Proceedings of the 8th International Conference on Structural Dynamics, EURODYN 2011
Leuven, Belgium, 4-6 July 2011
G. De Roeck, G. Degrande, G. Lombaert, G. Müller (eds.)
ISBN 978-90-760-1931-4

574

Surface Wave Propagation Analysis of
a Ground with Three-Dimensional Irregularities
1

Shoichi Nakai1, Hiroto Nakagawa1
Graduate School of Engineering, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba, Japanemail: nakai@faculty.chiba-u.jp, nakagawa-hiroto@graduate.chiba-u.jp

ABSTRACT: In this paper, a study is presented on the wave propagation in a three-dimensional (3-D) slope ground due toincident surface waves. The objective is to examine the effect of irregularities on the microtremor or ambient vibration wave
field. The analysis is based on the 3-D finite element method in conjunction with the 2.5-D thin layered and finite element
methods. It was found from the study that wave propagation characteristics change by a great deal when a 3-D irregularity suchas a tiny canyon exists in a slope ground with a fundamentally 2-D configuration. The frequency dependency of horizontal tovertical spectral ratios and phase velocities also changes according to the irregularity.
KEY WORDS: Irregular ground, Surface wave, Microtremor, Finite element analysis, 3-D analysis, 2.5-D analysis
1

INTRODUCTION

It is essential to know the condition of the ground when
considering earthquake disaster mitigation. It is well known
that the surface soil condition and micro topography, or
landform, influence the seismic intensity of the ground andhence impact structural damage to the buildings and civil
infrastructure during earthquakes. However, obtaining information on the ground condition, such as soil profiles, over awide area is not an easy task. One of the most popular
approaches to estimate the ground condition is to conduct
microtremor (ambient vibration) measurements on the groundsurface, from which natural frequencies of the ground are
obtained [1]-[3]. It is also possible to obtain soil profiles frommicrotremor array measurement results by applying inversion
techniques based on the surface wave propagation theory [4][7]. These approaches, however, are based on a parallel layer
assumption. The surface wave propagation theory based on aparallel layer assumption is also applied to the problem oftraffic induced vibration because of the location of excitation
being on the ground surface.
A difficulty arises when the ground has an irregularity,
which is often the case in an actual situation. For example,
Figure 1 shows the landform classification of Chiba, theeastern part of Tokyo metropolitan area. As can be seen in the
figure, this area consists of three major categories of landform,
i.e. terrace, lowland and reclaimed ground [8]. It is also notedthat one of the characteristic features of the landform is theexistence of a whole lot of narrow river valleys (lowland) thatpenetrate deep into terrace, which makes the landform of thisarea very complex. In addition, fairly steep slopes are formed
along most of the boundaries between terrace and lowland,
meaning that irregular ground is quite popular in this area.
A number of researches regarding wave propagation in an
irregular ground have been reported so far [9], [10]. Most ofthem, however, deal with body waves and only a few havelooked at surface waves [11], [12]. In addition, a completely
flat ground surface assumption for the far field is made inalmost all three-dimensional studies [13], [14].

In this paper, the effect of a three-dimensional groundirregularity on the surface wave propagation is studied. Morespecifically, a two-dimensional slope ground with a small
canyon (hollow) subject to an obliquely incident surface wave
is considered, as schematically illustrated in Figure 2. The
analysis method used in the study is a combination of threedimensional and two-and-a-half- (2.5-) dimensional finite
element methods [12] in conjunction with a substructuretechnique [15].

