Calculation Method of Time History Including Large-Scale Earthquake and Its Aftershock

PAPER

Calculation Method of Time History Including Large-scale Earthquake and Its Aftershock

Kimitoshi SAKAI Yoshitaka MURONO, Dr. Eng. Assistant Senior Researcher, Laboratory Head, Earthquake & Structural Engineering Laboratory, Structures Technology Division

Many aftershocks followed the 2011 off the Pacific coast of Tohoku earthquake after the main shock. There is a need to assess the seismic stability of railway facilities against large earthquakes in consideration of aftershocks. To this end prediction of aftershocks is necessary. This paper proposes a prediction method, which can calculate the magnitude and time lapses between aftershocks after a main shock, based on statistical processing on past earthquakes including aftershocks. Using the proposed method, preliminary calculations were conducted to obtain the time history for a main shock of magnitude 7.0, and its aftershocks. The effect of the aftershock on the damage to the structures was investigated using these waveform groups.

Keywords: main shock, aftershock, magnitude, elapsed time, number of aftershocks, earth- quake motion prediction

1. Introduction

The off the Pacific coast of Tohoku earthquake with a the Western Tottori eq. in 2000（M7.3） magnitude of 9.0 which occurred on March 11, 2011 was 0 the Mid Niigata eq. in 2004（M6.8） the largest earthquake ever to be observed in Japan. It is estimated that three or four focal areas slipped during this -0.5 earthquake. However, the occurrence of such a large earth- -1 quake was not expected in this area. After the earthquake, main

-M -1.5 the focal regions of the Tokai, Tonankai and Nankai earth- aft

quakes, each of which was a large scale earthquake along M -2 the Nankai Trough, were reassessed. This revision revealed -2.5 that a M9.0 earthquake might take place in the Nankai -3 area [1], the scale of which could almost be the same as that 0.01 0.1 1 10 100 1000 of the 2011 off the Pacific coast of Tohoku earthquake. To Elapsed time (hour) evaluate the stability of the railway structures against such a large-scale earthquake, it is necessary to precisely evalu- Fig. 1 Relationship between the elapsed time after the ate the earthquake motion of the main shock. main shock, and the magnitude of the aftershocks

Many smaller aftershocks are expected after a main (Maft-Mmain) (Western Tottori prefecture earthquake shock which could lead to significant further structural in 2000 and the Mid Niigata Prefecture Earth- damage. When evaluating the safety of structures against quake in 2004) large-scale earthquakes, it is therefore also desirable to consider the effect of these aftershocks. aftershocks. Finally, the paper discusses the effects of af- A combination of the Gutenberg-Richter law [2] and tershocks in relation to railway structure damage. the modified Omori formula [3] was proposed as a means to evaluate the aforementioned aftershocks. However, a de- terministic method for evaluating aftershocks as an input 2. Aftershock evaluation method motion to structures has yet to be proposed. The occurrence of aftershocks varies greatly depending on the earthquake, 2.1 Collection of earthquake records, and subse- and making prediction difficult. For example, the aftershock quent process conditions in the wake of the Western Tottori prefecture earthquake in 2000 and the Mid Niigata Prefecture Earth- First records of main shocks and aftershocks were quake in 2004 are shown in Fig. 1. Although the magnitude collected. Based on a list of space-time inter-relationship of both shocks is almost identical, the aftershock patterns in earthquake occurrence [4], the coordination was estab- are quite distinct. Accuracy of aftershock estimation none- lished between main shocks and aftershocks. Only inland theless is not essential for seismic design of structures; the earthquakes and subduction-zone earthquakes [5] with a only critical factors required are magnitude and frequency. main shock of magnitude over M6.0 were selected for the This study therefore proposed an aftershock model study along with aftershocks with a magnitude difference based on past earthquake data which can be used in struc- of less than three with the main shock. For earthquakes tural design. The proposed model was then applied to with a foreshock followed by a main shock, the foreshock calculate the time history waveforms of main shocks and was treated as the main shock, and subsequent earth-

72 QR of RTRI, Vol. 54, No. 2, May 2013 ：Subduction-zone earthquake Subduction-zone earthquake ：Inland earthquake Each data Average of each M Inland earthquake 45° 1 Each data Average of each M

