GROUND MOTION ATTENUATION RELATIONSHIP
BASED ON STOCHASTIC METHOD
Zhengru TAO ∗MEE09189

Supervisor: Tsuyoshi TAKADA ∗∗Advisor: Toshihide KASHIMA∗∗∗

ABSTRACTFor regions without adequate strong ground motion records, a method is developed to establish strong
ground motion attenuation relationships, based on the stochastic method. Sendai area of Japan(N36°~40°, E138°~143°) is selected as the target, since there are enough data from seismographic
observation for calculation, and those from strong ground motion observation and some empirical
relationships for result testing.
Three parameters related to regional source and crustal medium are inversed by themicro-Genetic Algorithm. Total of 240 records from 77 small events (Mw=3.5~4.5, focal depth≤30 km)
in two directions, recorded by F-net, are adopted for the inversion. Fourier spectra are adopted as theobjective function. These parameters are then taken into the stochastic method to estimate theroot-mean-square acceleration and velocity (Arms and Vrms) and the peak factor (γm), PGA or PGV is theproduct of Arms or Vrms and γm.
The regional ground motion attenuation relationships, for PGA and PGV, are comparedwith strong ground motion records from K-NET (Mw≥4.5, focal depth≤30 km, equivalent shear-wave
velocity≥500 m/s) and empirical relationships. There is a great agreement, which can illustrate thereliability of this method.
Keywords: Attenuation relationship, Inversion, Regional parameters.

1. INTRODUCTION
Ground motion attenuation relationship is essential in the field of earthquake engineering, which
describes the effects of source, path and local site condition on ground motion. Most relationships areempirical, strong ground motion observation records are the foundation. So, this kind of method is justeffective for regions with enough data, like in western US and Japan.
Nowadays, more and more instruments for strong ground motion observation have been orare being installed in some earthquake-prone countries. However, it is difficult to install these
instruments around the epicenters of future earthquakes, since it is impossible to predict the places.
Moreover, even this kind of instruments have been installed, it could be a long time before big
earthquakes occur, since it is impossible to predict the time. So, for most countries or regions without
enough strong ground motion observation records, it is necessary and urgent to develop a method forestablishing strong ground motion attenuation relationships. On the other hand, there are many digital
broadband seismographic networks, even in China. For instance, the Global Seismographic Network(GSN), which is a globally distributed digital seismic network, and real-time data can be obtained.
The purpose of this study is to develop a method, which can establish strong ground motion
attenuation relationships for regions with few or without strong ground motion records. Records fromF-net in small earthquakes are adopted to inverse by the micro-Genetic Algorithm.
∗Institute of Engineering Mechanics, China Earthquake Administration.
Professor, Graduate School of Engineering, University of Tokyo, Japan.
*∗∗Doctor, International Institute of Seismology and Earthquake Engineering, Japan.
∗∗12. DATA
Sendai area (N36°~40°, E138°~143°) is selected as the target region. In the target region, there are 12
F-net stations, Fourier spectra of last 10 records from each one, except some low-quality ones, areadopted as the objective function for inversion. These records are from 77 earthquakes (Mw=3.5~4.5,focal depth≤30 km, Nov. 2006~Apr. 2010). However, in this database, there is no record on nearfield (Hypocentral distance < 50 km), and there is no earthquake with shallow focal depth (D < 5 km),and those with focal depth 10 km~20 km are less than others.
For comparison, there are 2743 records from 1325 earthquakes (Mw≥4.5, focal depth≤30km, Jan. 1996~Apr. 2010), recorded by 29 K-NET bedrock stations ( V S 30 ≥500 m/s) in this region.
The database structures of F-net and K-NET are shown in Table 1, in which there is a little bitdifference.
Table 1(a). Database structure - Hypocentral distance
Hypocentral distance (km)F-net records
K-NET records<100171995
100~200
64567
200~300
34145
≥300
536
Table 1(b). Database structure - Depth
Earthquake Number (F-net)
Earthquake Number (K-NET)
36347
12666
29312

Depth (km)<1010~20
20~30

3. THEORY AND METHODOLOGY
3.1. Stochastic Method
Assuming that the far-field accelerations on an elastic half space are band-limited, finite-duration andwhite Gaussian noise, and the approach is based on Brune ω2 source spectrum, the Fourier spectrum
on a site can be presented as (Boore, 2003),
FA(M 0 , f , R ) =Rθφ FV
4πR0 ρ s β3s×

