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Research Interests
Investigation of crustal heterogeneity from scattered waves
Seismic properties of fault zone
Source properties of underground nuclear explosions
Development of a wavelet-based method for the
numerical simulation of elastic wave propagation
Scattering of elastic waves and numerical modelling
Seismic signal processing using wavelet transform
Investigation of crustal heterogeneity from scattered waves:
under construction!
Seismic properties of fault zone:
under construction!
Source properties of underground nuclear explosions:
under construction!
Development of a wavelet-based method for the
numerical simulation of elastic wave propagation:
Wavelet-based methods in signal processing have been
introduced relatively recently but are already
one of the most useful techniques.
One of the merits of wavelets is their confined response
in both the frequency and time domains and this offers
advantages in the field of numerical analysis.
In the last few years methods have been developed for
solving parabolic partial differential equations (PDEs)
in one space dimension and time.
This approach has been extended to elastic wave propagation
in two spatial dimensions.
In order to adapt the approach developed for a parabolic
PDE scheme to the elastic wave equation which is of
hyperbolic type, we rewrite the elastic wave equation
in a first order PDE system in time in terms of particle
velocities and displacements.
The spatial differentiation terms are regarded as linear
operators which can be represented through a wavelet transform.
The forcing terms in the elastic wave equation appears
as a non-linear term in the system of first order PDEs.
An explicit development in discretised time is made by
using a Taylor's series and a semigroup approach to the
representation of the linear operators using a hierarchy
of wavelets. The representation employs a matrix operator
on a vector composed of displacements and velocities.
For the case of two space dimensions there is a 2x2 matrix
operator for SH waves and a 4x4 matrix for each of 4 different
linear operators in the case of P-SV waves.
The wavelet representation provides high order accuracy spatial
differentiation and can avoid some of the problems
associated with a staggered-grid scheme in the
finite difference method.
Higher order accuracy can be introduced in the wavelet
representation by increasing the number of terms used in
the Taylor expansion with a higher order of spatial differentiation.
The source is most conveniently introduced by surrounding
the source point with a small zone of homogeneous
material for which the governing equations are simplified.
The source radiation is then passed on to the rest of
the computation as a boundary conditions on the perimeter
of the homogeneous zone.
A damping term is including in the governing equations
and is activated near the boundaries of the numerical
domain to provide an absorbing bounding condition.
Successful computations have been carried out for both
P-SV waves and SH waves in
layered and heterogeneous models.
Following examples illustrate the snapshot of the wavefields
by the numerical modelling of elastic wave propagation
in various media.
Ex.1) In 2D homogeneous media:
Ex.2) In 2D heterogeneous media:
Ex.3) In 2D homogeneous media with a topographic free surface:
Ex.4) Poster presented at the 2001 AGU fall meeting:
Scattering theory for elastic waves and numerical modelling:
A quantitative study for elastic waves which
accompanies a numerical modelling and a theory
is not so simple, since scattering of elastic waves
involves wavetype coupling, radiation patterns and
a traveltime variation from interactions among waves.
Some people have confined their scope to the scalar wave case.
However, there are significant difference between scalar wave scattering
and elastic wave scattering and
the use of the scalar wave approximation for the quantitative study for
elastic waves can lead to misleading conclusions.
Before one goes to study `more-earth-like' models
composed of random size of anelastic heterogeneities,
it is necessary to quantify the character of the energy loss
from scattering during wave propagation
in simple stochastic random media, which provides
an approximation to realistic heterogeneity.
Therefore, it is needed to formulate the
theoretical scattering attenuation factor (1/Q)
as a function of the normalized wavenumber (ka)
for elastic waves including traveltime corrections
and compare it with numerical results.
In mildly perturbed 2-D media
(10 % standard deviation of velocity perturbation),
the 1/Q-ka relationship
with traveltime correction based on the single scattering theory
can be formulated by considering only backscattering waves.
We also expect to give an empirical basis for the
theoretical minimum scattering angle for the 3-D scattering problem.
Fig 1) Accuracy test:
Comparison of the accuracy of differentiation between
4th order finite difference method (FDM)
and wavelet-based method:
(a) highly varying input signal which
is corresponding to the variation of physical parameters in
random media and (b) numerical results which show that
4th order FDM exhibits the attenuated
results for the analytic solutions but the wavelet-based method
generates the exact results when
the number of discretization of input signal
is decreased to 64 points.
[Figure]
Fig 2) Stability test:
A) Representation of (a) a pointwise random heterogeneous medium
with a standard deviation of velocity perturbation of 20 %
and (b) a stochastic random heterogeneous medium generated
by von Karman ACF with Hurst number of 0.25,
a correlation distance of 100 m
and a standard deviation of velocity perturbation of 52 %.
[Figure]
B) Time responses of the horizontal components of the displacement
at the free surface receivers for (a) the pointwise random medium
and (b) the stochastic random medium.
[Figure]
Fig 3) Theoretical attenuation curves:
Comparison of theoretical scattering attenuation (1/Q) curves
with the minimum scattering angle (theta_{\min}) of 30 degrees
for scalar waves and elastic waves with various ratios
(gamma=1.17, 1.75, 3.5, 5, 7) of P and S
wave velocities in von Karman random media with the Hurst number
of 0.25. Here the P wave velocity is set to be
constant by 6.74 km/s. The theoretical curves for elastic waves are
highly dependent on the velocity ratio.
[Figure]
Fig 4) Synthetic seismograms:
Synthetic seismograms in several random media.
The dominant frequency of incident waves is 4.5 Hz.
(a) the von Karman media with Hurst number 0.25 and
a=214.5 m, (b) the Gaussian media with a=3399 m,
(c) the exponential media with a=1353.2 m,
(d) the von Karman media with Hurst number 0.05 and
a=3399 m.
[Figure (a)]
[Figure (b)]
[Figure (c)]
[Figure (d)]
Fig 5) Comparisons of theoretical and numerical results:
Scattering attenuation factor 1/Q normalized for the
standard deviation e as a function of normalized wavenumber
ka in the von Karman stochastic random heterogeneous media with
Hurst number (a) 0.05, (b) 0.25 (c) 0.5 (corresponding
to exponential stochastic media) and (d) in the Gaussian stochastic media.
The minimum scattering angle is determined as 60-90 degrees.
[Figure (a)-(b)] |
[Figure (c)-(d)]
Seismic signal processing using wavelet transform:
under construction!
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