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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115, B07304, doi:10.1029/2009JB007109, 2010
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Multiple scattering and mode conversion revealed by an active
seismic experiment at Asama volcano, Japan
Mare Yamamoto1 and Haruo Sato1
Received 7 November 2009; revised 19 January 2010; accepted 17 February 2010; published 14 July 2010.
[1] Volcanoes are one of the most heterogeneous fields in the Earth’s crust, and the
understanding of such inhomogeneities may provide us important information on various
volcanic processes. In the previous studies on the seismic wave propagation at active
volcanoes, the diffusion model has been widely used to model the energy transportation and
the contribution of P and S waves and their mode conversion have not been well recognized
partly due to lack of dense observations capturing spatiotemporal pattern of energy
propagation. In this study, we present observational evidence of mode conversions and
multiple scattering at Asama volcano, Japan, revealed by an active seismic experiment with a
dense seismic network. The observed spatial distribution of propagating energy emitted from
the explosive P source shows a wavefront of direct P wave and a pattern exhibiting two
slopes which is indicative of multiple scattering and conversion scattering of two modes
having different scattering coefficients. These facts suggest that the radiative transfer theory
is more preferable than the diffusion theory to explain the observed characteristics. We thus
modeled the energy propagation using the radiative transfer theory assuming multiple
isotropic scattering including conversion scatterings and quantitatively estimated scattering
parameters. Estimated total scattering coefficients for P‐to‐S and S‐to‐S scattering are about
three times larger than that of P‐to‐P scattering, and the mean free path of S wave is
about 1 km for an 8–16 Hz band. These results suggest that the mode conversion and
multiple scattering have an indispensable effect on the seismic energy propagation in
heterogeneous volcanic environments.
Citation: Yamamoto, M., and H. Sato (2010), Multiple scattering and mode conversion revealed by an active seismic
experiment at Asama volcano, Japan, J. Geophys. Res., 115, B07304, doi:10.1029/2009JB007109.
1. Introduction
[2] In active volcanoes, various kinds of seismic signals
including volcanotectonic earthquakes, volcanic tremors, and
long‐period events have been observed [e.g., Chouet, 2003;
Kawakatsu and Yamamoto, 2007]. Volcanotectonic earthquakes are brittle failures occurring inside volcanoes due to
the concentration of stresses which may be caused by fluid
motions such as magma migrations. Volcanic tremors and
long‐period events are considered to be generated by the
dynamic interactions between volcanic fluids and volcanic
edifice. In spite of the difference in their generation
mechanisms, all of these signals thus can be seen as manifestations of passive and active processes of volcanic fluids
under active volcanoes. On the other hand, such interactions
along with more large‐scale tectonic activities make the
volcanic structure very complex, and volcano is considered to
be one of the most heterogeneous parts of the Earth’s crust.
The understanding of such inhomogeneities in volcanoes
1
Department of Geophysics, Graduate School of Science, Tohoku
University, Sendai, Japan.
Copyright 2010 by the American Geophysical Union.
0148‐0227/10/2009JB007109
may thus provide us important information on the various
volcanic processes as well as the volcanic structure.
[3] One way to infer the distribution and movement of
volcanic fluids beneath volcanoes is the use of tomography
for velocity and/or attenuation, which basically relies on the
direct waves. The tomographic method, however, has some
limitations in its resolution partly due to the wavelength of the
waves used in the analyses, and it is rather difficult to image
small‐scale heterogeneities like intruded dykes and fractures
using the method. An alternative approach to image small‐
scale heterogeneities is the use of scattered waves. In contrast
to the deterministic structure obtained by the tomographic
method, the analyses of scattered waves yield stochastic
images of the heterogeneous media which can be characterized by the background velocity (mean velocity) and the
scattering coefficients which represent the scattering power
per unit volume.
[4] So far, various models of scattering process have been
proposed and applied to the observed data. These models
include the single scattering model [e.g., Aki and Chouet,
1975; Sato, 1977], the multiple scattering model, and the
diffusion model. Among these, the diffusion model, which
reflects strong multiple scattering of the seismic energy, has
been widely used in the previous studies on the wave prop-
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Figure 1. Distribution of active sources (stars) and stations
(dots). Names of some stations are annotated. The topography
of Asama volcano is represented by contours at every 200 m.
The inset shows the location of Asama volcano in central
Japan.
agation at active volcanoes [e.g., Wegler and Luhr, 2001;
Wegler, 2003]. These studies indicated that the mean free
path of S waves beneath volcanoes is as short as the order of
hundreds of meters and suggested significant multiple scattering of energy due to the heterogeneous structure of volcanoes. Almost all of these studies, however, assume the
diffusion of a single mode (usually S wave), and the contribution of P and S waves and their conversion have not been
well recognized partly due to lack of dense seismic observations capturing spatiotemporal distribution of propagating energy.
[5] In this paper, we use the data of an active seismic
experiment at Asama volcano, Japan, conducted in 2006 and
analyze the wave propagation in the shallow heterogeneous
structure of the volcano to understand the scattering and
conversion processes due to the heterogeneity. First, we
briefly summarize the outlines of the active experiment
and describe the characteristics of the observed seismograms.
We then apply the multiple isotropic scattering model to
explain the observed spatiotemporal distribution of energy
and estimate the scattering parameters as well as the intrinsic
absorption. Finally, based on the estimation of the mode
conversion and multiple scattering of P and S modes,
we discuss some implications to the modeling and analysis
of seismic wave propagation in heterogeneous volcanic
environments.
2. Active Seismic Experiment at Asama Volcano
[6] Asama volcano is one of the most active volcanoes
in Japan. It is a medium‐sized, andesitic composite stratovolcano in central Japan, located at the junction of the Izu‐
Marianas and NE Japan volcanic arcs. The volcano has
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erupted generally in vulcanian style with a few exceptional
subplinian eruptions in 1108 and 1783, and the last eruption
occurred in 2004. The magma movement associated with the
2004 eruption had been successfully delineated from the
precise relocation of earthquakes and geodetic modeling
[e.g., Takeo et al., 2006], and the detected horizontal dyke
intrusion in the western part of the volcano and the vertical
alignment of the hypocenters enabled us to gain insights into
the magma pathway beneath the volcano.
[7] To delineate the seismic velocity structure around the
summit and an area of dike intrusion during the 2004 eruption
of the volcano, an active seismic experiment with artificial
shots and a dense seismic network was conducted in October
2006 as a part of the national project for the prediction of
volcanic eruptions [Aoki et al., 2009]. The experiment was
conducted with five active sources and 464 temporarily
deployed seismic stations around the volcano (Figure 1).
Most of the stations were deployed linearly to north–south
and east–west, transversing the volcano with a typical station
separation of 100 m. At each station, a seismometer (Mark
Products/Sercel L‐22D, natural frequency 2 Hz) was installed
on the ground surface to record seismic waves excited by
chemical explosions using 250–300 kg dynamite charges
at a depth of around 60 m. The seismometers are mainly
vertical ones except a few three‐component ones, and their
outputs were recorded by portable data loggers (Datamark
LS7000SH) with a sampling rate of 4 ms according to the
predefined schedules. The clock of each data logger was
locked with the GPS signal and kept enough accuracy
throughout the observation.
