Bulletin of the Seismological Society of America, Vol. 93, No. 1, pp. 253–263, February 2003
Laboratory Study on Scattering Characteristics of Shear Waves
in Rock Samples
by Yo Fukushima,* Osamu Nishizawa, Haruo Sato, and Masakazu Ohtake
Abstract To understand the scattering characteristics of seismic shear waves in
the Earth, a laboratory physical model experiment in ultrasonic frequency range was
performed. By analyzing waveform envelopes and particle motions of shear waves
in media with different correlation distance and fluctuation intensity, we studied the
strength of scattered wave excitation and the distortion of shear-wave polarization
as a function of ka (wavenumber times the correlation distance). We used gabbro
and granite as media having different small-scale random heterogeneities. The granite
we used contained preferred-oriented thin microcracks. The correlation distance a
and the standard deviation of the fractional fluctuation of the shear-wave velocity e
were estimated by fitting exponential-type autocorrelation functions. The estimated
values were a ⳱ 0.84 mm and e ⳱ 8.1% for the gabbro, and a ⳱ 0.39 mm and
e ⳱ 17.0% for the granite. Waveform measurements were performed in the frequency
range 0.25–1 MHz, which roughly corresponds to ka as 0.25–1. Envelope broadening
appeared when ka exceeds 0.4 for the granite and 1.4 for the gabbro. Shear-wave
polarizations were distorted by scattering when ka exceeds 0.2 for the granite and
0.7 for the gabbro. The results on envelope broadening suggest the importance of
large-angle scattering due to small-scale random heterogeneities and cracks in the
envelope broadening, in addition to diffraction effects due to large-scale heterogeneities. The effect of aligned microcracks on scattering was also examined. It was
found that scattering was prominent when shear wave propagates perpendicular to
the aligned crack plane.
Introduction
Coda waves in seismograms of local earthquakes are
mostly composed of back-scattered shear waves generated
from random heterogeneities in the crust (Aki and Chouet,
1975). Excitation of scattered waves from distributed heterogeneities has been extensively investigated to unravel the
structural characteristics of random heterogeneity in the
Earth (e.g., Sato, 1984, 1989; Benites et al., 1997; Margerin
et al., 2000). The random heterogeneities with scale lengths
less than 1 km are observed in logging data of deep wells
(e.g., Holliger, 1996; Shiomi et al., 1997), and those with
characteristic scale lengths of a few kilometers are found in
tomography with artificial seismic sources (e.g., Hammer et
al., 1994) and by reflection surveys (e.g., Tittgemeyer et al.,
1999).
Distortion of first-arriving elastic-wave polarization also
indicates random heterogeneity and has been investigated to
characterize the heterogeneities in the crust (Sato and Matsumura, 1980; Matsumura, 1981; Nishizawa et al., 1983;
Nishimura, 1996). The distortion provides additional information about random heterogeneity because the first-arriving
waves are affected by forward scattering and diffraction due
to the random heterogeneity around the seismic ray path.
Nishizawa et al. (1983) investigated the particle motions of
the first arrivals from microearthquakes and detected strong
scattering in the region where microcracks were presumably
created by water injection into hot crystalline rock. Fundamental studies about the excitation of scattered waves and
the distortion of shear-wave polarization are necessary to
understand the heterogeneities of the Earth’s interior.
Many researchers have studied seismic-wave propagation in random heterogeneous media. The studies can be
categorized into four different approaches: radiative transfer
theory approaches and those with the Born approximation
(e.g., Sato, 1984; Shang and Gao, 1988; Bal and Moscoso,
2000), an analytical approach using the parabolic approximation (Sato, 1989; Obara and Sato, 1995; Saito et al.,
2002), numerical simulations (e.g., Frankel and Clayton,
1986; Hestholm et al., 1994; Hoshiba, 2000; Fehler et al.,
2000), and laboratory experiments using physical models
*Present address: Laboratoire “Magmas et Volcans,” Université Blaise
Pascal, CNRS, 5, rue Kessler, 63038 Clermont-Ferrand Cedex, France.
253
254
(e.g., Vinogradov et al., 1989, 1992; Nishizawa et al., 1997).
The radiative transfer approaches treat the propagation of
energy packets. The analytical approach is established only
for scalar-wave propagation. Most of the numerical simulations of random media are limited to the two-dimensional
case so far, and it is still difficult to perform numerical simulations for three-dimensional media having short-wavelength random heterogeneities. Since the Earth’s crust contains random heterogeneities of broad scale lengths, it is
important to study the elastic-wave propagation in threedimensional random media with different scale lengths.
