A Wavelet-Based Method for Simulation of Seismic Wave Propagation

Seismic wave propagation (e.g., both P-SV and SH in 2-D) can be modeled using wavelets. 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. Absorbing boundary conditions are implemented by including attenuation terms in the formulation of the equations. The traction-free condition at a free surface can be introduced with an equivalent force system. Irregular boundaries can be handled through a remapping of the coordinate system. The method is based on a displacement-velocity scheme which reduces memory requirements by about 30% compared to the use of velocity-stress. The new approach gives excellent agreement with analytic results for simple models including the Rayleigh waves at a free surface. A major strength of the wavelet approach is that the formulation can be employed for highly heterogeneous media and so can be used for complex situations.