Pure Appl. Geophys. 170 (2013), 221–228 2012 Springer Basel AG DOI 10.1007/s00024-011-0445-5 Pure and Applied Geophysics
Aftershock Statistics of the 1999 Chi–Chi, Taiwan Earthquake and the Concept of Omori Times
1,2 1 1,3 2 YA-TING LEE, DONALD L. TURCOTTE, JOHN B. RUNDLE, and CHIEN-CHIH CHEN
Abstract—In this paper we consider the statistics of the after- the magnitude of the main shock and the largest shock sequence of the m = 7.65 20 September 1999 Chi–Chi, aftershock. This law states that the difference in Taiwan earthquake. We first consider the frequency-magnitude statistics. We find good agreement with Gutenberg–Richter scaling magnitude, Dm, between the main shock and its but find that the aftershock level is anomalously high. This level is largest aftershock is approximately constant inde- quantified using the difference in magnitude between the main pendent of the magnitude of the main shock shock and the largest inferred aftershock Dm : Typically, Dm is in the range 0.8–1.5, but for the Chi–Chi earthquake the value is Dm ¼ mms mmax as ð2Þ Dm = 0.03. We suggest that this may be due to an aseismic slow- earthquake component of rupture. We next consider the decay rate where mms is the magnitude of the main shock and of aftershock activity following the earthquake. The rates are well m is the magnitude of the largest aftershock, approximated by the modified Omori’s law. We show that the max as distribution of interoccurrence times between aftershocks follow a typically Dm & 1.2 (BA˚ TH 1965). Many studies have nonhomogeneous Poisson process. We introduce the concept of been carried out regarding the statistical variability of Omori times to study the merging of the aftershock activity with Dm (VERE-JONES 1969;KISSLINGER and JONES 1991; the background seismicity. The Omori time is defined to be the mean interoccurrence time over a fixed number of aftershocks. TSAPANOS 1990;FELZER et al. 2002, 2003;CONSOLE et al. 2003;HELMSTETTER and SORNETTE 2003; Key words: Earthquakes, aftershocks, Omori times, SHCHERBAKOV and TURCOTTE 2004). Perhaps Console interoccurrence times. et al.(2003) developed a mathematical model showing a substantial dependence of Dm on the 1. Introduction magnitude thresholds chosen for the main shocks and the aftershocks. This partly explains the large Thebehaviorofaftershocksequenceshasbeen Dm values reported in the past. (3) Omori’s law for discussed widely (i.e. KISSLINGER 1996). There are the temporal decay of aftershocks (OMORI 1894), and three applicable scaling laws: (1) The Gutenberg– the modified Omori’s law (the Omori–Utsu law) Richter (GR) law for frequency-magnitude scaling of which was proposed by Utsu (1961). We will utilize aftershocks (GUTENBERG and RICHTER 1954). The the modified Omori’s law in the form proposed by number N of earthquakes with magnitudes greater than SHCHERBAKOV et al.(2004) or equal to m is well approximated by the relation: 1 1 raðt; mcÞ¼ p ð3Þ log10 Nð mÞ¼ a bðmÞ: ð1Þ saðmcÞ ½1 þ t=cðmcÞ where b is the b-value and a is a measure of seismic where ra(t,mc) : dN/dt is the rate of occurrence of intensity. (2) Ba˚th’s law for the difference between aftershocks with magnitudes greater than or equal to
a lower cutoff mc, t is the time elapsed since the time of the main shock. In the limit t ? 0 we have -1 1 Department of Geology, University of California, Davis, ra = sa , the characteristic time sa(mc) is the reci- CA 95616, USA. E-mail: [email protected] procal of the initial (constant) rate of aftershock 2 Graduate Institute of Geophysics, National Central activity at early times. The characteristic time c(mc) University, Jhongli 320, Taiwan, ROC. 3 Department of Physics, University of California, Davis, is the time of transition from their rate to the power- CA 95616, USA. law decay of aftershock activity. 222 Y.- T. Lee et al. Pure Appl. Geophys.