Nishizawa et al. (1997) and Scales and van Wijk (1999)
utilized a laser Doppler vibrometer (LDV) to measure the
elastic waves in ultrasonic frequency range. In comparison
with the conventional piezoelectric transducers, a LDV has
many advantages such as a flat frequency response in a broad
band and detection of three-dimensional particle motions in
an area as small as 50 microns (Nishizawa et al., 1997,
1998). Fundamental techniques of shear-wave measurements with a LDV were shown by Nishizawa et al. (1998).
Using the same technique, Fukushima (2000) further measured the radiation patterns of piezoelectric transducers, both
compressional- and shear-type sources, and compared the
experimental results with theoretical calculations. By knowing the source radiation pattern and acquiring accurate waveforms, we can study the elastic-wave propagation in random
media for different scale-length relationships between elastic
wave and heterogeneity.
Employing compressional-type sources of several hundred kHz to MHz range, Nishizawa et al. (1997) and Sivaji
et al. (2001, 2002) made experiments to study scattering
characteristics in random heterogeneous media with different characteristic scales; their experiments correspond to the
ka (wavenumber times the correlation distance) range 0.1–
1. They observed strong coda-wave excitation when ka is
close to 1. This is probably originated from large-angle scattering of shear waves produced from the compressional-type
source (Tang et al., 1994; Fukushima, 2000). Because the
scattering characteristics of compressional and shear waves
are different, and because most of the coda-wave energy in
seismograms consist of scattered shear waves (Shapiro et al.,
2000), it is important to study shear-wave scattering by
physical model experiments using shear sources.
The present study is the first attempt to perform an experiment on scattering of shear waves in random heterogeneous media by measuring envelope broadening phenomena
and waveform particle motions. The purpose is to investigate
the scattering characteristics of shear waves in relation to the
scale of heterogeneity and to obtain insights into the scattering characteristics of seismic shear waves in the Earth. As
models of heterogeneous media, we selected natural rock
samples of which heterogeneities are well characterized by
analyzing the rock microstructures. Shear waves in the frequency range of 0.25–1 MHz are excited by a shear-type
piezoelectric transducer, and three-component waveforms
are recorded by using a LDV and analyzed.
Y. Fukushima, O. Nishizawa, H. Sato, and M. Ohtake
Experiments
We basically follow the setting of the experiment used
by Nishizawa et al. (1997). A shear-type piezoelectric transducer is attached on a surface of a sample as a source of
elastic waves. A single-cycle sine wave is generated by a
waveform synthesizer and amplified by an external amplifier. The electric signal is transmitted to the transducer to
excite elastic waves. The transducer mainly excites shear
wave (Fukushima, 2000). The vibration is detected by a
LDV; LV-1300 is manufactured by Ono-Sokki Inc., Japan.
The output velocity signal is A/D converted at a 40 nsec
sampling rate with 12-bit resolution and stored on a computer hard disk. Because the signals of ultrasonic waves are
as small as the electric noise level, signal averaging of repeated measurements is necessary to obtain a waveform with
sufficient signal-to-noise ratio (Nishizawa et al., 1997). In
this study, the waveforms averaged over 1000 measurements
are used in the analyses.
The samples are rectangular parallelepiped blocks of
300 mm ⳯ 300 mm ⳯ 80 mm (Fig. 1a). The source transducer is attached at the center of a 300 mm ⳯ 300 mm
surface. The wave field is monitored at the opposite side of
the block as shown in Figure 1. We use the same source for
all the experiments. This source transducer is disk shaped
(a)
Observation Points
300 mm
300 mm
y
z
x
80 mm
(b)
Aperture of the Array
80 mm
7
Transducer
Figure 1. (a) Sketch of the sample and array configuration. A shear-type source is attached on one surface, and an circular array is located at the opposite
side of the sample. We take the Cartesian coordinates
of which the x axis is the polarization direction of the
source and the z axis is normal to the sample surface.
(b) The aperture of the array is determined in such a
way that the array covers radiation angle of 7⬚.
255
Laboratory Study on Scattering Characteristics of Shear Waves in Rock Samples
with 2.5 mm in radius, and it has 1-MHz characteristic frequency. The frequencies of the input sine wave are 0.25, 0.5,
and 1 MHz.
Three components of elastic-wave particle motions are
measured for all the experiments. We follow the method
developed by Nishizawa et al. (1998) that works as follows:
First, we measure the vibrations in two directions, Ⳳ45⬚
with respect to the normal of the surface. Then, we decompose the obtained waveforms into the vertical and transverse
directions. The vibration for the other transverse component
can be obtained in the same manner by performing another
set of two independent measurements in the perpendicular
plane. Thus, a set of four independent measurements is made
to obtain the three-component waveforms at one observation
point. In order to focus the laser beam on the same point for
the four different measurements, the observation surface is
covered by a thin paper with holes of 0.5 mm in diameter,
through which the laser beam is irradiated. Thus, a spatial
resolution smaller than 0.5 mm is guaranteed.
