*4.2. Three-Layer Model*

Consider a three-layer structure defined by two interfaces placed at *z*1 = 200 m, *z*2 = 500 m, with a point source located at (*xs*, *ys*, *zs*) = (710 m, 710 m, 200 m). Assume that the velocity and density parameters are set as *v*1 = 1000 m/s, *ρ*1 = 1.5 g/cm3, *v*2 = 2000 m/s, *ρ*2 = 2.0 g/cm3, *v*3 = 3000 m/s, *ρ*3 = 3.0 g/cm3. The method in this paper is used to solve the layered model, and the symmetrical wavefield, as shown in Figure 4, is obtained. It can be seen from (a) and (d) that in the first layer, there are only up-going wave fields; in the third layer, only down-going wave fields; and in the second layer, there are upward and downward wave fields, which are mixed. Comparing with the FEM [24], the results are consistent in shape, and there are some numerical differences. The relative error of the real part is less than 0.09, and the imaginary part is less than 0.04. Figure 4 also indicates that it is correct to calculate the layered space wave field according to Formulas (11) and (12).

**Figure 4.** Comparison of the wavefield of 3-layer model (DE\_DCIM method and FEM method). (**a**) Real part of DE\_DCIM solution; (**b**) real part of FEM solution; (**c**) real part of relative error between (**<sup>a</sup>**,**b**). (**d**) Imaginary part of DE\_DCIM solution; (**e**) imaginary part of FEM solution; (**f**) imaginary part of relative error between (**d**,**<sup>e</sup>**).

To show the advantage of DE\_DCIM over DCIM for accuracy, consider the three-layer structure defined above with a point source located at (*xs*, *ys*, *zs*) = (0 m, 710 m, 200 m). Figure 5 compares the solutions of DE\_WA, DE\_DCIM, and DCIM in line (*<sup>x</sup>*, *ys*, *zs*). The three-level DCIM with surface wave extraction [26] is adopted. It can be observed in the figure that the DE\_DCIM result is more accurate than the three-level DCIM for about two orders of magnitude when both compared to DE\_WA, especially when the distance between source and field point is large. As discussed in reference [27], for multilayer media, it is very difficult to find surface wave poles, and the inaccurate extraction of the surface wave will bring unpredictable errors to the results.

**Figure 5.** The results of log10|*G*| and relative errors in line (*<sup>x</sup>*, *ys*, *zs*) of 3-layer model.