There are two limiting cases for the behavior of magnitude. The distribution of interoccurrence time
Omori’s law. In the first it is assumed that c0 is a is well approximated by a nonhomogeneous Poisson constant and that sa(mc) depends on the cutoff mag- (NHP) process driven by the modified Omori’s law nitude mc. In the second it is assumed that s0 is a over a finite time interval T (SHCHERBAKOV et al. 2005, constant and that c(mc) depends on the cutoff mag- 2006). At the end of this paper, we introduce the nitude mc.NANJO et al. (2007) has discussed these concept of Omori times to analyze the interoccur- limits in some detail with applications to four Japa- rence times for the Chi–Chi aftershock sequence. We nese earthquakes. In this paper we will assume c0 to calculate the mean interoccurrence interval over a be constant and sa(mc) is determined from the data. In fixed number of aftershocks N = 25, 50, 100, 200, this limit the modified form of Omori’s law given in and compare the Chi–Chi aftershocks with a combi- Eq. (3) becomes nation of Omori decay and a steady-state background. 1 1 raðt; mcÞ¼ p ð4Þ saðmcÞ ½1 þ t=c0 2. Scaling of Chi–Chi Aftershock Sequences For all values of time t the values of the after- shock rate ra(t, mc) for different mc are related by The Chi–Chi earthquake struck central Taiwan on 20 September 1999 at 17:47UTC. It ruptured along raðt; mc1Þ saðmc2Þ ¼ ð5Þ the Chelungpu fault, and the moment magnitude, mw, raðt; mc2Þ saðmc1Þ was 7.65. The hypocenter of the Chi–Chi earthquake However, the rates of aftershock occurrence for was located at 23.853 N, 120.816 E, at a depth of different values of mc satisfy the GR law. From 8 km. In this paper we study the aftershock sequence Eq. (1) we obtain of this earthquake. We combine three catalogs from 1994 to 2008 for our analyses. These three catalogs raðt; mc1Þ ¼ 10 bðmc1 mc2Þ ð6Þ include: (1) the Central Weather Bureau seismic r ðt; m Þ a c2 network (CWBSN) catalog from the Central Weather Combining Eqs. (5) and (6) we find Bureau (CWB) of Taiwan, (2) the Taiwan Strong Motion Instrumentation Program (TSMIP) (CHANG saðmc2Þ ¼ 10 bðmc1 mc2Þ ð7Þ et al. 2007) which provides good coverage for early s ðm Þ a c1 aftershocks, and (3) the Centroid-Moment-Tensor We will obtain a b-value from GR scaling and (CMT) catalog for large aftershocks. We use the will then fit aftershock statistics for several values of TSMIP catalog for the first hour of Chi–Chi after- mc to obtain preferred values of c0, p, and the sa(mc). shocks and the CMT catalog for aftershocks m [ 5. We illustrate it in the Sect. 2. The completeness magnitude for the area analyzed is In this study, we analyze the scaling laws for the m C 2.0. We combine these catalogs and then define aftershock sequence from the m = 7.65 20 Septem- the Chi–Chi aftershock region. ber 1999 Chi–Chi, Taiwan earthquake. The In Fig. 1 we define the spatial window for the aftershocks are in good agreement with the Guten- Chi–Chi aftershocks. We compare the spatial distri- berg–Richter law for frequency–magnitude scaling bution of seismic activity in Taiwan for 1,000 days and the aftershock decay rates satisfy the modified after the Chi–Chi earthquake with the activity for Omori’s