⎛πfR ⎞1z⎟⎟ ×× exp⎜⎜ −× (2πf ) × G (R )× A( f )
−1 2
⎛ f ⎞
⎝ Q( f )β s ⎠ ⎡ ⎛ f ⎞ 8 ⎤1 + ⎜⎜ ⎟⎟⎟⎟ ⎥⎢1 + ⎜⎜⎝ f0 ⎠⎢⎣ ⎝ f max ⎠ ⎥⎦M02(1)where, Rθ is the radiation pattern of the shear excitation, with the assumption that the energy was
equally partitioned into two horizontal components; F is the free surface effect; V is the vectorial
partitioning of shear wave energy into two components of equal amplitude; R0 is a reference distance;ρs is the density in the vicinity of the source; βs is the shear-wave velocity in the vicinity of the source;
M0 is the seismic moment; f is the frequency; f0 is the corner frequency, f0 = 4.9 ×106 × βs × (Δσ / M0 )1 3 ,∆σ is the stress drop; fmax is the high-frequency cutoff frequency; z is the index variable, z=0, 1 and 2 isfor displacement, velocity and acceleration; R is the hypocentral distance; Q(f) is the function offrequency, which describes the crustal medium, Q(f)=Q0fη; G(R) is the geometric spreading function;
A(f) is the amplification factor of near surface amplitude as a function of hypocentral distance R andfrequency f.

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According to the relation between Fourier spectrum and power spectrum and the definitionof spectral moment, the latter can be calculated by the following numerical integration,f max

mk = 2 ∫ (2πf ) FA(M 0 , f , R ) df
2k(2)0where, k=0, 2 or 4 denotes the zero-order spectral moment, the second-order spectral moment and thefourth-order spectral moment of ground motion
According to the Parseval’s Theorem and the definition of root-means-square (rms)
acceleration, Arms is1/ 2⎛m ⎞Arms = ⎜ 0 ⎟⎝T ⎠(3)where, T is the duration used in computing Arms, which is the sum of source duration Ts and oscillator
decay time (Boore and Joyner, 1984).
The peak factor γm, which describes the ratio of peak to rms motion, is calculated by thefollowing numerical integration (Boore, 2003),+∞{γ m = 2 ∫ 1 − [1 − ξ exp (− z 2 )]Ne}dz(4)0where, ξ is the bandwidth factor; Ne is the number of extrema.
Then, PGA or PGV can be calculated as a product of the rms acceleration Arms or the rms
velocity Vrms and the peak factor γm.
3.2. Inversion Strategy and Results
Micro-Genetic Algorithm (μGA) is adopted for the inversion. Premature phenomenon and local
optimum can be avoided and global optimum can be found quickly by this algorithm.
For large and shallow earthquakes, ∆σ varies from about 10 bars to 100 bars (Shearer,2009), which is one inverse parameter. In some researches, ∆σ is set as 80 bars (Atkinson and Silva,2000). Q is the frequency-dependent quality factor of shear waves, which describes the energy loss
from anelastic processes or internal friction during wave propagation. Usually, η is 0.5~1.0 (Wang,2007). For Tohoku area, Matsumoto (1989) suggests η equals to 0.6~1.2. Since Q is a factor relatedwith the crustal medium rather than the size of earthquakes, Q0 and η are other two inverse parameters.
The objective function is the residual sum of squares between the calculated Fourierspectrum and the objective spectrum from observed records, which describes the fitting. The objectivefunction j is2

φ j = ∑∑ [ FA0 (m, n) − FA j (m, n)]m(5)nwhere, m is observed records (one direction); n is the points on a Fourier spectrum; FA0 is theobjective Fourier spectrum; FAj is the calculated Fourier spectrum from generation j.
Fitness is calculated from the objective function by Eq. (6), which is between 0 and 1.

Fj = e− βφ jwhere, β is the fitness coefficient, which effects the evolution of the μGA to a large extent.

3(6)In this inversion, Fourier spectra on the frequency range of 0.1 Hz~10 Hz, from 120
records in each direction, are adopted as the objective. The ranges of inverse parameters aredetermined as ∆σ=70~150 bars, Q0=150~300 and η=0.6~1.0. After searching among 2000 generations
by μGA, the optimum solution is listed in Table 2.
Table 2. Inverse result∆σ (bars)
89.1095Q0η

173.1040

0.6201

By taking these parameters into the stochastic method, the strong ground motion
attenuation relationships for PGA and PGV are obtained, which are the maximum values in two
horizontal directions.

4. COMPARISONS
4.1. PGA Attenuation Relationship
Strong ground motion attenuation relationships obtained above (PGA and PGV) are compared withstrong ground motion records and some empirical relationships, as shown in Figure 1. Strong groundmotion records are from K-NET and empirical relations are from Kanno (2006), Si (2000) andFukushima (1990). The PGA and PGV of K-NET are the maximum values in two horizontal
directions.

