For our example we use a fairly sophisticated finite-difference method to time-step forward our test wavefield. Spatial derivatives are calculated using a 2-D pseudo-spectral method; odd-order FFT's are used to obviate the need for special treatment of the spatial Nyquist. We use the accurate time integration method of Tal-Ezer (1986) to avoid losing in the time derivatives the accuracy we purchased by our careful calculation of the space derivatives. A non-staggered grid is employed to center all spatial derivatives at the grid points. This allows a complete set of 2-D elastic constants to be used. (Specifically: ,,,,,, and .) This finite-difference method can take huge time steps; the wavefield can and does travel many gridpoint lengths in one time-update step. The wavefield can also be sampled quite close to the spatial Nyquist while still generating accurate model results.
The model medium consists of one homogeneous isotropic medium over a somewhat faster one (in fact but for the exact placement of the interface the model is just the one depicted in Figure 1). The model grid consists of gridpoints. The upper medium has elastic constants ,,and (isotropic). The lower medium has elastic constants ,,and (isotropic). The slope of the interface is 1/10.
Figure 2 shows the results for the model if we follow the usual practice and fit a best-approximating set of stair steps to the interface. The step every 10 horizontal gridpoints makes very noticeable diffractions in the scattered wavefield. Figure 3 shows the same example using our Schoenburg-Muir interpolation scheme. As advertised, the S&M model nearly eliminates the stair-step scattering.