RTM using effective boundary saving: A staggered grid GPU implementation

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## Effective boundary for regular grid finite difference

Assume -th order finite difference scheme is applied. The Laplacian operator is specified by

 (6)

where is given by Table 1, see a detailed derivation in Fornberg (1988). The Laplacian operator has and with same finite difference structure. For dimension only, the second derivative of order requires at least points in the boundary zone, as illustrated by Figure 2. In 2-D case, the required boundary zone has been plotted in Figure 3a. Note that four corners in in Figure 1 are not needed. This is exactly the boundary saving scheme proposed by Dussaud et al. (2008).

 -4 -3 -2 -1 0 1 2 3 4 1 -2 1 -1/12 4/3 -5/2 4/3 -1/12 1/90 -3/20 3/2 -49/18 3/2 -3/20 1/90 -1/560 8/315 -1/5 8/5 -205/72 8/5 -1/5 8/315 -1/560

Keep in mind that we only need to guarantee the correctness of the wavefield in the original model zone . However, the saved wavefield in is also correct. Is it possible to further shrink it to reduce number of points for saving? The answer is true. Our solution is: saving the inner layers on each side neighboring the boundary , as shown in Figure 3b. We call it the effective boundary for regular finite difference scheme.

After steps of forward modeling, we begin our backward propagation with the last 2 wavefield snap and and saved effective boundaries in . At that moment, the wavefield is correct for every grid point. (Of course, the correctness of the wavefield in is guaranteed.) At time , we assume the wavefield in is correct. One step of backward propagation means is shrunk to . In other words, the wavefield in is correctly reconstructed. Then we load the saved effective boundary of time to overwrite the area . Again, all points of the wavefield in are correct. We repeat this overwriting and computing process from one time step to another ( ), in reverse time order. The wavefield in the boundary may be incorrect because the points here are neither saved nor correctly reconstructed from the previous step.

fig2
Figure 2.
1-D schematic plot of required points in regular grid for boundary saving. Computing the laplacian needs points in the extended boundary zone, the rest points in the inner model grid. points is required for boundary saving.

fig3
Figure 3.
A 2-D sketch of required points for boundary saving for regular grid finite difference: (a) The scheme proposed by Dussaud et al. (2008) (red zone). (b) Proposed effective boundary saving scheme (gray zone).

 RTM using effective boundary saving: A staggered grid GPU implementation

Next: Effective boundary for staggered Up: Effective boundary saving Previous: Which part of the

2021-08-31