Benchmark Tests for Strong Ground Motion
Prediction Methods using Theoretical Methods
Y. Hisada
School of Architecture, Kogakuin University, Tokyo, Japan

M. Nagano
Tokyo University of Science, Chiba, Japan

A. Nozu
The Port and Airport Research Institute, Kanagawa, Japan

K. Miyakoshi
Geo-Research Institute, Osaka, Japan

K. Asano
Disaster Prevention Research Institute, Kyoto University, Kyoto, Japan

T. Matsumoto
Shinozuka Research Institute, Tokyo, Japan

T. Nakagawa
Fujita Corporation, Tokyo, JapanSUMMARY:
We performed benchmark tests of the three theoretical methods for simulating strong motions: the wavenumber
integration method, the discrete wavenumber method, and the thin layer method. We investigate the applicabilityof the methods through 6 Steps considering wide range of frequencies and hypocentral distances under the same
source, path and site condition. We started from simple point sources in simple media in Step1, and extended
sources in Step 2. Then, we used more complicated and realistic source and media models in Steps 3 and 4.
Finally, we tested actual sources in the Kanto sedimentary basin, such as the 1923 Kanto earthquake (M7.9) in
Steps 5 and 6. We obtained excellent agreements in almost all the cases, but found considerable discrepancies
under certain conditions, such as the cases for large epicentral distances in the media of different formulations ofQ (the quality factor), and the different approximations of surface sources.
Keywords: Benchmark Test, Strong Ground Motion Predication, Theoretical Methods

1. INTRODUCTION
Recently, various methods for predicting strong ground motions have been applied to engineering
purposes, such as the safety evaluations of high-rise buildings and nuclear power plants, and thedamage estimations of cities for large hypothesical earthquakes. Nonetheless engineers tend to be
sceptical about the reliability of the predicted results, because they occasionally show large
discrepancies using different methods, even under the same source, path, and site conditions. To
investigate the causes of those discrepancies, we have carried out a series of benchmark tests since2009, using the three-types of major methods for predicting strong ground motions; they are thetheoretical, numerical, and stochastic methods.
In this paper, we describe the results of the benchmark tests of the theoretical methods. Those of thenumerical and stochastic methods are presented in the subsequent papers (Yoshimura et.al., 2012;
Kato et.al, 2012 in this volume). Even though the theoretical methods have limited applicatiosn to flat
layered media, they are one of the most useful tools for simulating strong motions, because of the highaccuracies, the broad frequency band, and the short computing time. On the other hand, there are
numerous theoretical methods based on different formulation and coding. Thus, we have conductedthe benchmark tests using the three representative methods: the wavenumber integration, the discrete
wavenumber, and the thin layer methods (e.g., Hisada, 1995; Bouchon, 1981; Nagano et.al, 2007).
The benchmark tests consist of 6 steps. We started from very simple point and extended sources in
simple media in Steps 1 and 2. Then, we tested more complicated and realistic sources and media in
Steps 3 and 4. Finally, we applied them to actual source models, including the 1923 Kanto earthquake(M7.9), in the Kanto sedimentary basin in the Steps 5 and 6.

2. BENCHMARK TESTS OF THEORETICAL METHODS
2.1. The Participants and Methods
Five groups of researchers/engineers participated in the tests using their own methods/codes. Table 1
shows the participants and methods. The wavenumber integration method by Hisada & Matsumoto
and Nakagawa (Hisada, 1995), and the discrete wavenumber method by Nozu (2002) and Miyakoshi& Asano (Bouchon, 1981) are theoretically similar method on the basis of the wavenumber integration,
but the numerical integration schemes are different. The former uses general numerical integration
techniques, such as the Simpson and Filon formulas for smaller and larger wavenumbers, respectively,
whereas the latter uses the technique similar to the FFT algorism. As for the propagator matrix, all themethods use the R/T (Reflection/Transmission) matrix methods, which are numerically stable up tovery high frequencies. On the other hand, the thin layer method by Nagano (2007) combines thetheoretical and numerical wavefields in the horizontal and vertical directions, respectively. It uses thediscretization technique of the Galerkin method along the vertical direction, and superposes the eigen
modes to obtain ground motions. Even though it has to assume a rigid basement to compute the eigen
modes, the computing time is generally very short. On the other hand, a viscoelastic boundary and ahighly damping buffer zone are introduced to reduce the artificial reflections from the basement,
especially at the stations of large distances.
The one of the biggest differences among the participants is the formulation of Q (quality factor).
Through the tests, we asked the participants to use the frequency dependent Q, as shown in Tables 4.
7, and 9. The most participants introduce Q in the imaginary parts of Vs and Vp, using the following
equation.
−1
⎛
⎛
i ⎞
i ⎞
⎟⎟ ≈ V ⎜⎜1 +⎟V * = V ⎜⎜1 −
Q ⎟⎠
Q
2
2
⎝
⎠
⎝