Figure 1. Landform classification of Chiba area, Japan

2

PROBLEM UNDER STUDYAs mentioned earlier, the main objective of this study is to
examine the characteristics of surface wave propagation in an
irregular ground in three dimension in order to evaluate its
influence on the soil exploration based on microtremor
measurements. In order for this, a simplistic irregular groundthat consists of a two-dimensional slope ground with a slitlike narrow canyon that penetrates perpendicularly into the575

Proceedings of the 8th International Conference on Structural Dynamics, EURODYN 2011terrace (upland part of the landform), as shown in Figure 2.
The problem under study is the microtremor wave field of thislandform, which is assumed to be a synthesis of surface waves
propagating in a variety of directions. The problem consideredin this study is defined by:• The ground is basically a two-dimensional slope ground.
• The ground has a small canyon that penetrates perpendicularly into the terrace.
• The soil is either uniform or two layered.
• The ground is subject to a number of incident surface waves
of various modes coming from a variety of directions.

called a 2.5-dimensional analysis [16], [17]. Since irregularity(slope) is involved in this analysis itself, another substructuring is considered, i.e. the 2.5-dimensional thin layered
elements and 2.5-dimensional finite elements are combined toobtain the response due to an obliquely incident surface wave
to the slope. Figure 3 illustrates the method of analysis.

Figure 3. 2.5-dimensional analysis
3.3
Figure 2. A slope ground with a tiny canyon

3

METHOD OF ANALYSIS

The method of analysis is basically a three-dimensional finite
element method in conjunction with a substructure technique.
It features, however, a couple of points so that it can handle
the problem under study.
3.1

Substructure Method

There exist a variety of substructure approaches that deal witha wave propagation problem. The method used in this study
follows the following procedures [15]:(1) Subdivide the entire ground under study into two parts; anear field that involves three-dimensional irregularities,
and a far field which is basically a two-dimensional slope
ground.
(2) Compute an impedance matrix [Kc*] of the far field fromwhich the near field is excavated.
(3) Compute a displacement vector {uc} and traction vector{pc} of an equivalent far field which does not have an
excavation and is subject to an incident surface wave.
(4) Compute a driving force vector {fc*} at the boundary by
the following expression:{ f } = !"K #$ {u } + { p }
*
c

*
c

c

c

Response of a Near Field: Three-Dimensional Analysis

The substructure analysis of the target, i.e. a slope groundwith a tiny canyon, requires the impedance matrix [Kc*] andthe driving force vector {fc*} as described in 3.1. In this study,
the impedance matrix [Kc*] at the boundary of the analysis
model is computed as dashpots attached to the boundary. The
displacement vector {uc} of the equivalent far field can becomputed from 2.5-dimensional analysis described in 3.2 by
the following expression:{uc } = {u2.5 } exp ( !ikys y ), kys = ks sin !

in which, {u2.5} is a displacement vector of reference which isequal to the displacement field of the 2.5-dimensional analysis
described in 3.2, ks is the wave number of an incident surface
wave of s-th mode, and ! is the angle of incidence.
The traction vector {pc} of the equivalent far field can becomputed by differentiating the displacement field and by
applying the constitutive relationship. Finally, the driving
force vector { fc*} can be given by Eq. (1)Figure 4 illustrates the three-dimensional analysis.

(1)(5) Compute a response of the near field by attaching theimpedance matrix at its boundary and by applying thedriving force to its boundary.
3.2

Response of a Far Field: 2.5-Dimensional Analysis

In the substructure analysis described above requires a threedimensional analysis of a two-dimensional slope groundsubject to an incident surface wave. This type of analysis is(2)

Figure 4. Three-dimensional analysis

Proceedings of the 8th International Conference on Structural Dynamics, EURODYN 2011

4
4.1

576

SURFACE WAVE PROPAGATION IN A THREEDIMENSIONAL IRREGULAR GROUND
Slope Ground Composed of Uniform Soil

In order to verify the analysis method, a slope ground withuniform soil subject to an incident Rayleigh wave offundamental mode has been analyzed. Figure 5 shows thefinite element mesh layout. As shown in the figure, the angle
of inclination of the slope is 45º and the height is 12 meters.
Eight node linear elements are used in the three-dimensional
finite element analysis, while eight node quadratic elements
are used in the 2.5-dimensional finite and thin layered element
analysis.

