Scaling relationship of initiations for moderate to large earthquakes

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111,, doi:10.1029/2005jb003613, 2006 Scaling relationship of initiations for moderate to large earthquakes K. Sato 1 and J. Mori 1 Received 5 January 2005; revised 7 September 2005; accepted 21 December 2005; published 11 May 2006. [1] We investigate the initiation sizes of earthquakes over a wide range of moderate to large events. Using the model of Sato and Kanamori, which is based on Griffith crack theory for a circular fault, we estimate the initial crack sizes of earthquakes, using highquality data recorded at close hypocentral distances. Taking into account the local attenuation and the dynamic stress drop of the earthquakes, we estimate trigger factors and initial crack radii for 35 events in Japan and Taiwan with magnitudes of 3.5 to 7.9. Our results show that all of the earthquakes can be explained with a small trigger factor and initial crack sizes of 12 96 m. For the large range of earthquake sizes, this is a small range of initial crack sizes. There does not appear to be a clear scaling relationship between initial crack size and final earthquake size. Instead, our results show that all the moderate to large earthquakes initiate with initial crack radii on the order of tens of meters. Citation: Sato, K., and J. Mori (2006), Scaling relationship of initiations for moderate to large earthquakes, J. Geophys. Res., 111,, doi:10.1029/2005jb003613. 1. Introduction [2] The initiation process of earthquakes is currently not well understood and there are many unresolved issues such as, the triggering mechanisms, the rates of slip velocity, the rates of rupture speed, and the scaling with earthquake size. In this paper we look at the details of the beginnings of earthquake ruptures and try to determine whether or not there is a deterministic scaling of the initiation and the final earthquake size. [3] When investigating the earthquake initiation, two end-member models can be considered. In nucleation models there is a deterministic process from the beginning of the rupture initiation that scales with the final size of the earthquake. In contrast, for percolation or cascade models the rupture is thought of as a stochastic process and the growth or termination of the rupture is not predetermined. [4] In a nucleation model, a crack, which is called a nucleation zone, is generated by slow quasi-static rupture and high-speed rupture begins as the nucleation zone size becomes larger than some critical size. In this model, a quasi-static process precedes an earthquake. A larger earthquake requires a larger nucleation zone, so the final earthquake size can be detected at the time of the initiation. There are a number of laboratory results and earthquake observations that support a nucleation model. It has been shown in experiments that quasi-static rupture precedes high speed rupture of rock [Dieterich, 1978; Ohnaka and Kuwahara, 1990]. Ohnaka and Shen [1999] showed that nucleation zone size varies due to the roughness of a fault 1 Disaster Prevention Research Institute, Kyoto University, Kyoto, Japan. Copyright 2006 by the American Geophysical Union. 0148-0227/06/2005JB003613$09.00 surface. Iio [1992, 1995] analyzed the initial part of microeathquake waveforms and concluded that there were slow initial phases at the beginning of the waveforms and their durations scaled with their final earthquake sizes. Ellsworth and Beroza [1995] and Beroza and Ellsworth [1996] analyzed earthquakes in a wide magnitude range of M2.5 8.1 and reported that earthquakes have initial lowamplitude phases (called nucleation phases) and the seismic moment released during the process (nucleation moment) scaled with the final sizes. Umeda [1990, 1992] and Dodge et al. [1996] indicated that some earthquakes had weak initial phases and durations of these phases scaled with their final sizes. Shibazaki and Matsu ura [1998] considered that these phases were produced during the transitional process between quasi-static and high-speed ruptures. Hiramatsu et al. [2002] analyzed microearthquakes (M 0.3 to 2.2) which occurred during a water injection experiment in the Nojima fault, Japan and showed that some earthquakes could be explained by sources with a gradual initiation, while other earthquakes required an abrupt start. They also showed that initial crack radii of the events with the gradual initiations, scaled with the final earthquake sizes. [5] For the cascade models, the rupture is considered to be a more random process and the final size is not determined when the earthquakes begins. For example, Otsuka [1971a, 1971b] introduced a model where an earthquake fault consisted of small patches and rupture in a patch can be triggered if an adjacent patch failed. Otsuka [1971b] showed that this model generated the Gutenberg- Richter relationship of the magnitude-frequency distribution. Fukao et al. [1992] introduced a model where earthquakes of different sizes initiated from small regions of the same size, and therefore a large earthquake could not be separated from small earthquakes by observations of their beginnings. Mori and Kanamori [1996] analyzed earthquakes in the Ridgecrest, California, sequence and showed 1of11