The holes compose a small-aperture circular array: one
point at the center of the circle, and eight points along the
circle. The polarization direction of the shear source is defined as the x direction, and the orthogonal direction parallel
to the surface is defined as the y direction. The direction
perpendicular to the surface is defined as the z direction (Fig.
1a). The aperture of the array is determined in such a way
that the array covers a radiation angle of 7⬚ from the source
(Fig. 1b), which makes the aperture of the array about 20
mm. The observation points are in the nodal direction of the
compressional-wave radiation (Fukushima, 2000). Therefore, little compressional wave is expected to be recorded.
(a) Tamura Gabbro
(b) Oshima Granite
Samples
We use two rock samples: Tamura gabbro and Oshima
granite, as models of heterogeneous media. Figure 2 shows
the microstructures of the rock samples. The two rock samples apparently have the same order of grain sizes of about
1–3 mm. The gabbro consists mainly of two minerals: hornblende (dark) and plagioclase (light). The granite consists
mainly of three minerals: biotite (dark), quartz (gray), and
plagioclase (light). The mineral distributions for both rocks
are assumed isotropic. We also use a steel block as a reference model representing a homogeneous medium.
The gabbro has a very small microcrack density,
whereas the granite contains preferred-oriented thin microcracks in quartz grains (Sano et al., 1992). The size of the
microcracks is restricted by the grain size of quartz, thus the
maximum microcrack length is about 3 mm. Long microcracks can be visually identified, but many invisible microcracks are likely to exist (Hadley, 1976). The plane that is
parallel to most of the aligned microcrack planes is called
the rift plane (Sano et al., 1992). Roughly speaking, Oshima
granite has a transverse isotropy with respect to the rift plane.
To investigate the difference in the scattering characteristics
due to microcrack orientation, experiments were performed
10 mm
Figure 2.
Microstructures of the rock samples
used in this study: (a) Tamura gabbro, (b) Oshima
granite.
on three different configurations of the shear-wave polarization with respect to the rift plane. They are denoted in this
study as OS1, OS2, and OS3 as illustrated in Figure 3. In OS1
and OS2, the source is attached on the plane perpendicular
to the rift plane. The polarization of the source in OS1 is
parallel to the rift plane, and in OS2, it is perpendicular to
the rift plane. In OS3, the source is attached on the plane
parallel to the rift plane. We use the same sample for OS1
and OS2. Since the two granite samples, one for OS1 and
OS2 and the other for OS3, were quarried from the same
location, we can assume that the two samples have the same
heterogeneity and microcrack distribution characteristics.
Velocities and bulk densities for the used samples are
listed in Table 1. Velocities were measured in the present
experiment by using piezoelectric transducers as detectors
because they were more efficient for this purpose. Sano et al.
256
Y. Fukushima, O. Nishizawa, H. Sato, and M. Ohtake
(1992) reported orthorhombic velocity anisotropy in Oshima
granite. However, compressional- and shear-wave velocities
in axial directions of the orthorhombic symmetry are roughly
grouped into two velocities, depending on the propagation
and polarization directions with respect to the rift plane. Our
measured values are consistent with their result. The shearwave velocity for OS1 is larger than that for OS2. This is the
shear-wave splitting in cracked media (Crampin, 1981).
(a) OS1
Statistical Characterization of Rock Heterogeneities
(c) OS3
To investigate the effect of random heterogeneity on
scattering characteristics of seismic waves, it is important to
make statistical analysis on the random fluctuations of velocity or density of the media. Statistical character of random
heterogeneity is described by the autocorrelation function
(ACF) or the power spectral density function (PSDF) (Aki
and Richards, 1980; Sato and Fehler, 1998).
Spetzler et al. (2002) and Sivaji et al. (2001, 2002) investigated the ACF and PSDF of the compressional-wave velocity fluctuations in granitic rocks on the basis of microstructure analysis. In their analysis, the velocity fluctuations
were obtained by substituting the appropriate velocity values
to the distributed mineral grains. The same method is used
here to characterize the rock heterogeneities. We examine
the shear-wave velocity fluctuation on the assumption that
scattering is mainly caused by shear-wave velocity contrasts
between the mineral grains. The procedure of the analysis is
described as follows: (1) obtaining gray-scale images (600
dot per inch) of a sample surface by an image scanner;
(2) identifying the minerals by the darkness; (3) substituting
the shear-wave velocity values appropriate for the mineral
grains; (4) making velocity fluctuation by subtracting the
mean velocity; (5) calculating the PSDFs of 20 onedimensional velocity fluctuation profiles, and obtaining the
averaged PSDF; and (6) the ACF can be obtained by inverse
Fourier transform of the PSDF.