(1)
This formulation is simple, but is does not satisfy the causality condition. On the other hand,
Miyakoshi & Asano use the following formula (Aki and Richards, 1981),
−1

⎛
⎛
1
ω
i ⎞
1
ω
i ⎞
⎟⎟
⎟ ≈ V ⎜⎜1 ++V * = V ⎜⎜1 −ln REF −
ln
ω
2Q ⎟⎠
⎝ πQ
⎝ πQ ω REF 2Q ⎠(2).
where, ωREF is the reference circular frequency. On the contrary to equation (1), this formula satisfies
the causality condition, but Q is independent of frequencies (constant Q), and we have to set a certain
value for ωREF. In addition, the values of medium velocities are frequency dependent (see Figure 6),
which introduces the dispersion in waveforms.
Table 1. Participants and methods (○: Fully participating, △: Partially participating, ×: Not participating)No
Participant
MethodCode
Step1 Step2 Step3 Step4 Step5 Step6
1 Hisada & Matsumoto Wavenumber Integration Method Hisada
○
○
○
○
○
○
2
NaganoThin Layer Method
Nagano
○
○
○
○
○
○
3
Nozu
Discrete Wavenumber Method
Nozu
○
○
△
×
○
×
4 Miyakoshi & Asano Discrete Wavenumber Method O.Coutant* ○
○
△
×
○
○
5
Nakagawa
Wavenumber Integration Method Hisada
○
○
○
○
×
×

Note *: Dr. O.Coutant coded the original program based on Bouchon (1981), and Prof. Asano modified it.

2.2. Steps 1 and 2
Tables 2 & 3, and Figures 1 & 2 show the models of Steps 1 and 2, respectively. Those simple
starting models are basically same as those of Day et.al.(2000, 2003). Step 1 consists of 4 models (T11
to T14), by considering a point source of 2 or 20 km depths, and the 2 different media (homogeneous
bedrock, or two-layered half-spaces), whose material properties are shown in Table 4. As shown in
Figure 1, we locate the point source along the Z axis, and compute ground motions at the 6 stations
from 2 km to 100 km of the epicentral distances. The source model is a point source of strike slip
(strike=0 deg, dip=90 deg, rake=0, deg, M0=1018 Nm), and its moment rate function is the
exponential-type,t −t
M& (t ) = M 0 ⋅ 2 ⋅ e T(3)Twhere, we use T = 0.1 s. Figure 3 shows the moment, moment-rate functions, and the Fourier
amplitude spectrum of the moment-rate function.
On the other hand, Step 2 consists of 2 models of extended source models. T21 is the lateral slip fault
model, as shown in Figure 2 (strike=90 deg, dip=90 deg, rake=180, deg, L=8 km, W=4 km). The
shallowest edge of the faults is at located 2 km depth, and the hypocenter is at 4 km depth along the Z
axis. The rupture front is continuous using Vr = 3 km/s. The slip functions (D = 1 m) has the same
shape as those of Figure 3. T22 is the inverse slip fault model (see the detail in Day et.al (2000)).
Table 2. Model lists of Step 1

Table 3. Model lists of Step 2

Model
T11
T12
T13
T14
2 Layers ModelMedia
Bedrock
Infinite (No Damping)
Q
Table 1Infinite
Source
2 km
20 km
Depth
0
20
Hz
Frequencies
Output 6 points (+002, +006, +010, +030, +050, +100 km)

Model
T21
T22
2 Layers Model
Media
Q
Infinite (No Damping)
SourceLateral Slip Fault
Inverse Slip Fault
Model
Frequencies
0 - 5 Hz
Output 12 points (±002, ±006, ±010, ±030, ±050, ±100 km)

Reference

UHS.14)

LOH.14)

LOH.34)

LOH.14)

+030

+050

UD

tan-1(3/4)

1 km

+100

RadialTransverseXUD
12 stations from –100 to +100 km（+side）
-1
tan
(3/4)
Radial1 km
RadialTransverse（-side） 2 km

Figure 1. The point source and observation station for Step 1

Figure 2. The fault model and station for T21
Moment Rate T = 0.1 s 10
Functionin Eq (3)Moment 0.001 0.01 0.1 1 1
Function4
3.5

Thickness(m)
Sediment
1,000
Bedrock
Layer

Vp
(m/s)
4,000
6,000

Vs
Density
Qp=Qs(m/s) (kg/m3)
2,000 2,600
40f
3,464 2,70070f

Note 1: f of Q stands for frequency (Hz)Note 2: the Bedrock model is the homogenous half- space

