Absorbing boundary condition for modeling

Let us consider the explicit finite-difference scheme for the full wave equation

$$\frac{\partial^2p}{\partial t^2}=c^2 \left(\frac{\partial^2}{\partial x^2}+ \frac{\partial^2}{\partial z^2}\right)p +f,$$ (8)

where p is the wavefield and f is the force. We can extrapolate the wavefield along t using the following explicit finite-difference scheme  

$$\Delta_t^2p_{x,z}^t=c^2 (\Delta t)^2\left(\frac{\Delta_z^2}{... ...a_x^2}{(\Delta x)^2}\right)p_{x,z}^t +(\Delta t)^2 f_{x,z}^t,$$ (9)

where $\Delta_t^2$ and $\Delta_x^2$ are the second order central finite-difference operators

$$\Delta_t^2p_{x,z}^t=p_{x,z}^{t+1}+p_{x,z}^{t-1}-2p_{x,z}^{t},$$

$$\Delta_x^2p_{x,z}^t=p_{x+1,z}^t+p_{x-1,z}^t-2p_{x,z}^t.$$

Given the initial condition p^t-1_x,z and p^t_x,z, we can solve equation (9) to get the wavefield at time t+1, p^t+1_x,z from the wavefield at time t-1, p^t-1_x,z and the wavefield at time t,p^t_x,z, except for the wavefield at the boundaries p_{x=X<<177>>min,z}^t+1,p_{x=X<<179>>max,z}^t+1, p_{x,z=Z<<181>>min}^t+1,p_{x,z=Z<<183>>max}^t+1.

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:

$$\begin{eqnarray} \frac{\partial p}{\partial z}=-\frac{1}{c}\frac{\partial p}{\pa... ...} {\partial t^2}\right)q= \alpha \frac{\partial^2}{\partial x^2}p.\end{eqnarray}$$ (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

$$\begin{eqnarray} \frac{1}{2}\Delta_z^-\left(p^{t+1}_{x,Z_{max}}+p^{t}_{x,Z_{max}... ...t)^2}{(\Delta x)^2}\Delta_x^2 (p^t_{x,Z_{max}}+p^t_{x,Z_{max}-1}).\end{eqnarray}$$ (12) (13)

where $\Delta^-$ is the first order backward finite-difference operator, $\Delta^+$ is the first order forward finite-difference opertor:

$$\Delta^-_zp^t_{x,z}=p^t_{x,z}-p^t_{x,z-1}, \ \ \ \Delta^+_tp^t_{x,z}=p^{t+1}_{x,z}-p^t_{x,z},$$

and $\Delta^2$ is the second order central finite-difference operator:

$$\Delta^2_tq^t_{x,z}=q^{t+1}_{x,z}-2q^t_{x,z}+q^{t-1}_{x,z}, \ \ \ \Delta^2_{x}q^t_{x,z}=q^t_{x+1,z}-2q^t_{x,z}+q^t_{x-1,z}.$$

Assuming that the wavefield p_x,z^k for $k\leq t$ 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 $q_{x,Z_{max}-\frac{1}{2}}^{t+1}$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=Zmax

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.