law. An interesting result in this paper is that 1,000 days before the Chi–Chi earthquake. We we find that the Chi–Chi earthquake triggered an choose the area with the largest change in seismic anomalously large number of aftershocks. We also activity as the Chi–Chi aftershock region. Our pur- consider the statistics of interoccurrence times for pose is to exclude the region of high seismicity to the Chi–Chi aftershocks. The observed statistics of east. The epicenter of the mainshock is shown as a interoccurrence times for the Chi–Chi earthquake black star. The dashed lines show the region of sequence has a power-law dependence on the times seismic activity we associate with the Chi–Chi between successive aftershocks over several orders of aftershocks. During the 1,000 day time interval after Vol. 170, (2013) Aftershock Statistics of the 1999 Chi–Chi, Taiwan Earthquake 223
Figure 1 The spatial distribution of seismicity activity for 1,000 days before the 20 September 1999 Chi–Chi earthquake in (a), and 1,000 days after the Chi–Chi earthquake in (b). The magnitude of these earthquakes are great than 2.0. The epicenter of the Chi–Chi earthquake is shown as a black star. The black line shows the surface rupture of the Chi–Chi earthquake. The dash line shows the region of the aftershocks for the statistic of Chi–Chi aftershocks. EP Eurasian Plate, and PSP Philippine Sea Plate
the Chi–Chi earthquake, 42,952 aftershocks were 6 recorded with magnitudes m C 2.0, and nine of the y=-0.84x+6.4 aftershocks had magnitudes m C 6.0. A comparison 5 of Fig. 1a with b indicates that less than 5% of the earthquakes in our region were not aftershocks and 4 less than 5% of the aftershocks occurred outside the chosen region. Since the majority of aftershocks have N) (
10 3 been included we therefore expect the minor differ- g o ence in the spatial extent does not affect the following L analysis. 2 First, we consider the frequency–magnitude scal- ing for the Chi–Chi aftershock sequence. The 1 frequency–magnitude distribution of the seismicity in the Chi–Chi aftershock area (Fig. 1b) for 1,000 days 0 after the earthquake is given as a function of m in 1 2 3 4 5 6 7 8 m Fig. 2. The role over for small amplitudes m \ 2is attributed to lack of sensitivity of the networks for Figure 2 The cumulative frequency–magnitude distribution of Chi–Chi small aftershocks. The behavior for large amplitudes aftershocks for 1,000 days after the main shock. The dash line is m [ 5 is attributed to relatively few aftershocks. We the best fitting for data with the b = 0.84 by Gutenberg-Richter obtain the least square fit of the GR scaling relation relation Eq. (1) in the magnitude range 2 \ m \ 5 and find that b = 0.84, and a = 6.4 (the straight dashed line largest aftershock was mw = 6.70 so that Dm = 0.95. in Fig. 2). SHCHERBAKOV and TURCOTTE (2004) proposed a modified We next consider the application of Ba˚th’s law, version of Ba˚th’s law which is based on an extrapolation Eq. (2), to the Chi–Chi earthquake. The magnitude of of Gutenberg–Richter statistics for aftershocks. The the mainshock was mw = 7.65 and the magnitude of the inferred magnitude of the largest aftershock m is 224 Y.- T. Lee et al. Pure Appl. Geophys.