We note that these initiations are similar to the type described by Iio [1992, 1995]. Figure 1. Initiations of velocity waveforms for different initial crack radii. (a) Small trigger factor of d = 0.01. Initiation becomes slower when initial crack radius becomes larger. (b) Large trigger factor of d = 1. Waveforms have little differences. These synthetic waveforms are calculated for the event I318. that slow initial phases at the beginning of their P waves could be explained by attenuation effects. They concluded that slow propagation at the sources was not required. Ellsworth and Beroza [1998] also looked at the Ridgecrest sequence and concluded that the initiations did not have initial phases of the type described by Iio [1992, 1995] but had nucleation phases as described by Ellsworth and Beroza [1995]. [6] There is some confusion about the terms initiation, slow slip, slow initial phase, nucleation phase, and the timescales and slip velocities that they refer to. For example, nucleation phases of many earthquakes analyzed by Ellsworth and Beroza [1995] seem similar to the initial phases of Umeda [1990, 1992] but not like those described by Iio [1992, 1995]. We think it is important to specify the absolute amplitudes of the rupture propagation, when distinguishing between terms such as slow initial phase and high-speed rupture. [7] In this study, we look quantitatively at the rate at which smooth initiations begin for the very start of the rupture (0.1 s) for a large magnitude range of earthquakes. 2. Model [8] We use a source model developed by Sato and Kanamori [1999]. This is a circular crack model based on the Griffith s fracture criterion which can generate both sources with abrupt initiations and sources with more gradual growth in rupture speed. Rupture speed is controlled by the distribution of surface energy around the initial crack. Using this model, we can explain the beginning shape of a P wave for both abrupt beginnings and gradually increasing slopes. [9] In the case that surface energy on the edge of the initial crack is much larger than off the edge, rupture starts abruptly, and the pulse shape of the P wave shows a rapid linear growth in velocity. In the case that surface energy on the edge of the initial crack is close to the value off the crack, rupture propagates gradually, and the velocity pulse shape shows a gradual growth, with a concave upward shape. Contrast between surface energy on and off the initial crack is represented by a trigger factor, d, in the model. [10] Figure 1 shows how the velocity waveform (and therefore rupture pattern) changes according to the trigger factor and initial crack radius. We call these two parameters initiation-related parameters. As shown in Figure 1, smaller trigger factors generate slower initiations for the same initial crack radius. For a small trigger factor, larger initial crack radii generate slower initiations of the P wave (d 0.01). For a large trigger factor, different initial crack radii do not generate much differences in the shape of the P wave initiation (d 1). [11] In this study, we analyze the initial part of seismic waveforms of moderate to large earthquakes and estimate how these events initiated. In the analyses, we calculate synthetic waveforms based on the Sato and Kanamori [1999] model which include propagation effects and estimate the best initiation-related parameters. The analyses provide a convenient way to quantitatively compare the P wave shapes for a large range of earthquakes. [12] Although Sato and Kanamori [1999] used the two terms spontaneous model and trigger model for small and large trigger factor cases, respectively, we do not use such terms because both of them are described by the same equation. 3. Data [13] In this study, we analyze five data sets: (1) the 2000 Izu islands, Japan, earthquake swarm (M w 3.8 4.9), (2) the 1999 Chi-Chi, Taiwan, earthquake (M w 7.6), (3) the 2000 western Tottori, Japan, earthquake (M w 6.6), (4) the 2003 northern Miyagi, Japan, earthquakes (M w 3.1 6.1), and (5) the 2003 Tokachi-oki, Japan, earthquake and its aftershocks (M w 5.4 7.9). We describe these events and data characteristics below. These data cover a magnitude range between 3.1 and 7.9 and include both crustal and interplate earthquakes. Hypocentral locations and focal mechanisms are summarized in Table 1. All of these data sets were chosen because they had clear well-recorded waveforms at dis- 2of11

