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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:

Media 1.1 | SH waves | P-SV waves

Ex.2) In 2D heterogeneous media:

Media 2.1 | SH waves | P-SV waves
Media 2.2 | SH waves | P-SV waves

Ex.3) In 2D homogeneous media with a topographic free surface:

Media 3.1 | P-SV waves

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:

Fig 2) Stability test:

Fig 3) Theoretical attenuation curves:

Fig 4) Synthetic seismograms:

Fig 5) Comparisons of theoretical and numerical results:

Seismic signal processing using wavelet transform:

under construction!

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