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

Tomohiro Hasumi1, Chien-chih Chen2, Takuma Akimoto3 and Yoji Aizawa4

*1Division of Environment, Natural Resources and Energy, Mizuho Information and Research Institute, Inc., Tokyo 2Department of Earth Sciences and Graduate Institute of Geophysics, National Central University, Jhongli, Taoyuan 3Department of Mechanical Engineering, Keio University, Yokohama 4Department of Applied Physics, Advanced School of Science and Engineering, Waseda University, Tokyo 1,3,4Japan 2Taiwan* 

#### **1. Introduction**

Earthquakes are great complex phenomenon characterized by several empirical statistical laws (1). One of the most important statistical law is the Gutenberg - Richter law (2), where the cumulative number of *n*(> *m*) of magnitude *m* satisfy the following relation:

$$
\log n(>m) = a - bm,\tag{1}
$$

where *a* and *b* are constants. *b* is so-called *b*-value and is similar to unity. Another important statistical law is a power law decay of the occurrence of aftershocks, called Omori law (3).

The time intervals between successive earthquakes can be classified into two types: interoccurrence times and recurrence times (4). Interoccurrence times are the interval times between earthquakes on all faults in a region, and recurrence times are the time intervals between earthquakes in a single fault or fault segment. For seismology, recurrence times mean the interval times of characteristic earthquakes that occur quasi-periodically in a single fault. Recently, a unified scaling law of interoccurrence times was found using the Southern California earthquake catalogue (5) and worldwide earthquake catalogues (6). In Corral's paper (6), the probability distribution of interoccurrence time, *P*(*τ*), can be written as

$$P(\tau) = Rf(R\tau),\tag{2}$$

$$f(R\tau) = \mathcal{C} \frac{1}{(R\tau)^{1-\gamma}} \exp(-(R\tau)^{\delta}/B),\tag{3}$$

where *R* is the seismicity rate. He has found that *f*(*Rτ*) follows the generalized gamma distribution. In equation (3), *C* is a normalized constant and is *C* = 0.50 ± 0.05. *γ*, *δ*, and *B* are parameters estimated to be *γ* = 0.67 ± 0.05, *δ* = 0.98 ± 0.05, and *B* = 1.58 ± 0.15. It should


Table 1. Information on earthquake catalogues.

be noted that the interoccurrence times were analyzed for the events with the magnitude *m* above a certain threshold *mc* 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,

$$P(>\tau') = \frac{1}{(1+\varepsilon\tau')^\gamma} = e\_q(-\tau'/\tau\_0) = [\left(1+(1-q)(-\tau'/\tau\_0)\right)^{\frac{1}{1-q}}]\_+,\tag{4}$$

500 km

**2.1.1 Japan Metrological Agency (JMA) earthquake catalogue**

4195 8035

40840 16972

16926

6

(b)

5

4

3

log cumulative frequency

The Weibull – Log-Weibull Transition of Interoccurrence Time of Earthquakes 5

Fig. 1. Information on the Japan Metrological Agency (JMA) earthquake catalogues. (a) covering region. The number of each cell means the number of earthquakes. (b) the magnitude distribution. *b* = 0.84 is calculated from the slope of the distribution.

**2.1.2 Southern California Earthquake Data Center (SCEDC) earthquake catalogue**

**2.1.3 Taiwan Central Weather Bureau (TCWB) 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*<sup>0</sup>

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

TCWB catalogue is maintained by the Central Weather Bureau, covering from 21◦ to 26◦ N for 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 (b), the Gutenberg - Richter law is valid in a

2

1

0

1 2 3 4 5 6 7 8

0.84

*<sup>c</sup>* is

magnitude

1789

16834 2225

estimated to be 2.0.

distribution, and we obtain *b* = 0.97.

(a)

9301 13828 16707

16870 4259

2019

where *qt*, *τ*0, *γ*, and *�* 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" (13–16). 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.

#### **2. 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 *mmin* corresponds to the minimum magnitude in the catalogue and *m*<sup>0</sup> *<sup>c</sup>* 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*<sup>0</sup> *<sup>c</sup>* because events whose magnitudes are smaller than *m*<sup>0</sup> *<sup>c</sup>* are supposedly incomplete for recording.

2 Earthquake Research and Analysis / Book 5

Catalog Name Coverage Term Number of Earthquakes *mmin m*<sup>0</sup>

be noted that the interoccurrence times were analyzed for the events with the magnitude *m* above a certain threshold *mc* 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

/*τ*0)=[

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" (13–16). 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

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 *mmin* corresponds to the minimum magnitude in the

Gutenberg - Richter law holds. We basically consider events with magnitude greater than and

*<sup>c</sup>* because events whose magnitudes are smaller than *m*<sup>0</sup>

*<sup>c</sup>* is the magnitude of completeness, that is the lowest magnitude at which the

1 + (1 − *q*)(−*τ*�

/*τ*0) 1

<sup>1</sup>−*<sup>q</sup>* ]+, (4)

*<sup>c</sup>* are supposedly incomplete

distribution with *q* > 1, corresponding to a power law distribution (10), namely,

(<sup>1</sup> <sup>+</sup> *�τ*�)*<sup>γ</sup>* <sup>=</sup> *eq*(−*τ*�

where *qt*, *τ*0, *γ*, and *�* are positive constants and ([*a*]+ ≡ max[0, *a*]).

Table 1. Information on earthquake catalogues.

) = <sup>1</sup>

*P*(> *τ*�

big mainshocks.

catalogue and *m*<sup>0</sup>

equal to *m*<sup>0</sup>

for recording.

**2. Data and methodology**

**2.1 Earthquake catalogue**

JMA 25◦ –50◦ N and 125◦ –150◦ E 01/01/2001–31/1/2010 170,801 2.0 2.0 SCEDC 32◦ –37◦ N and 114◦ –122◦ W 01/01/2001–28/2/2010 116,089 0.0 1.4 TCWB 21◦ N–26◦ N and 119◦ –123◦ E 01/01/2001–28/2/2010 189,980 0.0 1.9

*c*

Fig. 1. Information on the Japan Metrological Agency (JMA) earthquake catalogues. (a) covering region. The number of each cell means the number of earthquakes. (b) the magnitude distribution. *b* = 0.84 is calculated from the slope of the distribution.

#### **2.1.1 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*<sup>0</sup> *<sup>c</sup>* is estimated to be 2.0.

#### **2.1.2 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.

#### **2.1.3 Taiwan Central Weather Bureau (TCWB) earthquake catalogue**

TCWB catalogue is maintained by the Central Weather Bureau, covering from 21◦ to 26◦ N for 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 (b), the Gutenberg - Richter law is valid in a

1

the information from previous studies (6; 12).

*Pln* (23), which are defined as

**2.2 Methodology (How to detect the appropriate distributions)**

Our method is similar to the that of previous works (17; 18) (see Figure 4).

3. We analyzed interoccurrence times greater than and equals to *h* day.

*Pw*(*τ*) = *<sup>τ</sup>*

(log *β*2)*α*<sup>2</sup>

*Plw*(*τ*) = (log(*τ*/*h*))*α*2−<sup>1</sup>

*β*1

*α*2 *<sup>τ</sup>* exp

than or equals to *h*.

latitude.

considered.

Fig. 4. Schematic diagram of the interoccurrence time of our analysis for different threshold of magnitude *mc*. Circles (◦) satisfy the condition. We analyze interoccurrence times greater

The Weibull – Log-Weibull Transition of Interoccurrence Time of Earthquakes 7

1. We divided the spatial areas into a window of *L* degrees in longitude and *L* degrees in

2. For each bin, earthquakes with magnitude *m* above a certain cutoff magnitude *mc* were

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 TCWB, we analyze earthquake covering the whole region. As shown in Figure 4, we 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

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 *Pw*, the log-Weibull distribution *Plw* (22), the power law distribution *Ppow* (10), the gamma distribution *Pgam* (in the case of *δ* = 1 in the paper (6)), and the log normal distribution

*<sup>α</sup>*1−<sup>1</sup> *<sup>α</sup>*<sup>1</sup>

*β*1

−

exp − *τ β*1

*Ppow*(*τ*) = <sup>1</sup>

log(*τ*/*h*) log *β*<sup>2</sup>

*α*<sup>1</sup>

*α*<sup>2</sup>

(1 + *β*3*τ*)*α*<sup>3</sup>

, (5)

, (6)

, (7)

*t*

2

3

*h*

Fig. 2. Southern California Earthquake Data Center (SCEDC) earthquake catalogue information. (a) covering region. (b) the magnitude distribution. *b* = 0.97 is calculated from the slope of the distribution.

Fig. 3. Information on the Taiwan Central Weather Bureau (TCWB) earthquake catalogue. (a) covering region. (b) the magnitude distribution. *b* = 0.90 is calculated from the slope of the distribution.

magnitude range, 1.9 ≤ *m* ≤ 6.7. *b*-value is calculated from the slope of the distribution, and is estimated to be 0.90.

4 Earthquake Research and Analysis / Book 5

246°

> 1 2 3 4 5 6 magnitude

1 2 3 4 5 6 7

0.90

magnitude

0.97

log cumulative frequency

(b) 6

5

4

3

log cumulative frequency

2

1

0

245°

238°

119°

119°

distribution.

is estimated to be 0.90.

120°

121°

21° 21°

22° 22°

23° 23°

24° 24°

25° 25°

122°

123°

Fig. 3. Information on the Taiwan Central Weather Bureau (TCWB) earthquake catalogue. (a) covering region. (b) the magnitude distribution. *b* = 0.90 is calculated from the slope of the

magnitude range, 1.9 ≤ *m* ≤ 6.7. *b*-value is calculated from the slope of the distribution, and

239°

240°

the slope of the distribution.

