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
where is the forcing function. We use a second-order finite-difference approximation for P-derivatives in equation (28) and a fourth-order approximation for F-derivatives. The solution for elliptically anisotropic media is obtained by setting =0. Since Alkhalifah (1997b) discusses in detail finite-difference application to a fourth-order equation closely resembling this one, no detailed discussion is included here.
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 .
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.