3
2.5
2

Fourier Spectra

Table 4. Material properties of media for Steps 1 and 2

Medium 2

Z(Down) hypocenter(0,1,4) km

Medium 2

Z

Medium 1

6 km

Medium 1(T14=20 km)

Y (East)4 km
Y (East)2 km Point Source

LOH.44)

UD Transvers

X+002 +006 +010

LOH.24)

Reference

1.5
1
0.5

Frequency (Hz)
10

100

0.1

0.01

0
0

0.2

0.4

0.6

time (s)

0.8

1

0.001

Figure 3. Moment, and moment-rate functions,and the Fourier amplitude spectra for Step 1

Since the amount of results is very large, and almost all the results show very good agreements, weonly show the selected results, especially showing the distinct differences. Figure 4 shows thecomparisons of the three components of velocities at the 100 km distance using the T11 model. The

numbers in the waves indicate the maximum/minimum values of amplitudes. As shown in Figure 4(a),the waves by the 4 researchers are practically identical (Nakagawa's results are not listed, because of
the use of Hisada's code). However, when we look at the close up waves as shown in Figure 4(b), we
see small differences. One of them is the different peak values, which caused by the rough timeinterval (0.01 s), as compared with sharp peak with very short duration (see Figure 3). The othernoises appear in the later phases in Nozu's results, which come from the artificial adjacent domains;
they are introduced by the formulation of the discrete wavenumber method.
Figure 5 shows the comparisons of the three components of velocities at the 100 km distance using
the T13 model. The top three waves are almost identical, which used Q of Table 4 based on equation(1), whereas the bottom waves are different from the others. This is caused by the use of the constant
Q based on equation (2), where ωREF= 1 (rad/s; fREF (the reference frequency) ≒ 0.16 Hz).
21

T11+100 (Transverse)

0.8

1.4
+0.0509‐0.2763

1.2

Nagano1+0.0523‐0.3076

0.8

Nozu

0.6
+0.0512‐0.2770

0.4

velocity (m/s)

Hisada0.8

Hisada+0.0574‐0.1156

Nagano0.5
0.4+0.0589‐0.1190

Nozu

0.3
0.2

Miyakoshi

0.2+0.0575‐0.1157

0.7
0.6+0.0579‐0.1173

20

30

0.6

Nagano+0.1289‐0.1486Nozu+0.1269‐0.1461

Miyakoshi

0.5
0.4
0.3

0
10

40

Hisada+01254‐0.14480.1

010+0.1253‐0.14480.7

0.2

Miyakoshi

0.1

0T11+100 (UD)0.9velocity (m/s)+0.0507‐0.2692

1.6
velocity (m/s)

1

T11+100 (Radial)

0.9

1.8

20

time (s)

30

40

10

20

30

40

time (s)time (s)(a) Full waveforms of velocities (the values are the maximum/minimum values of amplitudes)
T11+100 (Radial)

0.35

T11+100 (UD)Noises from the artificial
adjacent domains

0.3

Noises from the artificial
adjacent domains

0.4

Hisada0.25

0.3

Nagano

0.2

0.2

Nozu

0.1

Miyakoshi

0
28.7
-0.1

28.9

29.1

29.3

Difference ofpeak shapes

0.25

Hisada0.15

Nagano

0.1

Nozu

0.05

Miyakoshi

-0.05

30

35

HisadaNagano

0.1

Nozu

0.05

Miyakoshi-0.05 30

40
time (s)The reflected wave
from the rigid bottom-0.1

0.2
0.15

0

0

time (s)

-0.2
-0.3

velocity (m/s)

velocity (m/s)

0.3

velocity (m/s)

T11+100 (Transverse)

0.5

35

40
time (s)

-0.1
-0.15

(b) Close up of velocity waveforms (the close up areas are indicated in the boxes in Figure 4(a))
Figure 4. Comparisons of the three components of velocities at the 100 km distance using the T11 model
T13+100 (Transverse)

0.1

0.07
+00074‐0.0068

0.06

Nagano0.05+0.0069‐0.0077

0.04

Nozu

0.03
+0.0085‐0.0083

0.02

0.04

Hisadavelocity (m/s)

velocity (m/s)

0.09
0.08

Hisada0.035+00039‐0.0037

0.03

Nagano0.025
0.02+0.0044‐0.0038

Nozu

0.015
0.01

Miyakoshi

0.01+0.0043‐0.0038

velocity (m/s)+0.0069‐0.00770.08+0.0058‐0.0061

Miyakoshi

0.005
0

0
0

10

20

30

40 50
time (s)

60

70

80

T13+100 (UD)0.1

T13+100 (Radial)

0.05
0.045

0.09

+0.0089‐0.0086

Hisada0.07
0.06+00087‐0.0082

Nagano0.05+0.00890.04 ‐0.0086
0.03+0.0066
0.02 ‐0.0056
0.01

Nozu
Miyakoshi

0
0

10

20

30

40 50
time (s)

60

70

80

0

10

20

30

40 50
time (s)60

Figure 5. Comparisons of the three components of velocities at the 100 km distance using the T13 model70

80

It is interesting to check the effects of the different
constant Q based on equations (1) and (2). Figure 6
shows frequency vs. normalized medium velocities
using different fREF (=0.1, 1.0 and 10 Hz) in equation(2). The figure shows the dispersive characteristics; thevelocities keep the original values at f=fREF, but varies
from about -5% at 0.01 Hz to about + 5% at 100 Hz.