[8] Figure 2a shows an example of raw seismograms
observed by the east–west measurement line for the S3 shot
which is located at the cross point of north–south and east–
west measurement lines (see Figure 1). The seismograms
from the artificial shots, emitting mainly P wave around
10 Hz, clearly show the arrivals of the direct and refracted
P waves. Their travel times have been used to determine the
velocity structure beneath the volcano by Aoki et al. [2009].
The amplitude of these first arrivals are relatively small, and
the overall seismograms are characterized by spindle‐like
envelopes having small P onsets, missing clear S onsets, and
long‐lasting codas. The coda waves last for a few tens of
seconds, as shown in Figure 2b, and indicate significant
scattering of seismic energies due to the inhomogeneity
beneath the volcano.
3. Characteristics of Propagating Seismic Energy
[9] Figure 3 shows the root‐mean‐square (RMS) envelopes
of seismograms from the different artificial shots at one of the
stations. It is evident that the temporal decay of the coda wave
amplitudes for different shot are almost parallel at long lapse
time (about after 10 s). This characteristic of the coda wave
decay is also observed at other stations and suggests that the
seismic energy is uniformly distributed in the volume surrounding the source and stations. On the other hand, the
characteristics of the early portion of the envelopes differ
between artificial shots as well as between stations. Since this
difference reflects the transition between the transient direct
wave propagation and the quasi static energy diffusion at long
lapse time, we examined the spatiotemporal distribution of
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artificial shots to the original velocity seismograms and
compute the band‐pass‐filtered mean square (MS) envelope.
We then estimated the site amplification factor at each station
by applying the coda normalization method [e.g., Aki, 1980]
to a 5 s long time window starting from 15 s after the shot
origin time. We finally divided the MS envelope at each
station by the estimated site amplification factor and calculated temporal evolution of the mean normalized seismic
energy density at each station every 0.05 s to obtain the
spatiotemporal distribution of energy density. Since we are
focusing on the early portion of the wave propagation, we
used a linear phase symmetric finite impulse response filter
rather than a zero‐phase filter to preserve the characteristic of
the waveform of the wavefront and selected a very short time
window of 0.05 s.
[11] Figure 4 shows an example of the spatial distributions
of the propagating energy at different lapse times. We used
seismograms of the S3 shot observed at stations along measurement lines extended to the north, south, and east directions from the shot location. This is because the use of the
multiple measurement lines surrounding the source enhances
the apparent spatial density of the observation, which enables
us to study the details of the spatiotemporal energy distributions. Because of the rather complicated geological background, including repeated dyke intrusions beneath the west
measurement line between the S3 and S2 shots, we avoided
using the data observed along the line.
[12] The spatial distribution of the energy, which shows a
wide range of energy difference as symbolized by the logarithmic axis, clearly exhibits the accumulation of scattered
energy behind the wavefront of the P wave from the explosive
source. Such accumulation of the scattered energy is a direct
indication of the multiple scattering process and/or the diffusion process due to the medium heterogeneities because the
single scattering process generates the energy concentration
Figure 2. (a) Seismogram section of the raw vertical seismogram observed by east–west measure line for the S3 shot.
The amplitude of each trace is normalized by its maximum
amplitude. (b) Example of the raw seismogram, band‐pass‐
filtered (8–16 Hz) seismogram, and RMS envelope at F09 station which is located at 3390 m south of the S3 shot (see
Figure 1). The envelope is obtained with a smoothing window of 0.2 s. The vertical dashed line indicates the arrival
time of the direct wave.
propagating seismic energy to extract the information on the
wave scattering and mode conversions during the transient
stage.
[10] Although the temporal decay gradient of the coda
wave amplitude is similar between stations as mentioned
above, the absolute values of the amplitude sometimes differ
from station to station. To obtain the spatial distribution of
energy, we thus need to correct the effect of site amplification
due to the local shallow structure and station‐specific factors
such as the coupling between seismometers and the ground.
So, the spatiotemporal distribution of energy density is
obtained by the following procedure: We first applied an 8–
16 Hz band‐pass filter covering the dominant energy of the
Figure 3. Envelopes of 8–16 Hz band‐pass‐filtered seismograms for five shots observed at U20 station. The distance
to each shot is 11880 m (S1), 7370 m (S2), 3140 m (S3),
10880 m (S4), and 13660 m (S5), respectively. To see the overall behavior of the coda waves, the envelopes are obtained with
longer smoothing windows of 0.5 s. Thick lines show regression lines of the envelopes calculated with 7 s long time window starting from 13 s. All these regression lines have
almost the same gradient of RMS / 10(−0.090±0.002)t.
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mainly around the wavefront. On the basis of the similar
energy accumulation in the vicinity of the source, many
previous studies consider the diffusion of the energy and
estimate the diffusivity and scattering strength at active volcanoes [e.g., Matsumoto and Hasegawa, 1991; Wegler and
Luhr, 2001].
[13] Another important characteristic of the observed
spatial energy distribution is that each spatial distribution
exhibits a pattern having two slopes as shown in Figure 4h,
rather than a single smooth curve expected from the diffusion
model (see also Appendix A for examples of the spatial
energy density distribution expected from the diffusion
model). The pattern has a steeper slope behind the wavefront
of the direct P wave and has a moderate one in the vicinity
of the source; the kink of the pattern is situated around the
midpoint between the source and the wavefront of the direct
P wave. This characteristic at the early portion of the energy
propagation tends to disappear after a few seconds, and the
spatial distribution becomes like a single smooth curve at
long lapse time like the one in Figure 4g.
[14] Such spatial distributions of the energy showing two
slopes are reminiscent of the multiple scattering of two modes
having different scattering coefficients. Considering the initial energy generated by the explosion of buried dynamites,
which mainly generates P energy, this pattern can be seen as a
transient process of the mode conversion from P to S mode
and their multiple scattering. In sections 4 and 5, we thus
model the observed spatial energy distribution using the
radiative transfer theory, including the mode conversions,
and estimate the scattering coefficients using the observed
spatial pattern.
4. Spatiotemporal Energy Distribution Modeled
by the Radiative Transfer Theory
Figure 4. (a–g) Observed spatial distribution of energy density at difference lapse times for the S3 shot. The energy density at each station (dots) is evaluated by using a 0.05 s time
window after correcting the site amplification factor which is
obtained by taking the average of those estimated from the records of S4 and S5 shots. Black and gray arrows indicate the
wavefront of direct P wave and the kink of the spatial energy
density distribution. (h) Close‐up view of the spatial distribution at 1.2 s. As indicated by shadows, observed energy density distribution exhibits a pattern having two slopes.
[15] In media having strong random heterogeneity, like the
one beneath active volcanoes, the contribution of the multiple
scattering to the wavefield becomes larger and the scattered
wavefield cannot be approximated by a small disturbance of
waves. To model the multiple scattering process, the radiative
transfer theory which describes energy transportation through
heterogeneous media has been frequently used [e.g., Wu,
1985; Zeng, 1993]. The radiative transfer theory deals
directly with the transport of energy neglecting phase information rather than that of wave, and it is basically applicable
to model scattered energies at both short and long lapse time.
For the case of isotropic scattering in heterogeneous elastic
media, analytical solutions to the radiative transfer equation
had been derived for both one‐mode and two‐mode types
[Zeng, 1993; Sato, 1994], while numerical solutions using the
Monte Carlo method had been used to model more general
cases like the nonisotropic scattering of vector waves [e.g.,
Yoshimoto, 2000; Margerin et al., 2000].