The major mineral ingredients are listed in Table 2 with
the shear-wave velocities of the minerals. Hornblende and
biotite have strong velocity anisotropy. The velocity anisotropy in Oshima granite disappears under high confining
pressures, suggesting no significant preferred orientation of
minerals (Sano et al., 1992). When assigning velocity values
to each mineral grain, we change the velocities of hornblende and biotite by randomly selecting the velocity in the
ranges shown in Table 2. These ranges were calculated from
the Christoffel’s equation (Mavko et al., 1998). For the granite, the analysis is done on the surface perpendicular to the
rift plane, but the result should not depend on the direction
because the mineral distribution can be assumed isotropic
(Sano et al., 1992).
In Figures 4 and 5, the observed ACFs and PSDFs are
shown by solid curves for Tamura gabbro and for Oshima
granite, respectively. The observed ACFs are well approximated by the exponential-type ACF of one-dimensional
form,
(b) OS2
Rift Plane
e
lan
P
Rift
Rift Plane
Figure 3. Three different source configurations in
the granite sample. Shear wave propagates parallel
(OS1, OS2) and perpendicular (OS3) to the rift plane.
(a) OS1; (b) OS2; (c) OS3.
Table 1
List of the Elastic Parameters of the Samples
Used in the Experiment*
ST
GB
OS1
OS2
OS3
Vs
Vp
Bulk Density q(⳯103)
3.12 mm/ls
3.90
2.90
2.74
2.69
5.81 mm/ls
6.91
4.55
4.55
3.79
7.9 kg/m3†
3.0‡
2.7§
2.7§
2.7§
*Shear-wave velocities (Vs) and compressional-wave velocities (Vp) are
measured at a room pressure and a room temperature. Bulk densities are
typical values for the samples.
†National Astronomical Observatory of Japan (1999).
‡Birch (1961).
§Sano et al. (1992).
Table 2
List of the Shear-Wave Velocities for Mineral Composites
Used in the Calculations of ACF
Vs
Tamura gabbro
Hornblende
Plagioclase
3.2–4.2 mm/ls†
3.4‡
Oshima granite
Plagioclase
Quartz
Biotite
3.4‡
4.1‡
1.3–5.1†
*The gabbro is approximated by a composite of hornblende and plagioclase, while the granite was approximated by a composite of plagioclase,
quartz, and biotite. Hornblende and biotite have strong velocity anisotropy,
thus the velocities for them were randomly selected from the range stated
in the table, which were calculated from Christoffel’s equations.
†Calculated from the elastic constants in Bass (1995).
‡Simmons and Wang (1971).
257
Laboratory Study on Scattering Characteristics of Shear Waves in Rock Samples
(a)
–3
R(x) ⳱ e2 exp(ⳮ|x|/a)
Tamura Gabbro
(1)
x 10
ACF
6
where x is the spatial lag, a is the correlation distance, and
e is the standard deviation of the fractional fluctuation of the
shear-wave velocity. The observed ACFs are fitted to equation (1) (broken curves in Figs. 4a, and 5a) by the leastsquares method, and the values of a and e are estimated. The
correlation distances a are estimated as 0.84 mm for the
gabbro and 0.39 mm for the granite. The values for the gabbro is about twice as large as that for the granite, which
suggests a larger-scale heterogeneity for the gabbro. Estimated e values are 8.1% for the gabbro and 17.0% for the
granite. The one-dimensional Fourier transform of equation
(1), namely, the PSDF of the exponential-type random heterogeneity, is given by the following equation:
a = 0.84 mm
ε = 8.1 %
4
2
0
0
10
PSDF [mm]
10
10
10
10
1
(b)
–1
Lag [mm] 2
3
–2
–3
–4
P(k) ⳱
–5
1
10
0
10
1
10
2e2 a
1 Ⳮ k2 a2
(2)
2
10
where k is the wavenumber. The PSDFs calculated with the
estimated a and e are additionally plotted by broken curves
in Figures 4b and 5b. The observed ACF and PSDF generally
agree well with the parameterized functions of equations (1)
and (2), but the fall-off rates of the observed PSDF deviate
from equation (2) to the upper side for large wavenumbers
greater than 10 mmⳮ1 for the both rocks. This result suggests
that the rocks are rich in small-scale heterogeneities (scale
length less than 0.1 mm) compared to the exponential-type
random heterogeneity.