deduced by taking Log10N(Cm ) = 1 for a given is a function of mc so that we will use the form of aftershock sequence. From Eq. (1) we find Omori’s law given in Eq. (4) to fit the data in Fig. 3. Taking b = 0.84 from Fig. 2, we find from Eq. (7) a ¼ bm ð8Þ that The difference in magnitude between the main- s ðm ¼ 4Þ s ðm ¼ 3Þ shock and inferred largest aftershock Dm is given by a c ¼ a c ¼ 10b ¼ 6:92 ð11Þ saðmc ¼ 3Þ saðmc ¼ 2Þ Dm ¼ mms m ð9Þ The dashed lines in Fig. 3 have been obtained Substitution of Eqs. (8) and (9) into Eq. (1) gives from the least square best fit of Eq. (4) with condi- Log NðÞ¼ m bmðÞ¼ m bmðÞ Dm m tions given by Eq. (11). We find that p = 1.05 and 10 ms -3 -7 c0 = 6910 days. sa(mc = 2) = 6.71 9 10 days, ð10Þ -6 sa(mc = 3) = 4.70 9 10 days, and sa(mc = 4) = In Fig. 2, from the fit of the Gutenberg–Richter 3.30 9 10-5 days. For large times there is generally relation Eq. (1), we find from Eq. (8)thatm = good agreement. The deviations at small times for the a/b = 7.62 and Dm = 0.03. For the Chi–Chi earth- mc = 2 and mc = 3 data are can be attributed to a quake the magnitude of the largest actual aftershock, failure to record early weak aftershocks. mw = 6.70, is much smaller than the largest inferred SHCHERBAKOV et al.(2006) studied the interoc- aftershock, m = 7.62. For ten large earthquakes in currence time statistics for Parkfield aftershock California, SHCHERBAKOV and TURCOTTE (2004) found sequences. The distribution of interoccurrence times on average Dm = 1.11. If this value had been valid for is well approximated by a nonhomogeneous Poisson the Chi–Chi earthquake, the main shock magnitude (NHP) process driven by the modified Omori’s law would have been mms = 8.73. Clearly, the Chi–Chi over a finite time interval T (SHCHERBAKOV et al. earthquake had a large number of small aftershocks. 2005). In this study we also analyze the temporal NANJO et al.(2007) carried out a detailed study of aftershock statistics for four moderate sized Japanese 107 earthquakes. The relevant data were obtained from 6 the seismic catalog maintained by the Japan Meteo- 10 rological Agency. For the m = 7.3 1995 Kobe 105 earthquake m = 6.27 and Dm = 1.03, for the 104 m = 7.3 2000 Tottori earthquake m = 6.14 and -1 Dm = 1.16, for the m = 6.8 2004 Niigato earth- 103 quake m = 7.15 and Dm =-0.35, and for the 2 10 m =2.0 m = 7.0 2005 Fukuoka earthquake m = 6.27 and c dN/dt, days 1 m =3.0 10 c Dm = 0.73. Thus, the Niigato earthquake had a m =4.0 c large excess of aftershocks similar to the Chi–Chi 100 τ (4)= 3.30*10-5days a earthquake. τ (3)= 4.70*10-6days 10-1 a We next consider the decay rate of aftershock τ (2)= 6.71*10-7days a activity following the Chi–Chi earthquake. Figure 3 10-2 -3 -2 -1 0 1 2 3 shows the rates of occurrence of aftershocks with 10 10 10 10 10 10 10 t, days magnitudes greater than mc in numbers per day for a time period of 1,000 days after the Chi–Chi earth- Figure 3 The rates of occurrence aftershock with magnitude grater than mc quake. Different symbols corresponds to different in number per day for a time period of 1,000 days after Chi–Chi lower magnitude cutoffs, mc, which are taken to be earthquake. Difference symbol corresponds to a different lower magnitude cutoffs m , which were taken to be m = 2.0, 3.0, and mc = 2.0, 3.0, and 4.0. To quantify the observed c c 4.0. The dash line is the best fitting of the generalized Omori’s law aftershock occurrence scaling, we use the generalized -3 with p = 1.05 and c0 = 6 9 10 . The value of characteristic Omori’s law given in Eq. (3). As previously dis- times sa(mc) for mc = 2.0, mc = 3.0, and mc = 4.0 are -7 -6 -5 cussed we will assume that c0 is constant and sa(mc) sa = 6.71 9 10 days, 4.70 9 10 days, and 3.30 9 10 days Vol. 170, (2013) Aftershock