[16] Here, we follow the derivation of Sato [1994] to synthesize the time traces and the spatial distributions of the
energy densities in a 3‐D infinite medium taking into account
the multiple isotropic scattering including conversion scattering. We suppose a spherical radiation of P and S energies,
WP and WS, from a point source, and consider P‐to‐P, P‐to‐S,
S‐to‐P, and S‐to‐S isotropic scatterings by randomly and
uniformly distributed scatterers. The total energy density at
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PS SP
SS
Here, gPP
0 , g0 , g0 , and g0 are the total scattering coefficients
of the four isotropic scattering modes mentioned above, and
a0 and b0 are the P and S wave velocities of the background
medium, respectively. GP and GS are Green’s functions for
energies, which take into account the causality and the effects
of geometrical spreading, attenuation due to scattering, and
intrinsic absorption, and they are written as
PP
PS
1
r
eð0 g0 þ0 g0 þbP Þt
H
ð
t
Þ
t
4r2 0
0
1
r ð0 g0SS þ0 g0SP þbS Þt
e
GS ðx; t Þ ¼
H
ð
t
Þ
t
4r2 0
0
GP ðx; t Þ ¼
ð3Þ
where r = ∣x∣ and bP, bS represent the intrinsic absorptions for
P and S energies. These equations can be solved by means of
the Fourier‐Laplace transform, and the energy densities of
P and S waves, as well as the total energy density, can be
obtained as a function of space and time [e.g., Sato and
Fehler, 1998, chapter 7].
[17] Figure 5 shows an example of the spatial distribution
of the energy density evaluated by the radiative transfer
theory described above. In this example, we assumed that the
source is an isotropic P source and substituted WS = 0 into
equation (2). In the calculation hereafter, we also used the
relationship between the conversion scattering coefficients
Figure 5. Spatial energy distribution obtained by the radiative transfer theory considering the multiple isotropic scattering of P and S energies and their conversion scatterings in an
infinite scattering medium. The source is an isotropic P
energy source (WP = 1, WS = 0) with a Gaussian time function
of 0.04 s. (a) Spatial distribution of the total energy at 1.2 s
and those of P and S energies. The parameters used are
−1
PS PP
SS PP
a0 = 3.0 km s−1, gPP
0 = 1/3.0 km , g0 /g0 = 2.0, g0 /g0 =
3.0, and b = 0. (b) Spatial distribution of the total energy with
different scattering coefficients. The solid curve represents the
energy distribution using the same parameters as Figure 5a,
PP
and the other two curves are obtained with gSS
0 /g0 = 6.0
PS PP
(broken curve) and g0 /g0 = 4.0 (dashed‐dotted curve) while
the other parameters are fixed to those of Figure 5a.
arbitrary point x in the medium is written as a function of
time, t, as
E ðx; t Þ ¼ E P ðx; tÞ þ ES ðx; t Þ
ð1Þ
where EP and ES represent the energy density of P and S wave,
respectively, and they are expressed as sums of coherent
energy densities as
ZþZ1Z Z
E ðx; t Þ ¼ W G ðx; t Þ þ
P
P
P
EP ðx 0 ; t 0 Þ0 g0PP
1
þ E S ðx 0 ; t 0 Þ0 g0SP GP ðx x 0 ; t t 0 Þdt 0 dx 0
ZþZ1Z Z
S 0 0
E ðx ; t Þ0 g0SS
ES ðx; t Þ ¼ W S GS ðx; tÞ þ
1
þEP ðx 0 ; t 0 Þ0 g0PS GS ðx x 0 ; t t 0 Þdt 0 dx 0
ð2Þ
g0SP ¼
1 PS
g ;
202 0
0
0 0
ð4Þ
which is derived from the reciprocal theorem and valid for
any kind of heterogeneity [Aki, 1992]. The spatial distribution of the energy density is, therefore, controlled by six
PS SS
parameters: a0, g 0, gPP
0 , g0 , g0 , and b. In Figure 5, with the
strong heterogeneity beneath active volcanoes in mind, we
used the following scattering coefficients corresponding
to
pffiffiffi
a strong scattering medium: a0 = 3.0 km s−1, g 0 = 3, g0PP =
PP
SS PP
1/3.0 km−1, gPS
0 /g0 = 2.0, and g0 /g0 = 3.0.
[18] As shown in Figure 5a, the spatial distribution of
energy density obtained from the radiative transfer theory
considering the multiple isotropic scattering and mode conversion shows the energy accumulation in the vicinity of the
source and the clear wavefront of P wave. The modeled
energy density distribution also shows two slopes having
different inclinations and a kink between two slopes. The
location of the kink coincides with the wavefront of S wave
even though the source radiates only P wave. It is reasonable
considering that the mode conversion from P to S occurs most
efficiently where the energy density of P wave is the highest
(i.e., in the vicinity of the source) and the conversion back to
P from S occurs less efficiently (see equation (4)) due to the
uniformity of the heterogeneity assumed in the modeling. On
the other hand, the inclinations of the two slopes on both sides
of the kink are controlled by the value of scattering coefficients as shown in Figure 5b. For example, a larger value of
the conversion scattering coefficient, g0PS, yields steeper slope
between the wavefront of direct P having smaller energy and
the kink having larger energy because a larger value of gPS
0
means that the energy of the P wave emitted from the source
rapidly converted to S energy which travels with slower
velocity than P energy, and such P‐to‐S conversion occurs
most efficiently in the vicinity of the source. Similarly, a
larger value of S‐to‐S scattering coefficient, gSS
0 , yields a
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Table 1. Parameters Used and Estimated in the Fitting
−1
a0 (km s )
g0
1/gPP
0 (km)
PP
gPS
0 /g0
PP
gSS
0 /g0
b (s−1)
Search Range
Increment
Best Fit
Possible Range
p2.5–3.9
ffiffiffi
3 (fixed)
2.0–5.0
1.0–5.0
1.0–15.0
0.30–1.40
0.2
‐
0.2
0.2
0.5
0.05
2.7ffiffiffi
p
3
4.2
2.8
3.0
0.65
2.5–2.9
‐
3.2–4.8
1.8–3.8
2.0–7.5
0.50–0.70
bell‐shaped curve in the vicinity of the source due to stronger
multiple scattering of S energy.
5. Estimation of the Scattering Coefficients
[19] The spatiotemporal distribution of energy density
obtained by the radiative transfer theory exhibits similar
characteristics to that observed at Asama volcano. In addition,
each scattering coefficient has a different effect to the spatial
distribution of the energy density as described in section 4.
These facts suggest the possibility to estimate the scattering
coefficient from the observed spatiotemporal distribution of
energy based on the multiple isotropic scattering model.
[20] We used the spatial distributions of energy at six
time windows shown in Figures 4a–4f, and estimated the
medium parameters by minimizing the misfit, 2, between the
observed and model‐predicted energy densities defined by
2 ¼
Nw N
s ð iÞ
X
X
flog Eo ði; jÞ log Em ði; jÞg2
ð5Þ
i¼1 j¼1
where Eo(i, j) and Em(i, j) represent the observed and modeled
energy densities at station j and at time window i. Nw is the
number of time windows (Nw = 6 in this case) and Ns(i) is the
number of stations of which hypocentral distances are less
than a0t(i) where t(i) is the middle time of time window i.