0 main

40° -M -1 aft M -2

-3 35° 6 6.5 7 7.5 8

Magnitude of main shock (Mmain) Fig. 3 Relationship between the magnitude of the main 30° km shock and that of the maximum aftershock 0 500 the Pacific coast of Tohoku earthquake was 7.6 (up until December in 2012, and the difference in magnitude with 130° 135° 140° 145° the main shock is -1.4. This result is falls within the dis- Fig. 2 Epicenter locations of the selected main shocks persion illustrated in Fig. 3. quakes treated as aftershocks, even if their magnitude ex- 2.2.1 Number of aftershocks after the main shock ceeded the foreshock. Records for approximately 1,800,000 earthquakes [6] observed from 1923 to September, 2010 Figure 4 illustrate the relationship between elapsed were gathered, along with those for 50 inland earthquakes time after the main shock and the number of aftershocks. and 211 subduction-zone earthquakes. The epicenter loca- Figure 4 (a) indicates the total number of aftershocks. tions of the selected main shocks are shown in Fig. 2. Figure 4 (b) shows the number of aftershocks per unit of The magnitude and timing of the shocks are important 100 for evaluating structures. For example, liquefaction dam- 50 age may spread when a large-scale aftershock occurs im- mediately following the main shock when hydraulic pres- sure of the foundation disappears. If a major aftershock 10 occurs before structural repair can be carried out after the 5 main shock damage may be worsened. The modeling of the scale and frequency of accompanying aftershocks is a func- 1 0.5 Subduction-zone earthquake tion of the elapsed time from the main shock. Inland earthquake

2.2 Modeling of the aftershock Total number of aftershock 0.1 0.1 1 10 100 1000 1000 This section discusses the relationship between the Elapsed time from main shock (hour) scale of the maximum aftershock and the main shock. Fig- (a) Total number of aftershocks ure 3 illustrates the relationship between magnitude of the main shock and the maximum aftershock. The mean mag- nitude difference between the main shock and the maxi- 100 mum aftershock in relation to the main shock magnitude is Subduction-zone earthquake also shown in this figure. 10 Inland earthquake This figure shows that the variation in magnitude dif- ference is large. However, the mean value is likely to be 1 constant irrespective of the scale of the main shock. That is to say, the magnitude of the maximum aftershock in a subduction-zone earthquake is smaller than that of the 0.1 main shock by about a magnitude of one. The magnitude of the maximum aftershock of an inland earthquake however 0.01 is about the same as, or only slightly smaller than that of Number of aftershock per hour 0.1 1 10 100 1000 1000 the main shock. This shows that aftershocks are expressed Elapsed time from main shock (hour) not on the basis of the magnitude of the main shock but as (b) Number of aftershocks per hour the difference of magnitude between aftershocks with the main shock as a parameter. Fig. 4 Relationship between the elapsed time from the The magnitude of the maximum aftershock in the off main shock and the number of aftershocks

QR of RTRI, Vol. 54, No. 2, May 2013 73 0 Subduction-zone earthquake Mmain-3.0～-2.5 Mmain-2.4～-1.5 M -1.4～-0.5 M -0.4～ Subduction-zone earthquake main main Inland earthquake

Inland earthquake main -1 M -3.0～-2.5 M -2.4～-1.5 main main -M 100 Mmain-1.4～-0.5 Mmain-0.4～ max