Table 1. List of Earthquakes Analyzed in This Study Event Date Time, UT Latitude, deg Longitude, deg Depth, km M w Strike Dip Rake Focal Mechanism, deg Izu Islands I205 8 Jul 2000 0647:20.96 34.207 139.279 3.68 4.7 154 82 56 I262 8 Jul 2000 2129:01.05 34.212 139.254 10.65 4.0 150 81 28 I287 10 Jul 2000 1115:48.51 34.208 139.254 10.87 4.9 348 67 28 I308 11 Jul 2000 0855:24.73 34.216 139.263 10.99 4.3 334 78 63 I310 11 Jul 2000 0900:52.38 34.217 139.247 10.62 3.8 170 63 61 I318 11 Jul 2000 0944:28.64 34.214 139.268 9.31 4.7 156 54 63 I343 11 Jul 2000 1119:57.73 34.224 139.253 11.32 4.6 165 68 38 I350 11 Jul 2000 1142:57.19 34.205 139.271 11.48 4.1 95 67 151 I358 11 Jul 2000 1221:07.39 34.215 139.263 9.128 4.4 294 56 128 I361 11 Jul 2000 1243:33.37 34.219 139.250 9.678 3.8 96 89 179 I365 11 Jul 2000 1343:58.51 34.218 139.257 10.51 4.1 295 51 121 I390 11 Jul 2000 0248:27.78 34.218 139.259 10.76 3.9 329 72 76 Taiwan TAIWAN 20 Sep 1999 1747:00.00 23.853 120.816 8.00 7.6 5 34 65 Tottori TOTTORI 6 Oct 2000 0430:00.00 35.270 133.352 12.18 6.6 132 90 0 Northern Miyagi M001 25 Jul 2003 1513:08 38.431 141.168 11.60 5.5 209 54 92 M012 25 Jul 2003 1848:31 38.430 141.189 12.10 3.6 209 54 118 M013 25 Jul 2003 2022:04 38.383 141.155 11.10 3.5 199 40 80 M014 25 Jul 2003 2213:31 38.402 141.174 11.90 6.1 186 52 88 M054 26 Jul 2003 0205:36 38.462 141.223 11.10 3.7 159 67 92 M055 26 Jul 2003 0208:04 38.466 141.193 11.80 3.7 189 59 122 M057 26 Jul 2003 0316:24 38.472 141.224 11.00 3.5 189 50 84 M060 26 Jul 2003 0529:00 38.400 141.199 11.70 3.7 166 53 66 M063 26 Jul 2003 0603:39 38.465 141.190 11.50 3.6 187 70 60 M065 26 Jul 2003 0641:53 38.483 141.201 11.70 3.9 121 73 51 M067 26 Jul 2003 1056:44 38.497 141.193 12.00 5.3 131 39 101 M095 30 Jul 2003 1311:53 38.425 141.197 11.60 3.5 240 45 91 M101 31 Jul 2003 2136:41 38.460 141.193 12.20 3.4 244 58 119 M109 4 Aug 2003 2145:29 38.392 141.171 11.90 3.5 216 45 99 M114 6 Aug 2003 1626:31 38.465 141.221 11.60 3.1 11 87 153 M115 8 Aug 2003 0051:31 38.517 141.230 10.80 4.3 184 60 57 M118 8 Aug 2003 1754:51 38.457 141.171 13.20 3.7 164 52 55 M121 10 Aug 2003 0122:28 38.473 141.156 13.30 3.4 175 64 126 M122 12 Aug 2003 0027:58 38.495 141.181 12.10 4.1 113 63 58 M123 12 Aug 2003 0454:43 38.466 141.173 13.20 3.3 164 35 70 Tokachi-oki MAIN 25 Sep 2003 1950:7.42 41.776 144.082 23.0 7.9 249 15 127 tances relatively close to the hypocenters. The close distances of the stations are an important constraint for observing the details of the initiation of the ruptures. Because multiple stations with short hypocentral distances are available for the Taiwan and Tottori data sets, we analyze data of two stations and compare their results to check the reliability of our analyses. Stations used in this study are shown in Table 2. 3.1. The 2000 Izu Islands Earthquake Swarm [14] A strong earthquake swarm of thousands of events over several months occurred near Miyakejima, Kozushima, and Niijima (Izu islands region, Japan) from the end of June 2000, and continued to mid-september, with associated volcanic activity at Miyakejima. Seismic activity moved to the west and a M w 6.2 event occurred about 10 km east of Kozushima on 1 July 2000. This activity has been interpreted as due to a dyke intrusion [e.g., Ito and Yoshioka, 2000]. Figure 2 shows the hypocentral distribution. [15] We installed strong motion seismometers on Kozushima just after the M w 6.2 event and recorded many moderate Table 2. List of Stations Used in This Study Name Latitude Longitude Network Izu Islands KZA 34.198 139.139 Taiwan TCU078 23.8120 120.8450 CWB TCU079 23.8395 120.8942 CWB Tottori SMNH01 35.2931 133.2628 KiK-net TTRH02 35.2281 133.3936 KiK-net Northern Miyagi MYGH06 38.5878 141.0744 KiK-net MYGH11 38.5131 141.3456 KiK-net Tokachi-oki OBS1 41.6847 144.3991 JAMSTEC 3of11

Figure 2. Hypocenter distribution and focal mechanisms of the events in the Izu islands data set. Triangle represents the station KZA, where data we used in this study were recorded. Stars represent earthquakes analyzed in this study. Twelve earthquakes are analyzed in this data set. events between 4 and 13 July. We use data from the KZA station, which is sampled at 200 Hz. We analyze 12 closely recorded events in this data set. The magnitude range is from M3.8 to M4.9. We use hypocentral locations determined by the Earthquake Research Institute (ERI), University of Tokyo and focal mechanisms determined by F-net operated by the National Research Institute for Earth Science and Disaster Prevention (NIED). Hypocentral distances are between 10 and 20 km. 3.2. The 1999 Chi-Chi, Taiwan, Earthquake [16] We analyze the main shock of the 1999 Chi-Chi, Taiwan, earthquake (M w 7.6). This is a large shallow thrust earthquake that occurred