3. Azimuthal dependence of

1
Simulation of vector-wave envelopes in 3-D random
elastic media for non-spherical radiation source
based on the stochastic ray path method Kaoru
Sawazaki, Haruo Sato, and Takeshi Nishimura
(sawa_at_zisin.geophys.tohoku.ac.jp) Geophysics,
Science, Tohoku University, Japan
S41C-1866
1. Introduction
3. Azimuthal dependence of 3-component RMS
envelopes
ltParameters for the simulationgt
The Markov approximation is very useful for the
synthesis of wave envelopes near the direct wave
arrival in random media. However, this method has
not been precisely studied for an non-isotropic
radiation source. The motivation of our study is
to propose a method to synthesize vector-wave
envelopes for a point shear dislocation source by
using the stochastic ray path method, which
treats seismic ray bending as a stochastic random
process.
e5,a5km,k0.5,Dr2km VP6.0km/s,VS3.46km/s Fr
equency 10Hz ES/EP 23.4 Number of particles
500,000 Intrinsic absorption is not included
2. Simulation method
We describe the seismic energy propagation in a
random inhomogeneous medium as a ray bending
process under the assumption of the Markov
approximation (backward scattering and PS, SP
conversions are neglected. See Sato and Fehler
(1998) for the detail). Convolving the angular
spectrum of seismic ray at the distance r
with the scattering angle distribution , we
obtain the angular spectrum at the distance rDr
as

• The envelope amplitude appears at the Null-axis
  (q0,f0) direction, which is contribution of the
  detoured particles that have experienced
  scatterings.
• The largest P and S wave amplitudes appear at the
  A-axis (q90, f45) and the B-axis (q90, f0)
  directions, respectively, which reflects the
  original radiation pattern.
• The azimuthal dependence is clear at the maximum
  peak arrival, however, it becomes unclear as the
  lapse time increases.

Figure 1. von-Karman type PSDF (Saito et al.,
2005).
(1),
a Correlation distance e RMS value of the
velocity fluctuation k Parameter that controls
the decay of PSDF
(2),
Figure 3. Squared amplitude of the normalized
radiation pattern for P and S waves for a point
shear dislocation source.
(3),
Figure 4. Azimuthal dependence of the 3-comp. RMS
envelopes for P and S waves at the 100km distance.
where P is the power spectral density function
(PSDF) of the velocity fluctuation. We assume the
von-Karman type PSDF (figure 1) which is given by
4. Envelopes for different hypocentral distances
and frequencies
(4).
Solving eq. (1) by the Monte-Carlo method, we
chase the ray bending process.
ltStochastic ray path method (Williamson, 1972)gt

1. A random inhomogeneous medium from a source to a
   receiver is divided into N spherical layers.
2. Energy particles are shot from a point shear
   dislocation source with the weight of the
   radiation pattern, where the particles propagate
   with a constant velocity of VP or VS.
3. The particle is scattered at the layer boundaries
   following the scattering angle distribution given
   by eq. (2), which is treated by the Monte-Carlo
   method.
4. The oscillation direction of the energy particle
   at the receiver is projected into radial (r) and
   transverse (f and q) components.
5. The histogram of the accumulated travel times of
   the particles is calculated, which represents the
   3-component MS envelope.

Figure 5. Square root of 3-comp. sum S wave
envelopes for 10Hz at the different hypocentral
distances.
Figure 6. Square root of 3-comp. sum P and S
wave envelopes for 10Hz and 2Hz at the 100km
distance.

• The decay rate of the maximum amplitude against
  the hypocentral distance differs by azimuth.
• The azimuthal dependence of the maximum amplitude
  becomes more unclear for 10Hz envelopes than for
  2Hz ones.

5. Conclusion
We have synthesized three-component seismogram
envelopes in a random medium for a point shear
dislocation source by using the stochastic ray
path method. The envelopes synthesized show a
clear azimuthal dependence especially at the
maximum peak arrival for short distances
however, such an azimuthal dependence disappears
with travel distance or frequency increasing.
Those envelopes explain the characteristics of
observed seismograms of small earthquakes well in
short periods.
Figure 2. Schematic illustration of the
stochastic ray path method for a point shear
dislocation source.