Figure 5. Finite element mesh layout of a slope ground(a) 2.5-dimensional analysis(b) Three-dimensional analysis
Figure 6. Wave field due to an incident Rayleigh wave of fundamental mode of 8 Hz

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Proceedings of the 8th International Conference on Structural Dynamics, EURODYN 2011

Figure 6 compares three-dimensional and 2.5-dimensional
results of displacement distribution at the ground surface inthe case of Rayleigh wave incidence of fundamental mode of
8Hz with the incidence angle of 45º. The wave is coming fromthe upland part of the ground. Although two analyses mustcoincide with each other from the theoretical point of view,the three-dimensional analysis result is slightly different fromthe 2.5-dimensional counterpart especially for displacement
amplitudes. A possible reason for this can be addressed to an
insufficient accuracy of dashpots as the impedance function of
the far field. The fluctuation of displacement distribution
found in the three-dimensional results can thus be explainedby the interference of the incident wave and waves reflectedfrom the boundary. This fluctuation may be reduced byadopting a more efficient boundary (accurate impedance).
4.2

Slope Ground with a Tiny Canyon Composed ofUniform Soil

A similar slope ground with uniform soil subject to an
incident Rayleigh wave of fundamental mode was considered
next. The difference from the previous case is that there exists
a small canyon-like hollow that penetrates perpendicularly
into the slope, as shown in Figure 7(a). The configuration of
the canyon has the depth of 12 meters, the width of 20 meters
and the length of 20 meters. The ground surface inside thecanyon is on the same level as the ground surface of thelowland. The rest of the analysis conditions, including the soil
properties, are the same as those shown in Figure 5.
Figure 8 shows the displacement wave field for thefrequency of 8 Hz and the incidence angle of 45º. From thecomparison with Figure 6 which corresponds to the casewithout a canyon, it can be pointed out that:• Due to the existence of a canyon, the displacement fieldchanges by a great deal and becomes very complex.

(a) A slope ground with a canyon (uniform soil)(b) A slope ground with a canyon (two-layered soil)
Figure 7. Three-dimensional analysis model

Figure 8. Displacement amplitude due to an obliquely incident Rayleigh wave (uniform soil)

Figure 9. Displacement amplitude due to an obliquely incident Rayleigh wave (two-layered soil)

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Proceedings of the 8th International Conference on Structural Dynamics, EURODYN 2011

• Amplitude of displacement is large in the area locatedupstream with respect to the canyon and is small in the back.
• The affected area is fairy large when compared to the size ofthe canyon.
• Variation of the displacement amplitude is large on the
upland part of the ground but the influence on thedisplacement field is observed in a wide-ranging area of thelowland as well.
• Amplitude of displacement on the ground surface inside thecanyon is fairly small.
Although this characteristics varies depending on thefrequency, hence the wavelength, the influence appears overlarge areas even in the low frequency range.
3

H/V ratio

2

1-D
3-D

1

0

5
10
Frequency [Hz]H/V ratio

B-1
2

1-D
3-D

1

0

5
10
Frequency [Hz]H/V ratio

C-1
2

1-D
3-D

1

0

5
10
Frequency [Hz]

A-3

2

2

1

1

0

0

5
10
Frequency [Hz]

0

D-1
2

1-D
3-D

1
0

0

5
10
Frequency [Hz]

0

B-3

2

2

1

1

0

0

5
10
Frequency [Hz]

0

0

C-3

2

2

1

1

0

0

5
10
Frequency [Hz]0510Frequency [Hz]

3
D-2

D-3

2

2

1

1

0

5
10
Frequency [Hz]

3
C-2

0

5
10
Frequency [Hz]

3
B-2

3

3

H/V ratio

3
A-2

3

3

0

Finally, a two-layered ground is investigated. The analysis
model is the same as before except that the shear wave
velocity of the surface layer is half of that of the underlying
layer as shown in Figure 7(b). The thickness of the surface
layer is 12 meters, meaning that there exists no surface layer
in the innermost area of the canyon because the height of theslope is also 12 meters.
Figure 9 shows the displacement wave field for thefrequency of 8 Hz and the incidence angle of 45º. When
compared with Figure 8 for the case of uniform soil, it isnoted that, although the overall tendency is similar, the