on the Chelungpu fault in the fold and thrust belt of central Taiwan. There were large surface displacements of 1 to 8 m along the faulting length of about 70 km. We use strong motion accelerograms, sampled at 200 Hz, recorded at TCU078 and TCU079 by the Central Weather Bureau, Taiwan [Lee et al., 1999], with hypocentral distances of 9.8 and 11.6 km, respectively. For this study, the acceleration data were integrated to velocity. Since the depth is 7 km, the epicentral distances are only several kilometers, making these among the closest recorded data to the initiation of a large earthquake. We use the hypocentral location and a focal mechanism determined by Chang et al. [2000]. Figure 3 shows the distribution of hypocenters and stations. 3.3. The 2000 Western Tottori, Japan, Earthquake [17] We analyze the main shock of the 2000 western Tottori, Japan, earthquake (M w 6.6), which occurred on 6 October 2000. We use strong motion accelerograms, sampled at 200 Hz and recorded at about 100 m depth at borehole stations SMNH01 and TTRH02 of KiK-net, operated by NIED, with hypocentral distances of 14.9 and 13.6 km, respectively, The close hypocentral distances and borehole recordings produced clear waveforms for the onset of this moderate. earthquake. This was a shallow strike-slip earthquake with a rupture length of about 20 km that occurred on a previously unmapped fault in western Japan. We use the hypocentral location and a focal mechanism determined by Ohmi [2001]. Figure 4 shows the locations of the hypocenters and stations. 3.4. The 2003 Northern Miyagi, Japan, Earthquakes [18] We analyze 20 small to moderate events (M w 3.5 to 6.1) in the 2003 northern Miyagi, Japan, earthquake sequence. This was a series of shallow crustal thrust earthquakes, that included five events with magnitudes over M w 5.0 that occurred onshore near the Pacific coast on 26 July 2003. We use vertical component accelerograms sampled at 200 Hz and recorded at 100 to 200 m deep borehole stations MYGH06 and MYGH11 of KiK-net, NIED. Hypocenter distances ranged from 14.8 to 26.4 km. We use hypocentral locations from the Japan Meteorological Agency (JMA) and focal mechanisms of F-net, NIED. Figure 5 shows the distribution of hypocenters and stations. 3.5. The 2003 Tokachi-oki, Japan, Earthquake [19] We analyze the main shock of the 2003 Tokachi-oki, Japan, earthquake (M w 7.9). The main shock was a great interplate earthquake that occurred on the subduction Figure 3. Hypocentral location and focal mechanism of the 1999 Chi-Chi, Taiwan, earthquake. We analyzed data recorded at the nearest stations TCU078 and TCU079. Triangles and star represent stations and hypocenter, respectively. Dotted line represents the Chelungpu fault trace. 4of11

the waveforms, we also need to know the attenuation effects and dynamic stress drops, in addition to the parameters in order to model the waveforms. We estimate attenuation effects by the following scheme (following Masuda and Suzuki [1982] and Ide et al. [2003]). We assume a frequencyindependent Q and the omega square source model. The synthetic velocity spectrum calculated for an omega square model source with a frequency-independent Q path, is written as [Boatwright, 1978] j_uðfþj ¼ R P fm o 2ra 3 r n 1 þ ðf =f c Þ 4 o1 exp pft ð1þ 2 Q P Figure 4. Hypocenter location and focal mechanism of the 2000 western Tottori earthquake. We analyzed data recorded at the nearest stations SMNH01 and TTRH02. Triangles, black star, and white star represent stations, main shock, and an aftershock which we used to estimate attenuation effects, respectively. Aftershocks are plotted by gray dots. boundary between the Pacific and continental plates, southeast of Hokkaido. We use an ocean bottom accelerogram sampled at 100 Hz and recorded at OBS1 of an ocean bottom observation array operated by the Japan Marine Science and Technology Agency (JAMSTEC). The hypocentral distance is 36.9 km. This is the farthest distance used in the study, but we include it because it provides data for a very large earthquake. We use the epicentral location from JMA and focal mechanism and depth determined by F-net, NIED. Figure 6 shows the locations of the hypocenter and the station. 