[21] To calculate the modeled energy, Em, we assumed an
isotropic P energy source
with affi Gaussian source time
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
function given by WP(t) = 4= td2 exp(−(2t/td)2) where td =
0.08 s and considered the multiple isotropic scattering of P
and S energies in 3‐D infinite elastic medium ignoring
the
pffiffiffi
effect of the free surface. We further assumed g 0 = 3, and as
PS
a consequence the value of gSP
0 /g0 is fixed as 1/6 from
equation (4). The values of the intrinsic absorption for P and S
energies are also assumed to be the same (bP = bS ≡ b). The
remaining unknown parameters are thus P wave velocity (a0),
PS PP
isotropic scattering coefficients and their ratio (gPP
0 , g0 /g0 ,
SS PP
g0 /g0 ), initial P energy (WP), and intrinsic absorption (b).
Although these parameters are basically free parameters in
the fitting, the range of the scattering parameters was chosen
based on the scattering amplitude of each scattering process
on the basis of the Born approximation [e.g., Sato and Fehler,
1998, chapter 4]. As discussed also in section 6, the scattering
pattern of each scattering mode strongly depends on the
correlation distance, a, of the heterogeneity and wave number, k. In the case of scattering of elastic wave in a medium
having exponential autocorrelation, the ratio of scattering
PP
amplitudes, gSS
0 /g0 , obtained by integrating the angular‐
dependent scattering amplitude over whole solid angle varies
between 4 ∼ 11 for ak < 1 depending on the value of ak,
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whereas the other ratios take more or less constant values of
PP
SP PP
gPS
0 /g0 = 1 ∼ 3 and g0 /g0 ∼ 1. Using these results as a rough
PP
guide, we selected broader parameter range for gSS
0 /g0 than
other parameters (Table 1).
[22] In the estimation of the parameters, as the first step we
ignored the effect of the intrinsic absorption because its
contribution appears just as the constant shift of energy
density on the spatial distribution at fixed time as indicated by
equation (3) and it is rather difficult to resolve the trade‐off
between the intrinsic absorption, b, and the initial energy,
WP, using the short time interval used here. Under these
assumptions, we searched the best parameter set so that the
residual 2 becomes minimum by the grid search method. In
Table 1, along with the best fit parameter set, errors of the
estimated parameters obtained by the bootstrap method with
100 bootstrap samples are also listed.
[23] The spatial distributions of energy density modeled
with the best parameters are shown as the broken lines in
Figure 6 along with the observed ones. Although we only
used the time windows from 1.0 to 1.5 s in the estimation of
model parameters, the spatial energy distributions at longer
lapse times are also shown. It is seen that in each time window
used for the estimation, the observed spatial pattern having
the wavefront and two slopes are quantitatively well modeled
by the multiple isotropic scattering model. Although there are
small differences around the wavefront, they may be partly
due to the assumption in the radiative transfer model that
the wandering effect of the travel time fluctuations of different rays [e.g., Lee and Jokipii, 1975] is not taken into
consideration.
[24] In contrast to the good agreement at short lapse times,
the modeled energy density distributions drawn by the broken
lines show almost constant bias from the observed ones at
longer lapse time. To explain the constant bias, we next took
into consideration of the effect of the intrinsic absorption in
the modeling of the energy density distribution. The effect of
the intrinsic absorption is comparably small at short lapse
time; we thus fixed the values of the scattering coefficients
and the P wave velocity estimated above and search the
optimal value of the intrinsic absorption, b, and the initial
energy, WP.
[25] The obtained value of b using the time windows
starting from 5.0, 10.0, 15.0 sec and the distance range within
6 km is 0.65 sec−1, and the corresponding spatial distribution
of energy density is shown as the solid lines in Figure 6. The
obtained value of b corresponds to the quality factor Q of Q ∼
100 using the relationship b = Q−1w where w is the central
angular frequency of the frequency band (w = 2p × 12 Hz in
this case). Here, it is noticed that the value of b can be constrained only using the longer lapse time windows where the
multiple scattering of S energy dominates as discussed in
section 6 and Appendix A. The obtained value of b therefore
mainly reflects the attenuation for S energy, bS, and it is rather
difficult to constrain the attenuation for P energy, bP, using
the present method. In that sense, the methodology to separate
the effects of the scattering and attenuation described here
resembles the multiple lapse time analysis [e.g., Fehler et al.,
1992; Hoshiba, 1991] in which several time windows are
used to discriminate the scattering and intrinsic attenuation of
a single mode. It is also noted that the absorption parameter, b,
includes the effects of not only the intrinsic absorption but
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the following relationship between the isotropic scattering
coefficients:
lP ¼ 1= g0PP þ g0PS ; lS ¼ 1= g0SS þ g0SP ; tP ¼ lP =0 ; tS ¼ lS =0
ð6Þ
6.
Figure 6. The modeled spatiotemporal distribution of
energy using the best fit parameters. The broken lines represent the energy distributions assuming no intrinsic absorption
(b = 0), whereas the solid lines take into account the effect of
the intrinsic absorption. Black and gray arrows indicate the
wavefronts of P and S waves.
also the window effect corresponding to the one‐way energy
leakage to larger depth. The systematically small energy at
the vicinity of the source shown in Figure 6 is thus partly due
to the upward migration of a high‐velocity anomaly around
the S3 shot detected by Aoki et al. [2009], which causes the
rapid escape of the energy from the volume considered.
[26] The values of isotropic scattering coefficients and
corresponding mean free paths and mean free times derived
from the best fit parameters are summarized in Table 2.
Here, to calculate the mean free paths (lP, lS) and corresponding mean free times (tP, tS) of P and S waves, we used
Discussion
[27] Using the multiple isotropic scattering model including conversion scattering between P and S energies, we could
estimate quantitatively the scattering coefficients beneath
Asama volcano from the spatiotemporal distribution of
energy density observed by the dense seismic network of an
active source experiment. We also demonstrated that we
could estimate the scattering coefficients and the intrinsic
absorption separately by using the different time windows in
a manner slightly different from the multiple lapse time
analysis [e.g., Fehler et al., 1992; Hoshiba, 1991]. The estimated isotropic scattering coefficient in the 8–16 Hz band for
P‐to‐P scattering is 0.24 km−1 and those of S‐to‐S and P‐to‐
S scattering are 3 and 2.8 times larger than that of P‐to‐P
scattering. These scattering coefficients indicate that the
mean free paths of P and S waves are about 1 km in the
frequency band, and they correspond to the mean free times
of about 0.4 s and 0.8 s, respectively.
[28] The obtained value of the mean free path is much
shorter than the typical value of the usual Earth’s crust
[e.g., Sato and Fehler, 1998], and it may reflect the strong
heterogeneity at shallow depth of the active volcano. Very
short mean free path beneath active volcanoes has been also
reported by previous studies. Using the diffusion model for
single mode, Wegler and Luhr [2001] estimated a mean free
path of the order of 0.1 km for S wave at shallow depth of the
Merapi volcano in the frequency band of 4–20 Hz from an
active seismic experiment. Using the multiple lapse time
analysis, Londono [1996] estimated a mean free path of about
10 km in the frequency band of 6–24 Hz at Nevado Del Ruiz
volcano, Colombia, using volcanotectonic earthquakes.
