# Modeling GPR Wave Propagation in Complex Underground Structures Using Conformal ADI-FDTD Algorithm
## 3 Numerical Simulations  
### Two-Layer Medium Model with Circular Cavity  
As shown in Figure 5, the model area was set to 2.0 m \(\times\) 2.0 m. The upper layer of the model was a 0.2 m air layer, and the lower layer was a 1.8 m clay layer. The main frequency of the excitation source was a 1 GHz Ricker wavelet, which was located at a 0.2 m depth in the model, and the transmitter and receiver were 0.1 m apart. The relative dielectric constant of the clay layer was 12, and the conductivity was 2 mS/m. A circular cavity with a diameter of 0.1 m was set in the clay layer. The relative dielectric constant and conductivity were 30 and 0 mS/m, respectively. The relative magnetic conductivity of all the material was 1. We set the spatial step size to 0.005 m, the time step size to 0.01 ns, and the thickness of the UPML to 0.01 m.  
The time step of the conventional FDTD algorithm needs to satisfy the CFL stability condition. However, the ADI-FDTD algorithm is unconditionally stable and can save computation time by choosing a larger time step. Figure 6 shows a comparison of the \(E_{z}\) field distributions of the conformal ADI-FDTD with different CFLN, the non-conformal FDTD, and the non-conformal ADI-FDTD. The CFLN is the ratio of the time step of the ADI-FDTD algorithm to that of the FDTD algorithm. It can be seen that the results of the non-conformal FDTD, the non-conformal ADI FDTD algorithm, and the conformal ADI FDTD algorithm have good consistency in different time steps. Table 1 provides the calculation time of the different algorithms when simulating the circular cavity model, respectively. When CFLN is 5, compared with the non-conformal FDTD algorithm, the time step of the conformal ADI-FDTD algorithm is increased to 5 times, and the number of iterations is reduced to 1/5. At this point, the computation time of the conformal ADI-FDTD algorithm is 961 s and that of the non-conformal FDTD algorithm is 1849 s, saving about 48% of the time. Figure 7 shows the GPR profiles obtained by the non-conformal FDTD method, non-conformal ADI-FDTD method, and the conformal ADI-FDTD method. We indicate multiple reflections within the cavity with red boxes. As can be seen from Figure 7, the images of the non-conformal FDTD method and the non-conformal ADI-FDTD method are consistent. The conformal ADI-FDTD method has fewer multiple reflections than the non-conformal ADI-FDTD method. At the same time, the conformal ADI-FDTD method is closer to the conformal FDTD method than the non-conformal ADI-FDTD method. This indicates that the conformal ADI-FDTD method can effectively reduce the false reflection waves caused by conventional ladder approximation when simulating an underground circular cavity. Table 1 and Figure 7 show that the conformal ADI-FDTD algorithm can greatly improve the computational efficiency and can also effectively improve the computational accuracy.  
Figure 5: Schematic diagram of the circular cavity model.  
Figure 6: Comparison of \(E_{z}\) field distribution between ADI\(-\)FDTD and FDTD simulation.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Memory (Mb)** & \(\Delta t\) **(ns)** & **Iterations** & **Time (s)** \\ \hline Non-conformal FDTD & 10.16 & 0.01 & 5000 & 1849 \\ Conformal FDTD & 10.20 & 0.01 & 5000 & 1897 \\ Non-conformal ADI-FDTD & 22.39 & 0.01 & 5000 & 4765 \\ Conformal ADI-FDTD-CFLN = 1 & 22.48 & 0.01 & 5000 & 4812 \\ Conformal ADI-FDTD-CFLN = 2 & 22.48 & 0.02 & 2500 & 2403 \\ Conformal ADI-FDTD-CFLN = 3 & 22.48 & 0.03 & 1667 & 1608 \\ Conformal ADI-FDTD-CFLN = 4 & 22.48 & 0.04 & 1250 & 1196 \\ Conformal ADI-FDTD-CFLN = 5 & 22.48 & 0.05 & 1000 & 961 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Computational resource usage of different methods.  
Figure 7: _Cont._