*3.3. Undulating Interface Model*

The SGFD model will cause false diffraction due to the stepped grid when dealing with undulating interfaces. These diffraction waves will affect the accuracy of elastic wave modeling. The GFDM can simulate an elastic wave based on nodes adapting to the change of an undulating interface, so there will be no stepped grid diffraction. When the undulating interface model, as shown in Figure 10, was used, the velocity and density of the upper layer of the model were 4000 m/s, 2300 m/s, and 2.4 g/cm3, respectively, those of the middle layer were 5000 m/s, 2800 m/s, and 2.5 g/cm3, and those of the lower layer were 6000 m/s, 3500 m/s, 2.6 g/cm3. The time sampling interval was 0.5 ms, and the total computation time was 1.0 s. The source was located at (1000 m, 0 m), and the dominant frequency of the Ricker wavelet was 20 Hz. When the 4th-order 21 nodes GFDM was used, the proposed node generation algorithm was applied to discretize the model, with the node radius at 10 m, 12 m, and 15 m for the first, second, and third layers, respectively. For comparison, the SGFD model was used based on regular grids with a grid spacing of 10 m. The local grid distribution of the SGFD model is shown in Figure 11a, in which the stepped grid can be seen clearly, while the local node distribution of the GFDM is shown in Figure 11b, in which the nodes are directly and accurately distributed on the interface. It can be seen from the seismic records depicted in Figure 12a,b that there are a lot of diffracted waves behind the primary reflected wave when modeled by SGFD, while diffraction waves are practically non-existent in the GFDM record (Figure 12c,d). The results demonstrate that the forward modeling using GFDM avoids the effect of stepped grid diffraction, and is suitable for the forward modeling of a formation with undulating interfaces.

**Figure 10.** Velocity model of undulating interface: (**a**) longitudinal wave velocity; (**b**) shear wave velocity; (**c**) density.

**Figure 11.** Local grid (node) distribution: (**a**) staggered grid finite difference method, (**b**) generalized finite difference method.

**Figure 12.** Shot record of undulating interface model: (**a**) SGFD X component; (**b**) SGFD Z component; (**c**) 4th-order GFDM X component; (**d**) 4th-order GFDM Z component.
