Modeling of the acoustic wave equation with transform methods

Abstract

Numerical methods are described for the simulation of wave phenomena with application to the modeling of seismic data. Two separate topics are studied. The first deals with the solution of the acoustic wave equation. The second topic treats wave phenomena whose direction of propagation is restricted within ±90 degrees from a given axis. In the numerical methods developed here, the wave field is advanced in time by using standard time differencing schemes. On the other hand, expressions including space derivative terms are computed by Fourier transform methods. This approach to computing derivatives minimizes truncation errors. Another benefit of transform methods becomes evident when attempting to restrict propagation to upward moving waves, e.g., to avoid multiple reflections. Constraints imposed on the direction of the wave propagation are accomplished most precisely in the wavenumber domain. The error analysis of the algorithms shows that truncation errors are due mainly to time discretization. Such errors can be limited by the choice of the time step. Perhaps the most significant error phenomenon is related to aliasing. This becomes noticeable when a narrow pulse traverses a region with strong velocity variations. It is shown that aliasing errors can be limited by the choice of the pulse width. The feasibility of these modeling methods is demonstrated on numerical examples.