Let us consider the explicit finite-difference scheme for the full wave equation
(8) |
(9) |
Let us consider the wavefield on the boundary z=Z_{max}. There are only outgoing waves at z=Z_{max}, so the wavefield satisfies the downgoing wave equation, for which we can write its approximate equations:
(10) | ||
(11) |
For compatibility with the explicit finite-difference scheme at internal points, we apply the explicit finite-difference scheme for the boundaries using equation (10) and (11) and get
(12) | ||
(13) |
Assuming that the wavefield p_{x,z}^{k} for is known, then we solve the internal equation (9) to get the wavefield for the internal points X_{min}<x<X_{max}, Z_{min}<z<Z_{max} at time t+1, p_{x,z}^{t+1} first. Then, the auxiliary wavefield can be solved by equation (13) since the wavefield of the boundary at time t, p^{t}_{x,Z<<275>>max} and p^{t}_{x,Z<<276>>max-1} are known. Finally, we solve equation (12) to get the wavefield at the boundary p^{t}_{x,z=Z<<278>>max}. Figure 1 illustrates how the boundary conditions are solved.
boundary
Figure 1 solution at the boundary z=Z_{max} |
The method of solving the wavefield at the other three boundaries z=Z_{min}, x=X_{min}, and x=X_{max}, is similar to that of boundary z=Z_{max}. The only difference is that the boundary condition equation is an upgoing wave equation at z=Z_{min}, leftgoing wave equation at x=X_{min}, and right-going wave equation at x=X_{max}.
According to Zhang and Wei (1998), this absorbing boundary condition is stable.