| |

| |

A wavelet-based method is developed for the numerical modelling of acoustic and elastic wave propagation. Several techniques are implemented together to develop the method. Using a displacement-velocity formulation and treating spatial derivatives with linear operators based on wavelet transform, the wave equations are rewritten as a system of equations whose evolution in time is controlled by first-order derivatives. The linear operators for the spatial derivatives are treated in wavelet bases by projection with a wavelet transform. The discretized solution in time can then be represented in an explicit recursive form using a semigroup approach. Absorbing boundary conditions are considered implicitly by including attenuation terms in the governing equations, and the traction-free boundary conditions can be implemented by augmenting the system of equations with equivalent force terms at the boundaries.

This wavelet-based method is applied to acoustic,

The wavelet approach is then expanded to models with localised heterogeneity, with significant contrasts with their surroundings. We consider zones with both lowered wavespeed such as a fault gouge zone and elevated wavespeeds as in a subduction zone. In each of these situations the source lies within the heterogeneity. The representation of the source has therefore been adapted to work directly in a heterogeneous environment, rather than using a locally homogeneous zone around the source. This extension also allows the wavelet method to be used with a wider variety of sources, e.g., propagating sources. For the fault zone we consider both point and propagating sources through a moment tensor representation, and reveal significant trapped waves along the gouge zone as well as permanent displacements. For subduction zones a variety of effects are produced depending on the depth and position of the source relative to the subducting slab. A variety of secondary waves, such as reflected and interface waves, can be produced in wavetrains at regional distances and tend to be more important for greater source depth.

The high stability and accuracy of the wavelet-based method in highly perturbed media allows this approach to be exploited for the investigation of seismic-wave scattering in several stochastic (Gaussian, exponential, von Karman) random media. The scattering attenuation depends on the correlation distance of the random heterogeneities, the velocity ratio of

The characteristics of scattering attenuation patterns in elastic waves are investigated using the theoretical expressions for

From complementary studies on scatterings of acoustic and

Finally, elastic waves are modelled in media with randomly distributed fluid-filled circular cavities, which is a challenging problem for numerical techniques. The energy dissipation of the primary waves is proportional to the scale of the cavities. Theoretical attenuation variations for media with circular cavities, which may be filled with any materials (e.g., vacuum, fluid, elastic materials), have been formulated and they are compared with the numerical results for the media with fluid-filled cavities. The numerically estimated attenuation rates agree well with the theoretical variation. The attenuation rates increase linearly with normalized wavenumber, unlike those in stochastic random media that display a parabolic trend for the normalized wavenumber. Also, the normalized attenuation rates are identical between those measured from media with a same normalized wavenumber (i.e., same radius of cavities) even if the number densities (number of cavities per area in a medium) are different. It appears that random heterogeneities in a specific region can be described properly with combined use of stochastic random heterogeneities and random heterogeneities with high impedance by considering the scattering attenuation patterns.

[full text, PDF](36446k)| [full text, Gzipped PS](9511k)