Historically, the acoustic or elastic wave equation was solved on a
discrete grid using first or second-order differencing in time and
space (Kelly et al., 1976). The accuracy of the finite-difference
method improves by using longer, more accurate spatial differencing
operators, commonly 5-33 points long (Holberg, 1987). Taking this
approach to its limit, the pseudospectral method (Kosloff et al., 1985)
uses multiplication by *ik* in the wavenumber domain to compute spatial
derivatives that are accurate to spatial Nyquist. In addition to the
above improvements in the accuracy of spatial discretizations, Edwards
et al. (1987) describe a new time-update method that eliminates errors
due to time discretization.

These accurate modeling methods are efficient, especially in 3-D, because they allow the wave field and velocity model to be coarsely sampled. Unfortunately, there is also a drawback: if we sample the velocity model too coarsely it becomes difficult to represent curved or dipping boundaries accurately.

One solution is to use finite-element methods, which attack this problem by discretizing the model with elements that follow the boundaries. Fornberg (1988) introduced a mapping method that can be applied to pseudospectral finite-differences to convert curved-grid models to vertically layered models that can be more accurately represented on rectangular grids.

The main attraction of explicit finite-difference methods on rectangular grids is their efficiency and simplicity. Finite-element methods or other distorted-grid methods require complicated setup steps while the simpler rectangular-grid methods just need 2-D or 3-D arrays of values. So from a computational viewpoint, the real importance of the interpolation method we present in this paper is the ability to handle dipping interfaces without relinquishing regularly sampled grids.

1/13/1998