# 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
#### 3.1.1 The Solution of the Horizontal Plane of the Magnetic Dipole Source  
The horizontal plane where the magnetic dipole source is located has both an upward-propagating upper-decay solution \(\overline{F}^{-}_{Nb}e^{u_{Nb}z}\) and a downward-propagating lower-decay solution \(\overline{F}^{+}_{M0}e^{-u_{Nb}z}\). In this layer, in addition to the complementary solution, there is a special solution of nonhomogeneous differential equations. The special solution of the TE polarization mode through the entire space is [23]  
\[\overline{F}_{p}(k_{x},k_{y}) \tag{6}\]  
The vertical magnetic dipole with the magnetic moment \(mu_{z}\) can be expressed as \(\overline{F}_{p}~{}=~{}\hat{z}m/2u\), where \(\hat{z}\) and \(u\) are related to the source layer. Equation (6) is the scalar potential caused by the magnetic dipole source (primary field source); \(\overline{F}^{-}_{Nb}e^{u_{Nb}z}\) and \(\overline{F}^{+}_{M0}e^{u_{Nb}z}\)  
Figure 3: Solution of Fourier-transformed TE in \(M+N\) layers of earth.  
are the scalar potentials caused by the induced current (secondary field source) in the conductive medium, which are determined by the following formula:  
\[\overline{F}_{N0}^{-}\ =\ r_{TE_{N}}\overline{F}_{P}\ \ \ \overline{F}_{M0}^{+}\ =\ r_{TE_{M}} \overline{F}_{P} \tag{7}\]  
where \(r_{TE_{N}}\) and \(r_{TE_{M}}\) are the reflection coefficients from the below and above layers, respectively [20].  
The general solution in the transformation space can be obtained by adding the special solution of the magnetic dipole source to the complementary solution:  
\[\overline{F}_{0}\ =\ \overline{F}_{p}(1+r_{TE_{M}}+r_{TE_{N}}) \tag{8}\]  
When the magnetic dipole source is located on the surface, the \(m\ =\ 1,2,\cdots M\) layers represent the air layer, and there is an upward attenuation solution only at the source layer and above. Equation (8) can thus be abbreviated as follows:  
\[\overline{F}_{0}\ =\ \overline{F}_{p}(1+r_{TE_{N}})e^{\mu_{N0}z} \tag{9}\]  
1.2 The General Solution of the Intermediate Layers (\(n\ =\ 1,\cdots N-1\) and \(m\ =\ 1,\cdots M-1\))  
The general solution of the middle layers can be expressed as follows:  
\[\overline{F}_{Nn}\ =\ \overline{F}_{Nn}^{+}e^{-u_{Nn}(z-z_{Nn})}+\overline{F}_{ Nn}^{-}e^{\mu_{Nn}(z-z_{Nn})}\ \ \ \ \ \ \overline{F}_{Mm}\ =\ \overline{F}_{Mm}^{+}e^{-u_{Mm}(z-z_{Mn})}+\overline{F}_{Mm}^{-}e^{u_{Mm}(z- z_{Mn})} \tag{10}\]  
The downward amplitude coefficient \(\overline{F}_{Nn}^{+}\) and the upward amplitude coefficient \(\overline{F}_{Nn}^{-}\) below the source layer can be written as  
\[\overline{F}_{Nn}^{+}\ =\ \overline{F}_{Nn-1}^{+}e^{-u_{Nn-1}u_{Nn-1}}\frac{1+r _{TE_{Nn-1}}}{1+r_{TE_{Nn}}e^{-2u_{Nn}u_{Nn}}}\ \ \ \overline{F}_{Nn}^{-}\ =\ r_{TE_{Nn}} \overline{F}_{Nn}^{+}e^{-u_{Nn}u_{Nn}} \tag{11}\]  
where \(r_{TE_{Nn}}\) is the reflection coefficient of the \(n\)-th layer below the magnetic dipole source. The amplitude coefficient of the source layer can be expressed as  
\[\overline{F}_{N0}^{+}\ =\ \overline{F}_{P}\ =\ \frac{m\underline{z}_{N0}}{2u_{N0}} \tag{12}\]  
The downward amplitude coefficient \(\overline{F}_{Mm}^{+}\) and the upward amplitude coefficient \(\overline{F}_{Mm}^{-}\) above the source layer can be similarly obtained.  
3.1.3 The General Solution of the First Layer (\(m\ =\ M\)) and the Last Layer (\(n\ =\ N\))  
The first and the last layer contain only the electromagnetic wave propagating upward and downward through this perspective. The electromagnetic wave attenuates continuously when it propagates in the medium of this layer. A general solution is therefore,  
\[\overline{F}_{MM}\ =\ \overline{F}_{MM}^{-}e^{\mu_{MM}(z-z_{MM})}\ \ \ \overline{F}_{NN}\ =\ \overline{F}_{NN}^{-}e^{\mu_{NN}(z-z_{NN})} \tag{13}\]  
The upward amplitude coefficient \(\overline{F}_{MM}^{-}\) and the downward amplitude coefficient \(\overline{F}_{NN}^{-}\) can be obtained by Equation (11).