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A HYBRID METHOD FOR PREDICTING STRONG GROUND MOTIONS AT
BROAD-FREQUENCIES NEAR M8 EARTHQAUKES IN SUBDUCTION ZONES
Yoshiaki HISADA1SUMMARY
The near-field strong motions during the 1985 Michoacan earthquake (M8) are successfully
simulated at broadband frequencies. For the simulation at low frequencies (0 to 1 Hz), themodified k-square model (Hisada, 1999) is used by considering the k-squared (wavenumbersquared) distributions of slip (Mendoza and Hartzell, 1989) and the incoherent rupture time. For
modeling the slip velocity, the Kostrov-type function with fmax of 10 Hz is adopted. The
simulated results reproduced well the observed long-period ground motions, including the rampfunction-like displacements with the permanent offsets, and the near-field directivity pulses in thevelocities. On the other hand, for the simulation at high frequencies (0.5 to 10 Hz), the KamaeIrikura-Boore method (Kamae, et al, 1998) is used. The results reproduced well the observed
accelerations including the directivity effects. Accordingly, the simulated Fourier spectra agree
well with those observed at broadband frequencies (0 to 10 Hz). The source parameters of an M8
earthquake in a subduction zone can be easily applied to strong motion predictions in other areasusing appropriate Green’s functions.
INTRODUCTION
Since the 1994 Northridge and 1995 Hyogoken-Nanbu (Kobe) earthquakes, it is becoming possible to simulate
realistic near-field strong motions for M7 earthquakes (e.g., Somerville et al., 1999). However, we still do nothave a sound physical model for predicting near-field strong motions at broadband frequencies for M8
earthquakes. This is primarily because we lack good quality strong motion records near earthquakes. On theother hand, it is urgent to predict realistic near-field strong motions for M8 earthquakes in subduction zones,
such as, the hypothetical Tokai earthquake, Japan, for disaster prevention.

The purpose of this study is to construct a physical model for predicting near-field strong motions at broadband
frequencies for M8 earthquakes. For this, we use a hybrid method: the modified k-square model (Hisada, 1999)
at frequencies lower than 1 Hz, and Kamae-Irikura-Boore’s stochastic method (Kamae, et al., 1998) at
frequencies higher than 1 Hz. We choose the 1985 Michoacan, Mexico, earthquake to construct the source
parameters, because the near-field strong motion recorded during this earthquake is probably the only databasereliable at broadband frequencies (Figure 1). Somerville et al. (1991) and Dan and Sato (1999) carried out
similar strong motion studies using semi-empirical methods.
METHODIn this study, we combine the two methods: the modified k-square model (Hisada, 1999) at frequencies lower
than 1 Hz, and Kamae-Irikura-Boore’s method (Kamae, et al., 1998) at frequencies higher than 0.5 Hz.
(A) The Modified K-Square Model for Simulating Long-Period Strong Ground Motions

1Kogakuin University, Dept. of Architecture, Nishi-Shinjuku 1-24-2, Tokyo 163-8677,

E-mail: hisada@cc.kogakuin.ac.jp

The modified k-square model (Hisada, 1999) is a physical omega-square model, which is modified from the ksquare model of Bernard et al. (1996) to be more physically appropriate. This model uses a Kostrov-type slip
velocity and assumes the k-squared (wavenumber-squared) distributions for the slip and the incoherent rupture
time on a fault plane. The displacement in the frequency domain is expressed as follows,

Ui(X ) = ∫L0∫ {µ ⋅ D ( e n + e n )UWk0jjk*ik , j}⋅ exp (i ϖ ⋅ t r ) dxdy ,......... .......... .......... ....( 1)

where L and W are the fault length and width, µ is the rigidity, D is the slip, ek and nj are the slip and fault
normal vectors, U*ik is Green’s function, and tr is the rupture time. For the slip distribution, we use thoseobtained by source inversion studies.
Regarding the slip velocity function, we use the combination of 6 equilateral triangles (Hisada, 1999) as shownin Figure 2. The slip velocity used in this study has a decay of the inverse of the square root of time (i.e.,
Kostrov-type: Kostrov, 1964). The minimum duration of the first triangle is 0.1 second, which introduces fmaxof 10 Hz as shown in the spectrum in Figure 2.
The rupture time tr in equation (1) consists of the coherent rupture time (r/Vr) and the incoherent rupture time ∆tr(Figure 3; Hisada, 1999). The coherent rupture time is calculated using the average rupture velocity (Vr), which
generates the long-period ground motion. On the other hand, the incoherent rupture time ∆tr generates mediumto high-frequency waves. In order for equation (1) to represent the omega-square model, the spatial wavenumber
spectrum of ∆tr has to fall off as the inverse of the wavenumber squared (k-squared: Hisada, 1999). We use thefollowing function as the incoherent rupture time,NM∆t r ( x, y ) = ∑ ∑n =1 m =14 ⋅ ∆txycos  2π ⋅ m ⋅ + θ mn  cos  2π ⋅ n ⋅ + θ n ,......... .......( 2)
LW