4. Method [20] The purpose of this study is to investigate if the initiation pattern is different between small and large earthquakes. We would like to estimate initiation-related parameters (trigger factor and initial crack radius) for moderate to large earthquakes. However, to correctly model the shape of where R P, r, a, r, f, f c, and M o are P wave radiation pattern, density, P wave velocity, hypocentral distance, frequency, corner frequency, and seismic moment, respectively. We can estimate M o and Q P using a linear least square method because the logarithm of this equation is R P log j_uðfþj ¼ log 2ra 3 r pft Q P log e ( ) 4 þ log fm o 1 2 log 1 þ f f c We choose small earthquakes so that the sources can be considered to be close to point sources and the data in the observed passband are dominated by the attenuation effects. We estimate the corner frequency (f c ) with a grid search because it is not a linear parameter in this equation. [21] There is evidence that the Q value has a frequency dependence, especially in the range of 1 to 10 Hz [e.g., Adams and Abercrombie, 1998]. However, for this study we think it is adequate to describe the average apparent Q with a single frequency-independent value. Using the single parameter, we are able to model fairly well the frequency content of the small earthquake, as shown in Figure 7. [22] For some data sets, we did not have the waveforms from small earthquakes to estimate Q P. For these cases, we ð2þ Figure 5. Hypocenter location and focal mechanism of the events in the Miyagi data set. We analyzed 20 earthquakes recorded at MYGH06 or MYGH11. Triangles and stars represent stations and hypocenters, respectively. Figure 6. Hypocenter location and focal mechanism of the events in the Tokachi-oki data set. We analyzed the main shock recorded at OBS1. Triangle and star represent the station and hypocenter, respectively. 5of11

Figure 7. Observed and synthetic velocity spectra of the small earthquakes used to estimate attenuation effects. Synthetic spectra are calculated with assumptions of omega square model sources and frequencyindependent whole path Q. Bottom panels show the relationship between corner frequency and residual (shown by a ratio to the minimum residual). Inverted triangles in the bottom panels indicate the estimated corner frequency. used a range of typical values of Q P to estimate upper and lower values of the initial crack radius. [23] Using the determined Q P values or range, specific to each data set, we estimate dynamic stress drop and initiation-related parameters by a grid search method. Parameters are first searched for using a course grid (1 to 100 MPa with 1 MPa step for dynamic stress drop and 10 to 100 m with 10 m step for initial crack radius). Then a finer grid of values (0.5 MPa and 1m steps) is searched in the vicinity of the local minima. In some cases (e.g., too small stress drops), we tried to search the best solution using different search parameters. We find the best values which generate the smallest residuals between the observed and synthetic velocity waveforms calculated by equation (1), using an L2 norm. Because it is difficult to exactly pick the onset time for the observed waveform, due to background noise and instrumental limitations, we also search for the onset time simultaneously with the other parameters. There may be some trade between the onset time and the crack size, however, since the crack size is mostly controlled by the amount of curvature, we do not think there is a large problem. [24] Because there is a trade-off between the trigger factor and initial crack radius, we tested only two values of trigger factor: d = 0.01 for a small value and d = 1 for a large value. However, it is clear from looking at the seismograms, that the gradual start to the initiation indicates a small trigger factor. 5. Results 5.1. Attenuation Effects [25] We estimated Q P values for the Izu islands and Tottori data sets using waveforms of small events. Figure 7 6of11

Table 3. Initiation-Related Parameters Determined in This Study Event M w Station d l 0,m Ds, MPa r, km Izu Islands I205 4.7 KZA 0.01 45 18.0 14.3 I262 4.0 KZA 0.01 20 4.5 15.3 I287 4.9 KZA 0.01 18 2.5 15.4 I308 4.3 KZA 0.01 21 5.5 16.2 I310 3.8 KZA 0.01 24 7.5 14.9 I318 4.7 KZA 0.01 24 58.5 15.5 I343 4.6 KZA 0.01 51 81.5 15.9 I350 4.1 KZA 0.01 61 3.0 17.0 I358 4.4 KZA 0.01 48 21.0 15.0 I361 3.8 KZA 0.01 13 22 14.5 I365 4.1 KZA 0.01 63 7.55 15.5 I390 3.9 KZA 0.01 38 1.0 15.8 Taiwan TAIWAN 7.6 TCU078(Q P 50) 0.01 21 6.0 9.77 TAIWAN 7.6 TCU078(Q P 100) 0.01 26 3.5 9.77 TAIWAN 7.6 TCU078(Q P 200) 0.01 28 3.0 9.77 TAIWAN 7.6 TCU079(Q P 50) 0.01 12 6.0 11.6 TAIWAN 7.6 TCU079(Q P 100) 0.01 29 4.5 11.6 TAIWAN 7.6 TCU079(Q P 200) 0.01 34 4.0 11.6 [27] In the Miyagi data set, we used two small earthquakes (M063, M w 3.7 for MYGH06; M121, M w 3.4 for MYGH11) to estimate Q P values and obtained t* = 0.020 (Q P = 230) with a corner frequency of 2.2 Hz for MYGH06 and t* = 0.015 (Q P = 250) with a corner frequency of 4.0 Hz for MYGH11. [28] In other data sets, we could not directly estimate Q P values because of lack of similarly recorded small events. For the Taiwan data set we used a range of three different values (Q P = 50, 100, and 200) and two values (Q P = 100 and 200) for the Tokachi-oki data set. 5.2. Initiation-Related Parameters [29] We treated all of the data similarly and searched for the best fitting source