120°

241°

242°

121°

26° 26°

32° 32°

33° 33°

34° 34°

35° 35°

36° 36°

243°

122°

244°

245°

Fig. 2. Southern California Earthquake Data Center (SCEDC) earthquake catalogue

123°

(a)

246°

information. (a) covering region. (b) the magnitude distribution. *b* = 0.97 is calculated from

238°

239°

240°

241°

242°

37° 37°

243°

(a) (b)

244°

Fig. 4. Schematic diagram of the interoccurrence time of our analysis for different threshold of magnitude *mc*. Circles (◦) satisfy the condition. We analyze interoccurrence times greater than or equals to *h*.

#### **2.2 Methodology (How to detect the appropriate distributions)**

Our method is similar to the that of previous works (17; 18) (see Figure 4).


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 TCWB, we analyze earthquake covering the whole region. As shown in Figure 4, we 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 *Pw*, the log-Weibull distribution *Plw* (22), the power law distribution *Ppow* (10), the gamma distribution *Pgam* (in the case of *δ* = 1 in the paper (6)), and the log normal distribution *Pln* (23), which are defined as

$$P\_w(\tau) = \left(\frac{\tau}{\beta\_1}\right)^{a\_1 - 1} \frac{a\_1}{\beta\_1} \exp\left[-\left(\frac{\tau}{\beta\_1}\right)^{a\_1}\right],\tag{5}$$

$$P\_{lw}(\tau) = \frac{(\log(\tau/h))^{a\_2 - 1}}{(\log \beta\_2)^{a\_2}} \frac{a\_2}{\tau} \exp\left[-\left(\frac{\log(\tau/h)}{\log \beta\_2}\right)^{a\_2}\right],\tag{6}$$

$$P\_{pow}(\tau) = \frac{1}{(1 + \beta\_3 \tau)^{a\_3}},\tag{7}$$

$$P\_{\mathcal{S}am}(\tau) = \tau^{a\_4 - 1} \frac{\exp\left(-\tau/\beta\_4\right)}{\Gamma(a\_4)\beta\_4^{\alpha\_4}},\tag{8}$$

1

0.8

0.6

Cumulative distribution

The Weibull – Log-Weibull Transition of Interoccurrence Time of Earthquakes 9

Fig. 5. Cumulative distribution of interoccurrence time at Okinawa region different *mc*. The cumulative distribution is plotted by circles. (a) Several fitting curves are represented by lines (*mc* = 4.5). (b) The superposition of the Weibull and the log-Weibull distribution is

*n*� + 0.12 +

It is known that the preferred distribution shows the smallest value of DKS and the largest

First, we analyze the JMA data. Here, we consider the two region; Okinawa region (125◦–130◦E and 25◦–30◦N ), and Chuetsu region (135◦–140◦E and 35◦–40◦N). The total

The cumulative distributions of the interoccurrence times for different *mc* 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 (*mc* = 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

number of earthquakes in Okinawa and Chuetsu are 16,834 and 16,870, respectively.

0.11 <sup>√</sup>*n*� 

(a) (b) <sup>1</sup>

 Weibull distribution (rms=0.012) log-Weibull distribution (rms=0.029) power law distribution (rms=0.113) gamma distribution (rms=0.025) log-normal distribution (rms=0.028)

*<sup>λ</sup>* <sup>=</sup> *DKS* <sup>√</sup>

0 20 40 60 80 100 Interoccurrence time [day]

where *n*� stands for the number of data points.

**3.1 Interoccurrence time statistics in Japan**

follows that the Weibull distribution is preferred.

data

0.8

0.6

Cumulative distribution

where

value of *Q* (31).

**3. Results**

0.4

0.2

0

represented by line (*mc* = 2.0).

0.4

0.2

0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 Interoccurrence time [day]

data

Weibull + log-Weibull curve (p=0.41)

, (13)

$$P\_{ln}(\tau) = \frac{1}{\tau \beta\_5 \sqrt{2\pi}} \exp\left[-\frac{(\ln(\tau) - \alpha\_5)^2}{2\beta\_5^2}\right],\tag{9}$$

where *αi*, *βi*, and *h* 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*<sup>1</sup> = 1 and *α*<sup>1</sup> < 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*<sup>2</sup> = 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

$$\text{rms} = \sqrt{\frac{\sum\_{i=1}^{n} (\mathbf{x}\_i - \mathbf{x}\_i')^2}{n - k}},\tag{10}$$

where *xi* is actual data and *x*� *<sup>i</sup>* 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

$$\text{DKS} = \max\_{i} |y\_i - y\_i'| \,\text{}\tag{11}$$

where *yi* and *y*� *<sup>i</sup>* 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*, is defined as

$$Q = 2\sum\_{i=1}^{\infty} (-1)^{i-1} e^{2i^2 \lambda^2} \,\,\,\,\,\tag{12}$$

Fig. 5. Cumulative distribution of interoccurrence time at Okinawa region different *mc*. The cumulative distribution is plotted by circles. (a) Several fitting curves are represented by lines (*mc* = 4.5). (b) The superposition of the Weibull and the log-Weibull distribution is represented by line (*mc* = 2.0).

where

6 Earthquake Research and Analysis / Book 5

exp 

where *αi*, *βi*, and *h* 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

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*<sup>1</sup> = 1 and *α*<sup>1</sup> < 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*<sup>2</sup> = 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).

 ∑*n*

DKS = max

∞ ∑ *i*=1

*Q* = 2

rms =

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

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

*<sup>i</sup>*=1(*xi* − *x*�

*<sup>i</sup>* <sup>|</sup>*yi* <sup>−</sup> *<sup>y</sup>*� *i*

from the fitting distribution, respectively. Then, the significance level of probability of the

(−1)*i*−1*<sup>e</sup>*

*<sup>i</sup>* mean the actual data of the cumulative distribution and the data estimated

2*i* <sup>2</sup>*λ*<sup>2</sup>

*i* )2

*<sup>i</sup>* is estimated data obtained from *P*(*τ*). *n* and *k* indicate the

*<sup>n</sup>* <sup>−</sup> *<sup>k</sup>* , (10)


, (12)

*Pln*(*τ*) = <sup>1</sup>

normal distribution, respectively.

goodness-of-fit. The definition of the rms value is

where *xi* is actual data and *x*�

Kolomogorov-Smirnov statistic, as

goodness-of-fit, *Q*, is defined as

where *yi* and *y*�

*τβ*5 <sup>√</sup>2*<sup>π</sup>*

*Pgam*(*τ*) = *<sup>τ</sup>α*4−<sup>1</sup> exp (−*τ*/*β*4)

Γ(*α*4)*β*<sup>4</sup>

<sup>−</sup> (ln(*τ*) <sup>−</sup> *<sup>α</sup>*5)<sup>2</sup> 2*β*<sup>2</sup> 5

*<sup>α</sup>*<sup>4</sup> , (8)

, (9)

$$
\lambda = DKS \left( \sqrt{n'} + 0.12 + \frac{0.11}{\sqrt{n'}} \right) / \tag{13}
$$

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).

#### **3. Results**

#### **3.1 Interoccurrence time statistics in Japan**

First, we analyze the JMA data. Here, we consider the two region; Okinawa region (125◦–130◦E and 25◦–30◦N ), and Chuetsu region (135◦–140◦E and 35◦–40◦N). The total number of earthquakes in Okinawa and Chuetsu are 16,834 and 16,870, respectively.

The cumulative distributions of the interoccurrence times for different *mc* 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 (*mc* = 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.

*mc* Weibull distribution Distribution X RMS test KS test Region distribution X *<sup>α</sup>*<sup>1</sup> *<sup>β</sup>*<sup>1</sup> [day] *<sup>α</sup><sup>i</sup> <sup>β</sup><sup>i</sup>* [day] *<sup>p</sup>* rms [×10−3] DKS *<sup>Q</sup>*

The Weibull – Log-Weibull Transition of Interoccurrence Time of Earthquakes 11

*Plw* (*i* = 2) 0.82 ± 0.007 19.0 ± 0.13 − − 1 12 0.03 1 4.5 *Ppow* (*i* = 3) 0.82 ± 0.007 19.0 ± 0.13 − − 1 12 0.03 1 Okinawa *Pgam* (*i* = 4) 0.82 ± 0.007 19.0 ± 0.13 − − 1 12 0.03 1

*Pln* (*i* = 5) 0.82 ± 0.007 19.0 ± 0.13 − − 1 12 0.03 1 *Plw* (*i* = 2) 0.93 ± 0.009 8.47 ± 0.06 − − 1 13 0.03 1 4.0 *Ppow* (*i* = 3) 0.93 ± 0.009 8.47 ± 0.06 − − 1 13 0.03 1 Okinawa *Pgam* (*i* = 4) 0.93 ± 0.009 8.47 ± 0.06 − − 1 13 0.03 1

*Pln* (*i* = 5) 0.93 ± 0.009 8.47 ± 0.06 − − 1 13 0.03 1 *Plw* (*i* = 2) 1.07 ± 0.008 3.45 ± 0.02 2.07 ± 0.04 6.25 ± 0.09 0.77 ± 0.02 5.3 0.02 1 3.5 *Ppow* (*i* = 3) 1.07 ± 0.008 3.45 ± 0.02 1.81 ± 0.04 0.64 ± 0.04 0.94 ± 0.009 7.3 0.03 1 Okinawa *Pgam* (*i* = 4) 1.07 ± 0.008 3.45 ± 0.02 − − 1 8.8 0.04 1

*Pln* (*i* = 5) 1.07 ± 0.008 3.45 ± 0.02 0.85 ± 0.009 0.94 ± 0.01 0.65 ± 0.03 5.6 0.02 1 *Plw* (*i* = 2) 1.44 ± 0.02 1.77 ± 0.02 1.40 ± 0.03 2.57 ± 0.04 0.58 ± 0.01 3.8 0.01 1 3.0 *Ppow* (*i* = 3) 1.41 ± 0.02 1.63 ± 0.01 2.22 ± 0.04 0.55 ± 0.01 0.79 ± 0.01 8.8 0.07 0.73 Okinawa *Pgam* (*i* = 4) 1.41 ± 0.02 1.63 ± 0.01 − − 1 17 0.1 0.34

