*4.1. Half-Space*

It is assumed that the size of the study area is 1410 m ∗ 1410 m ∗ 710 m; the sampling interval in the horizontal and vertical directions is 10 m, taking the main frequency of 20 Hz Ricker wavelet as the source; and the simulation frequency is 10 Hz (all the study areas in the following are consistent with this study area). Consider a half-space defined by interface *z*1 = 350 m, with a point source located at (*xs*, *ys*, *zs*) = (710 m, 710 m, 360 m). Assume that the velocity and density parameters are set as *v*1 = 340 m/s, *ρ*1 = 0.00129 g/cm3, *v*2 = 2000 m/s, *ρ*2 = 1.5 g/cm3. The integration path is divided into three segments, the 

breakpoint *bk* = real(*ω*/*v*\$*i*), and the breakpoint *p* is set as *ki* 31 + (198 ∗ 5 ∗ *<sup>T</sup>*02/200) 2 .

Figure 2 shows the comparison of ESPRIT and GPOF methods to extract DCIM of the lower half-space. *g*(*<sup>m</sup>*2) is defined by (13) and (14); sampling point *t* is defined in (23). From Figure 2a, both methods gain a good fit with the original data, but the ESPRIT method is slightly more accurate, with a fitting error less than 0.008, while GPOF [25] is less than 0.014. Further, the time cost in computing half-space DCIM is also presented in Table 1, which shows ESPRIT also reduces the calculation time. The more layers there are, the more obvious time saving will be seen.

**Figure 2.** Comparison of ESPRIT and GPOF results with original data. (**a**) Fitting results; (**b**) fitting error.

**Table 1.** Computation time comparison with different methods.


Figure 3 shows the symmetrical wavefield of (*<sup>x</sup>*, *ys*, *z*) plane. Comparing the solution of DE\_DCIM and the numerical integration with DE\_WA, the relative errors are shown in (c) and (f). Excellent agreemen<sup>t</sup> is obtained with a relative error of real part less than 2.5 × 10−<sup>3</sup> and of image part less than 5.8 × <sup>10</sup>−4, which assesses the validity of the present method. The calculation time of the half-space with different parameters is shown in Table 2. It can be seen from Table 2 that the proposed DE\_DCIM method reduced the computational time by about 40% when compared to the DE\_WA method, with *ρ*1 = *ρ*2 = 1.0 g/cm3.

**Figure 3.** Comparison of DE\_DCIM and DE\_WA solution in half-space. (**a**) real part of DE\_DCIM solution; (**b**) real part of DE\_ WA solution; (**c**) real part of relative error between (**<sup>a</sup>**,**b**). (**d**) Imaginary part of DE\_DCIM solution; (**e**) imaginary part of DE\_ WA solution; (**f**) imaginary part of relative error between (**d**,**<sup>e</sup>**).


**Table 2.** Computation time comparison with different parameters (computation of (*<sup>x</sup>*, *ys*, *z*) plane).