50 aft -2 M 10 5 -3 0.1 1 10 100 1000 10000 1 Elapsed time from main shock (hour) 0.5 Fig. 6 Change in magnitude difference of the maximum aftershock following the time course after the Total number of aftershock 0.1 0.1 1 10 100 1000 10000 main shock Elapsed time from main shock (hour) frequency curve flattens around 100 hours after the main Fig. 5 Relationship between the elapsed time from the shock, indicating the higher probability of the maximum main shock and the number of aftershocks aftershock striking within four days of the main shock. (Ranked in order of difference of magnitude with the main shock) 2.2.3 Modeling of aftershock occurrence after the main shock time. As mentioned above, selected aftershocks are those with a difference in magnitude of less than three with the Based on the above analysis of aftershocks and their main shock. Figure 4 shows that the number of aftershocks frequency and magnitude in relation to lapse of time after per unit of time decreases over time. The total number of the main shock, these results were then modeled. This was aftershocks is over 50. The number of aftershocks from an done by first calculating a value to express expected the inland active fault earthquake is higher than for a subduc- earthquake frequency for each unit of time and each scale tion-zone earthquake for the first 10 hours. After this ten based on data in Fig. 5. The expectation value was then hour period however, the of the aftershocks between both discretized, in order to arrive at an estimation of the tim- types of quake is almost identical. ing of the aftershocks. Evaluation of the aftershock scale Figure 5 shows data from Fig. 4 (a) ranked according was carried out by a discrete manner as shown in Fig. 5. In to magnitude difference between the aftershock and main order to simplify the modeling, aftershocks magnitudes in shock. The figure shows that the larger the magnitude, the a given range were represented by a single magnitude. For smaller the number of aftershocks. It also reflects the same example, aftershocks with a magnitude difference of -2.4 tendency illustrated in Fig. 4 (a) of number of aftershocks to -1.5 were defined as aftershocks with a magnitude of decreasing over time. Relatively small scale aftershocks -2.0 in the model. Figure 7 shows the relationship between (magnitude difference of less than -1.5) are greater in num- elapsed time after the main shock and magnitude of after- ber due to fault activity. The figure also illustrates that the shocks in a subduction-zone earthquake, whereas Fig. 8 number of large aftershocks (magnitude difference over shows the same result in the case of an inland active fault -1.4) is greater in the case of subduction-zone earthquakes earthquake. than with an inland active fault earthquake. These figures show that that the proposed model can be used to express the number and scale of aftershocks 2.2.2 Maximum aftershock scale after the main appropriately: the inland active fault produces a large shock number of small aftershocks, whereas the oceanic trench earthquake generates aftershocks of relatively larger Figure 6 tracks the change in the scale of maximum magnitude. Large-scale aftershocks of a magnitude differ- aftershocks corresponding to elapsed time after the main ence with the main shock of about -1 are expected to occur shock. In the comparatively short period of less than about once within several hours (less than 10 hours) after the 100 hours after the main shock, there is higher probability main shock. It was also found that, in approximately 100 of a large aftershock in the case of an inland active fault hours after the main shock, a large-scale aftershock occurs earthquake than with a subduction-zone earthquake. On once again in case of the inland earthquake, and about the other hand, the probability of a large-scale aftershock two aftershocks occur an oceanic trench earthquake. The in the case of a subduction-zone earthquake is higher 100 proposed aftershock occurrence model may therefore be ap- hours after the main shock. The difference in magnitude plied to obtain a rough estimation of the frequency and the with the main shock of the final maximum aftershock, is scale of aftershocks. about 1.0 for both types of earthquakes. The gradient of the

74 QR of RTRI, Vol. 54, No. 2, May 2013 40 0

Mmain-3.0～-2.5 30 -1 Mmain-2.4～-1.5 main ～ 20 Mmain-1.4 -0.5 -M -2 aft

Mmain-0.4～ M 10 -3

Total number of aftershock 0 0.1 1 10 100 1000 0.1 1 10 100 1000 Elapsed time from main shock (hour) Elapsed time from main shock (hour) (a) Ranked in order of difference of magnitude with the main shock (b) Timing of aftershocks Fig. 7 Proposed aftershock occurrence model (Subduction-zone earthquake)

40 0

Mmain-3.0～-2.5 30 -1 Mmain-2.4～-1.5 main

20 -M ～ -2 Mmain-1.4 -0.5 aft M 10 -3

Total number of aftershock 0 0.1 1 10 100 1000 0.1 1 10 100 1000 Elapsed time from main shock (hour) Elapsed time from main shock (hour) (a) Ranked in order of difference in magnitude with the main shock (b) Timing of aftershocks Fig. 8 Proposed aftershock occurrence model (Inland earthquake)