*Pln* (*i* = 5) 1.41 ± 0.02 1.63 ± 0.01 0.20 ± 0.003 0.72 ± 0.003 0.04 ± 0.03 5.4 0.04 1 *Plw* (*i* = 2) 1.72 ± 0.02 1.14 ± 0.008 1.25 ± 0.01 1.68 ± 0.01 0.47 ± 0.006 2.1 0.007 1 2.5 *Ppow* (*i* = 3) 1.90 ± 0.05 1.01 ± 0.008 2.84 ± 0.04 0.51 ± 0.005 0.58 ± 0.02 9.2 0.04 1 Okinawa *Pgam* (*i* = 4) 1.90 ± 0.05 1.01 ± 0.008 1.07 ± 0.02 0.94 ± 0.02 0.99 ± 0.04 22 0.14 0.04

*Pln* (*i* = 5) − − −0.20 ± 0.004 0.52 ± 0.005 0 12 0.07 0.61 *Plw* (*i* = 2) 1.78 ± 0.03 0.76 ± 0.009 1.16 ± 0.01 1.44 ± 0.008 0.41 ± 0.008 2.0 0.007 1 2.0 *Ppow* (*i* = 3) 2.57 ± 0.10 0.77 ± 0.007 3.61 ± 0.05 0.48 ± 0.003 0.40 ± 0.02 7.0 0.03 1 Okinawa *Pgam* (*i* = 4) 2.57 ± 0.10 0.77 ± 0.007 1.09 ± 0.03 0.68 ± 0.03 0.96 ± 0.05 25 0.14 0.18

*Pln* (*i* = 5) − − −0.41 ± 0.005 0.39 ± 0.008 0 15 0.08 0.78

Table 4. Interoccurrence time statistics of earthquakes in Okinawa region. The error bars

another distribution, hereafter referred to as the distribution *PX*(*τ*),

Next we shall explain the parameter estimation procedures;

and the test function by varying five parameters, *α*1, *β*1, *αi*, *β<sup>i</sup>* and *p*.

However, the fitting accuracy of the Weibull distribution becomes worse with a gradual decrease in *mc*. We now propose a possible explanation which states that "the interoccurrence time distribution can be described by the superposition of the Weibull distribution and

*P*(*τ*) = *p* × Weibull distribution + (1 − *p*) × distribution X

where *p* is a parameter in the range, 0 ≤ *p* ≤ 1 and stands for the ratio of *Pw* divided by *P*(*τ*). The interoccurrence time distribution obeys the Weibull distribution for *p* = 1. On the other hand, it follows the distribution *PX*(*τ*) for *p* = 0. Here, the log-Weibull distribution, the power law distribution, the gamma distribution, and the log normal distribution are candidates for

(A); the optimal parameters are estimated so as to minimize the differences between the data

= *p* × *Pw*(*τ*)+(1 − *p*) × *PX*(*τ*) (14)

mean the 95% confidence level of fit.

the distribution *PX*(*τ*).


Table 2. Results of rms value, DKS, and *Q* for different distribution functions for Okinawa area (*mc* = 4.5). The error bars mean the 95% confidence level of fit.

Fig. 6. Cumulative distribution of interoccurrence time for Chuetsu area different *mc*. The cumulative distribution is plotted by circles. (a) Several fitting curves are represented by lines (*mc* = 4.5). (b) The superposition of the Weibull and the log-Weibull distribution is represented by line (*mc* = 2.0).


Table 3. Results of rms value, DKS, and *Q* for different distribution functions for Chuetsu area (*mc* = 4.5). The error bars mean the 95% confidence level of fit.


*mc* Distribution X RMS test KS test Region distribution X *<sup>α</sup><sup>i</sup> <sup>β</sup><sup>i</sup>* [day] rms [×10−3] DKS *<sup>Q</sup>*

Table 2. Results of rms value, DKS, and *Q* for different distribution functions for Okinawa

area (*mc* = 4.5). The error bars mean the 95% confidence level of fit.

1

0.8

0.6

Cumulative distribution

0.4

0.2

0

represented by line (*mc* = 2.0).

(a) (b) <sup>1</sup>

50 100 150

 Weibull distribution (rms=0.021) log-Weibull distribution (rms=0.039) power law distribution (rms=0.107) gamma distribution (rms=0.038) log-normal distribution (rms=0.039)

Interoccurrence time [day]

data

*Pw* (*i* = 1) 0.82 ± 0.007 19.0 ± 0.13 12 0.03 1 *Plw* (*i* = 2) 3.08 ± 0.06 35.3 ± 0.57 29 0.1 0.15 4.5 *Ppow* (*<sup>i</sup>* <sup>=</sup> <sup>3</sup>) 1.48 <sup>±</sup> 0.02 1.04 <sup>±</sup> 0.12 <sup>113</sup> 0.23 3.8×10−<sup>3</sup> Okinawa *Pgam* (*i* = 4) 0.96 ± 0.005 19.5 ± 0.24 24 0.07 0.55

*Pln* (*i* = 5) 2.45 ± 0.02 1.20 ± 0.02 28 0.09 0.33

0.8

0.6

Cumulative distribution

Fig. 6. Cumulative distribution of interoccurrence time for Chuetsu area different *mc*. The cumulative distribution is plotted by circles. (a) Several fitting curves are represented by lines (*mc* = 4.5). (b) The superposition of the Weibull and the log-Weibull distribution is

> *mc* Distribution X RMS test KS test Region distribution X *<sup>α</sup><sup>i</sup> <sup>β</sup><sup>i</sup>* [day] rms [×10−3] DKS *<sup>Q</sup>*

Table 3. Results of rms value, DKS, and *Q* for different distribution functions for Chuetsu

area (*mc* = 4.5). The error bars mean the 95% confidence level of fit.

*Pl* (*i* = 1) 0.75 ± 0.01 27.6 ± 0.40 21 0.06 0.94 *Plw* (*i* = 2) 3.12 ± 0.10 51.4 ± 1.39 39 0.14 0.06 4.5 *Ppow* (*<sup>i</sup>* <sup>=</sup> <sup>3</sup>) 1.47 <sup>±</sup> 0.03 1.51 <sup>±</sup> 0.21 <sup>107</sup> 0.19 4.8×10−<sup>3</sup> Chuetsu *Pgam* (*i* = 4) 0.94 ± 0.009 28.9 ± 0.67 38 0.11 0.27

*Pln* (*i* = 5) 2.78 ± 0.03 1.33 ± 0.04 39 0.12 0.15

0.4

0.2

0

0.5 1.0 1.5 2.0 2.5 3.0 Interoccurrence time [day]

Weibull + log-Weibull distribution (p=0.50)

data


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 *mc*. 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 *PX*(*τ*),

$$P(\tau) = p \times \text{Weibull distribution} + (1 - p) \times \text{distribution} \,\text{X}$$

$$= p \times P\_{\text{W}}(\tau) + (1 - p) \times P\_{\text{X}}(\tau) \tag{14}$$

where *p* is a parameter in the range, 0 ≤ *p* ≤ 1 and stands for the ratio of *Pw* divided by *P*(*τ*). The interoccurrence time distribution obeys the Weibull distribution for *p* = 1. On the other hand, it follows the distribution *PX*(*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 *PX*(*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, *αi*, *β<sup>i</sup>* and *p*.

500 km

*<sup>c</sup>* for each

*<sup>c</sup>* depends on the region and ranges

3.8 3.9

4.5

3.8

4.8

2.4

3.7 2.5

regime in Figure 7. As can be seen from the figure, *m*∗∗

**3.2 Interoccurrence time statistics in Southern California**

*<sup>c</sup>* , map around Japan.

denoted *m*∗∗

denoted by *m*∗∗

where we analyzed.

3.4 4.3 3.7

Fig. 7. The crossover magnitude from the superposition regime to the (pure) Weibull regime,

It has been shown that the interoccurrence time distribution of earthquakes with large mc obeys the Weibulldistribution with the exponent *α*<sup>1</sup> < 1. As shown in Tables 4 and 5, we stress the point that the distribution function of the interoccurrence time changes by varying *mc*. This indicates that the interoccurrence time statistics basically contains both Weibull and log-Weibull statistics, and a dominant distribution function is changed according to the ratio *p*. In this case, the dominant distribution of the interoccurrence time changes from the log-Weibull distribution to the Weibull distribution when *mc* is increased. Thus, the interoccurrence time statistics exhibit transition from the Weibull regime to the log-Weibull regime. The crossover magnitude from the superposition regime to the Weibull regime,

*<sup>c</sup>* , depends on the spatial area. We demonstrate the values of *m*∗∗

from 2.4 (140◦–145◦E and 45◦–50◦N) to 4.8 (145◦–150◦E and 40◦–45◦N). Comparing Figure 1 (a) and Figure 7, we have found that the Weibull - log-Weibull transition occurs in all region,

Second, we analyze the interoccurrence time statistics using the SCEDC data. The cumulative distributions of interoccurrence time for *mc* = 4.0 and *mc* = 2.0 are shown in Figure 8 (a) and (b), respectively. By the rms test and KS test, we confirmed that the Weibull distribution is preferred for large *mc* (*mc* = 4.0) [see Table 6], which is the same result as that from JMA

3.6

2.8

The Weibull – Log-Weibull Transition of Interoccurrence Time of Earthquakes 13

3.4


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 *Cv*, 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*<sup>1</sup> and *β*1, and the parameters of *PX*(*τ*), *α<sup>i</sup>* and *βi*, are optimized dependently and then *p* is estimated.