Wavenumber [1/mm]
Figure 4. (a) Observed ACF for the fractional fluctuation of shear-wave velocity for the gabbro (solid
curve) is fitted with an exponential-type ACF (broken
curve) by the least-squares method. (b) Observed
PSDF, the Fourier transform of the ACF, is plotted by
a solid curve. The PSDF for the exponential-type ACF
(broken curve) is also plotted by using the parameters
estimated in (a).
Results
(a)
Hereafter, we denote the experiments as ST, GB, OS1,
OS2, and OS3 combined with the source frequencies in MHz
shown in parentheses. ST and GB stand for steel and gabbro,
respectively, and OS1, OS2, and OS3 stand for the three dif-
Oshima Granite
a = 0.39 mm
ε = 17.0 %
ACF
0.02
0.01
0
0
10
PSDF [mm]
10
10
10
10
(b)
–1
1
Lag [mm] 2
3
–2
–3
–4
–5
1
10
0
10
1
10
Wavenumber [1/mm]
Figure 5.
Same as Figure 4, for the granite.
2
10
ferent experiments of the granite indicated in Figure 3. For
example, ST(0.5) denotes an experiment in the steel block
with 0.5 MHz source frequency.
Figure 6 shows the filtered x-component waveforms observed at the nine observation points for all the experiments.
This figure provides a general view of how the waveforms
differ depending on sample heterogeneity, source frequency,
and the shear-wave polarization with reference to the preferred orientation of microcracks. The bandpass filter we
used is the second-order butterworth type having corner frequencies at two-thirds and four-thirds of the source frequency. In the following analysis, the same filter is applied
to all the waveforms. The prominent phase observed at time
around 30 ls is the direct shear wave. Another prominent
phase is observed at around 80 ls in ST(0.25), ST(0.5), and
GB(0.25). This is the double-reflected shear wave, which is
first reflected from the observation surface, then reflected
from the source surface, and finally observed at the observation surface. We see from Figure 6 that the granite pro-
258
Y. Fukushima, O. Nishizawa, H. Sato, and M. Ohtake
0.5 MHz
1.7 mm/s
1 MHz
2.7 mm/s
2.6 mm/s
ST
0.25 MHz
2.7 mm/s
2.3 mm/s
1.6 mm/s
1.8 mm/s
0.9 mm/s
2.5 mm/s
1.5 mm/s
0.4 mm/s
1.8 mm/s
1.6 mm/s
0.7 mm/s
OS3
OS2
OS1
GB
2 mm/s
0
50
100
Time [µs]
0
50
100
Time [µs]
0
50
100
Time [µs]
Figure 6. Filtered x-component waveforms observed at all the observation points
for all the experiments.
duces more scattered waves compared to the gabbro in the
cases of 0.5- and 1-MHz source frequencies. The excitation
of scattered waves in OS3(1.0) is the most prominent.
Figure 7 shows the filtered three-component waveforms
observed at the center of the circular array for all the experiments. For ST, the amplitudes of the direct shear wave are
smaller in the y and z components compared with that of
the x component, showing very small distortion of the polarized shear wave. On the z component, weak compressional wave is identified at around 15 lsec for ST(0.5) and
ST(1.0). This weak radiation is presumably suggesting a
weak compressional-mode vibration of the transducer. By
employing the source model of Tang et al. (1994), Fukushima (2000) tried to fit the observed amplitudes from the
shear source to the calculated radiation patterns. They found
that the source model of Tang et al. (1994) could explain
the observed amplitudes for frequencies below 1 MHz but
also found that the shear-wave amplitude is smaller than
the calculated amplitude in 1 MHz when they assumed that
the observed and calculated compressional-wave amplitudes
agree. It suggests a difference of sensitivity between the vertical (z) and transverse (x, y) components in this frequency
range. Taking this fact into account, amplitude analyses are
only made on the transverse components.
259
Laboratory Study on Scattering Characteristics of Shear Waves in Rock Samples
ST
GB
x
y
z
OS2
x
y
z
x
y
z
OS3
x
y
z
OS1
0.25 MHz
x
y
z
0
0.5 MHz
1 MHz
1.7 mm/s
2.7 mm/s
2.6 mm/s
1.9 mm/s
1.8 mm/s
1.1 mm/s
1.2 mm/s
0.9 mm/s
0.5 mm/s
1.7 mm/s
1.5 mm/s
0.4 mm/s
1.4 mm/s
1.6 mm/s
0.5 mm/s
50
100
Time [µs]
0
50
100
Time [µs]
0
50
100
Time [µs]
Figure 7. Filtered three-component waveforms observed at the center of the circular
array, for all the experiments.