3. Calculation of main shock and aftershock wave- rupture process [12]. The path effect was evaluated using a forms recursive model obtained from past data records [13]. The site characteristics were evaluated from the site amplifi- 3.1 Calculation conditions cation factor, assuming that the group delay time of the transfer function can be expressed as a minimum phase Applying the model proposed above, estimations were shift function [12]. Amplification in the ground which was made of waveforms of a main shock and aftershocks. The shallower than the engineering bedrock was ignored. target main shock illustrated in Fig. 9, is an inland active The calculation of the aftershock waveforms was fault earthquake with a scale of 36 × 24 km. The main conducted in the same manner as for the main shock. Fo- shock has a magnitude of 7.0, and the dip angle is 45 de- cus spreading in aftershocks of M5.0 was ignored. Focus grees. The outer fault and inner fault parameters are fixed spreading was factored in however for calculation of M6.0 using the proposed method applied to data from past inland aftershocks, with the rupture spreading concentrically fault earthquakes [7, 8]. The proposed model was then used from the fault center. Figure 9 illustrates determination of to calculate the waveforms of aftershocks with a magnitude the aftershock origin. This location is based on the empiri- of between 5.0 and 6.0. The calculation period was set to cal assumption that many aftershocks are generated near 100 hours from the main shock. This time span was deter- the focus [14]. It is also assumed that the aftershocks oc- mined from the time generally required to repair structural cur in the order shown in Fig. 9. The target sites for which damage following the main shock, i.e. approximately three to five days [9]. Based on the above conditions and Fig. 8, ④ Depth of the fault top surface = 2km results gave an estimation of two aftershocks of M6.0 and ② ⑥ ③ ⑨ 10 aftershocks of M5.0 occurring during the 100 hour time ⑩ span. Corresponding waveforms were calculated for a total ⑪ 24km of 13 earthquakes including main shocks and aftershocks. ⑤ Stochastic green’s function method [10] was employed Hypocenter to evaluate the waveform. The site amplification factor ：M=7 (Main shock) for amplitude was fixed on the basis of the average earth- ① ⑫ ⑧ ⑦ ：M=6 ： quake amplification factor, evaluated at the seismic obser- M=5 vatories [11]. The phase characteristics of the earthquakes 36km were evaluated by independent modeling of source, path, Asperity and site characteristics. The source characteristics were Fig. 9 Fault model of the main shock, hypocenter of the expressed by superposition of impulses following the fault main shock and aftershocks

QR of RTRI, Vol. 54, No. 2, May 2013 75 20 Plane view 20 Side view 800 Mainshock M7.0 Max=-668.0(gal) Site B Site A 10 10 0

0 0 Acc(gal)

Y(km) Fault plane -800 -10 -10 0 5 10 15 20 25 30 35 40 Time(sec) -20 -20 (a) Site A -30 -20 -10 0 10 20 30 0 -10 -20 X(km) Z(km) 800 Fig. 10 Spatial relationship between the fault plane and Mainshock M7.0 Max=-729.4(gal) target sites 0

waveforms were calculated are sites A and B in Fig. 10. Acc(gal) -800 3.2 Ground motion of the main shock and the after- 0 5 10 15 20 25 30 35 40 shocks Time(sec)

Figure 11 shows the obtained waveforms. Although (b) Site B the distance between the fault and sites A and B are equal, Fig. 11 Calculated time histories of the main shock site B is located in the opposite direction to rupture trav- elling from the epicenter. Therefore, the duration of the earthquakes is small, the duration is very short. M6.0 earthquake motion in site B is longer and the wave groups earthquakes on the other hand last longer, and are ac- corresponding to the two asperities are clear. Since site A companied by larger amplitudes. The ground motion peak is located in the traveling direction of the rupture, asperity acceleration ratios differ clearly between No.3 and No.11 effects coincide and the duration of the earthquake motion even though they are on a similar scale. The latter is due to is shorter. the hypocenter location of the aftershock as shown in Fig. 9. The time histories obtained for aftershocks at site The distance between the target point and the hypocenter B are shown in Fig. 12. Since the rupture area of M 5.0 differs in each aftershock. This means that the aftershocks