According to those procedures (A) and (B), we obtain the fitting of results of *P*(*τ*). The results 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 *mc*, which means that the effect of the distribution *PX*(*τ*) is negligible. As observed in Table 4 and 5, the log-Weibull distribution is the most suitable distribution for the distribution *PX*(*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,

$$P(\tau) = p \times \text{Weibull distribution} + (1 - p) \times \text{log-Weibull distribution}$$

$$= p \times P\_{\text{W}} + (1 - p) \times P\_{\text{lw}} \tag{15}$$

*P*(*τ*) is controlled by five parameters, *α*1, *α*2, *β*1, *β*2, and *p*.

10 Earthquake Research and Analysis / Book 5

*mc* Weibull distribution Distribution X RMS test KS test Region distribution X *<sup>α</sup>*<sup>1</sup> *<sup>β</sup>*<sup>1</sup> [day] *<sup>α</sup><sup>i</sup> <sup>β</sup><sup>i</sup>* [day] *<sup>p</sup>* rms [×10−3] DKS *<sup>Q</sup>*

*Plw* (*i* = 2) 0.75 ± 0.01 27.6 ± 0.40 − − 1 21 0.06 0.94 4.5 *Ppow* (*i* = 3) 0.75 ± 0.01 27.6 ± 0.40 − − 1 21 0.06 0.94 Chuetsu *Pgam* (*i* = 4) 0.75 ± 0.01 27.6 ± 0.40 − − 1 21 0.06 0.94

*Pln* (*i* = 5) 0.75 ± 0.01 27.6 ± 0.40 − − 1 21 0.06 0.94 *Plw* (*i* = 2) 0.81 ± 0.01 10.6 ± 0.13 − − 1 17 0.03 1 4.0 *Ppow* (*i* = 3) 0.81 ± 0.01 10.6 ± 0.13 − − 1 17 0.03 1 Chuetsu *Pgam* (*i* = 4) 0.81 ± 0.01 10.6 ± 0.13 − − 1 17 0.03 1

*Pln* (*i* = 5) 0.81 ± 0.01 10.6 ± 0.13 − − 1 17 0.03 1 *Plw* (*i* = 2) 0.89 ± 0.006 4.84 ± 0.02 2.08 ± 0.06 8.70 ± 0.19 0.93 ± 0.02 6.1 0.03 1 3.5 *Ppow* (*i* = 3) 0.89 ± 0.006 4.84 ± 0.02 1.66 ± 0.04 0.62 ± 0.05 0.98 ± 0.009 6.3 0.03 1 Chuetsu *Pgam* (*i* = 4) 0.89 ± 0.006 4.84 ± 0.02 − − 1 6.9 0.04 1

*Pln* (*i* = 5) 0.89 ± 0.006 4.84 ± 0.02 1.12 ± 0.02 1.11 ± 0.02 0.90 ± 0.04 6.2 0.03 1 *Plw* (*i* = 2) 1.06 ± 0.02 2.00 ± 0.06 1.90 ± 0.16 5.13 ± 0.39 0.82 ± 0.03 3.9 0.012 1 3.0 *Ppow* (*i* = 3) 1.09 ± 0.008 2.14 ± 0.01 1.98 ± 0.04 0.52 ± 0.02 0.92 ± 0.009 5.0 0.02 1 Chuetsu *Pgam* (*i* = 4) 1.09 ± 0.008 2.14 ± 0.01 − − 1 6.5 0.03 1

*Pln* (*i* = 5) 1.09 ± 0.008 2.14 ± 0.01 0.39 ± 0.008 0.91 ± 0.02 0.63 ± 0.03 3.7 0.014 1 *Plw* (*i* = 2) 1.48 ± 0.02 1.21 ± 0.009 1.19 ± 0.02 1.91 ± 0.02 0.62 ± 0.008 2.5 0.01 1 2.5 *Ppow* (*i* = 3) 1.56 ± 0.03 1.16 ± 0.009 2.53 ± 0.04 0.49 ± 0.008 0.71 ± 0.01 6.4 0.03 1 Chuetsu *Pgam* (*i* = 4) 1.56 ± 0.03 1.16 ± 0.009 1.03 ± 0.009 1.09 ± 0.03 0.99 ± 0.04 15 0.01 0.38 *Pln* (*i* = 5) − − −0.11 ± 0.003 0.64 ± 0.004 0 5.1 0.04 1 *Plw* (*i* = 2) 1.85 ± 0.03 0.80 ± 0.009 1.18 ± 0.02 1.43 ± 0.01 0.50 ± 0.01 2.2 0.007 1 2.0 *Ppow* (*i* = 3) 2.46 ± 0.09 0.78 ± 0.007 3.51 ± 0.06 0.48 ± 0.004 0.47 ± 0.03 8.2 0.02 1 Chuetsu *Pgam* (*i* = 4) 2.46 ± 0.09 0.78 ± 0.007 1.19 ± 0.05 0.69 ± 0.02 0.97 ± 0.05 2.6 0.12 0.46

*Pln* (*i* = 5) − − −0.40 ± 0.005 0.40 ± 0.008 0 14 0.06 0.99

Table 5. Interoccurrence time statistics of earthquakes in Chuetsu area. The error bars mean

If there is a parameter, where *Cv*, the ratio of the standard deviation divided by the mean for

(B); the Weibull parameters, *α*<sup>1</sup> and *β*1, and the parameters of *PX*(*τ*), *α<sup>i</sup>* and *βi*, are optimized

According to those procedures (A) and (B), we obtain the fitting of results of *P*(*τ*). The results 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 *mc*, which means that the effect of the distribution *PX*(*τ*) is negligible. As observed in Table 4 and 5, the log-Weibull distribution is the most suitable distribution for the distribution *PX*(*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,

*P*(*τ*) = *p* × Weibull distribution + (1 − *p*) × log-Weibull distribution,

= *p* × *Pw* + (1 − *p*) × *Plw*, (15)

a parameter exceeds 0.1, another estimation procedure, (B), is performed.

*P*(*τ*) is controlled by five parameters, *α*1, *α*2, *β*1, *β*2, and *p*.

the 95% confidence level of fit.

namely,

dependently and then *p* is estimated.

Fig. 7. The crossover magnitude from the superposition regime to the (pure) Weibull regime, denoted *m*∗∗ *<sup>c</sup>* , map around Japan.

It has been shown that the interoccurrence time distribution of earthquakes with large mc obeys the Weibulldistribution with the exponent *α*<sup>1</sup> < 1. As shown in Tables 4 and 5, we stress the point that the distribution function of the interoccurrence time changes by varying *mc*. This indicates that the interoccurrence time statistics basically contains both Weibull and log-Weibull statistics, and a dominant distribution function is changed according to the ratio *p*. In this case, the dominant distribution of the interoccurrence time changes from the log-Weibull distribution to the Weibull distribution when *mc* is increased. Thus, the interoccurrence time statistics exhibit transition from the Weibull regime to the log-Weibull regime. The crossover magnitude from the superposition regime to the Weibull regime, denoted by *m*∗∗ *<sup>c</sup>* , depends on the spatial area. We demonstrate the values of *m*∗∗ *<sup>c</sup>* for each regime in Figure 7. As can be seen from the figure, *m*∗∗ *<sup>c</sup>* depends on the region and ranges from 2.4 (140◦–145◦E and 45◦–50◦N) to 4.8 (145◦–150◦E and 40◦–45◦N). Comparing Figure 1 (a) and Figure 7, we have found that the Weibull - log-Weibull transition occurs in all region, where we analyzed.

#### **3.2 Interoccurrence time statistics in Southern California**

Second, we analyze the interoccurrence time statistics using the SCEDC data. The cumulative distributions of interoccurrence time for *mc* = 4.0 and *mc* = 2.0 are shown in Figure 8 (a) and (b), respectively. By the rms test and KS test, we confirmed that the Weibull distribution is preferred for large *mc* (*mc* = 4.0) [see Table 6], which is the same result as that from JMA

*mc* Weibull distribution Distribution X RMS test KS test Region distribution X *<sup>α</sup>*<sup>1</sup> *<sup>β</sup>*<sup>1</sup> [day] *<sup>α</sup><sup>i</sup> <sup>β</sup><sup>i</sup>* [day] *<sup>p</sup>* rms [×10−3] DKS *<sup>Q</sup>*

The Weibull – Log-Weibull Transition of Interoccurrence Time of Earthquakes 15

*Plw* (*i* = 2) 0.91 ± 0.01 25.4 ± 0.21 − − 1 18 0.07 0.56 4.0 *Ppow* (*i* = 3) 0.91 ± 0.01 25.4 ± 0.21 − − 1 18 0.07 0.56 SCEDC *Pgam* (*i* = 4) 0.91 ± 0.01 25.4 ± 0.21 − − 1 18 0.07 0.56

*Pln* (*i* = 5) 0.91 ± 0.01 25.4 ± 0.21 − − 1 18 0.07 0.56 *Plw* (*i* = 2) 0.83 ± 0.006 9.29 ± 0.05 − − 1 11 0.03 1 3.5 *Ppow* (*i* = 3) 0.83 ± 0.006 9.29 ± 0.05 − − 1 11 0.03 1 SCEDC *Pgam* (*i* = 4) 0.83 ± 0.006 9.29 ± 0.05 − − 1 11 0.03 1

*Pln* (*i* = 5) 0.83 ± 0.006 9.29 ± 0.05 − − 1 11 0.03 1 *Plw* (*i* = 2) 1.01 ± 0.01 3.08 ± 0.04 1.37 ± 0.07 3.49 ± 0.19 0.80 ± 0.01 4.1 0.01 1 3.0 *Ppow* (*i* = 3) 0.98 ± 0.008 2.85 ± 0.02 1.85 ± 0.03 0.58 ± 0.03 0.91 ± 0.009 6.1 0.03 1 SCEDC *Pgam* (*i* = 4) − − 0.99 ± 0.002 2.84 ± 0.01 1 8.2 0.05 1

