In a general inhomogeneous medium, finite difference is the most practical method for solving the wave equation. Despite its enormous computational cost, finite-difference schemes provide a comprehensive solution of the wave equation, which includes an accurate representation of amplitude.

In this example, we use the second-order acoustic wave equation for VTI media in -domain, given by equation (27) and therefore need to solve simultaneously

and

where is the forcing function. We use a second-order finite-difference approximation forFigure 7

Figure 7 shows a velocity model in depth (on the top),
and its equivalent mapping in time (bottom).
Figure 8 shows the wavefield at 0.65 s resulting
a source igniting at time 0 s, that corresponds
to the isotropic velocity model in Figure 7. The wavefield is
computed using finite-difference approximations of equation (26).
The velocity model given in
the -domain is the input velocity model in the finite-difference
application. This same velocity model
is used to map the wavefield solution back to depth. The solid
curves in Figure 8 show the solution
of the conventional eikonal solver Vidale (1990)
implemented in the depth domain, and these curves nicely envelope the
wavefield solution. Therefore, computing the wavefield in the
-domain and in the conventional depth domain
are equivalent, regardless of the lateral inhomogeneity. However, the -domain implementation becomes
independent of vertical *P*-wave velocity when .

Figure 8

Is is also important to note that the apparent frequency of the time section is velocity independent, while waves in the depth section have wavelengths very much dependent on velocity.

10/9/1997