150 150 No.①（M5） Max=68.2(gal) No.⑦（M5） Max=20.4(gal) 0 0 0.15h after main shock 17h after main shock Acc(gal) Acc(gal) -150 -150 0 5 10 15 20 25 30 0 5 10 15 20 25 30 150 150 No.②（M5） Max=116.3(gal) No.⑧（M5） Max=32.4(gal) 0 0 0.65h after main shock 34h after main shock Acc(gal) -150 Acc(gal) -150 0 5 10 15 20 25 30 0 5 10 15 20 25 30 150 150 No.③（M6） Max=40.9(gal) No.⑨（M5） Max=-23.9(gal) 0 0 1.5h after main shock 46h after main shock Acc(gal) Acc(gal) -150 -150 0 5 10 15 20 25 30 0 5 10 15 20 25 30 150 150 No.④（M5） Max=-41.2(gal) No.⑩（M5） Max=56.2(gal) 0 0 2h after main shock 69h after main shock Acc(gal) -150 Acc(gal) -150 0 5 10 15 20 25 30 0 5 10 15 20 25 30 150 250 No.⑤（M5） Max=41.7(gal) No.⑪（M6） Max=216.6(gal) 0 0 4.5h after main shock 70h after main shock Acc(gal) Acc(gal) -150 -250 0 5 10 15 20 25 30 0 5 10 15 20 25 30 150 150 No.⑥（M5） Max=-61.2(gal) No.⑫（M5） Max=-53.7(gal) 0 0 9.5h after main shock 92h after main shock Acc(gal) Acc(gal) -150 -150 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Time(sec) Time(sec) Fig. 12 Time histories of the aftershocks (Site B)

76 QR of RTRI, Vol. 54, No. 2, May 2013 origin must be set in a way which does not lead to underes- timation of structural damage. DI (=DI1+DI2) DI1 DI 3.3 Evaluation of impact of main shock and after- 1.2 2 shocks on structures 1 0.8 Given the waveform sets obtained for the main shock 0.6 and aftershocks, railway structure damage can be inves- DI tigated. The amount of structural damage is evaluated by 0.4 the Damage Index (DI) determined as follows [15]. 0.2 δ β 0 DI =max + ∫ dE 800 δuQ yδu (1) Input EQ.

=DI1 + DI 2 0 Where d max is the maximum lateral displacement against Acc(gal) the design ground motion, d u is the ultimate displacement under monotonically increasing lateral deformation, ∫ dE is -800 the cumulative plastic energy dissipation due to the design No. No.No. No. No. No. No. No.No. No. No. No.

main ① ② ③ ④ ⑤ ⑥ ⑦ ⑧ ⑨ ⑩ ⑪ ⑫ ground motion, Qy is the yielding load and b is a non-nega- tive parameter to express the effect of repetitive loading on structural damage. Structural damage is expressed as the (a) Site A sum of DI1, damage from maximum deformation, and DI2, damage from repetitive motion. In this index, the structure is regarded as collapsed if DI is larger than one. In this DI (=DI1+DI2) calculation, structural conditions are assumed so that the DI1 equivalent natural period is 0.5 second, yield seismic inten- 1.2 DI2 µ= δ δ sity is 0.3, ductility demand u(/ u y ) is 6, and b is 0.15 1 [16]. d is a yielding displacement of the structure. y 0.8 Based on these conditions, DI results calculated for the 0.6 main shock and aftershocks are shown in Fig. 13. The com- DI parison of DI with and without considering the aftershocks 0.4 are shown in the Table 1. Structures in both site A and B 0.2 received extensive damage due to the main shock, follow- 0 ing that DI1 reaches a maximum under the main shock. It 800 follows that the damage by the maximum deformation is Input EQ. not extended by the aftershocks. On the other hand, DI2 gradually increases under the effect of aftershocks. In site 0

B in particular, DI2 increased greatly with aftershock No.11 Acc(gal) which occurred just beneath the site, causing DI to exceed -800 1.0. DI in sites A and B grew by 2% and 8% respectively No. No.No. No. No. No. No. No.No. No. No. No.

under the impact of aftershocks. Although this value is com- main ① ② ③ ④ ⑤ ⑥ ⑦ ⑧ ⑨ ⑩ ⑪ ⑫

Table 1 Change in DI after taking aftershocks into con- (b) Site B sideration Fig. 13 Result of evaluation of degree of structural dam- (a) Site A age (DI) Ratio main main+after ((main+after)/main) paratively small, it demonstrates that the influence of after- shocks may become significant, depending on the origin of DI 0.68 0.68 1.00 1 the main shock and aftershocks and structural conditions.