*Pln* (*i* = 5) 0.98 ± 0.008 2.85 ± 0.02 0.63 ± 0.009 1.00 ± 0.01 0.60 ± 0.03 5.0 0.03 1 *Plw* (*i* = 2) 1.32 ± 0.01 1.72 ± 0.01 1.35 ± 0.01 2.40 ± 0.02 0.57 ± 0.005 2.0 0.01 1 2.5 *Ppow* (*i* = 3) 1.33 ± 0.02 1.55 ± 0.009 2.27 ± 0.03 0.54 ± 0.008 0.76 ± 0.01 6.4 0.04 0.96 SCEDC *Pgam* (*i* = 4) 1.33 ± 0.02 1.17 ± 0.009 − − 1 12 0.10 0.04

*Pln* (*i* = 5) − − 0.13 ± 0.002 0.75 ± 0.003 0 4.4 0.04 0.92 *Plw* (*i* = 2) 1.88 ± 0.03 0.92 ± 0.008 1.15 ± 0.02 1.56 ± 0.01 0.47 ± 0.008 2.7 0.007 1.1 2.0 *Ppow* (*i* = 3) 2.18 ± 0.07 0.88 ± 0.008 3.14 ± 0.05 0.49 ± 0.005 0.52 ± 0.02 8.3 0.11 0.88 SCEDC *Pgam* (*i* = 4) 2.18 ± 0.07 0.88 ± 0.008 1.15 ± 0.04 0.80 ± 0.02 0.98 ± 0.05 26 0.13 0.72 *Pln* (*i* = 5) − − −0.31 ± 0.005 0.46 ± 0.007 0 7.5 0.007 1

Table 7. Interoccurrence time statistics of earthquakes in Southern California area. The error

the Weibull distribution and another distribution, we investigate the distribution *PX*(*τ*). As shown in Table 9, the log-Weibull distribution is preferred as the distribution on the basis of

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 *mc*, *P*(*τ*) obeys the Weibull distribution with *α*<sup>1</sup> < 1, indicating that the occurrence of earthquakes is not a Possion process. When the threshold of magnitude *mc* 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 *mc*. It follows that

the Weibull - log-Weibull transition exists in Japan, Southern California, and Taiwan.

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), we use JMA data we used is from

*<sup>c</sup>* to be 4.9.

the two goodness-of-fit tests. We estimated the crossover magnitude *m*∗∗

**3.4 Brief summary of the interoccurrence time statistics for earthquakes**

bars mean the 95% confidence level of fit.

**4. Discussion**

**4.1 Size dependency**

Fig. 8. Cumulative distribution of interoccurrence time in Southern California region different *mc* and distribution functions. (a) *mc* = 4.0 and (b) *mc* = 2.0.


Table 6. Results of rms value, DKS, and *Q* for different distribution functions in the case of *mc* = 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 *mc*. We propose the following hypothesis, "the interoccurrence time distribution can be described by the superposition of the Weibull distribution and the distribution *PX*(*τ*)" which is the same in 3.1. As shown in Table 7, the log-Weibull distribution is the most suitable for the distribution *PX*(*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*∗∗ *<sup>c</sup>* is estimated to be 3.3.

#### **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 *mc* = 4.5 and *mc* = 3.0, respectively. For large *mc*, the Weibull distribution is preferred on the basis of the rms and KS test [see Table 8]. As the threshold of magnitude *mc* 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


Table 7. Interoccurrence time statistics of earthquakes in Southern California area. The error bars mean the 95% confidence level of fit.

the Weibull distribution and another distribution, we investigate the distribution *PX*(*τ*). As 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*∗∗ *<sup>c</sup>* to be 4.9.

#### **3.4 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 *mc*, *P*(*τ*) obeys the Weibull distribution with *α*<sup>1</sup> < 1, indicating that the occurrence of earthquakes is not a Possion process. When the threshold of magnitude *mc* 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 *mc*. It follows that the Weibull - log-Weibull transition exists in Japan, Southern California, and Taiwan.

#### **4. Discussion**

12 Earthquake Research and Analysis / Book 5

1

0.8

0.6

Cumulative distribution

Fig. 8. Cumulative distribution of interoccurrence time in Southern California region

*mc* Distribution X RMS test KS test Region distribution X *<sup>α</sup><sup>i</sup> <sup>β</sup><sup>i</sup>* [day] rms [×10−3] DKS *<sup>Q</sup>*

Table 6. Results of rms value, DKS, and *Q* for different distribution functions in the case of *mc* = 4.0 for Southern California earthquakes. The error bars mean the 95% confidence level

data. Unfortunately, the fitting accuracy of the Weibull distribution gets worse by decreasing a threshold *mc*. We propose the following hypothesis, "the interoccurrence time distribution can be described by the superposition of the Weibull distribution and the distribution *PX*(*τ*)" which is the same in 3.1. As shown in Table 7, the log-Weibull distribution is the most suitable for the distribution *PX*(*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

Finally, the TCWB data was analyzed to investigate the interoccurrence time statistics in Taiwan. Figure 9 shows the cumulative distribution of interoccurrence time for *mc* = 4.5 and *mc* = 3.0, respectively. For large *mc*, the Weibull distribution is preferred on the basis of the rms and KS test [see Table 8]. As the threshold of magnitude *mc* 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

shows in Southern California earthquakes. The crossover magnitude *m*∗∗

**3.3 Interoccurrence time statistics in Taiwan**

*Pl* (*i* = 1) 0.91 ± 0.01 25.4 ± 0.21 18 0.07 0.56 *Plw* (*i* = 2) 3.66 ± 0.07 48.2 ± 0.68 31 0.14 0.11 4.0 *Ppow* (*<sup>i</sup>* <sup>=</sup> <sup>3</sup>) 1.44 <sup>±</sup> 0.02 1.15 <sup>±</sup> 0.15 <sup>133</sup> 0.49 1.6×10−<sup>9</sup> SCEDC *Pgam* (*i* = 4) 0.97 ± 0.004 25.6 ± 0.22 20 0.08 0.36

*Pln* (*i* = 5) 2.79 ± 0.02 1.10 ± 0.02 33 0.13 0.02

0.4

0.2

0

0.6 0.8 1.0 1.2 1.4 1.6 1.8

Weibull + log-Weibull distribution (p=0.47)

*<sup>c</sup>* is estimated to be

Interoccurrence time [day]

data

(a) (b) <sup>1</sup>

 Weibull distibution (rms=0.018) log-Weibull distribution (rms=0.031) power law distribution (rms=0.132) gamma distribution (rms=0.02) log-normal distribution (rms=0.033)

different *mc* and distribution functions. (a) *mc* = 4.0 and (b) *mc* = 2.0.

20 40 60 80 100 120

Interoccurrence time [day]

data

0.8

0.6

Cumulative distribution

0.4

0.2

0

of fit.

3.3.

#### **4.1 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), we use JMA data we used is from

(a) (b) <sup>1</sup>

1

0.8

0.6

Cumulative distribution

The Weibull – Log-Weibull Transition of Interoccurrence Time of Earthquakes 17

Fig. 9. Cumulative distribution of interoccurrence time for Taiwan for different *mc* and distribution function. (a) and (b) represent interoccurrence time when *mc* = 4.5 and *mc* = 3.0,

*L* Region *mc α*<sup>1</sup> *β*<sup>1</sup> [day] rms *m*∗∗

*L* = 3 140◦-143◦ E and 35◦-38◦ N 3.9 0.88 ± 0.01 19.4 ± 0.18 0.011 4.6 *L* = 5 140◦-145◦ E and 35◦-40◦ N 4.0 0.75 ± 0.02 10 ± 0.19 0.014 4.7 *L* = 10 140◦-150◦ E and 35◦-45◦ N 4.2 0.94 ± 0.005 8.36 ± 0.04 0.0077 4.9 *L* = 25 125◦-150◦ E and 25◦-50◦ N 5.0 0.93 ± 0.01 17.8 ± 0.10 0.041 5.7

Table 10. Interoccurrence time statistics for different system size *L* by analyzing the JMA

1st January 2001 to 31st October 2007. We use the data covering the region 140◦-143◦ E and 35◦-38◦ N for *L* = 3, 140◦-145◦ E and 35◦-40◦ N for *L* = 5, 140◦-150◦ E and 35◦-45◦ N for *L* = 10, and 125◦-150◦ E and 25◦-50◦ N for *L* = 25. As for *L* = 25, the data covers the whole region of the JMA catalogue. The result of fitting parameters of *P*(*τ*), the crossover magnitude

*<sup>c</sup>* , and the rms value are listed in Table 10. It is demonstrated that in all the cases, the Weibull

*<sup>c</sup>* = 4.0 for *L* = 5, *m*∗∗

for *L* = 25. Therefore we can conclude that the interoccurrence time statistics, namely the

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,

*<sup>c</sup>* is proportional to the maximum magnitude of an earthquake in a region, where we analyzed, denoted here *mmax* [see Figure 10]. We then obtain a region-independent relation

exponent *α*<sup>1</sup> is less than unity and the Weibull - log-Weibull transition appears. *m*∗∗

Weibull - log-Weibull transition, presented here hold from *L* = 3 to *L* = 25.

*m*∗∗

*<sup>c</sup>* **and** *mmax*

0.4

0.2

0

0.5 1.0 1.5 2.0 2.5 Interoccurrence time [day]

*<sup>c</sup>* = 4.2 for *L* = 10, and *m*∗∗

*<sup>c</sup>* /*mmax* = 0.56 ± 0.08 (16)

Weibull + log-Weibull distribution (p=0.51)

*c*

*<sup>c</sup>* depends

*<sup>c</sup>* = 5.0

data

20 40 60 80 100

 Webull distribution (rms=0.0084) log-Weibull distribution (rms=0.024) power law distribution (rms=0.111) gamma distribution (rms=0.016) log-normal distribution (rms=0.024)

Interoccurrence time [day]

data from 1st January 2001 to 31st October 2007 (17).

*<sup>c</sup>* = 3.9 for *L* = 3, *m*∗∗

data

0.8

0.6

Cumulative distribution

0.4

0.2

0

respectively.