1 + (m + n )222

where the phases θnm and θn are randomly generated. The lower modes, especially, the first mode (n=m=1) of ∆tr
is probably strongly related to the spatial slip variation rather than random distribution. The numerical results forsource dynamic by Day (1982) showed that spatial variations of peak slip velocity are strongly coupled to spatial
variations of rupture velocity. Since we use the same Kostrov-type slip velocity on the whole fault plane, the slip
variations could be directly coupled to those of the rupture velocity. Thus, we calculated ∆tr of the fundamental
mode by assuming that the local rupture velocity is proportional to the local slip. The incoherent rupture timeused in this study is shown in the bottom of Figure 3, where we used ∆t=6 (sec) in equation (2).
(B) Kamae-Irikura-Boore’s Method for Simulating Short-Period Strong Ground MotionsTo simulate the strong motion at frequencies higher than 1 Hz, we use Kamae-Irikura-Boore’s method (Kamae etal., 1998). In this method, we distribute Boore’s stochastic point sources (Boore, 1983) on sub-faults in a mannerthat the total Fourier spectrum follows the omega-square model (Irikura, 1986, Aki, 1972).
BROADBAND STRONG MOTION SIMUTATIONS FOR THE 1985 MICHOACAN EARTHQUAKE
We apply the hybrid method mentioned above to the near strong motions recorded during the 1985 Michoacanearthquake. Figure 1 shows the locations of the fault and the observation stations, CAL (Caleta de Campos), VIL(La Villita), UNI (La Union), and ZIH (Zihuatanejo). We compute the strong motions at the 4 stations using themodified k-square model at frequencies from 0 to 1 Hz, and those using the Kamae-Irikura-Boore method from0.5 to 10 Hz. Then, we superpose them using the highcut filter for the former (tapering from 0.5 to 1 Hz) and thelowcut filter for the latter (tapering from 1 to 0.5 Hz).
Table 1 shows the structure model used by Somerville et al. (1991). Regarding the k-squared model, since weneeded to compute about 18,000 Green’s functions at each station to express a continuous wavefront, we used
the Green’s function of the full-space with material properties shown in the bottom of Table 1. Then we
multiplied them by the amplification factor using the top two layers. When we compared those results with theresults using the complete Green’s function of the layered half-space (Hisada, 1995), we found that the waves
coming from far sources were overestimated up to about 20%. Therefore, we corrected the amplitudes of thefull-space Green functions on the basis of the incident angles. We confirmed that the corrected Green’s functions
agree well with the complete Green’s functions.

