Simulation of Seismic Wave Propagation Using a Wavelet-Based Method

A wavelet-based method is developed for numerical simulation of seismic wave propagation in heterogeneous media. The method is not only accurate in obtaining seismic responses, but also stable during computation by treating spatial derivative terms in equations using a wavelet transform. We expand the method to seismic wave studies in global scale using parallel computing on a supercomputer.

Principal Investigator

Brian Kennett



Facilities Used



Tae-Kyung Hong

RFCD Codes


Significant Achievements, Anticipated Outcomes and Future Work

The wavelet-based method has been introduced for numerical modeling of an elastic wave propagation in two dimensional media problems. The scheme represents spatial differentiation operators through wavelet bases and the resulting second-order differential equations for time are treated by a displacement-velocity formalism and a semigroup approach. The wavelet method for a spatial differentiation is not a grid-based scheme in the physical domain like a Fourier method although sampling is needed at given points. Therefore, we can maintain an accuracy of computation of spatial derivative uniformly throughout a whole domain in contrast to usual grid methods such as finite difference scheme that cumulates numerical errors during computation of derivative terms from grid to grid. For classical 2-D problems, the numerical results exhibit a high accuracy compared to analytic solutions. The method is not only stable during numerical computation, but also has achieved quite good results in various comparisons. Also, the method works well for both simple heterogeneous media and highly perturbed random velocity media. The wavelet-based method is very effective not only in the case of sudden variation of physical parameters at a boundary, but also for linear gradients where physical parameters are changing continuously. Moreover, the method provides stable time responses in a highly perturbed medium and obeys the energy conservation law. We expect the method can be extended to complex stochastic media problems where accurate treatments of spatial derivatives are essential for stable modeling. Also, the method can be used successfully in the seismic quantitative studies such as measurement of energy loss during wave propagation in attenuating media. Future work will concentrate on characterization of the influence of different styles of heterogeneity and the extension of the technique to 3-D where parallel techniques are essential.


Computational Techniques Used

The principle of the wavelet-based method is described as follows. The governing elastic wave equations are transformed to a first-order differential equation system in time with a displacement-velocity formulation. Spatial derivatives are represented with a wavelet expansion using a semigroup approach. The evolution equations in time are derived from a Taylor expansion in terms of wavelet operators. The wavelet representation allows high accuracy for the spatial derivatives. Due to the use of a displacement-velocity scheme we reduce memory requirements by about 30% compared to the use of a velocity-stress scheme. Although the wavelet approach is highly accurate for the heterogeneous media, the method is still expensive in computational cost compared to other methods such as finite difference method or finite element method. With application of parallel computing using several supercomputers with high speed computational performance, we not only reduce the computational time, but also obtain good results from larger domains.