# Electromagnetic Field Distribution and Data Characteristics of SUTEM of Multilayer Aquifers
## 3 Forward Modeling Mathematical Method of SUTEM
### TEM Field Excited by Rectangular Transmitting Loop
#### 3.2.1 Frequency Domain Electromagnetic Field in TE Polarization Mode  
* Total field in layered media  
The scalar potential obtained in the TE mode is brought into the inverse Fourier transform to obtain the relationship \(F(x,y,z)\) between the vertical magnetic dipole and the potential function. The double Fourier transform can be converted into the Hankel transform [26]. 1. Electromagnetic field in the plane perpendicular to the magnetic dipole source  
Bringing Equation (8) into the inverse Fourier transform and applying the Hankel transform, we can obtain  
\[F(\rho,z)\ =\ \frac{\hat{z}_{0}m}{4\pi}\int_{0}^{\infty}\left(1+r_{TE_{M}}+r_{ TE_{N}}\right)J_{0}(\lambda\rho)d\lambda \tag{14}\]  
where \(\rho\ =\ \sqrt{(x^{2}+y^{2})}\); \(J_{0}(\lambda\rho)\) is the zero-order Bessel function. For the plane where the vertical magnetic dipole source is located, the primary field contains only the horizontal component of the electric field and the vertical component of the magnetic field; there is no vertical component of the electric field or horizontal component of the magnetic field. According to the relationship between the electromagnetic field component and the potential, the electromagnetic field frequency domain expression generated by the vertical magnetic dipole source in its horizontal plane is given by  
\[ \begin{array}{rcl}E_{x}&=&\frac{\hat{z}_{0}m}{4\pi}\int_{0}^{\infty}\frac{ \lambda^{2}}{u_{0}}(1+r_{TE_{M}}+r_{TE_{N}})\frac{\
u}{\rho}\underline{J}_{1 }(\lambda\rho)\mathrm{d}\lambda&E_{y}\ =\ -\frac{\hat{z}_{0}m}{4\pi}\int_{0}^{\infty}\frac{ \lambda^{2}}{u_{0}}(1+r_{TE_{M}}+r_{TE_{N}})\frac{\
u}{\rho}\underline{J}_{1 }(\lambda\rho)\mathrm{d}\lambda\\ H_{z}&=&\frac{m}{4\pi}\int_{0}^{\infty}\frac{\lambda^{3}}{u_{0}}(1+r_{TE_{M} }+r_{TE_{N}})I_{1}(\lambda\rho)\mathrm{d}\lambda\end{array}  \tag{15}\]  
where \(J_{1}(\lambda\rho)\) is the first-order Bessel function. 2. The frequency domain electromagnetic field component of the intermediate layer (\(n=1,2,\ldots N-1\) and \(m=1,2,\ldots M-1\)) The intermediate layer below the source can be described by the following: \[F(\rho,z)\ =\ \frac{\hat{z}_{0}m}{4\pi}\int_{0}^{\infty}(F_{Nn}^{+}e^{-u_{ Nn}(z-z_{Nn})}+F_{Nn}^{-}e^{u_{Nn}(z-z_{Nn})})J  
3. The frequency domain electromagnetic field components of the first (\(m=M\)) and last layers (\(n=N\))  
Bringing Equation (13) into the inverse Fourier transform, and we can obtain the following formula:  
\[F(\rho,z) = \frac{\hat{z}_{MM}m}{4\pi}\int_{0}^{\infty}\frac{\lambda^{2}}{u_{0}}F_{ MM}^{-}e^{u_{MM}(z-z_{MM})}J_{0}(\lambda\rho)\mathrm{d}\lambda \tag{18}\]  
The electromagnetic field components in the frequency domain of the first layer are obtained:  
\[E_{x} = \frac{\hat{z}_{MM}m}{4\pi}\int_{0}^{\infty}\frac{\lambda^{2}}{u_{0 }}F_{MM}^{-}e^{u_{MM}(z-z_{MM})}\frac{y}{\rho}I_{1}(\lambda\rho)\mathrm{d}\lambda\] \[E_{x} = -\frac{\hat{z}_{MM}m}{4\pi}\int_{0}^{\infty}\frac{\lambda^{2}}{u_ {0}}F_{MM}^{-}e^{u_{MM}(z-z_{MM})}\frac{x}{\rho}I_{1}(\lambda\rho)\mathrm{d} \lambda\quad H_{x}=\frac{m}{4\pi}\int_{0}^{\infty}\lambda^{2}F_{MM}^{-}e^{u_{ MM}(z-z_{MM})}\frac{x}{\rho}I_{1}(\lambda\rho)\mathrm{d}\lambda \tag{19}\] \[H_{y} = \frac{m}{4\pi}\int_{0}^{\infty}\lambda^{2}F_{MM}^{-}e^{u_{MM}(z- z_{MM})}\frac{y}{\rho}I_{1}(\lambda\rho)\mathrm{d}\lambda\quad H_{z}=\frac{m}{4\pi}\int_{0}^{ \infty}\frac{\lambda^{3}}{u_{0}}F_{MM}^{-}e^{u_{MM}(z-z_{MM})}J_{0}(\lambda\rho )\mathrm{d}\lambda\]  
Similarly, the frequency domain electromagnetic field of the last layer excited by the magnetic dipole source can be obtained. * The secondary field in layered media  
Through the solution process of the scalar potential in the TE polarization mode, it can be seen that the scalar potential of the source layer contains both the scalar potential caused by the magnetic dipole source (i.e., the primary field source) and the scalar potential caused by the induced current (i.e., the secondary field source) in the conductive medium.