parameters. Parameters were deter- Tottori TOTTORI 6.6 SMNH01 0.01 30 0.6 14.87 TOTTORI 6.6 TTRH02 0.01 19 0.7 13.59 Northern Miyagi M001 5.5 MYGH06 0.01 17 13.0 22.5 M012 3.6 MYGH06 0.01 37 6.0 23.6 M013 3.5 MYGH06 0.01 66 0.5 26.4 M014 6.1 MYGH06 0.01 46 11.5 25.5 M054 3.7 MYGH06 0.01 79 0.7 22.1 M055 3.7 MYGH06 0.01 33 0.5 20.8 M057 3.5 MYGH06 0.01 32 6.5 21.4 M060 3.7 MYGH06 0.01 39 0.6 26.4 M063 3.6 MYGH06 0.01 16 0.8 20.5 M065 3.9 MYGH06 0.01 15 98.5 19.9 M067 5.3 MYGH06 0.01 47 0.5 18.8 M095 3.5 MYGH11 0.01 56 1.5 20.0 M101 3.4 MYGH06 0.01 34 4.0 21.4 M109 3.5 MYGH06 0.01 27 4.5 26.3 M114 3.1 MYGH06 0.01 31 4.5 22.1 M115 4.3 MYGH11 0.01 30 3.0 14.8 M118 3.7 MYGH11 0.01 22 14.0 21.1 M121 3.4 MYGH11 0.01 34 0.8 21.7 M122 4.1 MYGH11 0.01 43 0.7 18.9 M124 3.3 MYGH11 0.01 17 1.5 20.7 Tokachi-oki MAIN 7.9 OBS1(Q P 100) 0.01 86 2.0 36.9 7.9 OBS1(Q P 200) 0.01 96 1.0 36.9 shows the best fit observed and synthetic spectra for the various events analyzed. In the Izu islands data set, We used two small events (I310, M w 3.8 and I365, M w 4.1) to estimate the Q P value and obtained t* = 0.026 (Q P =111) for I310 with a corner frequency of 5.6 Hz and t* = 0.038 (Q P = 79) for I365 with a corner frequency of 4.2 Hz. From these results, we use Q P = 100 for all events in the Izu islands data set. [26] In the Tottori data set, we used a small aftershock (M w 3.3) recorded at the two stations, SMNH01 and TTRH02, and obtained t* = 0.011 (Q P = 195) for SMNH01 with a corner frequency of 5.0 Hz and t* = 0.009 (Q P = 203) for TTRH02 with a corner frequency of 3.6 Hz. This event occurred near the hypocenter of the main shock, and we assumed the attenuation effects are the same as for the main shock (shown by a white star in Figure 4). Figure 8. Observed waveforms and synthetic waveforms of the Izu islands data set calculated with the best fit values of the initiation parameters. Upper traces show the whole P wave (observed waveforms) and lower traces show magnified plots of the initial portions (observed and synthetic). Vertical lines in the upper traces show the times of the onset for the expanded traces. Amplitudes are velocity in m/s. 7of11

for Q P = 200 and a lower limit of 12m for Q P = 50, with a small trigger factor for both cases. The difference in the values from the two stations may be related to directivity effects. [32] Figure 10 shows observed and synthetic waveforms for the 2000 western Tottori (M6.6) earthquake. We estimated initiation related parameters for the main shock using two stations in this data set. We obtained a small trigger factor (d = 0.01) for both cases. An initial crack radius is estimated as 23 m from SMNH01 and 19 m from TTRH01 (Figure 11). [33] Figure 12 shows observed and synthetic waveforms for 20 events (M w 3.5 to 6.1) of the 2003 northern Miyagi sequence. We estimated initiation-related parameters using data recorded at MYGH06 or MYGH11for. We found a small trigger factor (d = 0.01) for all the events. Initial crack radii are 15 m to 79 m. Figure 9. Observed waveforms and synthetic waveforms of the Taiwan data set calculated with the best fit values of the initiation parameters. Upper traces show 2 s of the P wave (observed waveform) and lower traces show magnified plots of the initial portions (observed and synthetic). Amplitudes are velocity in m/s. mined using grid searches with the results summarized in Table 3 and described for each data set below. [30] Figure 8 shows the observed waveforms and synthetic waveforms calculated with parameters associated with the smallest residuals for 12 earthquakes (M w 3.8 to 4.9) of the 2000 Izu islands swarm. We found a small trigger factor (d = 0.01) and initial crack radii of 13 to 63 m for these events. [31] Figure 9 shows the observed and synthetic waveforms for the Chi-Chi, Taiwan (M w 7.6), earthquake. Because we could not estimate Q P values for this data set, we tested three different values for Q P. Using data from the station TCU078, we obtained an upper limit for the initial crack radius of 28 m for Q P = 200 and a lower limit of 21 m for Q P = 50, with a small trigger factor (d = 0.01) for both cases. We also used data from the station TCU079 and obtained an upper limit for the initial crack radius of 34 m Figure 10. Observed waveforms and synthetic waveforms of the Tottori data set calculated with the best fit values of the initiation parameters. Upper traces show about 1 s of the P wave (observed waveform) and lower traces show magnified plots of the initial portions (observed and synthetic). Amplitudes are velocity in m/s. Figure 11. Observed waveforms and synthetic waveforms of the northern Miyagi data set calculated with the best fit values of the initiation parameters. Upper traces show the whole P waves (observed waveforms) and lower traces show magnified plots of the initial portions (observed and synthetic). Amplitudes are velocity in m/s. 8of11