*m*∗∗

*m*∗∗

between *m*∗∗

on *L*, namely *m*∗∗

**4.2 Relation between the** *m*∗∗

*<sup>c</sup>* and *mmax*,


Table 8. Results of rms value, DKS, and *Q* for different distribution functions in the case of *mc* = 5.0 for Taiwan earthquakes. The error bars mean the 95% confidence level of fit.


Table 9. Interoccurrence time statistics of earthquakes in Taiwan area. The error bars mean the 95% confidence level of fit.

14 Earthquake Research and Analysis / Book 5

*mc* Distribution X RMS test KS test Region distribution X *<sup>α</sup><sup>i</sup> <sup>β</sup><sup>i</sup>* [day] rms [×10−3] DKS *<sup>Q</sup>*

Table 8. Results of rms value, DKS, and *Q* for different distribution functions in the case of *mc* = 5.0 for Taiwan earthquakes. The error bars mean the 95% confidence level of fit.

*mc* Weibull distribution Distribution X RMS test KS test Region distribution X *<sup>α</sup>*<sup>1</sup> *<sup>β</sup>*<sup>1</sup> [day] *<sup>α</sup><sup>i</sup> <sup>β</sup><sup>i</sup>* [day] *<sup>p</sup>* rms [×10−3] DKS *<sup>Q</sup>*

*Plw* (*i* = 2) 0.86 ± 0.006 15.0 ± 0.08 − − 1 8.4 0.02 1 5.0 *Ppow* (*i* = 3) 0.86 ± 0.006 15.0 ± 0.08 − − 1 8.4 0.02 1 TCWB *Pgam* (*i* = 4) 0.86 ± 0.006 15.0 ± 0.08 − − 1 8.4 0.02 1

*Pln* (*i* = 5) 0.86 ± 0.006 15.0 ± 0.08 − − 1 8.4 0.02 1 *Plw* (*i* = 2) 0.88 ± 0.004 5.34 ± 0.02 − − 1 4.3 0.01 1 4.5 *Ppow* (*i* = 3) 0.88 ± 0.004 5.34 ± 0.02 − − 1 4.3 0.01 1 TCWB *Pgam* (*i* = 4) 0.88 ± 0.004 5.34 ± 0.02 − − 1 4.3 0.01 1

*Pln* (*i* = 5) 0.88 ± 0.004 5.34 ± 0.02 − − 1 4.3 0.01 1 *Plw* (*i* = 2) 1.00 ± 0.01 2.30 ± 0.04 1.82 ± 0.05 4.06 ± 0.13 0.68 ± 0.01 2.8 0.009 1 4.0 *Ppow* (*i* = 3) 1.08 ± 0.01 2.30 ± 0.02 1.95 ± 0.04 0.55 ± 0.02 0.89 ± 0.01 6.9 0.03 1 TCWB *Pgam* (*i* = 4) 1.08 ± 0.01 2.30 ± 0.02 − − 1 10 0.05 0.99 *Pln* (*i* = 5) 1.08 ± 0.01 2.30 ± 0.02 0.46 ± 0.005 0.92 ± 0.006 0.39 ± 0.02 2.8 0.007 1 *Plw* (*i* = 2) 1.44 ± 0.03 1.29 ± 0.02 1.32 ± 0.05 2.15 ± 0.05 0.61 ± 0.02 4.5 0.01 1 3.5 *Ppow* (*i* = 3) 1.52 ± 0.03 1.25 ± 0.01 2.41 ± 0.05 0.50 ± 0.01 0.74 ± 0.02 8.6 0.07 0.94 TCWB *Pgam* (*i* = 4) 1.52 ± 0.03 1.25 ± 0.01 − − 1 18 0.09 0.73 *Pln* (*i* = 5) − − −0.04 ± 0.003 0.66 ± 0.004 0 5.4 0.03 1 *Plw* (*i* = 2) 2.07 ± 0.06 0.75 ± 0.009 1.18 ± 0.03 1.47 ± 0.01 0.51 ± 0.01 3.2 0.01 1 3.0 *Ppow* (*i* = 3) 2.62 ± 0.09 0.78 ± 0.006 3.58 ± 0.08 0.48 ± 0.005 0.53 ± 0.02 7.5 0.02 1 TCWB *Pgam* (*i* = 4) 2.62 ± 0.09 0.78 ± 0.006 1.26 ± 0.08 0.66 ± 0.03 0.97 ± 0.05 23 0.10 0.76 *Pln* (*i* = 5) − − −0.41 ± 0.004 0.38 ± 0.006 0 12 0.05 1 *Plw* (*i* = 2) 5.35 ± 0.41 0.60 ± 0.005 0.86 ± 0.05 1.16 ± 0.007 0.43 ± 0.04 16 0.037 1 2.5 *Ppow* (*i* = 3) 5.35 ± 0.41 0.60 ± 0.005 6.54 ± 0.09 0.49 ± 0.001 0.10 ± 0.04 10 0.041 1 TCWB *Pgam* (*i* = 4) 5.35 ± 0.41 0.60 ± 0.005 1.69 ± 2.04 0.33 ± 0.61 0.94 ± 0.07 46 0.17 0.41

*Pln* (*i* = 5) − − −0.58 ± 0.006 0.19 ± 0.01 0 35 0.11 0.87

Table 9. Interoccurrence time statistics of earthquakes in Taiwan area. The error bars mean

the 95% confidence level of fit.

*Pl* (*i* = 1) 0.86 ± 0.006 15.1 ± 0.08 8.4 0.02 1 *Plw* (*i* = 2) 3.01 ± 0.06 27.9 ± 0.42 24 0.10 0.24 5.0 *Ppow* (*<sup>i</sup>* <sup>=</sup> <sup>3</sup>) 1.51 <sup>±</sup> 0.03 0.93 <sup>±</sup> 0.11 <sup>111</sup> 0.25 3.6×10−<sup>6</sup> TCWB *Pgam* (*i* = 4) 0.97 ± 0.003 15.3 ± 0.14 16 0.05 0.95

*Pln* (*i* = 5) 2.23 ± 0.02 1.15 ± 0.02 24 0.08 0.53

Fig. 9. Cumulative distribution of interoccurrence time for Taiwan for different *mc* and distribution function. (a) and (b) represent interoccurrence time when *mc* = 4.5 and *mc* = 3.0, respectively.


Table 10. Interoccurrence time statistics for different system size *L* by analyzing the JMA data from 1st January 2001 to 31st October 2007 (17).

1st January 2001 to 31st October 2007. We use the data covering the region 140◦-143◦ E and 35◦-38◦ N for *L* = 3, 140◦-145◦ E and 35◦-40◦ N for *L* = 5, 140◦-150◦ E and 35◦-45◦ N for *L* = 10, and 125◦-150◦ E and 25◦-50◦ N for *L* = 25. As for *L* = 25, the data covers the whole region of the JMA catalogue. The result of fitting parameters of *P*(*τ*), the crossover magnitude *m*∗∗ *<sup>c</sup>* , and the rms value are listed in Table 10. It is demonstrated that in all the cases, the Weibull exponent *α*<sup>1</sup> is less than unity and the Weibull - log-Weibull transition appears. *m*∗∗ *<sup>c</sup>* depends on *L*, namely *m*∗∗ *<sup>c</sup>* = 3.9 for *L* = 3, *m*∗∗ *<sup>c</sup>* = 4.0 for *L* = 5, *m*∗∗ *<sup>c</sup>* = 4.2 for *L* = 10, and *m*∗∗ *<sup>c</sup>* = 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* = 3 to *L* = 25.

#### **4.2 Relation between the** *m*∗∗ *<sup>c</sup>* **and** *mmax*

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*∗∗ *<sup>c</sup>* is proportional to the maximum magnitude of an earthquake in a region, where we analyzed, denoted here *mmax* [see Figure 10]. We then obtain a region-independent relation between *m*∗∗ *<sup>c</sup>* and *mmax*,

$$m\_{\rm c}^{\ast \ast} / m\_{\rm max} = 0.56 \pm 0.08 \tag{16}$$

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, *Mw* is directly proportional to the plate-velocity, *V*, and is directly inversely proportional to

The Weibull – Log-Weibull Transition of Interoccurrence Time of Earthquakes 19

The relation *mc*∗∗/0.56 = *mmax* 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 *mc* 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

**4.4 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

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

where *m*<sup>1</sup> and *m*<sup>2</sup> are arbitrary values of *m*. This implies that the quantity defined by

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

*α* = *fα*(*m*, *b*)

where the functional forms of *f<sup>α</sup>* and *f<sup>β</sup>* characterize the time series of earthquakes under

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<sup>α</sup>* and *f<sup>β</sup>* by a perturbational

= *β*(*m*2)*e*−*bm*2Γ

). We assume that these two statistics are correct over wide ranges, and

 1 + 1 *α*2 

*β* = *fβ*(*m*, *b*), (19)

is a universal constant when we consider the local earthquakes in a

, (18)

*Mw* = −0.000953*T* + 1.43*V* + 8.01. (17)

plate-age *T*, namely,

damaging mechanics.

<sup>−</sup>*α*−<sup>1</sup> · *<sup>e</sup>*

 1 + <sup>1</sup> *α*1 

(*τ*/*β*)*<sup>α</sup>*

earthquakes whose magnitude is larger than *m* ,

magnitude *m* as well as on the GR parameter *b*,

*β*(*m*1)*e*−*bm*1Γ

 1 + 1 *α*1 

(*P*(*τ*) ∝ *t*

*β*(*m*1)*e*−*bm*1Γ

consideration.

relatively small area.

Maximum magnitude

Fig. 10. The crossover magnitude *m*∗∗ *<sup>c</sup>* v.s the maximum magnitude *mmax* in 16 regions (14 region in Japan, Southern California, and Taiwan).


Table 11. List of the crossover magnitude, *m*∗∗ *<sup>c</sup>* 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.

