull transition shows in Southern California earthquakes. The crossover magnitude m * * c is estimated to be 3.3.

Interoccurrence time statistics in Taiwan
Finally, the TCWB data was analyzed to investigate the interoccurrence time statistics in Taiwan. Figure 9 shows the cumulative distribution of interoccurrence time for m c = 4.5 and m c = 3.0, respectively. For large m c , the Weibull distribution is preferred on the basis of the rms and KS test [see Table 8]. As the threshold of magnitude m c decreases, the fitting accuracy of the Weibull distribution is getting worse, as is common in JMA and SCEDC. According to a hypothesis that the interoccurrence time distribution can be described by the superposition of the Weibull distribution and another distribution, we investigate the distribution P X (τ).A s shown in Table 9, the log-Weibull distribution is preferred as the distribution on the basis of the two goodness-of-fit tests. We estimated the crossover magnitude m * * c to be 4.9.

Brief summary of the interoccurrence time statistics for earthquakes
Taken all together, we clarified that distribution of interoccurrence time is well fitted by the superposition of the Weibull distribution and log-Weibull distribution. For large m c , P(τ) obeys the Weibull distribution with α 1 < 1, indicating that the occurrence of earthquakes is not a Possion process. When the threshold of magnitude m c decreases, the ratio of the Weibull distribution of P(τ) gradually increases. We suggest that the Weibull statistics and log-Weibull statistics coexist in interoccurrence time statistics, where the change of the distribution means the change of a dominant distribution. In this case, the dominant distribution changes from the log-Weibull distribution to the Weibull distribution by increasing the m c . It follows that the Weibull -log-Weibull transition exists in Japan, Southern California, and Taiwan.

Size dependency
To investigate the region-size, L dependency of the Weibull -log-Weibull transition, we change the window size L is varied from 3 • to 25 • (17). In (17) Table 9. Interoccurrence time statistics of earthquakes in Taiwan area. The error bars mean the 95% confidence level of fit.  Table 10. It is demonstrated that in all the cases, the Weibull exponent α 1 is less than unity and the Weibull -log-Weibull transition appears. m * * c depends on L,namelym * * c = 3.9 for L = 3, m * * c = 4.0 for L = 5, m * * c = 4.2 for L = 10, and m * * c = 5.0 for L = 25. Therefore we can conclude that the interoccurrence time statistics, namely the Weibull -log-Weibull transition, presented here hold from L = 3toL = 25.

Relation between the m * * c and m max
To study the feature of the Weibull -log-Weibull transition, we summarize our results obtained from 16 different regions (14 regions in Japan, Southern California, and Taiwan.) Interestingly, m * * c is proportional to the maximum magnitude of an earthquake in a region, where we analyzed, denoted here m max [see Figure 10]. We then obtain a region-independent relation between m * * c and m max , m * * c /m max = 0.56 ± 0.08 (16) Table 11. List of the crossover magnitude, m * * c and the plate velocity (32; 33). The notation of PH, EU, PA, and NA represent PHilippine Sea plate, EUrasian plate, PAcific plate, and North American plate, respectively. This relation can be useful to interpret the Weibull -log-Weibull transition of geophysical meaning.

Interpretation of the Weibull -log-Weibull transition
Although the scaled crossover magnitudes m c * * /m max is region-independent, the crossover magnitude m * * c from the superposition regime to the pure Weibull regime probably depends on the tectonic region (Figure 7). To investigate the Weibull -log-Weibull transition further, we consider the plate velocity with m * * c , which can shed light on the geophysical implication of the region-dependent m * * c . As shown in Table 11, m * * c is on the average proportional to the plate velocity. That means that the maximum magnitude m max for a tectonic region is more or less proportional to the plate velocity since m * * c /0.56 = m max . Such an interesting consequence is reminiscent of the early study by Ruff and Kanamori (1980) (34). They showed a relation that the magnitude of characteristic earthquake occurred in the subduction-zone, M w is directly proportional to the plate-velocity, V, and is directly inversely proportional to plate-age T,namely , The relation m c * * /0.56 = m max can thus be explained on the basis of their early observation about the velocity-dependence of the characteristic earthquake magnitude. The physical interpretation of the Weibull -log-Weibull transition remains open. However, it might suggest that the occurrence mechanism of earthquake could probably depend on its magnitude then, inevitably, the distribution of the interoccurrence time statistics changes as the threshold of magnitude m c is varied. It is well known that the Weibull distribution for life-time of materials can be derived in the framework of damage mechanics (4; 24; 35-37). Our present results thus suggest that larger earthquakes might be caused by the damage mechanism driven by the plate motion, whereas the effect of the plate-driven damaging process might become minor for smaller earthquakes. Hence, the transition from the Weibull regime to the log-Weibull regime could be interpreted from the geophysical sense as the decrement of the plate-driven damaging mechanics.

