# Electromagnetic Field Distribution and Data Characteristics of SUTEM of Multilayer Aquifers
## 3 Forward Modeling Mathematical Method of SUTEM
### Electromagnetic Field Excited by a Vertical Magnetic Dipole Source  
The primary electric field excited by the horizontally positioned and vertically magnetic dipole source contains only the horizontal component, so there is no accumulated charge at the interface of each layer of the medium. The secondary electric field excited in the layered medium also contains only a horizontal component, which represents the TE polarization mode. With the vector potential in TE polarization mode defined as \(\mathbf{F}\), the electromagnetic field can be expressed as follows:  
\[\mathbf{E}\ =\ -\
abla\times\mathbf{F}\quad\mathbf{H}=-\hat{y}\mathbf{F}+\frac{1}{\hat{z}} \
abla(\
abla\cdot\mathbf{F}) \tag{2}\]  
It is assumed that the vector potential \(\mathbf{F}\) has only one vertical component \(\mathbf{F}\); then, the components of the electric field and the magnetic field are described in [PERSON]'s book [23].  
As long as the vertical component \(\mathbf{F}\) of the vector potential \(\mathbf{F}\) is obtained, the components of the electric field and magnetic field excited by the vertical magnetic dipole source can be obtained. In the Cartesian coordinate system, the vector potential vertical component of the TE polarization mode satisfies the following formula:  
\[\
abla^{2}\mathbf{F}+\mathbf{k}^{2}\mathbf{F}\ =\ 0 \tag{3}\]  
where \(\mathbf{k}^{2}\ =\ -\hat{z}\hat{y}\). The wavenumber domain equation is obtained by using the double Fourier transform to Equation (3):  
\[\frac{\mathrm{d}^{2}\overline{\mathbf{F}}}{\mathrm{d}\mathbf{z}^{2}}-\mathbf{u}^{2} \overline{\mathbf{F}}\ =\ 0 \tag{4}\]where \(\overline{F}\) is the scalar potential in the wavenumber domain; \(u^{2}~{}=~{}k_{x}^{2}+k_{y}^{2}-k^{2};\lambda^{2}~{}=~{}k_{x}^{2}+k_{y}^{2}\); and \(k_{x}\) and \(k_{y}\) are the components of the wavenumber on the \(x\) and \(y\) axes, respectively. The solution of (4) is similar to that of the plane wave:  
\[\overline{F}(k_{x},k_{y},z)~{}=~{}\overline{F}^{+}(k_{x},k_{y})e^{-uz}+ \overline{F}^{-}(k_{x},k_{y})e^{uz} \tag{5}\]  
where the superscripts \"\(+\)\" and \"\(-\)\" denote the downward and the upward decaying solution, respectively.  
Figure 3 shows the solution of the potential \(\overline{F}\) under the TE polarization mode of the Fourier transform in the \(M+N\) layers. The vertical magnetic dipole source is located in any layer underground, with \(M\) layers above and \(N\) layers below. The values \(h\) and \(z\) represent the thickness of each layer and the vertical distance between the layer interface and the source layer, respectively. In the \(M^{-th}\) layer, only the upward solution decays \(\overline{F}^{-}_{MM}(k_{x},k_{y})e^{u_{MM}z}\); in the \(N^{-th}\) layer, only the downward solution decays \(\overline{F}^{+}_{NN}(k_{x},k_{y})e^{-u_{NN}z}\); in the intermediate layers (\(n~{}=~{}1,2,\cdots N-1\) and \(m~{}=~{}1,2,\cdots M-1\)), the upward \(\overline{F}^{-}(k_{x},k_{y})e^{uz}\) and downward decaying \(\overline{F}^{+}(k_{x},k_{y})e^{-uz}\) solutions are both present.  
The general solution for potential \(F\) in Figure 3 is divided into the three domains shown in the following subsections.