[37] The line in Figure 14 shows a constant ratio between initial and final crack radii derived from the data of Hiramatsu et al. [2002]. Our results should fall close to the line if earthquakes are to be explained by the nucleation model and their final sources sizes are scaled to the initiation size. However, the results of our study show that initial crack sizes are much smaller than this relationship. For example, an initial crack radius of tens of kilometers is necessary for M7 events to fit this relationship. [38] The result that the initiation crack sizes are about the same over a wide range of moderate to large earthquakes, can also readily be seen in the waveforms of Figures 8 to 12. In these waveforms, all the earthquakes have reached normal rupture growth within about 0.1 to 0.2 s. By normal, we mean that the dynamic stress drops are on the order of megapascals, the rupture speed is on the order of kilometers per second, and the slip velocity is on the order of centimeters per second. Even for the large M7 earthquakes, there is no evidence of a slow initial phases with durations longer than 0.2 s. Figure 12. Observed waveforms and synthetic waveforms of the northern Miyagi data set calculated with the best fit values of the initiation parameters. Upper traces show the whole P waves (observed waveforms) and lower traces show magnified plots of the initial portions (observed and synthetic). Amplitudes are velocity in m/s. [34] Figure 13 shows observed and synthetic waveforms for the 2003 Tokachi-oki (M7.9) earthquake. We tested two different Q P values for this data set. We obtained for the main shock an upper limit of the initial crack radius of 96 m for Q P = 200 and a lower limit of 86 m for Q P = 100, with a small trigger factor (d = 0.01) for both cases. [35] We found small trigger factors for all of the initiations in this study. Looking at the waveforms, we always see a curvature to the beginning, which means we cannot model the data with a large trigger factor that produces a more abrupt beginning (Figure 1). There may be intermediate trigger value that trades off directly with the crack size. So these crack sizes can be considered to be maximum estimates of the dimensions. [36] Figure 14 is the main result of this paper and shows the relationship between initial crack sizes and final earthquake sizes for all data in this study. We can see that initial crack radii have values of the same order (about 15 to 50 m) for large earthquakes that span many orders of final sizes. We also plotted the results of Hiramatsu et al. [2002], which are for microearthquakes in the Nojima fault, Japan. They concluded that initial crack radii were proportional to their final sizes for some events. However, our results show that such a proportional relationship does not fit the larger earthquakes and we conclude that all events that we analyzed in this study initiated from similar sizes regardless of their final sizes. 6. Discussion [39] We consider some of the uncertainties in our results. Because we estimated initiation-related parameters using Q P results, our results of initiation-related parameters will have errors if Q P values are incorrect. However, we can say that the uncertainty of Q P does not affect the basic results of this study, because our tests for different Q P values in the Taiwan and Tokachi data sets result in initial crack radii of different sizes by a factor of less than 2. [40] If different results are obtained from different station data for the same event, the results would be considered unreliable. We tested this issue for the Taiwan and Tottori data sets. For these data sets, we estimated initiation-related parameters using data from different stations. For the case of the main shock of the Chi-Chi, Taiwan earthquake, parameters estimated for Q P = 200 are d = 0.01, l 0 =28m for the station TCU078 and d = 0.01, l 0 = 34 m for the station TCU079. For the case of the main shock of Tottori, estimated parameters are d = 0.01, l 0 = 30 m for the station SMNH01 and d = 0.01, l 0 = 19 m for the station TTRH02. Again the results show differences of less than a factor of 2. Figure 13. Observed waveforms and synthetic waveforms of the Tokachi-oki data set calculated with the best fit values of the initiation parameters. Upper traces show about 1.5 s of the P waves (observed waveforms) and lower traces show magnified plots of the initial portions (observed and synthetic). Amplitudes are velocity in m/s. 9of11

