# Modeling GPR Wave Propagation in Complex Underground Structures Using Conformal ADI-FDTD Algorithm
## 2 Methodology
### Surface Conformal Technology  
The surface conformal technique was used to simulate the mesh generation of circular underground structures. Figure 1 shows schematic diagrams of the grid division of the circular structure model constructed by different methods. Figure 1a shows the actual subdivision grid of the circular structure, where the white squares are ordinary and the red squares are the actual subdivisions. Figure 1b shows the conventional ladder approximation subdivision grid for a circular structure, where the green squares are the subdivisions obtained by the step approximation method. The subdivision grid of a circular structure based on surface conformal technology is shown in Figure 1c, where the orange squares are conformal areas and the blue squares are non-conformal areas. A conformal grid point in Figure 1c is taken as an example in order to demonstrate the selection of equivalent medium parameters for the conformal grid region in 2D TM waves. As shown in Figure 2, where \(\mathbf{F}\) is the sampling point of the electric field \(E_{z}\); \(A\), \(B\) and \(C\), \(D\) are sampling points of the magnetic field \(H_{x}\) and \(H_{y}\), respectively; \(\Delta y\) and \(\Delta x\) are the height and width of the grid, respectively; \(S_{xy1}\) and \(S_{xy2}\) are the areas of media 1 and 2 in the conformal grid, respectively; \(L_{x1}\) and \(L_{x2}\) are the lengths occupied by media 1 and 2 on the edge of magnetic field node \(B\), respectively; and \(L_{y1}\) and \(L_{y2}\) are the lengths occupied by media 1 and 2 on the edge of magnetic field node \(D\), respectively. Figure 1: Grid dissection of the circular structure: (**a**) Actual grid dissection of a circular structure; (**b**) Grid dissection of circular structure by ladder approximation method; (**c**) Dissection of circular structure by conformal grid technology. Figure 2: Equivalent schematic of the media parameters at the conformal grid points: (**a**) Actual grids; (**b**) Conformal grids. Considering the two-dimensional TM wave, for the conformal grid of an ideal conductor (metal conductor), the recursive formula of a magnetic field node remains unchanged and is still calculated according to conventional ADI-FDTD. For the recursive formula of the electric field node, we make the following changes:  
In the \(n\)\(\rightarrow\)\(n+1/2\) time step, Equation (6) is changed to  
\[ \begin{split} E_{z}^{n+\frac{1}{2}}(i,j)=CA\cdot& E_{z}^{n}(i,j)+CB \cdot\bigg{\{}\frac{l_{y}}{5}\cdot\bigg{[}H_{y}^{n+\frac{1}{2}}(i+\frac{1}{2}, j)-H_{y}^{n+\frac{1}{2}}(i-\frac{1}{2},j)\bigg{]}\\ &-\frac{l_{x}}{5}\cdot\bigg{[}H_{x}^{n}(i,j+\frac{1}{2})-H_{x}^{n }(i,j-\frac{1}{2})\bigg{]}\bigg{\}}.\end{split}  \tag{21}\]  
where \(l_{x}\) and \(l_{y}\) represent the length of the corresponding edge of the electric field node outside the conductor, and \(S\) represents the area of the cell outside the conductor. We substitute \(H_{y}^{n+1/2}\) in Equation (5) into Equation (21) and obtain the update equation about \(E_{z}^{n+1/2}\)  
\[a_{i}^{\prime}E_{z}^{n+\frac{1}{2}}(i-1,j)+b_{i}^{\prime}E_{z}^{n+\frac{1}{2} }(i,j)+c_{i}^{\prime}E_{z}^{n+\frac{1}{2}}(i+1,j)=d_{i}^{\prime}, \tag{22}\]  
where  
\[ \begin{split} a_{i}^{\prime}=-\frac{\Delta t}{2\mu\Delta x} \cdot\frac{l_{y}}{5}\cdot CB\quad b_{i}^{\prime}=1+\frac{\Delta t}{\mu\Delta x} \cdot\frac{l_{y}}{5}\cdot CB\quad c_{i}^{\prime}=-\frac{\Delta t}{2\mu\Delta x} \cdot\frac{l_{y}}{5}\cdot CB\\ d_{i}^{\prime}=CA\cdot E_{z}^{n}(i,j)+CB\cdot\frac{l_{y}}{5} \cdot\bigg{[}H_{y}^{n}(i+\frac{1}{2},j)-H_{y}^{n}(i-\frac{1}{2},j)\bigg{]}-CB \cdot\frac{l_{z}}{5}\cdot\bigg{[}H_{x}^{n}(i,j+\frac{1}{2})-H_{x}^{n}(i,j- \frac{1}{2})\bigg{]}.\end{split}  \tag{23}\]  
In the \(n+1/2\)\(\rightarrow\)\(n+1\) time step, the difference discretization schemes of Equation (15) are  
\[