1 We take an average using three regions; 25◦–30◦N, 140◦–145◦E (*m*∗∗ *<sup>c</sup>* = 3.8), 30◦–35◦N, 140◦–145◦E (*m*∗∗ *<sup>c</sup>* = 3.9), 35◦–40◦N, 140◦–145◦E (*m*∗∗ *<sup>c</sup>* = 4.5).

2 We take an average using five regions; 25◦–30◦N, 125◦–130◦E (*m*∗∗ *<sup>c</sup>* = 3.7), 25◦–30◦N, 130◦–135◦E (*m*∗∗ *<sup>c</sup>* = 3.4), 30◦–35◦N, 130◦–135◦E (*m*∗∗ *<sup>c</sup>* = 4.3), 30◦–35◦N, 135◦–140◦E (*m*∗∗ *<sup>c</sup>* = 3.7), 35◦–40◦N, 135◦–140◦E (*m*∗∗ *<sup>c</sup>* = 3.6).

This relation can be useful to interpret the Weibull - log-Weibull transition of geophysical meaning.

#### **4.3 Interpretation of the Weibull - log-Weibull transition**

Although the scaled crossover magnitudes *mc*∗∗/*mmax* is region-independent, the crossover magnitude *m*∗∗ *<sup>c</sup>* 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*∗∗ *<sup>c</sup>* , which can shed light on the geophysical implication of the region-dependent *m*∗∗ *<sup>c</sup>* . As shown in Table 11, *m*∗∗ *<sup>c</sup>* is on the average proportional to the plate velocity. That means that the maximum magnitude *mmax* for a tectonic region is more or less proportional to the plate velocity since *m*∗∗ *<sup>c</sup>* /0.56 = *mmax*. Such an interesting 16 Earthquake Research and Analysis / Book 5

 25-30N and 125-130E 25-30N and 130-135E 25-30N and 140-145E 30-35N and 125-130E 30-35N and 130-135E 30-35N and 130-135E 30-35N and 135-140E 35-40N and 130-135E 35-40N and 135-140E 35-40N and 140-145E 40-45N and 135-140E 40-45N and 140-145E 40-45N and 145-150E 45-50N and 140-145E California Taiwan

3 4 5 6 7 8

Maximum magnitude

Region relative plate motion velocity [mm/yr] *m*∗∗

Taiwan PH-EU 71 4.90 East Japan PA-PH 49 4.07 <sup>1</sup> West Japan PH-EU 47 3.74 <sup>2</sup> California PA-NA 47 3.30

PH, EU, PA, and NA represent PHilippine Sea plate, EUrasian plate, PAcific plate, and North

This relation can be useful to interpret the Weibull - log-Weibull transition of geophysical

Although the scaled crossover magnitudes *mc*∗∗/*mmax* is region-independent, the crossover

on the tectonic region (Figure 7). To investigate the Weibull - log-Weibull transition further,

the plate velocity. That means that the maximum magnitude *mmax* for a tectonic region is

*<sup>c</sup>* . As shown in Table 11, *m*∗∗

*<sup>c</sup>* from the superposition regime to the pure Weibull regime probably depends

*<sup>c</sup>* , which can shed light on the geophysical implication

*<sup>c</sup>* is on the average proportional to

*<sup>c</sup>* /0.56 = *mmax*. Such an interesting

*<sup>c</sup>* = 4.3), 30◦–35◦N, 135◦–140◦E (*m*∗∗

*<sup>c</sup>* v.s the maximum magnitude *mmax* in 16 regions (14

*c*

*<sup>c</sup>* and the plate velocity (32; 33). The notation of

*<sup>c</sup>* = 3.8), 30◦–35◦N, 140◦–145◦E

*<sup>c</sup>* = 3.7), 25◦–30◦N, 130◦–135◦E

*<sup>c</sup>* = 3.7), 35◦–40◦N,

4.5

4

3.5

Crossover magnitude

Fig. 10. The crossover magnitude *m*∗∗

region in Japan, Southern California, and Taiwan).

Table 11. List of the crossover magnitude, *m*∗∗

1 We take an average using three regions; 25◦–30◦N, 140◦–145◦E (*m*∗∗

2 We take an average using five regions; 25◦–30◦N, 125◦–130◦E (*m*∗∗

**4.3 Interpretation of the Weibull - log-Weibull transition**

more or less proportional to the plate velocity since *m*∗∗

*<sup>c</sup>* = 4.5).

American plate, respectively.

*<sup>c</sup>* = 3.9), 35◦–40◦N, 140◦–145◦E (*m*∗∗

*<sup>c</sup>* = 3.6).

*<sup>c</sup>* = 3.4), 30◦–35◦N, 130◦–135◦E (*m*∗∗

we consider the plate velocity with *m*∗∗

of the region-dependent *m*∗∗

(*m*∗∗

(*m*∗∗

135◦–140◦E (*m*∗∗

magnitude *m*∗∗

meaning.

3

2.5

2

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, *Mw* is directly proportional to the plate-velocity, *V*, and is directly inversely proportional to plate-age *T*, namely,

$$M\_{\overline{w}} = -0.000953T + 1.43V + 8.01. \tag{17}$$

The relation *mc*∗∗/0.56 = *mmax* 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 *mc* 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.

#### **4.4 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* <sup>−</sup>*α*−<sup>1</sup> · *<sup>e</sup>* (*τ*/*β*)*<sup>α</sup>* ). 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* ,

$$
\beta(m\_1)e^{-bm\_1}\Gamma\left(1+\frac{1}{a\_1}\right) = \beta(m\_2)e^{-bm\_2}\Gamma\left(1+\frac{1}{a\_2}\right),
\tag{18}
$$

where *m*<sup>1</sup> and *m*<sup>2</sup> are arbitrary values of *m*. This implies that the quantity defined by *β*(*m*1)*e*−*bm*1Γ 1 + <sup>1</sup> *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*,

$$\begin{aligned} \mathfrak{a} &= f\_{\mathfrak{a}}(m, b) \\ \mathfrak{b} &= f\_{\mathfrak{B}}(m, b), \end{aligned} \tag{19}$$

where the functional forms of *f<sup>α</sup>* and *f<sup>β</sup>* 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<sup>α</sup>* and *f<sup>β</sup>* by a perturbational

1.0 0.8 0.6 0.4 0.2 0

1.0 0.8 0.6 0.4 0.2 0

0 20 40 60 80 100 Interoccurrence time

show the Weibull distribution.

**5. Conclusion**

0 5 10 15 20 25 30

 Data Weibull distribution log Weibull distribution power law gamma distribution log normal distribution

(a) 1.0

0.8 0.6 0.4 0.2 0

0 300 600 900 1200 Interoccurrence time

Fig. 12. Interoccurrence time statistics for different magnitude *mc* by analyzing catalogue

to the stretched exponential distribution because *α*<sup>1</sup> 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 *mc*, 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

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 *mc* 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

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*<sup>1</sup> < 1, indicating that a large earthquake is not a Poisson process but a phenomenon exhibiting a long-tail distribution. As the threshold of magnitude *mc* 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 *mc* is increased. Therefore, we concluded that the interoccurrence time statistics exhibit a Weibull - log-Weibull

synthetic catalogs produced by the spring-block model [see Figure 12] (29).

 data super position of Weibull and log weibull

(b) 1.0

The Weibull – Log-Weibull Transition of Interoccurrence Time of Earthquakes 21

0.8 0.6 0.4 0.2 0 (c)

0 1000 2000 3000 4000 5000 Interoccurrence time

0.8 0.6 0.4 0.2 0

 Data Weibull distribution log Weibull distribution power law gamma distribution log normal distribution

0 400 800 1200

Cumulative distribution

Cumulative distribution

produced by the two-dimensional spring-block model (29).

Cumulative distribution

Fig. 11. Schematic picture of universal behavior of *fα*(*m*, *b*) and *fβ*(*m*, *b*) near *m* � *mc*.

approach. Here we consider a particular solution of equation (19) which satisfies the following conditions; *fβ*(*m*, *b*) = exp [*b*(*m* − *mc*) + *c*] and *fα*(*mc*, *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* = *mc*, where *b*� and *c* are constant parameters. By use of this simplification, equation (18) is rewritten by putting *m*<sup>1</sup> = *m* and *m*<sup>2</sup> = *m*,

$$\begin{split} (b'-b)(m-m\_c) &= -\log \Gamma \left( 1 + \frac{1}{a(m)} \right) \\ &\cong \frac{1}{2}\Delta - \frac{3}{4}(\Delta)^2 + \cdots, \ (\Delta = a(m-1). \end{split} \tag{20}$$

Here we used the Taylor expansion near *m* ∼= *mc* (i.e., *α*(*m*) ∼= *α*(*mc* )). 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* − *mc* ) + *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).

#### **4.5 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 *mc*, the distribution in a long time domain is similar

Fig. 12. Interoccurrence time statistics for different magnitude *mc* by analyzing catalogue produced by the two-dimensional spring-block model (29).

to the stretched exponential distribution because *α*<sup>1</sup> 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 *mc*, 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 *mc* 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).

#### **5. Conclusion**

18 Earthquake Research and Analysis / Book 5

Fig. 11. Schematic picture of universal behavior of *fα*(*m*, *b*) and *fβ*(*m*, *b*) near *m* � *mc*.

simplification, equation (18) is rewritten by putting *m*<sup>1</sup> = *m* and *m*<sup>2</sup> = *m*,

<sup>∼</sup><sup>=</sup> <sup>1</sup> 2 <sup>Δ</sup> <sup>−</sup> <sup>3</sup> 4 (Δ)

(*b*� − *b*)(*m* − *mc* ) = − log Γ

(*m* − *mc* ) + *c* is a little bit accelerated.

be studied in our forthcoming paper (38).