Figure 14. Relationship between seismic moment and initial crack radius for all events analyzed in this study. Results of Hiramatsu et al. [2002] are also plotted (shown by crosses). The dashed line shows a constant ratio between initial and final radii for a constant stress drop. From these analyses, our results appear to have uncertainties less than a factor of 2. [41] Hiramatsu et al. [2002] indicated that some small events in their data set could be explained well with a rapidly propagating circular crack model of Sato and Hirasawa [1973] plus attenuation effects; however, it is difficult to explain our data in the same way. To model our data in this way, an extremely low Q P value (lower than 50) is required. This Q P value is probably much too low for an average value in the crust for the distance ranges that we use. [42] There is a question of the applicability of the Sato and Kanamori [1999] model, which is based on static Griffith theory for a circular model. The use of this model enables the interpretation of the curvature of the P wave onset as a measure of the initial crack size. In the present study, we did not consider other geometrical effects (e.g., elliptical sources, line sources, asperity models, directivity effects) but assumed that the circular model was reasonable for a first-order estimate. We use this model because it provides a simple way for quantitatively comparing the waveform onsets for a variety of data sets. Using the same time window for earthquakes from M4 to M8, we can make consistent comparisons of the shape of the initiation of the waveforms. [43] There are many considerations when interpreting the onset curvature as an absolute initiation size. Other processes are likely occurring at the rupture initiation which affect the shape of the waveform initiation. For example, in a heterogenous material, rupture initiation is likely occurring on very small scale preexisting cracks. Also, an important issue to investigate is, what are the dimensions of these cracks and are they similar for the initiations of small and large earthquakes. The basic observational point we make in this study is that over a large range of magnitudes from M4 to M8, the initiation onsets of the waveforms for the first 0.1 to 0.2 s (which likely correspond to dimensions of tens of meters) look very similar. Thus we infer that the initiation sizes are similar. [44] There may be some important features in the way the rupture grows in the subsequent few seconds, after the 0.1 to 0.2 s time window that is studied in this paper. If heterogeneities on the fault, with dimensions of tens to hundreds of meters, are important in controlling the growth of an earthquake rupture, there may be some size-dependent differences in the rupture process during the first few seconds. This issue is examined in more detail in a related paper [Sato and Mori, 2006]. [45] The results of this study do not completely exclude the nucleation model. It is reasonable that the size of a crack from which high speed rupture starts, is related to the energy of the rupture process, so that earthquakes with larger nucleation zones may statistically tend to have a larger final sizes. Also, we are studying only the very beginning of rupture, and not do consider longer period processes that may last for several seconds. However, the results of this study show that the initiation sizes appear not to be proportional to the final size of the earthquake. 7. Conclusions [46] We investigated the initiation sizes of small to large earthquakes based on a circular crack model developed by Sato and Kanamori [1999]. We estimated initiation-related parameters, trigger factor and initial crack radius, for moderate to large earthquakes in five data sets. The results showed that all events analyzed in this study could be explained with a small trigger factor, or a gradually growing crack, rather than a large trigger factor, or a rapidly growing crack. Initial crack radii range from 12m to 96m but most are within 15 to 50m. Although reliability tests showed that our results might contain uncertainties of about a factor of 2, the initial crack radii of these events are much smaller than those expected if there is a proportional scaling between the initial crack size and the final earthquake size. More important than the actual value of the size estimate, is the observational result that the onsets of the waveforms for the first 0.1 to 0.2 s look quite similar for a wide range of earthquake sizes. This result implies that the small-scale initiation is similar for a large range of earthquake sizes, and that further growth of the rupture may be best explained by cascade-type models. [47] Acknowledgments. Tomotaka Iwata and Hironori Kawakata contributed helpful comments. Takuo Shibutani, Yasuhiro Umeda, and Koji Matsunami helped our observation in Kozushima. We used seismic data of KiK-net, NIED, and JAMSTEC; hypocenter locations of JMA and ERI, the University of Tokyo; and focal mechanisms of F-net, NIED. We thank 10 of 11

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