On a Wavelet-Based Method for the Numerical Simulation of Wave Propagation

Abstract

A wavelet-based method for the numerical simulation of acoustic and elastic wave propagation is developed. Using a displacement-velocity formulation and treating spatial derivatives with linear operators, the wave equations are rewritten as a system of equations whose evolution in time is controlled by first-order derivatives. The linear operators for spatial derivatives are implemented in wavelet bases using an operator projection technique with nonstandard forms of wavelet transform. Using a semigroup approach, the discretized solution in time can be represented in an explicit recursive form, based on Taylor expansion of exponential functions of operator matrices. The boundary conditions are implemented by augmenting the system of equations with equivalent force terms at the boundaries. The wavelet-based method is applied to the acoustic wave equation with rigid boundary conditions at both ends in 1-D domain and to the elastic wave equation with a traction-free boundary conditions at a free surface in 2-D spatial media. The method can be applied directly to media with plane surfaces, and surface topography can be included with the aid of distortion of the grid describing the properties of the medium. The numerical results are compared with analytic solutions based on the Cagniard technique and show high accuracy. The wavelet-based approach is also demonstrated for complex media including highly varying topography or stochastic heterogeneity with rapid variations in physical parameters. These examples indicate the value of the approach as an accurate and stable tool for the simulation of wave propagation in general complex media.