4Figure 1. Comparison on attenuation relationships
It is shown that the results are lower than empirical relations on medium and far fields(hypocentral distance > 30 km), and on the near field, they are higher than Si’s and Fukushima’srelations, close to Kanno’s. However, these results just pass through the strong ground motion records.
It is reasonable, for instance, the relation for Mw=5 is compared with the records from earthquakes
with Mw=4.5~5.5.
Comparisons between the mean residuals, defined by Eq. (7), from this thesis and empirical
relations are shown in Table 3. These data are separated into three groups, 4.5≤Mw<5.5 (Mw=5), 5.5≤
Mw <6.5 (Mw=6) and 6.5≤Mw≤7.5 (Mw=7).
It is obvious that the residuals from this method are the lowest. So, compared with otherrelations, results from this study are closer to the observation data.
residual =PGA

PGV

1 N
predicted∑ log10 ( observed
N i=1value)
value

Table 3. Comparison of mean residuals
Relations
Mw =5
Mw =6
Kanno (2006)
0.37930.3896Si (2000)
0.09470.1324
Fukushima (1990)
0.0500-0.1189
This study
0.02550.0172
Relations
Mw =5
Mw =6
Kanno (2006)
0.59820.5709Si (2000)
0.33380.2142This study
0.20870.1732(7)

Mw =7-1.0288-1.1666-1.8428-0.8191
Mw =70.2075-0.6674-0.1695

5. CONCLUSIONSFor the regions with few or without strong ground motion records, a method to establish strong groundmotion attenuation relationships, based on the stochastic method, is developed. Sendai area in Japan isselected as the target. Regional source and crustal medium parameters, the stress drop ∆σ, and theparameters indicating quality factor (Q0 and η), are inversed by μGA from small earthquakes(Mw=3.5~4.5), recorded by a seismographic network F-net. The database involves 120 records on eachhorizontal component, from 77 earthquakes with focal depths less than 30 km. Then, these inverse
parameters are taken into the stochastic method to calculate the root-mean-square acceleration andvelocity, Arms and Vrms, as well as the peak factor γm. Attenuation relationships for PGA and PGV areobtained. These results are compared with strong ground motion records from K-NET and some5

Japanese empirical relationships. There is an excellent agreement between predicted values andobserved data.

6. RECOMMENDATIONThis method is proved primarily that it can establish strong ground motion attenuation relationships
for regions without abundant strong ground motion records. However, there are still some parts can beimproved:(a) Observed records on the near-field or with shallow focal depth can be added to thedatabase for the inversion;(b) Values on the highest one or five frequencies can be adopted as the objective function,
since PGA is mainly dominated by high-frequency values;(c) Geometric parameters used in this thesis are empirical, so they can be other inverse
parameters;(d) The selection of the objective function can be more reasonable;(e) Uncertainty from the selection and determination of parameters can be considered.
AKNOWLEDGEMENTMy deepest gratitude goes first to Professor Tsuyoshi Takada, my supervisor, and Dr. Tatsuya Itoi ofthe University of Tokyo, for their constant encouragement, discussions and comments. And also, I
would like to express my heartfelt gratitude to Dr. Toshihide Kashima of Building Research Institute(BRI), Dr. Hongjun Si of Kozo Keikaku Engineering Inc. and Dr. Toshiaki Yokoi, for their help toread F-net data.

REFERENCES
Atkinson, G.M. and W., Silva, 2000, Bull. Seism. Soc. Am., 90, 255–274.
Boore, D.M., 2003, Pure Appl. Geophys, 160, 635-676.
Boore, D.M. and W.B., Joyner, 1984, Bull. Seism. Soc. Am., 74, 2035-2039.
Fukushima, Y. and T., Tanaka, 1990, Bull. Seism. Soc. Am., 96, 757–783.
Kanno, T., Narita, A., Morikawa, N., Fujiwara, H. and Fukushima, Y., 2006, Bull. Seism. Soc. Am.,96, 879–897.
Matsumoto, S. and A., Hasegawa, 1989, Geophys. J. Int., 99, 101-108.
National Research Institute for Earth Science and Disaster Prevention., F-net Broadband Seismic
Network Laboratory, http://www.fnet.bosai.go.jp/REGS/dataget/?LANG=en, 2010.6
National Research Institute for Earth Science and Disaster Prevention., K-NET Kyoshin Network,
http://www.k-net.bosai.go.jp/k-net/search/index_en.html, 2010.6
Shearer, P.M., 2009, Cambridge University Press, London.
Si, H.J. and S., Midorikawa, 2000, Proc. of 12th World Conference on Earthquake Engineering,
No.0532.
Wang, M., 2007, Doctor dissertation, Graduate School of Engineering, The University of Tokyo.

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