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Table 2 and 3 show the static source parameters. Those are based on Mendoza and Hartzell (1989), andSomerville et al. (1991). Table 4 shows the dynamic source parameters used for the modified k-square model.
We use the rupture velocity of 2.8 km/sec to calculate the coherent rupture time. As mentioned above, wecompute the fundamental mode of the incoherent rupture time (∆tr) by assuming that the local rupture velocity
on each sub-fault is proportional to the static slip. In our simulations, the max and min local rupture velocities
are 3.5 and 1.0 km/s, respectively. For the second and higher modes for ∆tr, we used equation (2) and ∆t=6
second. Figure 3 shows the incoherent rupture time. Regarding the slip velocity function, we use afac= 2 (theratio between the areas of subsequent triangles, see Hisada, 1999, and Figure 2). This value makes the decay ofthe slip velocity to be the inverse of root square of time (i.e., the Kostrov-type function). As shown in Figure 2
and Table 2, we used the fmax=10 Hz and slip duration=3.2 second.
Table 5 shows the source parameters for the Kamae-Irikura-Boore method. For details of those parameters, seeBoore (1983) and Kamae et al. (1998).
Figures 4 to 6 show the simulated and observed accelerations, velocities, and displacements at the 4 stations. The
accelerations are mainly composed of the results by Kamae-Irikura-Boore’s method. The directivity effects arewell-reproduced; the accelerations near the epicenter show relatively longer duration (CAL and UNI), and thoseat the forward directivity sites show short duration (UNI and ZIH). Regarding the displacements, the rampfunction-like permanent offsets are also well-represented by the simulations. Accordingly, the corresponding
long-period velocity pulses are also reproduced well. These waves are mainly modeled by the slip distribution
and the coherent rupture time. On the other hand, the ripples in the velocities are generated mainly by theincoherent rupture time; drastic changes in the rupture front create a series of the near-source long-period pulses.
Those features of the observation records are also reproduced well.
Finally, Figure 7 shows the simulated and observed Fourier amplitude spectra. The simulated spectra agree wellwith those of the observations at broad frequencies.
CONCLUSIONS
We successfully simulated the near-field strong motions during the 1985 Michoacan earthquake at broadband
frequencies. For the simulation at frequencies lower than 1 Hz, we adopted the modified k-square model using
the k-squared slip distribution by Mendoza and Hartzell (1989) and the k-squared incoherent rupture time. For
modeling the slip velocity, we used the Kostrov-type function with fmax of 10 Hz. The simulated results
reproduced well the observed long-period ground motions including the ramp-function-like permanent offsets inthe displacements (Figure 6) and the near-field directivity pulses in the velocities (Figure 5). On the other hand,for the simulation at frequencies higher than 1 Hz, we used Kamae-Irikura-Boore method. The results
reproduced well the observed accelerations including the directivity effects (Figure 4). Accordingly, thesimulated Fourier spectra agree with those observed at broadband frequencies (Figure 7). These source
parameters can be easily applied to strong motion predictions in the area near a M8 earthquake in a subduction
zone.
ACKNOWLEDGEMENTS
This research was partly supported by a special project of “US-Japan Cooperative Research for Urban
Earthquake Disaster” by the Ministry of Education, Science, Sports and Culture, and by Disaster Prevention
Research Institute of Kyoto University. The advice and discussion with J. Bielak, K. Irikura and K. kamae are
greatly improved this research and manuscript.
REFERENCESAkl, K. (1967), “Scaling Law of Seismic Spectrum”, J. Geophys. Res., Vol.72, pp.1217-1231.
Anderson, J.G., et al. (1986), “Strong Ground Motion from the Michoacan, Mexico, Earthquake”, Science, Vol.
233, pp.1043-1049
Bernard, J. A. Herrero, and C. Berge (1996), “Modeling Directivity of Heterogeneous Earthquake Ruptures”,
Bull. Seismo. Soc. Am., Vol.86, pp.1149-1160.
Boore, D. (1983), “Stochastic Simulation of High-Frequency Ground Motions Based on Seismological Modelsof Radiated Spectra”, Bull. Seismo. Soc. Am., Vol.73, pp.1865-1894.
Dan, K. and T. Sato (1999), “A Semi-Empirical Method for Simulating Strong Ground Motions Based on
Variable-Slip Rupture Models for Large Earthquake”, Bull. Seismo. Soc. Am., Vol.89, No.1, pp.36-53.

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Day, S.M. (1982), “Three-Dimensional Simulation of Spontaneous Rupture: The Effect of NonuniformPrestress”, Bull. Seismo. Soc. Am., Vol.72, No.6, pp.1881-1902.
Hisada, Y. (1995), “An Efficient Method for Computing Green’s Function for a Layered Half-Space with
Sources and Receivers at Close Depths”, Bull. Seismo. Soc. Am., Vol.85, pp.1080-1093.
Hisada, Y. (1999), “A Physical Model for Constructing the Omega-Square Model – Modification of the KSquare Model including Spatial Variation in Slip Distribution, Slip Velocity, and Rupture Velocity –“,submitted to Bull. Seismo. Soc. Am.
Irikura, K. (1986), “Prediction of Strong Acceleration Motion using Empirical Green’s Function”, Proc. 7th Japan
Earthq. Engng. Sym., pp.151-156
Kamae, K., K. Irikura, and A. Pitarka (1998), “A Technique for Simulating Strong Ground Motion using Hybrid
Green’s Function”, Bull. Seismo. Soc. Am., Vol.88, No.2, pp.357-367.
Kostrov, B.V. (1964), “Self-Similar Problems of Propagation of Shear Cracks”, J. Appl. Math. Mech., Vol.28,
pp.1077-1087.
Mendoza, C. and S. H. Hartzell (1989), “Slip Distribution of the 19 September 1985 Michoacan, Mexico,
earthquake: Near-Source and Teleseismic Constraints”, Bull. Seismo. Soc. Am., Vol.79, pp655-669.
Somerville, P.G., M. Sen, and B. Cohee (1991), “Simulation of Strong Ground Motions Recorded during the1985 Michoacan, Mexico and Valparaiso, Chile Earthquakes”, Bull. Seismo. Soc. Am., Vol.81, pp.1-27.
Somerville, P. et al. (1999), “Characterizing Crustal Earthquake Slip Models for Predicting of Strong Motion”,
Seism. Res. Lett., Vol.70, pp.59-80.
Table 1: The layered structure model for the Michoacan area (after Somerville et al. 1991)
ρ(g/cm3)
2.50
2.68
2.78