1.06
Normalized Velocity

f-REF = 0.1 Hz
f-REF = 1 Hz

1.04

f-REF = 10 Hz

1.02
1
0.98
0.96
0.94

Figure 7 shows the comparisons of the radial
0.01
0.1
1
10
100
components of velocities at 100 km using the different
frequency (Hz)constant Q (=40) for T13. In Fig.7(left), the top is the
Figure 6. Frequency vs. normalized medium
velocity of eq. (1), and the next two are the velocities
velocities using different fREF based on Eq (2)of eq. (2) using fREF = 1.0 and 0.16 Hz, respectively.
Those waves are computed by Hisada's code, and the bottom is Miyakoshi's result using fREF = 0.16
Hz;. The last two waves are almost identical. Fig.7(center) & (right) are close ups of the initial andcoda parts. In the initial parts, the wave of eq. (1) starts before signal's arrival; it does not satisfy thecausality condition. On the contrary, the waves of eq. (2) show the sharp rises just after signal's
arrivals. However, the arrival times are different between fREF = 1.0 and 0.16 Hz. The former is slower,
but close to the wave of eq. (1). On the other hand, we see completely different phases in the coda
parts in Fig.7 (right). Since the waves of eq. (1) and those of eq. (2) using fREF =1.0 Hz show similarphases,. f REF =1.0 Hz in eq. (2) seems the one of the best choice in this particular case.
Figure 8 shows the comparisons of the velocities at 100 km using the T14 model. All the waves
except Nagano (the thin layer method) are almost identical. Nagano's results are smaller than the other,because it computed only up to 7 Hz, but not 25 Hz. The thin layer method is very accurate andefficient for shallow sources at close distance, but may not be efficient for deep sources at long
distances at very high frequencies.
0.004

T13+100 (Radial)+0.0058‐0.0052

0.03+0.0059‐0.0058

ffref=
REF 1.0 Hz

0.025
0.02

+0.0058‐0.0062

ffref=
REF 0.16 Hz

0.015
0.01

+0.0058‐0.00610.005

Eq 虚数のみ(2) usingfref= 1.0 Hz
fREFfref== 0.160.16 HzHz
Miyakoshi（Q一定）

0.003

Eq (1)
虚数のみ

0.035

velocity (m/s)

velocity (m/s)

0.04

0.002

Eq (1)
15

0

10

20

30

40 50
time (s)

60

70

16

17

0.000361

62

63
time (s)-0.000718

80-0.0012time (s)-0.001

Eq (1)
fREF =
1.0 Hz0.0008-0.0002 60

00Range 60 - 63 (s)Eq (2 using fREF
= 0.16 Hz

0.0013

fREF=1.0 Hz

0.001

Miyakoshi

0.0018Range 15 - 18 (s)velocity (m/s)

0.05
0.045

Figure 7. Comparisons of the radial velocities at 100 km using the T13 model and the different constant Q=40
T14+100 (Transverse)

1+0.0297‐0.0949

0.6

Nagano0.5+0.0282‐0.1724

0.4

Nozu

0.3

0.35
+0.0391‐0.0291

0.3

Nagano0.25+0.0718‐0.0547

0.2

Miyakoshi

0.1

Nozu

+0.0577‐0.0431

0.1

25

27

29
31
time (s)

33

35

0

10

20
30
time (s)40

Hisada0.35+0.0446‐0.0380

0.3

Nagano0.25+0.0772‐0.0477

0.2

Nozu

0.15

Miyakoshi

0.05

0+0.0710‐0.0425

0.4

Hisada0.15+0.0291‐0.13900.2+0.0615‐0.04620.4

Hisada0.7

velocity (m/s)+0.0277‐0.1478

velocity (m/s)

velocity (m/s)

0.45

0.45

0.8

T14+100 (UD)0.5

T14+100 (Radial)

0.5

0.9

50

+0.0678‐0.0437

0.1

Miyakoshi

0.05
0

10

20
30
time (s)Figure 8. Comparisons of the three components of velocities at 100 km using the T14 model