Excitation of Scattered Waves
We examine the scattered-wave excitation by analyzing
the root mean square (rms) envelopes. Here, “mean” indicates the running time average. We calculated rms envelopes
for each waveform in the following way: (1) calculating
square amplitude; (2) averaging the square amplitude trace
with a moving window of time length 5.0 lsec (MS envelope); and (3) calculating the square root of (2).
Figure 8 shows the space-averaged x-component rms
envelopes for all the experiments. They were calculated in
the way that the MS envelopes of the nine observation points
are first averaged, and then the square root of the outcome
is taken. For the source frequency 0.25 MHz, the envelope
shapes look similar to each other, whereas for the frequency
1 MHz, the direct shear waves are strongly attenuated
through the rock samples (compare the amplitudes with
those for 0.25 MHz), and scattered waves are excited by the
heterogeneities in the rock samples. The amount of the
scattered-wave excitation varies with the rock sample and
the orientation of the granite.
The strength of the excitation of scattered waves can be
described by measuring the envelope width tq, which is defined by the lapse time from the onset of shear wave to the
time when the rms envelope amplitude decreases to the half
of its peak value (see Fig. 8). The envelope width was first
introduced in the study to characterize the random heterogeneities in the crust (Sato, 1989; Obara and Sato, 1995). If
we consider the same source-observation distance, tq depends on the intensity of velocity fluctuation and the characteristic scale length of random heterogeneity as well as the
attenuation factor Qⳮ1 (Sato, 1989). We measured tq for
examining the scattered-wave excitation with respect to the
intensity of velocity fluctuation, scaled wavenumber ka
(wavenumber times the correlation distance), and shearwave polarization with reference to microcrack orientation.
The result looks clearer if we normalize tq by dividing
by the value of ST at the corresponding frequency. Figure 9
shows the normalized tq values (hereafter we use tq for normalized tq) for the rock samples plotted against the scaled
wavenumber ka. The symbols and the error bars indicate the
average and the standard deviations of those obtained from
the nine traces, respectively. For GB(0.25) and GB(0.5) (ka
⬃0.3 and 0.7), the tq values are nearly equal to 1. This means
that the gabbro can be assumed as an equivalent homogeneous medium in this ka range. The value for GB(1.0) (ka
⬃1.4) becomes about 2, which indicates a considerable ex-
260
Y. Fukushima, O. Nishizawa, H. Sato, and M. Ohtake
0.25 MHz
0.5 MHz
1 MHz
0.5
ST
0.5
Spaceaveraged RMS Amplitude [mm/s]
OS2
OS1
GB
0
0.5
0
tq
0
0.5
0.5
0.5
0
0
0.5
0.5
0
0
0.5
0.5
0
0
0.5
0.5
0
0.5
0
0.5
0
OS3
0.5
0
0
50
100
Time [µs]
Figure 8.
15
50
100
Time [µs]
0
0
50
100
Time [µs]
Space-averaged rms envelopes for all the experiments.
pears at larger ka values in the gabbro (ka ⬃1.4) than in the
granite (ka ⬃0.4). The difference may be attributed to the
difference of the intensity of velocity fluctuation between
the two rocks (e ⳱ 8.1% for the gabbro and e ⳱ 17.0% for
the granite), and also to the high microcrack density in the
granite. The tq value for OS3 is significantly larger than those
for OS1 and OS2 at ka ⬃0.8 (1 MHz). It suggests that strong
scattering appears when shear-wave incidence is normal to
the crack plane.
GB
OS1
OS2
OS3
tq
10
0
0
5
Distortion of Shear-Wave Polarization
1
0
0
0.5
1
1.5
ka
Figure 9.
Plot of the normalized envelope width
tq against ka for the rock samples. The symbols and
the error bars indicate the average and the standard
deviations of those obtained from the nine traces, respectively.
citation of scattered waves at this ka. For the three granite
experiments, the normalized envelope widths tq are close to
1 for ka ⬃0.2, indicating an equivalent homogeneous medium. The values increase in the range ka ⬃0.4 and 0.8,
indicating a strong scattering medium. The boundary of
equivalent homogeneous medium and scattering medium ap-
In a homogeneous isotropic medium, compressional and
shear waves are linearly polarized parallel and perpendicular
to the propagating direction respectively, if the wave source
is pure compression or pure shear. However, in a heterogeneous medium, particle motions are distorted and polarizations deviate from linear motions. Wave distortion can be
characterized by tracing the trajectory of particle motions
and determining the shape of the particle-motion spheroids.