**4.5 Comparison with previous works**

log *β*(*m*) � *b*�

approach. Here we consider a particular solution of equation (19) which satisfies the following conditions; *fβ*(*m*, *b*) = exp [*b*(*m* − *mc*) + *c*] and *fα*(*mc*, *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* = *mc*, where *b*� and *c* are constant parameters. By use of this

> 1 +

Here we used the Taylor expansion near *m* ∼= *mc* (i.e., *α*(*m*) ∼= *α*(*mc* )). 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*,

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

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 *mc*, the distribution in a long time domain is similar

1 *α*(*m*) <sup>2</sup> <sup>+</sup> ··· , (<sup>Δ</sup> <sup>=</sup> *<sup>α</sup>*(*<sup>m</sup>* <sup>−</sup> <sup>1</sup>). (20)

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*<sup>1</sup> < 1, indicating that a large earthquake is not a Poisson process but a phenomenon exhibiting a long-tail distribution. As the threshold of magnitude *mc* 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 *mc* is increased. Therefore, we concluded that the interoccurrence time statistics exhibit a Weibull - log-Weibull

[11] J. K. Gardner, L. Knopoff, Is the sequence of earthquakes in southern California with aftershocks removed, Poissonian?, *Bulletin of the Seismological Society of America*, vol. 64,

The Weibull – Log-Weibull Transition of Interoccurrence Time of Earthquakes 23

[12] B. Enescu, Z. Struzik, and K. Kiyono, On the recurrence time of earthquakes: insight from Vrancea (Romania) intermediate-depth events, *Geophysical Journal International*,

[13] A. Bunde, J. F. Eichner, J. W. Kantelhardt, and S. Havlin, Long-Term Memory: A Natural Mechanism for the Clustering of Extreme Events and Anomalous Residual Times in

[14] V. N. Livina, S. Havlin, and A. Bunde, Memory in the Occurrence of Earthquakes,

[15] S. Lennartz, V. N. Livina, A. Bunde, and S. Havlin, Long-term memory in earthquakes and the distribution of interoccurrence times, *Europhysics Lettters*, vol. 81, 69001, 2008. [16] T. Akimoto, T. Hasumi, and Y. Aizawa, "Characterization of intermittency in renewal processes: Application to earthquakes", *Physical Review E* vol. 81, 031133, 2010. [17] T. Hasumi, T. Akimoto, and Y. Aizawa, "The Weibull - log-Weibull distribution for interoccurrence times of earthquakes", *Physica A*, vol. 388, pp. 491-498, 2009. [18] T. Hasumi, C. Chen, T. Akimoto, and Y. Aizawa, "The Weibull - log-Weibull transition of interoccurrence times for synthetic and natural earthquakes", *Techtonophysics*, vol. 485,

[22] T. Huillet and H. F. Raynaud, "Rare events in a log-Weibull scenario-Application to earthquake magnitude data", *The European Physical Journal B*, vol. 12, pp. 457-469, 1999. [23] M. V. Matthews, W. L. Ellsworth, and P. A. Reasenberg, "A brownian model for recurrent earthquakes", *Bulletin of the Seismological Society of America*, vol. 92, pp. 2233-2250, 2002. [24] W. Weibull, A statistical distribution function of wide applicability, *Journal of Applied*

[25] Y. Hagiwara, "Probability of earthquake occurrence as obtained from a Weibull distribution analysis of crustal strain", *Tectonophys*, vol. 23, pp. 313-318, 1974. [26] W. H. Bakun, B. Aagard, B. Dost, *et al.*, "Implications for prediction and hazard assessment from the 2004 Parkfield earthquake", *Nature*, vol. 437, pp. 969-974, 2005. [27] S. G. Abaimov, D. L. Turcotte, and J. B. Rundle, "Recurrence-time and frequency-slip statistics of slip events on the creeping section of the San Andreas fault in central

[28] K. Z. Nanjo, D. L. Turcotte, and R. Shcherbakov, "A model of damage mechanics for the deformation of the continental crust", *Journal of Geophysics Research*, vol. 110, B07403,

[29] T. Hasumi, T. Akimoto, and Y. Aizawa, "The Weibull - log-Weibull transition of the interoccurrence statistics in the two-dimensional Burridge-Knopoff earthquake model",

California", *Geophysical Journal International*, vol.170, pp. 1289-1299, 2007.

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[20] Southern California Earthquake Data Center: http://www.data.scec.org

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transition. We also find the region-independent relation, namely, *m*∗∗ *<sup>c</sup>* /*mmax* = 0.56 ± 0.08. In addition, the crossover magnitude *m*∗∗ *<sup>c</sup>* 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 Weibull log-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.

#### **6. Acknowledgments**

The authors thank the editor of this book and Intech publisher for giving us the opportunity to take part in this book project. We would like to thank the JMA, SCEDC and TCWB for allowing us to use the earthquake data. The effort of the Taiwan Central Weather Bureau for maintaining the CWB Seismic Network is highly appreciated. TH is supported by the Japan Society for the Promotion of Science (JSPS) and the Earthquake Research Institute cooperative research program at the University of Tokyo. This work is partly supported by the Sasagawa Scientific Research Grant from The Japan Science Society by the Waseda University.. CCC is also grateful for research supports from the National Science Council (ROC) and the Department of Earth Sciences at National Central University (ROC).

#### **7. References**


20 Earthquake Research and Analysis / Book 5

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 Weibull log-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

The authors thank the editor of this book and Intech publisher for giving us the opportunity to take part in this book project. We would like to thank the JMA, SCEDC and TCWB for allowing us to use the earthquake data. The effort of the Taiwan Central Weather Bureau for maintaining the CWB Seismic Network is highly appreciated. TH is supported by the Japan Society for the Promotion of Science (JSPS) and the Earthquake Research Institute cooperative research program at the University of Tokyo. This work is partly supported by the Sasagawa Scientific Research Grant from The Japan Science Society by the Waseda University.. CCC is also grateful for research supports from the National Science Council (ROC) and the

[1] I. G. Main, "Statistical physics, seismogenesis, and seismic hazard", *Review of Geophysics*,

[2] B. Gutenberg and C. F. Richter, "Magnitude and energy of earthquakes", *Annals of*

[3] F. Omori, "On the after-shocks of earthquakes", *Journal of the College of Science, Imperial*

[4] S. G. Abaimov, D. L. Turcotte, R. Shcherbakov, and J. B. Rundle, "Recurrence and interoccurrence behavior of self-organized complex phenomena", *Nonlinear Processes in*

[5] P. Bak, K. Christensen, L. Danon, and T. Scanlon, "Unified Scaling Law for Earthquakes",

[6] A. Corral, "Long-Term Clustering, Scaling, and Universality in the Temporal Occurrence

[7] R. Shcherbakov, G. Yakovlev, D. L. Turcotte, and J. B. Rundle, "Model for the Distribution of Aftershock Interoccurrence Times", *Physical Review Letters*, vol. 95, 218501, 2005. [8] A. Saichev and D. Sornette, ""Universal" Distribution of Inter-Earthquake Times

[9] A. Saichev and D. Sornette, "Theory of earthquake recurrence times", *Journal of*

[10] S. Abe and N. Suzuki, "Scale-free statistics of time interval between successive

*<sup>c</sup>* /*mmax* = 0.56 ± 0.08. In

*<sup>c</sup>* is proportional to the plate velocity, which is consistent

transition. We also find the region-independent relation, namely, *m*∗∗

theoretical and geophysical understanding of this transition.

Department of Earth Sciences at National Central University (ROC).

addition, the crossover magnitude *m*∗∗

**6. Acknowledgments**

**7. References**

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*Geophysics*, vol. 14, pp. 455-464, 2007.


**2** 

Ayten Yiğiter

*Turkey* 

**Change Point Analysis in Earthquake Data** 

*Hacettepe University, Faculty of Science, Department of Statistics Beytepe-Ankara* 

Earthquake forecasts are very important in human life in terms of estimating hazard and managing emergency systems. Defining of earthquake characteristics plays an important role in these forecasts. Of these characteristics, one is the frequency distribution of earthquakes and and the other is the magnitude distribution of the earthquakes. Each

There are various statistical distributions used to model the earthquake numbers. As is well known, these are binomial, Poisson, geometric and negative binomial distributions. It is generally assumed that earthquake occurrences are well described by the Poisson distribution because of its certain characteristics (for some details, see Kagan, 2010; Leonard & Papasouliotis, 2001). In their study, Rydelek & Sacks (1989) used the Poisson distribution of earthquakes at any magnitude level. The Poisson distribution is generally used for earthquakes of a large magnitude, and the earthquake occurrences with time/space can be modeled with the Poisson process in which, as is known, the Poisson distribution is one that

There is a significant amount of research on the change point as applied to earthquake data. Amorese (2007) used a nonparametric method for the detection of change points in frequency magnitude distributions. Yigiter & İnal (2010) used earthquake data for their method developed for the estimator of the change point in Poisson process. Aktaş et al. (2009) investigated a change point in Turkish earthquake data. Rotondi & Garavaglia (2002) applied the hierarchical Bayesian method for the change point in data, taken from the Italian

Recently, much research in the literature has focused on whether there is an increase in the frequency of earthquake occurrences. It is further suggested that any increase in the frequency of earthquakes, in some aspects, is due to climate change in the world. There is considerable debate on whether climate change really does increase the frequency of natural disasters such as earthquakes and volcano eruptions. In many studies, it is emphasized that there is serious concern about impact of climate change on the frequencies of hazardous events (Peduzzi, 2005; Lindsey, 2007; Mandewille, 2007 etc.). In Peduzzi's study (2005), there are some indicators about increasing number of the earthquakes especially affecting human settlements, and it is also reported that there is an increase in the percentage of earthquakes affecting human settlements from 1980 onwards. The change point analysis can be used to

study the increase or decrease in the frequency of the earthquake occurrences.

statistical distribution has many parameters describing the actual distribution.

counts the events that have occurred over a certain period of time.

**1. Introduction** 

NT4.1.1 catalogue.