40

50

2.2. Steps 3 and 4
Tables 5 and 6 show the model lists of Steps 3 and 4, respectively, which are more complicated andrealistic for sources and media than those of Steps 1 and 2. We use the 4 layered model shown in
Table 7, and the 2 layered model same as Table 4. We use point sources in Step 3 with 2 km and 0
km depths. The source parameters are the same as those of Step 1, except the moment-rate function.
We use the following the Gaussian-type function,•⎛ (t − μ )2 ⎞1⎟
M (t ) = M 0 ⋅⋅ exp⎜⎜ −2σ 2 ⎟⎠
2π σ(4)⎝where, σ is the standard deviation, and μ is the peak time. We useσ=0.2 (s) and μ=0.8 (s),which are the same as those of Day et.al. (2000).
On the other hand, we use the 4 models of the lateral slip fault in Step 4, where the basic source
parameters are the same as T21, except the slip function. We use the pseudo-dynamic slip function(Nakamura and Miyatake, 2000), as shown in Figure 9, using Fmax=6 Hz, td≒0.05305 s, tr≒0.6667 s,
ts=1.0 s, and Vm≒5.168 m/s. Among 4 models (T41-T44), the first three are the buried models sameas Step 2 (see Figure 2), and the last model is the surface fault, whose top edge reaches the freesurface. We also introduce three different rupture models using Vr= 3 km/s. T41 and T42 assume thediscrete ruptures at the intervals of 1 km2. The former uses the constant Vr, whereas the latter uses
random values in the rupture times, using the following equation,t ij =

η ij+ ε ij ( w / 2Vr > ε > − w / 2Vr )ij(5)where, ηij is the distance between the hypocenter and the centres of sub-faults, w is the size of the
sub-faults, and εij is the random numbers. We generate three sets of the random numbers. On theother hand, we assume the continuous rupture fronts using constant Vr in T41 and T42.

Vr

Table 5. Model lists ofr Step 3
T33
Model
T31
T32
4 Layers Model
2 Layeres ModelMedia
Table 7
Q
Infinite
Source2 km (Buried)
0 km (Suface)Depth
0 - 5 Hz
Frequencies
6 points (+002, +006, +010, +030, +050, +100 km)
OutputTable 6. Model lists of Step 4
T41
T42
T41
T42
Model
2 Layers ModelMedia
Table 4
InfiniteQ
Lateral Slip Fault (Buried)(Surface)
Fault
Model 1 km2 (constant) 1 km2 (random)
continunous
0 - 5 Hz
Frequencies
12 points (±002, ±006, ±010, ±030, ±050, ±100 km)
Output1 sets1 sets3 sets
6

Table 7. Material properties of the 4 layered model
Thickness
Vp
Vs
Density
No
Qp=Qs
(m)
(m/s) (m/s) (kg/m3)
1
200
1,600
400
2,000
20f
2
400
2,600 1,000 2,400
30f
3
1,000
4,000 2,000 2,600
40f
46,000 3,464 2,70070f∞

Slip velocity
function

54

1.2

Fourier Spectraof slip velocity
function1
0.8

Slip function

3

0.6

2

0.4

1

0.2
0

0
0

0.5

1

1.5
2
time (S)

0

2

4

6
8 10
frequency (Hz)Figure 9. The pseudo-dynamic slip and velocity functions

Again, the amount of results is large, and almost all the comparisons show exellent agreements. Thus,
we only show the results showing the distinct differences. Figure 10 shows the comparisons of theradial components of velocities at the 10 km distance using T31. The waves seem almost identicaleach other, whereas Figure 10 (b) and (c) show the close up of the initial (0-5 s) and the coda waves(30-35 s). Even though the amplitudes are very small, we see the base line drifts except Nagano'results. Those permanent drifts are probably caused by the use of the unphysical frequency dependent
Q. Q becomes 0 (i.e., infinite damping), when f goes to 0 Hz. The thin layer method by Nagano did
not show these drifts, because those low frequency noises are probably absorbed tin the dampingbuffer zone at the bottom.
Figure 11 shows the comparisons of velocities at the 100 km distance using T33 (a surface point
source). The results between Hisada and Nagano are almost identical, but those of Miyakoshi and
Nakagawa are small. Those small amplitudes are caused by the approximated depth (50 m). Thisapproximation was needed for carrying out the numerical wavenumber integration; in the case of thesame depths of the source and observation points, the convergences of integrants are extremely slow
for increasing wavenumbers. On the contrary, the methods by Hisada and Nagano eliminated those
problems, theoretically (see Hisada, 1995; Nagano and Watanabe, 2007).
Figures 12 and 13 show the comparisons of velocities and their Fourier amplitudes at 10 km using theT41, T42, and T43 models. T41 and T42 use the discrete rupture processes at the intervals of 1km2,with the constant and random rupture times, respectively. T42 use three sets of random numbers. On
the other hand, T43 use the continuous rupture considering very small sizes of sub-faults, whichcaused very small amplitudes at higher frequencies. The discrepancies of the results among threeresearchers are very small for T41 and T43. On the other hand, those of T42 show randomcharacteristics, but not very large differences. It is noted that the Fourier amplitudes of T41 shows the
sharp peaks around 3 Hz, whereas those of T42 are not clear. Those peaks are caused by the artificial
rupture interval times between adjacent sub-faults (Vr/w=3/1= 3 Hz).
Figure 14 shows the comparisons of the displacements at 2 km using T44. The displacements arecomputed by integrating velocities in the time domain. Since the site is very close to the surface fault,
we see the fling steps (the permanent offsets with large amplitudes). Beside the benchmark tests, we
also computed the static terms, which are the convolution of the slip function and static Green's
function (Hisada and Bielak, 2003). The results by three researchers are almost identical, and thevalues of the permanent offsets agree with the static values. While those values were computed by nodamping (Q) media, the displacements showing "Including Q of Eq (2)" in Fig.14 are the results using
the frequency dependent Q. Their amplitudes are increasing or decreasing in time, and do not show thecorrect values. This is probably caused by the unphysical Q, that is, the value becomes 0 (infinite
damping), when frequency goes to 0 Hz
0.05