3

3

0

Slope Ground with a Tiny Canyon Composed of TwoLayered Soil

3
A-1

0

4.3

0

5
10
Frequency [Hz]Figure 10. Comparison of H/V spectra

0

0

5
10
Frequency [Hz]

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Proceedings of the 8th International Conference on Structural Dynamics, EURODYN 2011

Phase Velocity [m/s]

displacement distribution shows finer fluctuation. This may be
resulted from the existence of higher modes of Rayleigh andLove waves which have larger values of phase velocity.
Figure 10 shows the horizontal to vertical Fourier spectral
ratios, or H/V spectra, while Figure 11 shows the phasevelocity dispersion curves at various locations near the canyon
in this slope ground. Both H/V spectrum and dispersion curve
were computed based on the aggregated displacement wave
field at each point indicated in Figure 7(b) by summing up thedisplacements due to all possible surface (Rayleigh and Love)waves including higher modes with the incidence angles of!135º, !45º, 0º, 45º, 135º and 180º. Summation was done in
the form of the power of displacements. The dispersion curve
500

500

500

400

400

400

300

300

300

200

200

200
1-D
3-D

100

Phase Velocity [m/s]00Phase Velocity [m/s]

A-1

5
10
Frequency [Hz]

100
0

A-2
0

5
10
Frequency [Hz]

100
0

500

500

500

400

400

400

300

300

300

200

200

200

100
0
0

1-D
3-D

B-1

5
10
Frequency [Hz]

100
0
0

B-2
5
10
Frequency [Hz]

0

0

500

400

400

400

300

300

300

200

200

200

1-D
3-D

C-1

5
10
Frequency [Hz]

100
0
0

C-2
5
10
Frequency [Hz]

100

500

500

400

400

400

300

300

300

200

200

200

1-D
3-D

0

0

D-1

5
10
Frequency [Hz]

100
0

D-2
0

5
10
Frequency [Hz]

100
0

5
10
Frequency [Hz]510Frequency [Hz]

C-3

0
0

500

100

0

B-3

500

100

A-3

100

500

0
0

Phase Velocity [m/s]

was computed by the centerless circular array method
proposed by Cho et al. [6] by considering the surrounding four
nodes of the finite element mesh as censor locations. From
Figures 10 and 11, it is observed that:• The difference between 1-D and 3-D results is fairly large
for both H/V spectra and dispersion curves, meaning that the
influence of a three-dimensional irregularity on the microtremor wave field is very large.
• The difference is large especially in the frequency range
near the natural frequency of the surface soil (4.17Hz).
• This may suggest that the parallel layer assumption may
result in some errors when conducting an inversion analysis
based on it.

510Frequency [Hz]D-3
0Figure 11. Comparison of dispersion curves (phase velocity)

510Frequency [Hz]

Proceedings of the 8th International Conference on Structural Dynamics, EURODYN 2011

Fluctuation of H/V spectra and dispersion curves may be
resulted from insufficient number of incident waves andinsufficient capability of dashpots as impedance functions of
the far field ground. This subject will be addressed in the
future work.
5

CONCLUSIONSIn order to examine the effect of irregularity of a ground on
the microtremor wave field by taking a slope ground with atiny canyon as a target and conducting a three-dimensional
finite element analysis in conjunction with a 2.5-dimensional
thin layered element analysis. It was found from the study
that:• It is possible to conduct a three-dimensional analysis of aground with basically a two-dimensional topography.
• The wave field becomes very complex when there exists a
small canyon that penetrates into the upland through theslope, causing a big difference between the results in 2.5 andthree dimensions.
• The microtremor wave field is also affected by the existenceof a small canyon and the frequency dependency of H/Vspectra and dispersion curves in the area close to the canyon
show significant fluctuation because of this.
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