[29] The model used in this study is, of course, more or less
too simplified, and we need to investigate several points
including the effects of the existence of the free surface and
the nonisotropic scattering as well as the three‐dimensional
structure of the volcanic edifice. These subjects would be
addressed by using numerical methods like the finite difference modeling and Monte Carlo simulation in the near future.
Nevertheless, the characterization of the strong scattering
Table 2. Parameters Derived From the Best Fit Parameters for an
8–16 Hz Band
Property
Symbol
Value
Background P wave velocity
Background S wave velocity
Intrinsic absorption
P‐to‐P isotropic scattering coefficient
P‐to‐S isotropic scattering coefficient
S‐to‐P isotropic scattering coefficient
S‐to‐S isotropic scattering coefficient
Mean free path of P wave
Mean free path of S wave
Mean free time of P wave
Mean free time of S wave
a0
b0
Q
gPP
0
gPS
0
gSP
0
gSS
0
lP
lS
tP
tS
2.7 km s−1
1.56 km s−1
100
0.24 km−1
0.67 km−1
0.11 km−1
0.71 km−1
1.10 km
1.21 km
0.41 s
0.78 s
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Figure 7. Energy partitioning modeled by the multiple isotropic scattering model in an infinite scattering
medium using the estimated best fit parameters. The source is an isotropic P energy source (WP = 1, WS = 0)
with a Gaussian time function of 0.04 s. (a) Time traces of the total (solid lines), P (broken lines), and S
(dashed‐dotted lines) energy densities and the ratio between P and S energy densities at two different hypocentral distances. Thin broken lines represents the theoretical energy ratio at equilibrium state. (b) Spatial
distribution of energy densities and their ratio at two fixed times.
phenomena beneath the volcano and the quantitative estimation of the scattering coefficients enable us to have a few
implications important for the interpretations and modelings
of the seismic waves observed at active volcanoes. These
implications include the energy partitioning between P and S
modes, the effect of the scattering and intrinsic attention on
the amplitude of direct waves, and the scattering characteristics of volcanic environments.
6.1. Energy Partitioning
[30] It has been considered that the coda waves consist of
incoherent scattered waves and S energy dominates in the
coda waves. This is because the P‐to‐S conversion scattering occurs more effectively than S‐to‐P scattering (see
equation (4)) and S‐to‐S scattering is generally stronger than
P‐to‐P scattering. On the basis of such considerations, many
previous studies assumed the scattering and diffusion of S
energy to explain the observed energy propagation even
when the source itself is the P energy source like artificial
shots. For example, from the experiment with a marine air
gun source, Matsumoto and Hasegawa [1991] suggested a
large contribution of S waves converted from P waves in
seismograms recorded in a volcanic area. It is thus meaningful to discuss the transition of the energy partition among
P and S modes based on the scattering parameters estimated
in this study.
[31] As described in section 5, the radiative transfer theory
considering the multiple isotropic scattering predicts energy
densities EP and ES individually as functions of space and
time once the scattering parameters are given. Figure 7a
shows an example of time traces of the total energy density
Et (= EP + ES), EP, and ES at two different hypocentral distances. Here we assumed a P energy source and used the
scattering coefficients estimated in this study. The intrinsic
absorption is neglected for simplicity. The modeled energy
densities clearly show that the S energy density becomes
larger than P energy density just after the arrival of the direct
wave which is mainly formed by P energy. It is also shown
that the local energy balance at a fixed observation point
comes close to an equipartition about 1 s after the direct wave
arrival. These results suggest that the incoherent energy in
coda waves near the source is dominated by S energy after a
few seconds regardless of the source radiation in a strongly
homogeneous medium and support the assumption of single‐
mode diffusion process used by Wegler and Luhr [2001] and
other previous studies, although the use of the single‐mode
diffusion model tends to yield stronger scattering strength as
discussed in Appendix A.
[32] It should also be noted that the modeled ES/EP
ratio once exceeds the value of equipartition
in the full space
pffiffiffi
ES/EP ≡ rS/P = 2g 30 (= 10.39 for g 0 = 3) represented by thin
broken lines and that it takes much longer time to steady
this local energy balance down to the equilibrium one. For
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the dependence of the duration on hypocentral distance are
qualitatively consistent with the establishment of equipartition shown in Figure 7a.
[34] Of course, the discussion above is based on the
scattering coefficients obtained under the assumption of the
multiple isotropic scattering model. So, we need to have more
direct evidence of the energy partitioning by independent
ways like the wavefield decomposition by array observations
[e.g., Shapiro et al., 2000] to verify the discussions and
indications obtained here.
Figure 8. Examples of the observed three‐component MS
envelopes at two different hypocentral distances (see Figure 1).
The solid lines represents the vertical component, and the broken and dashed‐dotted lines represent two horizontal components. The energy of each trace is individually normalized by
using the coda level of the component.
example, at a hypocentral distance of 1500 m, it takes about
8.8 s to fall within ±5% of the value of equipartition, and the
duration is about 13 times longer than the mean free time of
S wave (see also Appendix A for different examples). Similar
slow convergence toward equipartition is also reported by
Margerin et al. [2000], in which they numerically studied the
multiple scattering of elastic waves using the Monte Carlo
method. They found that equipartition is established after at
least 15 mean free times at the hypocentral distance of one
mean free path in the Rayleigh regime where scattering
anisotropy is weak. Figure 7b shows the spatial distributions
of P and S energies at fixed time and gives an account for the
phenomenon; that is, the strong multiple scattering of S
energy in the vicinity of the source prevents the S energy to
diffuse outward, whereas multiple scattering of P energy is
less effective and P energy propagates more easily.
[33] The signature of the rapid mode conversion can also
be seen on the observed three‐component seismograms at
Asama volcano. Figure 8 shows examples of the MS envelopes of the three‐component data which are band‐pass filtered between 8 Hz and 16 Hz and individually normalized
using the coda level. As shown in Figure 8, energy densities
of the horizontal components rapidly grow up just after the
arrival of the direct wave and show almost the same temporal
decay as that of the vertical component after a few seconds
from the shot origin time. Furthermore, the duration to reach
the same decay rate seems to become longer as hypocentral
distance increases. Although the energies of P and S modes
and those of vertical and horizontal components obviously
have different physical meanings [e.g., Margerin et al.,
2009], the observed same decay rate at short lapse time and
6.2. Attenuation of the Direct P Waves
[35] The distribution of the intrinsic attenuation is one of
the most important keys to infer the thermal state and composition of Earth’s media as well as the existence of fluid. The
observed attenuation is, however, caused by both the intrinsic
attenuation and the attenuation due to scattering, and it is
needed to separate the contributions of these two different
attenuation mechanisms. One way to separate these two
attenuation mechanisms is the use of the multiple lapse time
analysis of the whole S waves as mentioned before. While
such analyses yield the averaged intrinsic and scattering
attenuation over some volume surrounding the raypath, analyses using the direct wave give information of more localized
volume. So, it is also meaningful to examine the amplitude of
direct waves using the scattering and attenuation parameters
estimated in this study.