T31+010 (Radial)
+0.42
‐0.69

3
2.5
2
1.5

+0.42
‐0.69

NAGANO

+0.40
‐0.68

NOZU

+0.42
‐0.71

1
0.5
0

5

10

15

20

25

30

time(s)

0.02

0.03

0.01
0
-0.01

NAKAGAWA0

HisadaNakagawa

0.03

HISADAvelocity (m/s)

Velocity (m/s)

3.5

-0.02
35

30 -35 s

0.04

0.04

4.5
4

0.05

0-5s

velocity (m/s)

5

0

1

2

3
4
time (s)

5

Nozu

0.01
0
-0.01

30

31

-0.02

Nagano

Nozu

HisadaNakagawa

0.02

32

33

34 35
time (s)Nagano

-0.03

-0.03

(a) Whole wave forms(b) Close up between 0 to 5 s (c) Close up between 30 -35 s
Figure 10. Comparisons of the radial components of velocities at the 10 km distance using the T31 model

+0.44
‐0.42
+0.43
‐0.42

3
2

+0.36
‐0.33

1

+0.27
‐0.26

NAGANOMIYAKOSHI&ASANONAKAGAWA

+0.74
‐0.86
+0.73
‐0.85

8

HISADA
Velocity (m/s)

Velocity (m/s)

4

T33+100(Transverse)

10

6
4

+0.73
‐0.85

2

+0.62
‐0.72

NAGANOMIYAKOSHI&ASANONAKAGAWA

0

20

40

60

80

100
time (s)

0

20

40

60

80

100
time (s)

+0.77
‐0.77
+0.76
‐0.76
+0.59
‐0.59
+0.47
‐0.47

8

HISADA00

T33+100(UD)10Velocity (m/s)

T33+100(Radial)

5

6
4
2

HISADA
NAGANOMIYAKOSHI&ASANONAKAGAWA

0
0

20

40

60

Figure 11. Comparisons of the three components of velocities at the 100 km distance using the T33 model

80

100
time (s)

0.8
0.6
0.4
0.2

T41

Velocity (m/s)

Velocity (m/s)

1

1.2

T4-010-HISADA (NS)
+0.14
‐0.13
+0.14
‐0.15
+0.10
‐0.14
+0.19
‐0.17
+0.01
‐0.05

T42-1
T42-2
T42-3
T43

T4-010-NAGANO (NS)

+0.14
‐0.13
+0.11
0.8
‐0.14
+0.18
‐0.18
0.4 +0.11
‐0.16
+0.01
‐0.05

1.2

T41

Velocity (m/s)

1.2

T42-1
T42-2
T42-3
T43

0

0
0

5

10

0

15

time (s)5

10time (s)T4-010-NAKAGAWA (NS)

+0.13
‐0.13
+0.13
0.8
‐0.14
+0.14
‐0.20
+0.10
0.4
‐0.13
+0.01
‐0.05

T41
T42-1
T42-2
T42-3
T43

0

15

0

5

10

time (s)15

Figure 12. Comparisons of velocities at the 10 km distance using the T41, T42, and T43 models
11frequency (Hz)T42-010-HISADA(NS)

T41(3 Hz)

0.01

0.1

T43

0.001

1

1
10

0.1

T41
T42-1
T42-2
T42-3
T43

1

frequency (Hz)

0.1

10

T42-010-NAGANO(NS)T41(3 Hz)

0.01

Fourier Amplitude (m)

0.1

T41
T42-1
T42-2
T42-3
T43

Fourier Amplitude (m)Fourier Amplitude (m)