Several studies have been performed to investigate the underground scattering properties by analyzing the particle
motions of seismic waves (Sato and Matsumura, 1980; Matsumura, 1981; Nishizawa et al., 1983; Nishimura, 1996). It
is interesting to compare the distortion of shear-wave particle
motions and the characteristics of random heterogeneities.
Figure 10 shows the particle motions of the direct shearwave portion in the x-y plane at the center of the array, for
all the experiments. The time window starts from the onset
261
Laboratory Study on Scattering Characteristics of Shear Waves in Rock Samples
of the shear-wave packet (shown as a dot) and has a length
of three cycles of the source sine wave (12 lsec for 0.25
MHz, 6 lsec for 0.5 MHz, and 3 lsec for 1 MHz). The
polarization direction for GB(0.25) which is slightly different
from the x axis is considered to be originated from a small
misalignment of the source transducer from the x axis, but
it does not seriously affect the results. For ST, the motions
for the three source frequencies are all strongly polarized to
the x axis. For GB, particle motions are gradually distorted
from linear motions with increasing frequency. For the granite experiments, the particle motions deviate from linear motions in all frequency range.
When the distortion of shear-wave polarization occurs,
the ratio of the rms amplitudes of direct shear-wave portion
between the y and x components can be an indicator of waveform distortion. Here we define the following ratio:
v⳱
冪
兺i Ayi2
,
2
A
兺i xi
0.25 MHz
0.5 MHz
1
0
0
–1
–1
–1
0
–2
–2
1
0
2
0.5
0
0
0
–1
–1
–0.5
GB
1
–1
0
1
–1
0
1
–0.5 0 0.5
0.5
0.5
0
0
0
–0.5
–0.2
–0.5
OS2
–0.5 0 0.5
1
0
0
–1
1
–1
–1
1
0
0
–1
0
1
As shown in the previous section, significant excitation
of scattered waves was observed in envelope broadening
phenomena when ka exceeds 1 for the gabbro and 0.4 for
the granite (Fig. 9). This difference suggests that the validity
limit of equivalent homogeneous approach depends on the
fluctuation intensity and/or the existence of cracks. By observing the vertical component from compressional sources,
0.2
–0.5
1
Envelope Broadening
–1 0 1
1
OS3
y [mm/s]
Discussion
0
0
0.5
2.5
–0.2 0 0.2
GB
OS1
OS2
OS3
2
0.2
0
1.5
–0.2
0
1
χ
ST
1 MHz
2
1
OS1
(3)
where Axi and Ayi are the rms amplitudes of the x and y
components at time ti, respectively, and the summation 兺 is
taken for the time window that is three times the cycle of
the source sine wave from the onset. Since the x direction is
the polarization direction of the source, large v indicates
large deviation from a linear polarization of shear wave.
When v ⳱ 1, it means that the same amount of energy is
partitioned into the x and y components.
Figure 11 shows the rms amplitude ratio v against ka
for the rock samples. The symbols and the error bars stand
for the average and the standard deviations of those obtained
from the nine observation points, respectively. The average
v values for ST are: 0.03 (0.25 MHz), 0.05 (0.5 MHz) and
0.13 (1 MHz). We see from Figure 11 that the values for the
rock samples are all larger than the corresponding values of
ST. The smallest value among the rock samples is 0.16
(GB(0.25)). In this case, the particle motions are almost identical to those for ST(0.25) (note the misalignment of the
source that was mentioned above). For the three granite
cases, the v values settle around 1 for ka ⬃0.8, indicating
that the particle motions are completely distorted. The standard deviations show large values in OS2 (ka ⬃0.4) and OS3
(ka ⬃0.8).
–0.2 0 0.2
1
0.2
0
0.5
–0.2
–1
–1
0
1
–1
–1
0
1
–0.2 0 0.2
x [mm/s]
Figure 10.
Particle motions of direct shear-wave
portion in the x-y plane for all the experiments. The
time window is set from the onset of the shear-wave
packet and has a time length three times the cycle of
the source sine wave. Black (solid) dots indicate the
starting points of the particle motions.
0
0
0.5
1
ka
Figure 11.
Plot of the rms amplitude ratio v
against ka for the rock samples. The symbols and the
error bars indicate the average and the standard deviations of those obtained from the nine observation
points, respectively.
1.5
262
Y. Fukushima, O. Nishizawa, H. Sato, and M. Ohtake
Sivaji et al. (2002) found that scattered waves become strong
and waveforms become more complicated in the range ka
⬎ 0.2–0.3. The present study of shear waves in the granite
shows a similar result in terms of the relation between ka
and the appearance of scattering.