[36] Figure 9 shows the distance dependence of the
observed peak amplitudes of the direct wave which are
measured after applying the 8–16 Hz linear phase filter to the
raw velocity seismograms. As shown in Figure 9, the peak
amplitude of the direct wave decays as distance increases, and
the decay of the peak amplitude is almost inversely propor-
Figure 9. Distance dependence of the observed and modeled amplitude of direct waves. Solid circles represent the
peak amplitude of the observed direct P wave in an 8–
16 Hz band, and solid and broken lines represent the peak
amplitude obtained from the modeled energy density assuming the parameters estimated in this study. The modeled peak
amplitude is obtained by taking the square of the calculated
energy density from an isotropic P energy source. The broken
line represents the amplitude decay due to the intrinsic
absorption (b = 0.65), whereas the solid line takes into
account the effects of both the scattering attenuation and
intrinsic absorption.
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Asama volcano. Different parameters quantifying the medium
heterogeneity are the characteristic scale and strength of the
randomness. For that purpose, the correlation distance, a, and
the magnitude of fractional fluctuation, , of the medium have
been widely used as a statistical measure of the randomness
[e.g., Sato and Fehler, 1998, chapter 4]. Of course, the
scattering coefficients and these statistical parameters (a and
) are not independent, and it is possible to relate each other
once the statistical characteristic of the medium is assumed.
[39] Figure 10 shows the ratios of scattering coefficients for
different correlation distance, a, and wave number, k, in case
of an isotropically random medium having exponential
autocorrelation. Even though each scattering process generally exhibits a nonisotropic scattering pattern, we obtained the
values of the ratios shown as solid lines by integrating the
angular‐dependent scattering coefficients over whole solid
PP
angle. That is, the ratio gPS
0 /g0 is calculated from
Figure 10. Plot of the ratios of scattering coefficients
against normalized wave number, ak. The medium is
assumed to have exponential autocorrelation
with correlation
pffiffiffi
length of a. We assumed g 0 = 3, and the fractional fluctuation of P and S wave velocities are assumed to be the same.
Solid lines represent the ratios obtained by integrating the
scattering coefficient of each scattering mode over whole
solid angle, and broken lines represent the ratios of integrated
momentum transfer scattering coefficients.
tional to the square of the distance (a / r−2) instead of the
distance itself (a / r−1) expected from the geometrical
spreading of the spherical body wave. This relatively strong
distance dependence of the amplitude suggests that both
intrinsic and scattering attenuation affect the amplitude of the
direct wave.
[37] To explain the observed distance dependence of the
direct wave amplitude, we calculated the amplitude using the
estimated scattering parameters and the multiple isotropic
scattering model. In the modeling, we first calculated the time
trace of energy density at every hypocentral distance
assuming a Gaussian source pulse of 0.08 s and then measure
the peak energy density of the direct wave. We then obtained
the peak amplitude of the direct wave by taking the square
root of the measured peak energy density. Figure 9 also shows
the distance dependencies of the modeled peak amplitudes
assuming two cases: one only considers the contribution of
the intrinsic attenuation (broken line) and the other takes into
consideration the effect of both the intrinsic and scattering
attenuation (solid line). It is shown that it seems not to be
possible to explain the observed distance dependence by
merely considering the intrinsic attenuation because of the
very short travel time of the direct wave. In contrast, the total
contribution of the intrinsic and scattering attenuation well
capture the distance dependence of inverse square of distance.
Although the use of the isotropic scattering model is not
suitable to model the direct wave, nevertheless, these results
suggest that the scattering attenuation has an indispensable
effect on the modeling of not only coda waves but also direct
waves in heterogeneous volcanic environments.
6.3. Scattering Coefficients and Medium Heterogeneity
[38] In this study, we estimated the isotropic scattering
coefficient as a measure of the medium heterogeneity beneath
g0PS
¼
g0PP
R R
1 2 PS
g ð
4 0
R
R0
1 2 PP
0 g ð
4 0
; Þ sin d d
; Þ sin d d
ð7Þ
where y is the angle from the propagation direction of the
incident wave and z is the angle in the plane perpendicular to
PP
SS PP
the propagation direction. Other ratios gSP
0 /g0 and g0 /g0
are also calculated in a similar manner. Here, we used the
scattering amplitudes calculated by the Born approximation
assuming the same fractional fluctuation both for P and S
PP
waves. As shown by solid lines in Figure 10, the ratio gSS
0 /g0
varies greatly depending on the value of ak, whereas the other
ratios are more or less constant in the range of ak < 1. The
ratios are also dependent on the value of g 0 (not shown here),
but the overall characteristic of the ratios is almost the same.
The regime where scattering process becomes isotropic corPP
PS
responds to the range ak 1, and the ratios gSS
0 /g0 and g0 /
PP
g0 are around 11 and 3 in that range, respectively. When
multiple scattering is dominant, scattering process can be
effectively described by isotropic scattering even if each
scattering process is nonisotropic. In Figure 10, the ratios of
the equivalent isotropic coefficients
(momentum transfer
R R
1 2 scattering coefficient gm ≡ 4
0
0 (1 − cos y)g(y, z) sin
ydydz) calculated like
gmPS
¼
gmPP
R R
1 2 ð1 cos
4 0
R
R0
1 2 ð
0 1 cos
4 0
Þg PS ð ; Þ sin d d
Þg PP ð ; Þ sin d d
ð8Þ
are also shown as broken lines. Although the ratios of
momentum transfer scattering coefficients, gm, show slightly
different values than the ratios of the total scattering coefficients, g0, due to the suppression of the strong forward scattering, overall characteristics of the ratios is kept unchanged.
PP
PS PP
[40] Our estimation of gSS
0 /g0 and g0 /g0 at Asama volcano are 3.0 and 2.8, respectively. These values are almost
the same as those of ak ∼ 1. This seems to somehow
contradict our assumption of isotropic scattering which is
expected in the regime ak 1. To figure out the puzzle,
systematic estimation of the scattering coefficients at different frequency bands would be needed in the future. In addition, to gain details of the scattering process and the medium
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parameters, it may also be necessary to examine the scattering patterns and the ratios between scattering coefficients in
the case of different velocity fluctuations for P and S waves
which is expected in a volcanic environment where volcano‐
specific inhomogeneities such as fluid‐filled cracks exists.
7. Conclusion
[41] From an active seismic experiment with a dense network, we found an observational evidence of mode conversions and multiple scattering in the shallow heterogeneous
structure at Asama volcano, Japan. Spatial distributions of
observed energy density suggest the dominant P‐to‐S conversion scattering at short lapse time and following S‐to‐S
multiple scattering. Assuming the multiple isotropic scattering model including conversions scattering in an infinite
uniformly random medium, we modeled the observed spatiotemporal energy distribution on the basis of the radiative
transfer theory and quantitatively estimated the scattering
coefficients. The mean free paths for P and S waves are
estimated as short as about 1 km for the 8–16 Hz band. These
values of the mean free paths are much shorter than values for
the usual Earth’s crust, and we may say that the scattering
coefficients are useful parameters to quantify the medium
heterogeneity beneath active volcanoes.
[42] From the estimated scattering coefficients, we obtain
the following implications which might be important in the
analyses of seismic signals using the seismic interferometry
[e.g., Wapenaar et al., 2008] and the attenuation tomography.