Vp(km/s)
4.6
5.8
6.4

Qp
200.0
200.0
200.0

Vs(km/s)
2.7
3.4
3.7

Qs
100.0
100.0
100.0

Thickness (km)0.4
5.6-Table 2: The static source parameters (after Mendoza and Hartzell, 1989)
length (km) hypocenter along strike (km)hypocenter along dip (km) deepest edge (km)175.0width (km)
140.0

125.0
Strike (deg)dip(deg)
295.0

14.0

rake (deg)

90.0

45
Seismic Moment (dyne-cm)

70.0

1.48 x 1028

Table 3: The Slip Distributions for the 1985 Michoacan Earthquake (after Mendoza and Hartzell,
1989; Somerville et al. 1991). The size of subfaults is 25 x 20 km2, and the bottom left of the tablecorresponds to the northeast of the fault plane (see Figure 1).
1.0
1.6
1.8
1.5
1.4
1.5
0.5
2.5
3.5
1.5
4.0
5.0
3.0
0.5
3.0
4.0
1.7
1.5
3.0
2.5
0.8
2.12.50.7
1.0
1.3
1.5
0.5
0.8
1.5
1.0
0.7
0.7
1.0
0.5
0.8
2.0
1.7
1.1
0.8
0.5
0.8
1.3
1.3
1.3
0.9
0.8
0.7
1.2
Table 4: The dynamic source parameters for simulating the ground motions during
the Michoacan Earthquake using the modified k-squared model (Hisada, 1999)
N and M
Green func.
fmax
Slip
MinMax
Coherent∆tin eq.(2)perafac(Hz)
duration
Vr
Vr
Vr
(s)
sub- fault(sec)(km/s) (km/s)(km/s)
2.8
3.5
1.0
6.0
1.414
10.0
3.2
62
19 x 19Table 5: The dynamic source parameters for simulating the ground motions during theMichoacan Earthquake using the Kamae-Irikura-Boore’s method (Kamae et al., 1998)
PRTITNVr (km/s)
fmax (Hz)∆σ (bar)Rθφ10010.0
0.63
1/1.4142.8

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Figure 1. Geographic map of the Michoacan coast showing the surface projection of the fault (175 km x
140 km with the dip angle of 14 degree) used in this study. The fault plane is divided into 7 x 7 sub-faults.
The earthquake epicenter and the four strong motion stations are also indicated (after Somerville et al.,1991).

Figure 2. Waveforms (Top) and their Fourier Spectra (Bottom) of the slip velocity and displacement usedin this study. The slip velocity consists of 6 equilateral triangles with different durations (0.1, 0.2, 0.4, 0.8,
1.6 and 3.2 seconds), as shown in the dashed and dotted lines in the top figure. Its amplitude decays as theinverse of the square root of time (i.e., the Kostrov-type function). The minimum duration of the first
triangle is 0.1 second, and it creates the fmax of 10 Hz as shown in the spectra.

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Figure 3. The coherent rupture time (top) and the incoherent rupture time (bottom) on the fault plane of
the 1985 Michoacan earthquake model, viewing from the southwest edge to the northeast edge).
Equation (2) and the values of Table 4 are used. The wavenumber spectrum amplitude of thedistribution of the incoherent rupture time (bottom) falls of as the inverse of wavenumber squared.

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Figure 4. Simulated and observed accelerations at the four stations, CAL, VIL, UNI, and ZIH.

Figure 5. Simulated and observed velocities at the four stations, CAL, VIL, UNI, and ZIH.

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Figure 6. Simulated and observed displacements at the four stations, CAL, VIL, UNI, and ZIH.

Figure 7. Simulated and observed Fourier acceleration spectra from 0.01 to 10 Hz at the four stations

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