Envelope broadening is often found in field observations for the frequency range of 2–30 Hz. It has been explained by a model-based analytical approach on diffraction
and/or multiple forward scattering due to long-wavelength
components of velocity fluctuation (Sato, 1989; Obara and
Sato, 1995; Fehler et al., 2000; Saito et al., 2002). These
theoretical approaches are established in the framework of
the parabolic approximation, which is valid for ka k 1; however, our results show that the envelope broadening happens
even in the ka ⬃1 range in strong scattering media. Large
angle scattering may play a major role in the present case
because this type of scattering is expected instead of multiple
forward scattering when the heterogeneity scale is comparable or smaller than the wavelength.
The large tq in OS3 compared to those in OS1 and OS2
for ka ⬃0.8 can be explained qualitatively by the difference
of scattering patterns of cracks. Neerhoff and van der Hijden
(1984) and van der Hijden and Neerhoff (1984) theoretically
investigated the scattering characteristics of shear waves by
a plane crack of finite length for the kac values 1.5 and larger
(ac is the crack half width). According to their result, backangle scattered energy becomes the largest when shear wave
propagates perpendicular to the crack plane. In our experiments, OS3 corresponds to this case. If we assume that backangle scattering causes reverberation and thus leads to longer
envelope width, our result is consistent with the theory.
Distortion of Shear-Wave Particle Motion
The distortion of shear-wave particle motions in the
both rock samples appears in small ka where the envelope
broadening indicates only weak scattering (Figs. 9 and 11).
This means that shear-wave polarization is more sensitive to
the heterogeneity than the envelope broadening. It might be
mainly due to the preferred-oriented microcracks in the granite cases. The scattering strength indicated by the particlemotion analysis increases with ka, as in the envelope analysis.
As ka becomes large, the averaged v values and the
standard deviations of the values for OS2 and OS3 become
larger than those for OS1. This indicates an influence of polarization direction with respect to the microcrack orientation. Our results show that distortion of shear-wave polarization is stronger in the slow shear wave (OS2, OS3) than in
the fast shear wave (OS1).
The envelope width tq at ka ⬃0.8 in OS3 shows a large
value, indicating strong scattering. This corresponds to the
large standard deviation of v at ka ⬃0.8 in OS3. It suggests
large waveform fluctuation when shear wave propagates perpendicular to the aligned crack plane.
Distortion of shear waves caused by scattering also affects the arrival-time measurements of the fast and slow
shear waves. It was hard to detect the slow shear-wave ar-
rival time from the particle motions in OS2 and OS3 because
of the complication of the particle motions. Effect of scattering on measurements of shear-wave splitting is one of the
important subjects left for future studies.
Conclusion
Scattering of shear waves is one of the major issues
when we work on local earthquakes in frequencies higher
than 1 Hz. The present study was the first experimental study
to investigate the relationships between the scattering characteristics of shear waves and the characteristics of the random heterogeneous media.
Envelope broadening appeared when ka exceeds 0.4 for
the strong-fluctuation medium (e ⳱ 17.0%) with microcracks and 1.4 for the weak-fluctuation medium (e ⳱ 8.1%).
The shear-wave polarization was distorted when ka exceeds
0.2 for the strong-fluctuation medium with microcracks and
0.7 for the weak-fluctuation medium. Appearance of scattered shear waves in a direction different from the source
polarization was more prominent compared with the envelope broadening phenomena.
Envelope broadening of shear waves in the Earth has
been explained by a model that assumes diffraction and/or
multiple forward scattering under the assumption of ka k
1. It was found in the present study that envelope broadening
occurs even when ka ⬃1 in strong scattering media. Our
experiments suggest the importance of large-angle scattering
due to small-scale random heterogeneities and cracks for the
envelope broadening in addition to diffraction effects due to
large-scale heterogeneities.
The effect of crack alignment in scattering was also
found in both envelope broadening and distortion of shearwave polarization. The results suggest that scattering is
prominent when shear wave propagates perpendicular to the
aligned crack plane.
Acknowledgments
We would like to thank Dr. X. Lei of National Institute of Advanced
Industrial Science and Technology for helping around the experimental
apparatus, especially for the software environments. We would also like to
thank Prof. Michael Fehler and the two anonymous reviewers for giving
useful comments to improve the quality of this article.
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Department of Geophysics
Graduate School of Science
Tohoku University
Aoba-ku, Sendai, 980-8578, Japan
(Y.F.)
National Institute of Advanced Industrial Science and Technology
Central 7, 1-1-1 Higashi
Tsukuba, 305-8567, Japan
(O.N.)
Department of Geophysics
Graduate School of Science
Tohoku University
Aoba-ku, Sendai, 980-8578, Japan
(H.S., M.O.)
Manuscript received 1 March 2002.