The local energy partitioning between P and S energies
modeled with the obtained scattering coefficients approaches
the theoretical equipartition in the full space within a few
seconds after the arrival of the direct wave, although it takes
much more time to reach equipartition (about 10–15 times
longer than the mean free time of S wave). The amplitude
of direct wave is controlled by the scattering attenuation
EP ðk; sÞ ¼
ES ðk; sÞ ¼
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Appendix A: Comparison With the Diffusion
Model
[44] When multiple scattering dominates like in this study,
we may expect that the wave propagation becomes like a
diffusion process because the spatial distributions of the
directional distribution of P and S energy densities become
smooth and their angular dependence is very small. In this
section, using the estimated scattering parameters at Asama
volcano, we thus compare the characteristics of the diffusion
model with the multiple isotropic scattering model we used in
this study.
[45] In the case of the dominant multiple scattering, the
propagation of P and S energy densities from an impulsive
isotropic radiation of P and S energies (WP and WS) are
described by a set of diffusion equations as [e.g., Trégourès
and van Tiggelen, 2002]
@
EP DP DEP þ g0PS 0 EP þ bEP g0SP 0 ES ¼ WP ðxÞðt Þ
@t
@
ES DS DES þ g0SP 0 ES þ bES g0PS 0 EP ¼ WS ðxÞðt Þ
@t
ðA1Þ
where we further suppose that the temporal change is small.
Diffusion constants for P and S energy densities, DP and DS,
are related to the isotropic scattering coefficients as
1
0
3 g0PP þ g0PS
1
0
DS ¼
3 g0SS þ g0SP
DP ¼
ðA2Þ
Coupled equations (A1) can be solved by using the Fourier‐
Laplace transform in space and time, and P and S energy
densities are obtained as
s þ b þ DS k 2 þ 0 g0SP WP þ 0 g0SP WS
þ 0 g0SP þ DP k 2 þ DS k 2 s þ b2 þ b0 g0PS þ b0 g0SP þ bDP þ bDS þ 0 DS g0PS þ 0 DP g0SP k 2 þ DP DS k 4
s þ b þ DP k 2 þ 0 g0PS WS þ 0 g0PS WP
þ 0 g0SP þ DP k 2 þ DS k 2 s þ b2 þ b0 g0PS þ b0 g0SP þ bDP þ bDS þ 0 DS g0PS þ 0 DP g0SP k 2 þ DP DS k 4
s2 þ 2b þ 0 g0PS
s2 þ 2b þ 0 g0PS
ðA3Þ
rather than the intrinsic absorption in the frequency range we
analyzed.
[43] The strong scattering characteristics estimated in this
study and the implications above suggest that mode conversion and multiple scattering have an indispensable effect
in the modeling and analysis of seismic wave propagation in
a heterogeneous volcanic environment. Further observation
and analysis using, e.g., the three‐component seismic arrays
may give us more information on the scattering and conversion processes, and the characterization of the stochastic
small‐scale heterogeneity using the multiply scattered wave
along with that of the deterministic structure using the direct
waves undoubtedly help us to infer more details of the existence and transportation of volcanic fluids beneath active
volcanoes.
[46] Figure A1 shows the comparison of spatial distributions of energy densities at different lapse times obtained
by the multiple isotropic scattering model with mode conversions (solid lines) and the diffusion model (broken lines)
for P energy source (Figure A1a) and S energy source
(Figure A1b). The values of the four isotropic scattering
coefficients and P and S wave velocities are the same as the
best fit parameters obtained in this study (see Table 1), and
the intrinsic absorption is neglected for simplicity. The corresponding diffusivities for the diffusion model are DP =
1.0 km2/s and DS = 0.63 km2/s. As clearly seen in Figure A1,
the diffusion solution breaks the causality and shows the
propagation of energy ahead of the wavefront of P energy
which is properly evaluated by the multiple isotropic scattering model and the acausal propagation of the energy results
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Figure A1. Spatial distribution of energy densities at different lapse times modeled by the multiple isotropic scattering (solid lines) and the coupled diffusion model (broken lines) using the estimated best fit parameters. (a) Energy distributions in the case of P energy source. (b) Energy distributions in the case of S energy
source.
in the gentle slope of the energy distribution especially in the
case of P energy source. The result implies that the diffusion
model yields stronger scattering strength than the multiple
isotropic scattering model for the same spatial energy distribution and suggests the adequacy of the use of the multiple
isotropic scattering model in this study.
[47] It is also meaningful to see the difference in the energy
balance between P and S energy densities obtained by the two
models. The spatial distributions of the energy density ratio at
different lapse times calculated with the same parameters are
shown in Figure A2. In the case of P energy source
(Figure A2a), similarly to Figure 7, the spatial distribution of
energy density ratio modeled by the multiple isotropic scattering model (solid lines) shows a steep slope corresponding
to the P‐to‐S conversion scattering behind the wavefront of
P wave, and the local energy density ratio in the vicinity of
the source once exceeds the theoretical ratio of equipartition
in the full space represented by thin dotted lines, and reach
to equipartition. On the other hand, the spatial energy density distribution modeled by the diffusion model (broken
lines) shows more smooth spatial variation and slowly
changes with time because the energy flux is proportional to
the gradient of the spatial distribution of energy density. In
the case of S energy source (Figure A2b), the spatial energy
density distributions obtained by the two models are more
or less similar except the existence of the Gaussian peak
corresponding to the wavefront of S wave. The local energy
density ratio behind the S wavefront shows a relatively flat
spatial variation and monotonically decreases approaching
equipartition.
[48] The local energy density ratio is found to approach the
theoretical equipartition regardless of the energy balance at
the source due to the existence of conversion scatterings,
although it takes a long time to achieve equilibrium. The
temporal changes of the local energy balance at fixed hypocentral distances are shown in Figure A3 for P and S energy
radiations. In each plot of Figure A3, as a comparison, the
energy ratio of total P and S energies in the full space modeled
by the multiple isotropic scattering model, which is determined from the Fourier transform at wave number k = 0, is
also shown as a dashed‐dotted line. In the case of P energy
radiation shown in Figure A3a, the local energy density ratio
obtained by both the multiple scattering model and the diffusion model increases and becomes larger than the theoretical ratio. The ratio then decreases with increasing lapse time.
It takes about 10 s for the local energy density ratios obtained
by these two models to reach within a 5% error of the theoretical ratio, while it takes only about 3 s for the total energy
ratio to reach within the 5% error. As the estimated mean free
time of S wave at the volcano is about 0.7 s, the time interval
needed to steady down to the error range corresponds to about
14 times as large as the mean free time for the local energy
density ratio and about 4 times as large as the mean free time
for the total energy density ratio in the full space. At lapse
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Figure A2. Spatial distribution of energy density ratios at different lapse times modeled by the multiple
isotropic scattering (solid lines) and the coupled diffusion model (broken lines) using the estimated best
fit parameters. Thin broken line represents the theoretical ratio of the equipartition. (a) Energy distributions
in the case of P energy source. (b) Energy distributions in the case of S energy source.
Figure A3. Temporal change of energy density ratios at different hypocentral distances modeled by the
multiple isotropic scattering (solid lines) and the coupled diffusion model (broken lines) using the estimated
best fit parameters. Thin broken line represents the theoretical ratio of the equipartition, and dashed‐dotted
line represents the energy ratio in the full space. (a) Energy distributions in the case of P energy source.