Statistics of the 1999 Chi–Chi, Taiwan Earthquake 225 correlation properties of the Chi–Chi aftershock tRþDt rðuÞdu sequence. We treat all aftershocks with magnitudes Fðt; DtÞ¼1 e t ð12Þ equal to or greater than mc as a point process (SHCHERBAKOV et al. 2006). The interoccurrence times where r(u) is the rate of occurrence of aftershocks at between successive aftershocks are defined as time t. The probability density function of interoc- currence times over a finite time interval T is given by Dti = ti - ti-1, i = 1, 2…. In Fig. 4 we give the probability distribution function of the interoccurrence P ðDtÞ T 2 3 times between aftershocks, P(Dt, mc), each symbol sRþDt RDt TZ Dt rðuÞdu rðuÞdu corresponds to a different magnitude cutoff, mc = 2.0, 1 6 7 ¼ 4 rðsÞrðsþDtÞe 0 dsþrðDtÞe 0 5 3.0 and 4.0. We consider the time interval following the N main shock of T = 1,000 days. We apply the data bin- 0 ning technique proposed by CHRISTENSEN and MOLONEY ð13Þ (2005) which is using the log span for the P(Dt, m )to c where N is the total number of events during a time reduce the noise effect of long interoccurrence times. period T. In order to evaluate the rate r(t)wetakethe The observed statistics of interoccurrence times for the values from Eq. (4) that have been given as the dashed Chi–Chi aftershock sequence has a power-law depen- lines in Fig. 3. Substituting the values for m = 2, 3, and dence on the times between successive aftershocks for c 4 in Eq. (13) and integrating gives the dashed lines different magnitude cutoffs. plotted in Fig. 4.InFigs.3 and 4 we compare our the- The distribution of interoccurrence times can be oretical derivations of aftershock rates and distributions approximated by a distribution derived from the of interoccurrence times from Chi–Chi aftershocks, we assumption that aftershocks follow the NHP process see excellent agreement with mc = 4.0. This is evidence (SHCHERBAKOV et al. 2005). The probability distribu- that the catalogue is nearly complete for m C 4.0 with tion function of interoccurrence times at time t, until few missing early aftershocks. But for the smaller the next event in accordance with the NHP hypothesis aftershocks with m \ 4.0, it is clear that significant has the following form: numbers early in the aftershock sequence are lost.
-1 We next introduce the concept of Omori times for 10 the Chi–Chi region (Fig. 1) from 1994 to 2008. The mc =2.0
-2 mc =3.0 Omori time sN is defined as the mean interoccurrence 10 m =4.0 c interval over a fixed number N of earthquakes
-3 10 1 Xn 1
) sN ðtnÞ¼ ðtmþ1 tmÞð14Þ c -4 N 10 m¼n N t,m Δ We assign the values of s to the end tn of the
P ( -5 10 subsequence of N events. The Omori time is the
-6 reciprocal of the average of the rate of aftershock 10 activity r for N aftershocks. In Fig. 5 we give the
-7 Omori times for the 1999 Chi–Chi Taiwan earth- 10 quake considered above. The earthquake magnitudes
100 102 104 106 are greater than mc = 2.0. The Omori times sN are Δt (second) given for N = 25, 50, 100, and 200. For the 6 years before the Chi–Chi earthquake, the data are reason- Figure 4 The probability distribution function of the interoccurrence times able well represented by the mean Omori time sb = between aftershocks, P(Dt, mc). Each symbol corresponds to a 0.1753 days (the grey dashed lines in Fig. 5), corre- different magnitude cutoff mc, mc = 2.0, 3.0 and 4.0. A time -1 sponding to a background rate rb = 5.7045 day . interval following the mainshock T = 1,000 days in considered. The resulting theoretical distribution from the nonhomogeneous During the aftershock sequence, the modified form of Poisson (NHP) process is shown by dashed curves Omori’s law is given by Eq. (4). We consider the 226 Y.- T. Lee et al. Pure Appl. Geophys.
