## 2 Theoretical Foundations
### Wave Propagation in a Complex Anisotropic Medium  
Finally, we used the frequency domain method based on M-PML to perform wavefield simulation for a more complicated cross-well model. The cross-well model is shown in Figure 9. The corresponding parameters selected are as follows: computing area, \(0\leq x\leq 108\) m, \(0\leq z\leq 289\) m; spatial step, \(h=\Delta x=\Delta z=0.5\) m; grid parameter, \(578~{}\times~{}216\); the number of M-PML layers, 30; dominant frequency of the source, 130 Hz, the location of the source: \((135~{}\text{m},~{}0~{}\text{m})\); receiver alignment positions: \(x=108\) m, \(0.5\leq z\leq 288.5\) m; receiver interval, 2 m. The cross-well model has eight layers. There are numerical constants in red in Figure 8, which label the layers in the figure. The red constants 1-8 in the figure correspond to layers 1-8 in Table 2, respectively. The asterisk in the figure represents the source. These anisotropic medium parameters are taken from the true sedimentary and are available in the study of [PERSON] [28].  
Figure 8: Viscoelastic anisotropic wavefield of medium 2 calculated by M-PML at (**a**) 0.1222 s, (**b**) 0.1889 s, and (**c**) 0.6 s.  
Figure 9: Multi-layer cross-well model with single-excited source well on the left and multiple-received receiver well on the right, with red constants for medium numbers.  
Figure 10 shows the mono-frequency wavefield snapshots of elastic and viscoelastic anisotropy obtained by the proposed method at 163.2 Hz, 329.8 Hz, and 429.5 Hz. Furthermore, the time-domain snapshots at the time point of 0.0199 s, 0.0299 s, and 0.0399 s computed by the proposed method are shown in Figure 11, respectively. It is easy to see that there exist no artificial edge effects near the truncated boundary in either the elastic or viscoelastic anisotropic wavefield. Then, we give the single-shot seismograms of the cross-well model, which are shown in Figure 12, and we also see that there exist no numerical dispersion or artificial edge effects in either elastic or viscoelastic anisotropic seismograms. Thus, we may conclude that the M-PML for the finite-difference frequency domain is suitable for wavefield simulation in a viscoelastic anisotropic and complicated medium.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Layer Media** & **Vp (m/s)** & **Vs (m/s)** & **Qp** & **Qs** & \(\epsilon\) & \(\delta\) & \(\rho\) **(kg/m3)** \\ \hline Layer 1 & 1875 & 826 & 10 & 10 & 0.225 & 0.100 & 2000 \\ Layer 2 & 2202 & 969 & 10 & 10 & 0.015 & 0.060 & 2250 \\ Layer 3 & 2868 & 1350 & 15 & 15 & 0.970 & \(-\)0.090 & 1860 \\ Layer 4 & 3368 & 1829 & 21 & 18 & 0.110 & \(-\)0.035 & 2500 \\ Layer 5 & 3688 & 2774 & 30 & 19 & 0.081 & 0.057 & 2730 \\ Layer 6 & 3901 & 2682 & 38 & 25 & 0.137 & \(-\)0.012 & 2640 \\ Layer 7 & 4296 & 2471 & 42 & 35 & 0.081 & 0.129 & 2660 \\ Layer 8 & 4529 & 2703 & 50 & 40 & 0.034 & 0.211 & 2520 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Physical parameters used for the cross-well model.  
Figure 10: Mono-frequency horizontal elastic anisotropic wavefield snapshots of different frequencies for the cross-well model at (**a**) 163.2 Hz, (**b**) 329.8 Hz, (**c**) 429.5 Hz, and viscoelastic anisotropic wavefield at (**d**) 163.2 Hz, (**e**) 329.8 Hz, and (**f**) 429.5 Hz.  
Figure 11: Time-domain horizontal elastic anisotropic wavefield snapshots for the cross-well model at (a) 0.02 s, (b) 0.03 s, (c) 0.04 s, and viscoelastic anisotropic wavefield at (d) 0.02 s, (e) 0.03 s, and (f) 0.04 s.  
Figure 12: Time-domain horizontal (**a**) elastic anisotropic and (**b**) viscoelastic anisotropic seismogram corresponding to Figure 11.