0.1

T43

0.001

0.1

T41
T42-1
T42-2
T42-3
T43

1

frequency (Hz)10

T42-010-NAKAGAWA(NS)

T41(3 Hz)

0.01

T43

0.001

Figure 13. Comparisons of the Fourier amplitudes at the 10 km distance using the T41, T42, and T43 models
HisadaNo Q Nagano
Nakagawa

0
0

2

4

6

8
time (s)

-0.05

-0.1

-0.15

10

Displacement (m)Static0.05
Displacement (m)0.6

T44+002 (NS)0.06

T44+002 (EW)

0.05

No Q (Hisada, Nagano,and Nakagawa)

0.5
0.4

0.04
0.03
Displacement (m)

0.1

0.3
0.2

Including Q of Eq(2)Static

0.1

0
-0.1

2

4

6

8

10
time (s)HisadaNo Q Nagano
StaticNakagawa

0.01
0-0.01 0

2

4

6

-0.02

8

10
time (s)-0.03

0Including Q of Eq(2)0.02

T44+002 (UD)

-0.04
-0.05

Including Q of Eq(2)Figure 14. Comparisons of the tree components of displacements at the 2 km distance using the T44 model

2.4. Steps 5 and 6
Table 8 shows the model lists of Steps 5 and 6, where we use the actual source models in the Kanto
sedimentary basin. In Step 5, we use relatively small earthquakes, and compare the simulated waves
and observation records. As an example, Table 9 and Figure 15 show the source parameters and thelocations of the the 1990 West Kanagawa earthquake (the Odawara earthquake in Sato et.al., 1998),
together with the observation stations. In Step 6, we simulate long-period ground motions of the 1923
Great Kanto earthquake. Table 10 and Figure 16 show the source parameters and the location of thefault (see for more detail in Sato et.al., 2005)
Table 9 shows the material properties and the locations of the top boundaries of the layers of the
Kanto basin model. Although the original data are for the 3-D basin model (The headquarters for
Earthquake Research Promotion, 2009), we use the values just under the observation stations as theflat layered models. The table shows examples of the boundary depths at the four sites.
Figure 17 shows the comparisons of the NS components of velocities for the 1990 West Kanagawa
earthquake at the KNO, ASK, JSK, and FUT sites. The top waves are observed records (AIJ, 1996),
and the next three waves are the simulations by three researchers. The excellent agreements are
obtained among the simulations. The simulations at KNO, which is located at the rock site closest to
the epicenter, show excellent agreements with the records. However, the other results do not agree

well with the records, especially the surface waves excited in the Kanto basin. These results clearly
indicate the limitations of the theoretical methods based on the flat layered structures.
Table 8. Model lists of Steps 5 and 6
Step 5
Step 6
T51
T52
T53
T61
1990 West Kanagawa 1990 Near Izu-Oshima 1992 Tokyo Bay EQ 1923 Great Kanto EQ
Earthquake
EQ (Mj 5.1)
EQ (Mj 6.5)
(Mj 5.7)(Mj 7.9)
Source Model Point （Sato, T. et al. Extended（Yamada and Point（Yamada and Extended（Sato, H.,
1998 )
Yamanaka, 2003)
Yamanaka, 2003)
et. al., 2005)(Reference)
Flat layered half-spaces at the observation stations using the 3D Kanto basin modelMedia
Table 9Q
0～0.33 Hz
Frequencies
19 sites in the Kanso basin, where the strong motion data are available (AIJ, 1996)
OutputModel

Table 9. The source parameters of the 1990 West Kanagawa EQLongitude of Epicenter （°）
Latitude of Epicenter （°）
Depth of Epicenter (km)Strike Angle（°）

35.2133
139.0998
15.3
215

Dip Angle（°）
Rake Angle（°）
Sesmic Moment（Nm）

3540
3.3×1016

JSK

ASK
KNO
Epicenter

FUT
N
E

Figure 15. Locations of the epicenter of the 1990
West Kanagawa EQ and observation stations

N

Table 10. The source parameters of the 1923 Great Kanto EQ
Fault Length（km）

130

Fault Width（km）

70

Longitude of Epicenter （°）
Latitude of Epicenter （°）

35.3847139.1144

Depth of Epicenter (km)10.651

Strike Angle（°）

294

Dip Angle（°）
Rupture Velocity（km/s）

163

Sesmic Moment（Nm）

9.94×1020

E

Epicenter ASK
JSK

b

a
KNO

FUT

Figure 16. Locations of the fault and epicenter ofthe 1923 Great Kanto EQ and observation stations
Table 11. The material properties and the depths of the top boundaries of the Kanto basin model
No.
1
2
3
4
5
6
7
8
9
10
11

