Absorbing boundary condition for modeling

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

408#408 (163)

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

 409#409 (164)

where 410#410 and 411#411 are the second order central finite-difference operators

412#412

413#413

Given the initial condition p^t-1_x,z and p^t_x,z, we can solve equation () 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<<8128>>min,z}^t+1,p_{x=X<<8130>>max,z}^t+1, p_{x,z=Z<<8132>>min}^t+1,p_{x,z=Z<<8134>>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:

      414#414 (165)

For compatibility with the explicit finite-difference scheme at internal points, we apply the explicit finite-difference scheme for the boundaries using equation () and () and get

      415#415 (166)

where 416#416 is the first order backward finite-difference operator, 417#417 is the first order forward finite-difference opertor:

418#418

and 419#419 is the second order central finite-difference operator:

420#420

Assuming that the wavefield p_x,z^k for 421#421 is known, then we solve the internal equation () 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 422#422can be solved by equation () since the wavefield of the boundary at time t, p^t_{x,Z<<8226>>max} and p^t_{x,Z<<8227>>max-1} are known. Finally, we solve equation () to get the wavefield at the boundary p^t_{x,z=Z<<8229>>max}. Figure 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 (), this absorbing boundary condition is stable.