# Electromagnetic Imaging for Buried Conductors Using Deep Convolutional Neural Networks
## 2 Forward Problem  
Consider a perfect conductor buried in a lossy half-space, as shown in Figure 1. The permittivity and conductivity in Region I and Region II are \((\varepsilon_{1},\sigma_{1})\) and \((\varepsilon_{2},\sigma_{2})\), respectively. In each region, the permeability of free space is set to \(\mu_{0}\); that is, only non-magnetic substances are considered. Let the scatterer be a cylindrical conductor extending infinitely in the \(z\)-axis. Its cross-sectional area in the \(xy\)-plane can be expressed by the equation \(\rho=F(\theta)\). The time-varying relation of the incident wave is set to \(e^{j\omega t}\) and its incident angle is \(\varnothing_{1}\), as depicted in Figure 1.  
For simplicity, the direction of the incident electric field is assumed to be parallel to the \(z\)-axis, which is the TM polarization wave direction. Using \(E_{i}\) to represent the electric field distribution when the conductor is not present, \(E_{i}\) can then be expressed as follows:  
\[\overset{\rightarrow}{E}_{i}(\overset{\rightarrow}{r})=E_{i}(x,y)\hat{z}, \tag{1}\]  
where  
\[E_{i}(x,y)=\left\{ \begin{array}{cc}E_{1}(x,y)=e^{-\vec{k}_{1}[xsin\varpi_{1} +(y+A)cos\varpi_{1}]}+R_{1}e^{-\vec{k}_{1}[xsin\varpi_{1}-(y+A)cos\varpi_{1}]} &,y\leq-A\\ E_{2}(x,y)=Te^{-\vec{k}_{2}[xsin\varpi_{2}+(y+A)cos\varpi_{2}]}&,y>-A\end{array} \right. \tag{2}\]  
\[R_{1}=\frac{1-n}{1+n}\text{, }T=\frac{2}{1+n}\text{, }n=\frac{cos\varpi_{2}}{ cos\varpi_{1}}\sqrt{\frac{\varepsilon_{2}-j\varpi_{2}/\omega}{ \varepsilon_{1}-j\sigma_{1}/\omega}} \tag{3}\]  
\[k_{1}sin\varpi_{1}=k_{2}sin\varpi_{2} \tag{4}\]  
\[k_{i}^{2}=\omega^{2}\varepsilon_{1}\mu_{0}-jw\mu_{0}\sigma_{i},\text{ }i=1,2\text{ }Im(k_{i})\leq 0. \tag{5}\]  
The first term, \(E_{1}(x,y)\), is the so-called incident field \(E^{inc}\). If Region I and Region II are the lossless media, then \(\varnothing_{1}\) and \(\varnothing_{2}\) represent the angle of incidence and refraction, respectively. On the contrary, if Region I and Region II are the lossy media, then \(\varnothing_{1}\) and \(\varnothing_{2}\) become more complex. The wave form will be very complicated. Its propagation direction will be different with the decay direction. The total electric field at any point in the space can be tabulated as follows:  
\[E(x,y)= \begin{cases}E_{1}(x,y)+E_{s}(x,y)\text{, }y\leq-A\\ E_{2}(x,y)+E_{s}(x,y)\text{, }y>-A\end{cases}  \tag{6}\]  
where \(E_{s}(x,y)\) is the scattered field. Since the size of our interested object is about one wavelength (i.e., in the resonance region), the scattered field will have a severe diffraction effect. In order to solve for a correct scattered field, we must first carefully calculate the Green's function \(G(x,y;x\prime,y\prime)\) for this half-space problem. This can be generated by a line current source at \((x\prime,y\prime)\) and the scattered field at \((x,y)\). Via the Fourier transform technique, \(G(x,y;x\prime,y\prime)\) can be expressed as follows:  
\[G\left(x,y;x^{\prime},y^{\prime}\right)=\left\{ \begin{array}{cc}G_{1}(x,y;x ^{\prime},y^{\prime})&,y\leq-A\\ G_{2}(x,y;x^{\prime},y^{\prime})=G_{f}(x,y;x^{\prime},y^{\prime})+G_{s}(x,y;x^ {\prime},y^{\prime})&,y>-A\end{array} \right. \tag{7}\]  
Figure 1: Schematic diagram of a two-dimensional perfect buried conductor in half-space.