Synthesis of Wave Envelopes in 3D Random Media Characterized by a Nonisotropic Gaussian ACF based on

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Synthesis of Wave Envelopes in 3-D Random Media
Characterized by a Nonisotropic Gaussian ACF
based on the Markov Approximation Haruo SATO
Tohoku University, Sendai, JAPAN
Envelope broadening with travel distance
increasing Multiple scattering due to random
velocity inhomogeneity
Seismogram of a local earthquake (Hi-net)
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Kubanza et al. (2006)
Energy partition into the transverse component
as a measure of lithospheric heterogeneity
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Existence of Nonisotropic
Randomness KTB Well-log (Wu et al. 1994)
Geological Survey (Holliger Levander, 1992)
Quasi Laminate Structure of the Lower Crust
(Campillo and Paul, 1992
Nielsen and Thybo, 2003) Scattering Oceanic
Slab Model (Furumura Kennett, 2005)
Statistical Characterization ACF of Random
Velocity Inhomogeneity
Vector wave envelopes in nonisotropic random
media?
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Mathematical Approaches

• Numerical Simulations of Waves (Hong and Wu,
  2005 Saito, 2006)
• Radiative Transfer Theory
• Scalar Waves in Nonisotropic Random Media
  (Margerin, 2006)
• Markov Approximation
• Scalar Waves in Nonisotropic Random Media
  (Saito, 2006)Vector Waves in Isotropic Random
  Media (Korn and Sato, 2005 Sato, 2006, 2007
  Sato and Korn, 2007)

Objective is to derive vector envelopes for a
point source radiation in random media
characterized by nonisotropic Gaussian ACF on the
basis of the Markov approximation.
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Wavelength ltlt Correlation Distance gt No PS
Conversion
P-wave Radiation from a Point Source at the Origin
Potential for Spherically Outgoing P-waves
(1)
Parabolic Wave Equation
Nondim. Velocity Fluctuation
(2)
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Statistical Characterization

• Ensemble of Random Media

Homogeneous and Nonisotropic (Nonisometric)
Gaussian ACF
(3)
Define Two-Frequency Mutual Coherence Function
(TFMCF) on the Transverse Plane as
(z-??)
(4)
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Markov Approximation
º Extension of phase screen method º Causality,
neglect of backscattering º Focus on incoherent
waves only
Impulsive Radiation of P-Wavelet from a Point
Source

• Master Equation for TFMCF for Global Ray Parallel
  to the z Axis

(5)
(6)
Travel Time Fluctuation
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Fourier Transform of TFMCF gives Intensity
Spectral Density (MS Envelopes)
(7)
Initial Condition for Isotropic Radiation from
Point Source
(8)
When the medium is uniform,
.
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TFMCF Solution for Nonisotropic Gaussian ACF
(9)
Nonisotropic in 3D gt Isotropic in 3D (Shishov,
1974)
Nonisotropic in 2D (Saito, 2006) gt Isotropic
in 2D (Fehler et al., 2000)
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Intensity Spectral Density Functions of P-Wavelet
for Point Source Radiation In the Case of
Isotropic Gaussian ACF
(10)
Analytic Solution
(11)
Elliptic Theta function
(12)
Characteristic Time
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Variation of Intensity Spectral Density of
P-Wavelet with Travel Distance Increasing in the
Case of Isotropic Gaussian ACF
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Intensity Spectral Density of P-Wavelet in the
Case of Nonisotropic Gaussian ACF
Envelope broadening becomes longer and the
transverse component amplitude increases when the
transverse correlation distance becomes smaller.
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Nonisotropic Random Media (Laminated Random Media)

• Choose the global ray to the z direction

When the ACF is rotationally symmetric around the
global ray direction, characteristic time becomes
proportional to the square of the aspect ratio,
Envelope broadening is larger for the ray
parallel to the lamination (H) compared with that
for the ray perpendicular to the lamination (V).
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Conclusion

• Vector-wave envelopes are statistically derived
  for a point source radiation of P-wavelet in
  random media characterized by nonisotropic
  Gaussian ACF on the basis of the Markov
  approximation when the wavelength is shorter than
  the correlation distance. (The same for S-wave
  vector envelopes )
• When ACF is rotationally symmetric around the
  global ray, the characteristic time is
  proportional to the square of aspect ratio.
• Envelope broadening and transverse component
  amplitude become larger when the transverse
  correlation distance becomes smaller.
• In laminated random media, envelope broadening is
  larger for the ray along the lamination compared
  with that for the ray perpendicular to the
  lamination.

Future Tasks

• TheoryEnvelope synthesis for the case of oblique
  incidenceNonisotropic random media having a
  power-law spectrumNonspherical source radiation
  (Point shear dislocation)Comparison of coda
  excitation and envelope broadening
• ObservationQuantitative analysis of 3-component
  vector seismogramsAngular spectrum measurement
  by using a seismic array

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