Accurate wave equation modeling
, by John T. Etgen
The study of complex wave fields requires precise modeling of the acoustic or elastic
wave equation in two and three dimensions. I solve the elastic wave equation on a discrete
grid using an accurate-time-update pseudospectral method. Spatial derivatives are calculated
using the pseudospectral method; time evolution is based upon an orthogonal polynomial
expansion of the formal solution of the wave equation. The resulting algorithm, like
traditional finite-difference techniques, is a time-stepping solution to the wave equation.
It differs from finite-difference methods because it is accurate to arbitrary precision for
large time step sizes and all spatial frequencies up to Nyquist. Most importantly, numerical
dispersion and numerical anisotropy are eliminated. For models with curved boundaries
between regions of different material parameters, I construct an irregular grid that follows
the boundaries, and a mapping that maps the irregular model grid onto a regularly sampled
computational grid. The accurate-time-update pseudospectral method is used to calculate the
solution of an appropriately modified wave equation on the computatonal grid. The resulting
solution is mapped back onto the original irregular model grid. Mapping the interfaces to
follow grid lines eliminates the errors associated with "stair-step" discretization of
interfaces onto rectangular grids. The accurate-time-update pseudospectral method is useful
whenever precise solutions to the elastic wave equation are needed in heterogeneous media.
STANFORD