DI2 0.28 0.30 1.06 DI 0.97 0.99 1.02 4. Conclusion

(b) Site B This study proposed a model, using past earthquake Ratio data, for calculating the occurrence of aftershocks with a main main+after view to verifying the seismic performance of railway struc- ((main+after)/main) tures. Application of this model enables estimation of the DI1 0.55 0.55 1.00 scale and frequency of aftershocks, within a given period after the main shock. As an example, the time history for a DI2 0.40 0.47 1.19 main shock (M7.0) and its aftershocks was calculated. The DI 0.95 1.02 1.08 waveform groups obtained from this calculation were em-

QR of RTRI, Vol. 54, No. 2, May 2013 77 ployed to investigate the effects of the aftershocks on struc- kyoshindo/08apr_kego/recipe.pdf, 2008 (in Japanese). tural damage. Results confirmed that aftershocks may ag- [9] Sakai, K., Murono, Y., Sato, T.,“ Method of Priority gravate structural damage. Judgment of the Seismic Countermeasure Based on Life Cycle Cost,”RTRI report, Vol.25, No.2, 2011 (in Japanese). References [10] Kamae, K., Irikura, K., Fukuchi, Y.,“Prediction of strong ground motion based on scaling law of earth- [1] Central Disaster Prevention Council,“Interim draft quake: by stochastic synthesis method,”Journal of report of the giant earthquake model of the Nankai Struct. Constr. Engng., AIJ, No.430, pp. 1-9, 1991 (in Jap- Trough investigative commission,”2011 (in Japanese). anese). [2] Gutenberg, B. and Richter C.F.,“Frequency of earth- [11] Nozu, A., Nagao, T., Yamada, M.,“Site Amplifica- quakes in California,”Bull. Seism. Soc. Am., Vol.34, tion Factors for Strong-Motion Sites in Japan Based pp.185-188, 1944. on Spectral Inversion Technique and Their Use for [3] Utsu, T.,“Magnitude of earthquakes and occurrence Strong-Motion Evaluation,”Journal of JAEE, Vol.7, of their aftershocks,”Zisin(2), Vol.10, No.1, pp.35-45, No.2, pp.215-234, 2007 (in Japanese). 1957 (in Japanese). [12] Sato, T., Murono, Y., Nishimura, A.,“Modeling of [4] Hoshiba, M., Seino, M., Okada, M. and Ito, H.,“A list phase characteristics of strong earthquake motion,” of spacetime inter-relationship in earthquake occur- Journal of JSCE, No.612/I-46, pp.201-213, 1999 (in Japa- rence and its applications,”Papers in Meteorology and nese). Geophysics, Vol.44, , No.3, pp.83-90, 1993 (in Japanese). [13] Murono, Y., Kawanishi, T., Sakai, K.,“Simulation of [5] The Headquarters for Earthquake Research Promotion Earthquake Motion Based on Inversion of Phase Spec- Earthquake Research Committee,“Regarding meth- trum,”RTRI report, Vol.23, No.12, 2009 (in Japanese). ods for estimating aftershock probabilities,”1998. [14] Hamada, N., Yoshikawa, K., Nishiwaki, M., Abe, M., [6] Japan meteorological agency,“Earthquake Catalog of Kusano, F.,“A Comprehensive Study of Aftershocks Japan,”2010. of the 1923 Kanto Earthquake,”Zisin(2), Vol.54, No.2, [7] Irikura, K.,“Recipe for Predicting Strong Ground Mo- pp.251-265, 2001 (in Japanese). tion from Future Large Earthquake,”Annuals of Disas. [15] Young Ji Park and Alfredo H. S. Ang“Mechanistic Prev. Inst., Kyoto Univ., No.47 A, 2004. Seismic Damage Model for Reinforced Concrete, ”J. [8] The Headquarters for Earthquake Research Promotion Struct. Eng., 111:4, pp.722-739, 1985. Earthquake Research Committee,“Strong ground mo- [16] Fajfar, P.“Equivalent ductility factors taking into ac- tion prediction method (″Recipe″) for earthquakes with count low-cycle fatigue,”Earthquake Engineering and specified source faults,”http://www.jishin.go.jp/main/ Structural Dynamics, Vol.21, No.10, pp.837-848, 1992.

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