-3 Omori time behavior during the 9 years after the sb = 0.1753 days, p = 1.05, c0 = 6910 days, and -7 Chi–Chi earthquake, and assume that the rate of sa(mc = 2.0) = 6.71 9 10 days (s is the red occurrence r is the sum of the rate of background dashed lines in Fig. 5). These parameters are from the seismicity, rb, and the rate of aftershock seismicity ra Omori’s law correlation given in Fig. 3. (KISSLINGER 1996). With ra given by Eq. (4) we obtain 1 r ¼ rb þ ra ¼ rb þ p ð15Þ 3. Discussion and Conclusions sað1 þ t=c0Þ
The Omori times are given by Seismicity is a complex system, and within this complexity there are several scaling laws. The GR 1 s þ s ð1 þ t=c Þp 1 s ¼ ¼ b a 0 ð16Þ scaling is equivalent to fractal scaling between the r s s ð1 þ t=c Þp b a 0 number of earthquakes and their rupture area, and this The Omori times for the Chi–Chi region (Fig. 5) scaling may be the consequence of a fractal distri- are in good agreement with Eq. (16) taking bution of fault sizes. Presumably aftershocks satisfy
0.4 N=25
τ 0.2 0 1994 1996 1998 2000 2002 2004 2006 2008
0.4 N=50
τ 0.2 0 1994 1996 1998 2000 2002 2004 2006 2008
0.4 N=100
τ 0.2 0 1994 1996 1998 2000 2002 2004 2006 2008
0.4 N=200
τ 0.2 0 1994 1996 1998 2000 2002 2004 2006 2008
8 7 6 5 M 4 3 2 1994 1996 1998 2000 2002 2004 2006 2008 Time
Figure 5 The Omori times for the Chi–Chi region (Fig. 1) from 1994 to 2008. The earthquake magnitudes are greater than 2.0. The Omori times sN are given for N = 25, 50, 100, and 200. The gray dashed lines are the mean Omori times before the Chi–Chi earthquake. The red dashed lines are -3 the theoretical Omori times for Chi–Chi aftershocks from Eq. (12) with s0 = 0.1753 days, p = 1.05, c = 6910 days, and -7 s(mc = 2.0) = 6.71 9 10 days. The magnitude of these earthquakes are also given Vol. 170, (2013) Aftershock Statistics of the 1999 Chi–Chi, Taiwan Earthquake 227
GR scaling for the same reason that all earthquake do. decay, RUNDLE (1989), RUNDLE and KLEIN (1993), and However, there is no accepted theory for the expla- RUNDLE et al. (1999) have proposed the spinodal line nation of the scale-invariant nature of this for critical point nucleation to relate aftershock distribution. Generally, two models have been pro- sequences to the power–law scaling. posed: (1) each fault has a GR distribution of The temporal evolution of the rate of occurrence earthquake magnitudes and (2) there is a power-law of aftershocks is quantified using Omori’s law frequency-area distribution of faults and each fault Eq. (3). This paper also introduces another quantity, has recurring characteristic earthquakes (TURCOTTE Omori times, to quantify the temporal evolution of et al. 2007). interoccurrence times. In Fig. 5 we give the Omori SHCHERBAKOV et al.(2006) analyzed the scaling times for the period from 1994 to 2008 for the Chi– laws for the aftershock sequence from the 2004 Chi earthquake. It includes the Chi–Chi aftershocks Parkfield earthquake, and the aftershocks satisfy the with a combination of Omori decay and a steady-state GR relation with b = 0.88 and a = 4.4. They also background. The result shows that during the period considered Ba˚th’s law for Parkfield earthquake. The 1999 after the Chi–Chi earthquake to 2008, the rate of magnitude of the Parkfield earthquake was mms = 6.0 seismicity decayed systematically with the exception and the magnitude of the largest aftershock was of a burst of seismicity in 1999. The distribution of mmaxas = 5.0 so that Dm = 1.0 which is close to the Omori times during the period 2000–2008 is in good value of Dm = 0.95 for the Chi–Chi earthquake. 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(Received February 16, 2011, revised November 22, 2011, accepted November 28, 2011, Published online March 23, 2012)