(b) Energy distributions in the case of S energy source.
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times larger than about 7 s the temporal decay of the local
energy density ratios of the two models are almost the same,
and we may say that the coupled diffusion model is a good
approximation of the multiple isotropic scattering model with
conversion scatterings. In the case of S energy radiation
(Figure A3b), the local energy density ratios by the two
models monotonously decrease with lapse time and they
show almost the same temporal decay at lapse times larger
than about 7 s. It again takes about 10 s for the local energy
ratios obtained by the two models to reach within the 5%
error, whereas about 3 s of lapse time is needed for the total
energy ratio to reach the error range. The mean free time of S
wave in short periods is about 25 s in the lithosphere, which is
35 times longer than the mean free time at the volcano we
analyzed. If the scaling is applicable, these results suggest that
it takes about 350 s for the local energy density ratio to reach
the theoretical equipartition.
[49] Acknowledgments. We are grateful to the participants of the
active seismic experiment at Asama volcano. We would especially like
to thank Yosuke Aoki and Minoru Takeo for all the effort they put into the
project. Careful and constructive comments from the Associate Editor and
two anonymous reviewers significantly improved this manuscript. This
study was partly supported by the G‐COE program of Tohoku University
“Global Education and Research Center for Earth and Planetary Dynamics.”
References
Aki, K. (1980), Attenuation of shear‐waves in the lithosphere for frequencies from 0.05 to 25 Hz, Phys. Earth Planet. Inter., 21(1), 50–60,
doi:10.1016/0031-9201(80)90019-9.
Aki, K. (1992), Scattering conversions P to S versus S to P, Bull. Seismol.
Soc. Am., 82(4), 1969–1972.
Aki, K., and B. Chouet (1975), Origin of coda waves: Source, attenuation
and scattering effects, J. Geophys. Res., 80(23), 3322–3342, doi:10.1029/
JB080i023p03322.
Aoki, Y., et al. (2009), P wave velocity structure beneath Asama volcano,
Japan, inferred from active source seismic experiment, J. Volcanol.
Geotherm. Res., 187(3–4), 272–277, doi:10.1016/j.jvolgeores.2009.09.004.
Chouet, B. (2003), Volcano seismology, Pure Appl. Geophys., 160(3–4),
739–788, doi:10.1007/PL00012556.
Fehler, M., M. Hoshiba, H. Sato, and K. Obara (1992), Separation of scattering and intrinsic attenuation for the Kanto‐Tokai region, Japan, using
measurements of S wave energy versus hypocentral distance, Geophys. J.
Int., 108(3), 787–800, doi:10.1111/j.1365-246X.1992.tb03470.x.
Hoshiba, M. (1991), Simulation of multiple‐scattered coda wave excitation
based on the energy conservation law, Phys. Earth Planet. Inter., 67(1–2),
123–136, doi:10.1016/0031-9201(91)90066-Q.
Kawakatsu, H., and M. Yamamoto (2007), Volcano seismology, in Earthquake Seismology, Treatise Geophys., vol. 4, edited by H. Kanamori,
chap. 13, Elsevier, New York.
B07304
Lee, L. C., and J. R. Jokipii (1975), Strong scintillations in astrophysics. II.
A theory of temporal broadening of pulses, Astrophys. J., 201, 532–543.
Londono, J. M. (1996), Temporal change in coda Q at Nevado Del Ruiz
volcano, Colombia, J. Volcanol. Geotherm. Res., 73(1–2), 129–139,
doi:10.1016/0377-0273(95)00084-4.
Margerin, L., M. Campillo, and B. V. Tiggelen (2000), Monte Carlo simulation of multiple scattering of elastic waves, J. Geophys. Res., 105(B4),
7873–7892, doi:10.1029/1999JB900359.
Margerin, L., M. Campillo, B. A. V. Tiggelen, and R. Hennino (2009),
Energy partition of seismic coda waves in layered media: Theory
and application to Pinyon Flats Observatory, Geophys. J. Int., 177(2),
571–585, doi:10.1111/j.1365-246X.2008.04068.x.
Matsumoto, S., and A. Hasegawa (1991), Estimation of relative site effect,
coda Q and scattering strength from the records obtained by a large airgun
experiment, Phys. Earth Planet. Inter., 67(1–2), 95–103, doi:10.1016/
0031-9201(91)90063-N.
Sato, H. (1977), Single isotropic scattering model including wave conversions:
Simple theoretical model of the short period body wave propagations,
J. Phys. Earth, 25, 163–176.
Sato, H. (1994), Multiple isotropic scattering model including P‐S conversions for the seismogram envelope formation, Geophys. J. Int., 117(2),
487–494, doi:10.1111/j.1365-246X.1994.tb03946.x.
Sato, H., and M. C. Fehler (1998), Seismic Wave Propagation and Scattering in the Heterogeneous Earth, Springer, New York.
Shapiro, N. M., M. Campillo, L. Margerin, S. K. Singh, V. Kostoglodov,
and J. Pacheco (2000), The energy partitioning and the diffusive character
of the seismic coda, Bull. Seismol. Soc. Am., 90(3), 655–665, doi:10.1785/
0119990021.
Takeo, M., Y. Aoki, T. Ohminato, and M. Yamamoto (2006), Magma
supply path beneath Mt. Asama volcano, Geophys. Res. Lett., 33,
L15310, doi:10.1029/2006GL026247.
Trégourès, N. P., and B. A. van Tiggelen (2002), Generalized diffusion
equation for multiple scattered elastic waves, Waves in Random Media,
12(1), 21–38, doi:10.1088/0959-7174/12/1/302.
Wapenaar, K., D. Draganov, and J. O. A. Robertsson (2008), Seismic Interferometry: History and Present Status, Geophys. Reprint Ser. 26, Soc. of
Explor. Geophys., Tulsa, Okla.
Wegler, U. (2003), Analysis of multiple scattering at Vesuvius volcano,
Italy, using data of the TomoVes active seismic experiment, J. Volcanol.
Geotherm. Res., 128(1–3), 45–63, doi:10.1016/S0377-0273(03)00246-4.
Wegler, U., and B.‐G. Luhr (2001), Scattering behaviour at Merapi volcano
(Java) revealed from an active seismic experiment, Geophys. J. Int.,
145(3), 579–592, doi:10.1046/j.1365-246x.2001.01390.x.
Wu, R.‐S. (1985), Multiple scattering and energy transfer of seismic
waves ‐ separation of scattering effect from intrinsic attenuation: I.
Theoretical modeling, Geophys. J. R. Astron. Soc., 82(1), 57–80,
doi:10.1111/j.1365-246X.1985.tb05128.x.
Yoshimoto, K. (2000), Monte Carlo simulation of seismogram envelopes
in scattering media, J. Geophys. Res., 105(B3), 6153–6162,
doi:10.1029/1999JB900437.
Zeng, Y. (1993), Theory of scattered P and S wave energy in a random
isotropic scattering medium, Bull. Seismol. Soc. Am., 83(4), 1264–1276.
H. Sato and M. Yamamoto, Department of Geophysics, Graduate School
of Science, Tohoku University, Sendai 980‐8578, Japan. (sato@zisin.
geophys.tohoku.ac.jp; [email protected])
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