# Modeling GPR Wave Propagation in Complex Underground Structures Using Conformal ADI-FDTD Algorithm
## 2 Methodology  
### ADI-FDTD Algorithm  
For transverse magnetic (TM) waves in two-dimensional lossy media, the Maxwell equations [37] are expressed as  
\[\mu\frac{\partial H_{x}}{\partial t}=-\frac{\partial E_{z}}{\partial y}, \tag{1}\]  
\[\mu\frac{\partial H_{y}}{\partial t}=\frac{\partial E_{z}}{\partial x}, \tag{2}\]  
\[\epsilon\frac{\partial E_{z}}{\partial t}+\sigma E_{z}=\frac{\partial H_{y}}{ \partial x}-\frac{\partial H_{x}}{\partial y}, \tag{3}\]  
where \(H_{x}\) and \(H_{y}\) are the magnetic field intensities in the \(x\) and \(y\) directions; \(E_{z}\) is the electric field intensity in the \(z\) direction; \(\sigma\) is the electrical conductivity; \(\epsilon\) is the dielectric constant; \(\mu\) is the magnetic permeability. In the ADI-FDTD algorithm, we divide a time step into two sub-steps and discretize them in different ways. In the sub-step \(n\)\(\rightarrow\)\(n+1/2\), the \(x\)-direction derivative adopts the implicit difference format, and the \(y\)-direction derivative adopts the explicit difference format. In the sub-step of \(n+1/2\)\(\rightarrow\)\(n+1\), the \(x\)-direction derivative adopts the explicit difference format, and the \(y\)-direction derivative adopts the implicit difference format. In the \(n\)\(\rightarrow\)\(n+1/2\) time step, the difference scheme of Equation (1) is as follows:  
\[H_{x}^{n+\frac{1}{2}}(i,j+\frac{1}{2})=H_{x}^{n}(i,j+\frac{1}{2})-\frac{\Delta t }{2\mu}\cdot\frac{1}{\Delta y}\cdot[E_{z}^{n}(i,j+1)-E_{z}^{n}(i,j)]. \tag{4}\]  
Similarly, the difference discretization scheme of Equation (2) is  
\[H_{y}^{n+\frac{1}{2}}(i+\frac{1}{2},j)=H_{y}^{n}(i+\frac{1}{2},j)+\frac{\Delta t }{2\mu}\cdot\frac{1}{\Delta x}\cdot\bigg{[}E_{z}^{n+\frac{1}{2}}(i+1,j)-E_{z}^ {n+\frac{1}{2}}(i,j)\bigg{]}. \tag{5}\]  
For Equation (3), the \(E_{z}\) component takes the mean value of time \(n\Delta t\) and time \((n+1/2)\Delta t\), where the difference discretization scheme of Equation (3) is  
\[ \begin{array}{l}E_{z}^{n+\frac{1}{2}}(i,j)=CA\cdot E_{z}^{n}(i,j)+CB\cdot \bigg{\{}\frac{1}{\Delta x}\cdot\bigg{[}H_{y}^{n+\frac{1}{2}}(i+\frac{1}{2},j )-H_{y}^{n+\frac{1}{2}}(i-\frac{1}{2},j)\bigg{]}\\ \hskip 142.26378 pt-\frac{1}{2W}\cdot\bigg{[}H_{x}^{n}(i,j+\frac{1}{2})-H_{x}^ {n}(i,j-\frac{1}{2})\bigg{]}\bigg{\}}\end{array}  \tag{6}\]  
where  
\[CA=\frac{4\epsilon-\sigma\Delta t}{4\epsilon+\sigma\Delta t}\ \ \ CB=\frac{2\Delta t }{4\epsilon+  
In the \(n+1/2\!\!\rightarrow\!\!n+1\) time step, the difference discretization schemes of Equations (1)-(3) are  
\[H_{x}^{n+1}(i,j+\frac{1}{2})=H_{x}^{n+\frac{1}{2}}(i,j+\frac{1}{2})-\frac{\Delta t }{2\mu}.\frac{1}{\Delta y}.\Big{[}E_{z}^{n+1}(i,j+1)-E_{z}^{n+1}(i,j)\Big{]}, \tag{13}\]  
\[H_{y}^{n+1}(i+\frac{1}{2},j)=H_{y}^{n+\frac{1}{2}}(i+\frac{1}{2},j)+\frac{ \Delta t}{2\mu}.\frac{1}{\Delta x}.\Big{[}E_{z}^{n+\frac{1}{2}}(i+1,j)-E_{z}^{n +\frac{1}{2}}(i,j)\Big{]}, \tag{14}\]  
\[ \begin{array}{l}E_{z}^{n+1}(i,j)=CA.E_{z}^{n+\frac{1}{2}}(i,j)+CB.\bigg{\{} \frac{1}{\Delta x}.\bigg{[}H_{y}^{n+\frac{1}{2}}(i+\frac{1}{2},j)-H_{y}^{n+ \frac{1}{2}}(i-\frac{1}{2},j)\bigg{]}\\ -\frac{1}{\Delta y}.\bigg{[}H_{x}^{n+1}(i,j+\frac{1}{2})-H_{x}^{n+1}(i,j-\frac {1}{2})\bigg{]}\bigg{\}}.\end{array}  \tag{15}\]  
The time domain advance of the electromagnetic field in sub-step \(n+1/2\!\!\rightarrow\!\!n+1\) can be calculated by Equations (13)-(15), as Equations (13) and (15) contain the simultaneous field value (i.e., implicit format) and cannot be used to calculate the display time advance. From Equations (13) and (15), we can derive  
\[ \begin{array}{l}-\frac{\Delta t}{2\mu\Delta y^{2}}.CB.E_{z}^{n+1}(i,j-1)+ \left[1+\frac{\Delta t}{\mu\Delta y^{2}}.CB\right]E_{z}^{n+1}(i,j)-\frac{ \Delta t}{2\mu\Delta y^{2}}.CB.E_{z}^{n+1}(i,j+1)=\\ CA.E_{z}^{n+1}(i,j)-\frac{CB}{\Delta y}.\bigg{[}H_{x}^{n+\frac{1}{2}}(i,j+\frac {1}{2})-H_{x}^{n+\frac{1}{2}}(i,j-\frac{1}{2})\bigg{]}+\frac{CB}{\Delta x}. \bigg{[}H_{y}^{n+\frac{1}{2}}(i+\frac{1}{2},j)-H_{y}^{n+\frac{1}{2}}(i-\frac {1}{2},j)\bigg{]},\end{array}  \tag{16}\]  
where  
\[