ρ
(kg/m3)
1,950
2,050
2,100
2,150
2,250
2,300
2,450
2,600
2,650
2,700
2,800

Vp
(m/s)
1,800
2,100
2,300
2,400
3,000
3,200
4,200
5,000
5,500
6,000
6,700

Depth of Top Boundary（m）
Vs
Qp=Qs
(m/s)
Kuno (KNO) Asakawa (ASK) Hongo (JSK) Futtsu(FUT)
500
500f*
0
0
0
0
700
700f
－
－
－
－
900
900f
152
6
406
260
1,000 1000f
－
－
－
－
384
51
1,736
1,426
1,500 1500f
1,700 1700f－－－－
2,400 2000f
616
－
－
－
2,900 2000f
676
－
－
－
3,200 2000f
1,317
201
3,067
3,347
3,530 2000f
1,798
7,107
7,830
6,792
3,940 2000f
4,494
17,76819,575
16,980Finally, Figure 18 shows the comparisons of the NS components of velocities for the 1923 Great Kanto
earthquake at the KNO, ASK, JSK, and FUT sites. Even though the source model and the media arevery complicated, the excellent agreements are obtained among the researchers. This clearly shows thereliability of the theoretical methods under the same source, path and site conditions.
+0.205‐0.259+0.159‐0.179+0.159‐0.183+0.160‐0.181+0.160‐0.180Observation
HisadaNagano
Nozu
Asano

Obs.
‐0.051 Hisada+0.031‐0.029Nagano+0.031‐0.029+0.032Nozu‐0.030+0.031
Asano+0.045‐0.029+0.008‐0.012+0.022‐0.014+0.022‐0.014+0.024‐0.016+0.022+0.014+0.056‐0.058+0.016‐0.020+0.016‐0.020+0.017‐0.021+0.016‐0.021

Observation
HisadaNaganoNozu
Asano+29.3 Hisada‐42.1+28.8 Nagano‐43.5+28.7 Asano‐42.4+6.67 Hisada‐18.3+6.78 Nagano‐17.7+6.91 Asano
‐17.2

Obs.
HisadaNaganoNozu
Asano+57.9 Hisada‐71.7+57.7 Nagano‐72.6+58.1 Asano‐71.8+30.9 Hisada‐33.1+30.9 Nagano‐33.0+30.8 Asano‐33.3

Figure 17. Comparisons of velocities for the 1990
West Kanagawa earthquake (NS components)

Figure 18. Comparisons of velocities for the 1923
Great Kanto earthquake (NS components)

3. CONCLUSIONS
We performed benchmark tests of 6 Steps, using the three different theoretical methods, and obtained
excellent agreements in almost all the results. At the same time, we found discrepancies, such as thecases for large epicentral distances in the media with different formulations of Q (medium damping),
and the different approximations of surface sources.
AKCNOWLEDGEMENT
This work was supported by Grant-in-Aid for Scientific Research (B) of MEXT (Japan Ministry of Education,
Culture, Sports, Science and Technology), and Research Centre of Urban Disaster Mitigation (UDM) in
Kogakuin University. We also thank Prof. N. Yamada of Fukuoka University of Education, and Prof. R.
Kobayashi of Kagoshima University for providing their source parameters, and Sub-Committee on Strong
Motion Observation of Architectural Institute of Japan for providing the strong motion data.

REFERENCES
AIJ (Architectual Institute of Japan, 1996). Journal of architecture and building science, April.
Aki, K., and Richards, P. G. (1980). Quantitative seismology: Theory and methods, pp.167-185.
Bouchon, M.(1981). Bulletin of Seismological Society of America, Vol.71, pp.959-971.
Day, S. M. et.al. (2000). Final Report to PEER Center, Lifelines Program TASK 1A01, pp.1-24.
Day, S. M. et.al. (2003). Final Report to PEER Center, Lifelines Program TASK 1A02, pp.1-32.
Hisada, Y. (1995). Bulletin of Seismological Society of America, Vol.85, No.4, pp.1080-1093.
Hisada, Y. and Bielak, J. (2003). Bulletin of Seismological Society of America, Vol.93, No.3, pp.1154-1168,.
Nakamura, H. and Miyatake, T. (2000), Journal of the Seismological Society of Japan, Vol.53, No.1, pp.1-11.
Nagano, M. and Watanabe, T. (2007). AIJ Journal of Technology and Design, Vol.13, No.26, pp.451-456
Nozu, A. (2002). Technical Note of Port and Airport Research Institute, Japan, No.1037.
Sato, H. et al. (2005). Earthquake source fault beneath Tokyo, Science, Vol. 309, pp.462-464.
Sato, T. et al. (1998). Bulletin of Seismological Society of America, Vol.88, No.1, pp.183-205.
Yamada, N. and Yamanaka, H.. (2003), Journal of the Seismological Society of Japan, Vol.56, pp.111-123.