A universal relation and intrinsic meanings of the Gutenberg-Richter parameter
Here we consider the interrelation between the Gutenberg-Richter law, denoted in this subsection P(m) ∝ e −bm and the Weibull distribution for the interoccurrence time (P(τ) ∝ t −α−1 · e (τ/β) α ). We assume that these two statistics are correct over wide ranges, and the parameters (α, β) are depending on the magnitude, i.e., α(m) and β(m), then the following relation is easily obtained from the calculation of the mean interoccurrence time between two earthquakes whose magnitude is larger than m , where m 1 and m 2 are arbitrary values of m. This implies that the quantity defined by β(m 1 )e −bm 1 Γ 1 + 1 α 1 is a universal constant when we consider the local earthquakes in a relatively small area.
One of the most important results derived from equation (18) is that the GR parameter b is determined by two parameters, in other words, the parameters (α, β) depend on the magnitude m as well as on the GR parameter b, where the functional forms of f α and f β characterize the time series of earthquakes under consideration.
It is difficult to determine those forms completely from any seismological relations known so far, but it is possible for us to obtain the universal aspects of f α and f β by a perturbational approach. Here we consider a particular solution of equation (19) which satisfies the following m c , b), namely, the characteristic time β is a exponentilally increasing function of m, and the interoccurrence time distribution is an exponential one (α = 1) at m = m c ,w h e r eb ′ and c are constant parameters. By use of this simplification, equation (18) is rewritten by putting m 1 = m and m 2 = m, Here we used the Taylor expansion near m ∼ = m c (i.e., α(m) ∼ = α(m c )). Figure 11 shows the schematic result of equation (20). One can see that the universal relation is recognized in many cases treated in this chapter (section 4.3.3.), though the exponential growth of β, log β(m) ≃ b ′ (m − m c )+c is a little bit accelerated.
We have to remind that the solution mentioned above is not unique, but many other solutions for equation (19) are possible under the universal relation of equation (18). Further details will be studied in our forthcoming paper (38).

Comparison with previous works
Finally, we compared our results with those of previous studies. The unified scaling law shows a generalized gamma distribution [see in equations (2), and (3)] which is approximately the gamma distribution, because δ in Corral's paper (6) is close to unity (δ = 0.98 ± 0.05). For a long time domain, this distribution decays exponentially, supporting the view that an earthquake is a Poisson process. However, we have demonstrated that the Weibull distribution is more appropriate than the gamma distribution on the basis of two goodness-fit-tests. In addition, for large m c , the distribution in a long time domain is similar to the stretched exponential distribution because α 1 is less than unity, suggesting that an occurrence of earthquake is not a Poisson process but has a memory. We provide the first evidence that the distribution changes from the Weibull to log-Weibull distribution by varying m c , i.e., the Weibull -log-Weibull transition. Recently, Abaimov et al. showed that the recurrence time distribution is also well-fitted by the Weibull distribution (4) rather than the Brownian passage time (BPT) distribution (23) and the log normal distribution. Taken together, we infer that both the recurrence time statistics and the interoccurrence time statistics show the Weibull distribution.
In this chapter, we propose a new insight into the interoccurrence time statistics, stating that the interoccurrence statistics exhibit the Weibull -log-Weibull transition by analyzing the different tectonic settings, JMA, SCEDC, and TCWB. This stresses that the distribution function can be described by the superposition of the Weibull distribution and the log-Weibull distribution, and that the predominant distribution function changes from the log-Weibull distribution to the Weibull distribution as m c is increased. Note that there is a possibility that a more suitable distribution might be found instead of the log-Weibull distribution. Furthermore, the Weibull -log-Weibull transition can be extracted more clearly by analyzing synthetic catalogs produced by the spring-block model [see Figure 12] (29).

Conclusion
In conclusion, we have proposed a new feature of interoccurrence time statistics by analyzing the Japan (JMA), Southern California, (SCEDC), and Taiwan (TCWB) for different tectonic conditions. We found that the distribution of the interoccurrence time can be described clearly by the superposition of the Weibull distribution and the log-Weibull distribution. Especially for large earthquakes, the interoccurrence time distribution obeys the Weibull distribution with the exponent α 1 < 1, indicating that a large earthquake is not a Poisson process but a phenomenon exhibiting a long-tail distribution. As the threshold of magnitude m c increases, the ratio of the Weibull distribution in the interoccurrence time distribution p gradually increases. Our findings support the view that the Weibull statistics and log-Weibull statistics coexist in the interoccurrence time statistics. We interpret the change of distribution function as the change of the predominant distribution function; the predominant distribution changes from the log-Weibull distribution to the Weibull distribution when m c is increased. Therefore, we concluded that the interoccurrence time statistics exhibit a Weibull -log-Weibull transition. We also find the region-independent relation, namely, m * * c /m max = 0.56 ± 0.08. In addition, the crossover magnitude m * * c is proportional to the plate velocity, which is consistent with an earlier observation about the velocity-dependence of the characteristic earthquake magnitude (34). Although the origins of both the log-Weibull distribution and the Weibulllog-Weibull transition remain open questions, we suggest the change in the distribution from the log-Weibull distribution to the Weibull distribution can be considered as the enhancement in the plate-driven damaging mechanics. We believe that this work is a first step toward a theoretical and geophysical understanding of this transition. The study of earthquakes plays a key role in order to minimize human and material losses when they inevitably occur. Chapters in this book will be devoted to various aspects of earthquake research and analysis. The different sections present in the book span from statistical seismology studies, the latest techniques and advances on earthquake precursors and forecasting, as well as, new methods for early detection, data acquisition and interpretation. The topics are tackled from theoretical advances to practical applications.