EXAMPLE

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: $\rho$,${\hbox{\rm C}}_{11}$,${\hbox{\rm C}}_{13}$,${\hbox{\rm C}}_{15}$,${\hbox{\rm C}}_{33}$,${\hbox{\rm C}}_{35}$, and ${\hbox{\rm C}}_{55}$.) This finite-difference method can take huge time steps; the wavefield can and does travel many gridpoint lengths in one $\Delta T$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 $243\times 243$ gridpoints. The upper medium has elastic constants $\rho = 1.$,${\hbox{\rm C}}_{11} = 4.41$,and ${\hbox{\rm C}}_{44} = .81$ (isotropic). The lower medium has elastic constants $\rho = 1.25$,${\hbox{\rm C}}_{11} = 8.45$,and ${\hbox{\rm C}}_{44} = 2.45$ (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.