RTM using effective boundary saving: A staggered grid GPU implementation

Which part of the wavefield should be saved?

To reconstruct the modeled source wavefield in backward steps rather than read the stored history from the disk, one can reuse the same template by exchanging the role of $p^{k+1}$ and $p^{k-1}$ , that is,

$$p^{k-1}=2p^{k}-p^{k+1}+v^2\Delta t^2 \nabla^2 p^{k}.$$ (5)

We conduct the modeling (and the backward propagation in the same way due to template reuse):

$$\begin{split} &for\;ix,iz... \quad p_0(:)=2p_1(:)-p_0(:)+v^... ...=p_0;p_0=p_1;p_1=ptr;// \mathrm{exchange\; pointer} \end{split}$$

where $(:)=[ix,iz]$ , $p_0$ and $p_1$ are $p^{k+1}/p^{k-1}$ and $p^k$ , respectively. When the modeling is finished, only the last two wave snaps ($p^{nt}$ and $p^{nt-1}$ ) as well as the saved boundaries are required to do the backward time recursion.

As you see, RTM begs for an accurate reconstruction before applying the imaging condition using the backward propagated wavefield. The velocity model is typically extended with sponge absorbing boundary condition (ABC) (Cerjan et al., 1985) or PML and its variants (Komatitsch and Martin, 2007) to a larger size. In Figure 1, the original model size $A_1A_2A_3A_4$ is extended to $C_1C_2C_3C_4$ . In between is the artificial boundary ( $C_1C_2C_3C_4\backslash A_1A_2A_3A_4$ ). Actually, the wavefield we intend to reconstruct is not the part in extended artificial boundary $C_1C_2C_3C_4\backslash A_1A_2A_3A_4$ but the part in the original model zone $A_1A_2A_3A_4$ . We can reduce the boundary load further (from whole $C_1C_2C_3C_4\backslash A_1A_2A_3A_4$ to part of it $B_1B_2B_3B_4$ ) depending on the required grids in finite difference scheme, as long as we can maintain the correctness of wavefield in $A_1A_2A_3A_4$ . We do not care about the correctness of the wavefield neither in $A_1A_2A_3A_4$ nor in the effective zone $B_1B_2B_3B_4$ (i.e. the wavefield in $C_1C_2C_3C_4\backslash B_1B_2B_3B_4$ ). Furthermore, we only need to compute the imaging condition in the zone $A_1A_2A_3A_4$ , no concern with the part in $C_1C_2C_3C_4\backslash A_1A_2A_3A_4$ .

fig1
Figure 1. Extend the model size with artificial boundary. $A_1A_2A_3A_4$ indicates the original model size ( $nz\times nx$ ). $C_1C_2C_3C_4$ is the extended model size $(nz+2nb)(nx+2nb)$ . $B_1B_2B_3B_4\backslash A_1A_2A_3A_4$ is the effective boundary area.

RTM using effective boundary saving: A staggered grid GPU implementation