# Petrophysics Parameter Inversion and Its Application Based on the Transient Electromagnetic Method
## 2 Theory  
### Forward Modeling  
The transient electromagnetic (TEM) method is a time-domain electromagnetic exploration technique that uses an ungrounded loop source to transmit a step current, thereby generating a primary magnetic field in the surrounding space. This primary magnetic field induces eddy currents in subsurface geological formations, resulting in a time-varying secondary magnetic field. The secondary field is detected using a receiving coil. The temporal attenuation characteristics of the secondary field are primarily governed by the conductivity, volume, and burial depth of the anomalous body, as well as the waveform and frequency of the transmitted current. By analyzing the spatial variation of the secondary field recorded by the receiving coil, it is possible to infer the spatial distribution of subsurface anomalies. The working principle of the TEM method is illustrated in Figure 1[31].  
According to [PERSON]'s equations, the propagation of electromagnetic fields in conductive media can be described by diffusion equations. Assuming a time-harmonic field \(\mathrm{e}^{-i\omega t}\), [PERSON]'s equations can be simplified as follows [32]:  
\[\
abla\times\mathbf{E}=i\omega\mu\mathbf{H} \tag{1}\]  
\[\
abla\times\mathbf{H}=\sigma\mathbf{E}+\mathbf{J}_{s} \tag{2}\]  
where \(\mathbf{J}_{s}\) is the emission current density. For an axisymmetric loop source, the field quantity depends only on the radial distance \(r\) and vertical direction \(z\). In the cylindrical coordinate system, the vertical component \(H\)z of the magnetic field satisfies the following scalar wave equation:  
\[\Big{(}\
abla^{2}+k^{2}\Big{)}H_{z}=0 \tag{3}\]  
where \(k^{2}=i\omega\mu\sigma\). By separation of variables, the solution can be expressed as an integral of the wave number \(\lambda\):  
\[H_{z}(r,\omega)=\int_{0}^{\infty}\widetilde{H}_{z}(\lambda,\omega)J_{0}( \lambda r)\lambda d\lambda \tag{4}\]  
where \(J_{0}\) is the zero-order Bessel function and is the field response in the wavenumber domain.  
For layered media, the reflection coefficient \(r_{TE}\) of the transverse electric (TE) mode is calculated using a recursive formula. The wave impedance \(Z_{k}\) of the \(k\)th layer is expressed as  
\[Z_{k}=\frac{i\omega\mu_{k}}{\gamma_{k}},\ \gamma_{k}=\sqrt{\lambda^{2}+i \omega\mu_{k}\sigma_{k}} \tag{5}\]  
The interface reflection coefficient \(r_{k}\) is expressed as follows:  
\[r_{k}=\frac{Z_{k}-Z_{k+1}}{Z_{k}+Z_{k+1}} \tag{6}\]  
The total reflection coefficient \(r_{TE}^{(k)}\) is obtained through the following recursive relationship:  
\[r_{TE}^{(k)}=\frac{r_{k}+r_{TE}^{(k+1)}e^{-2\gamma_{k}h_{k}}}{1+r_{k}r_{TE}^{( k+1)}e^{-2\gamma_{k}h_{k}}} \tag{7}\]  
where \(h_{k}\) is the thickness of the kth layer.  
The field response \(\widetilde{H}_{z}(\lambda,\omega)\) in the wavenumber domain is related to the reflection coefficient \(r_{TE}\), and the spatial domain expression of the vertical magnetic field component can be obtained as:  
\[H_{z}(r,\omega)=\int_{0}^{\infty}\frac{r_{TE}(\lambda,\omega)}{i\omega\mu}J_{ 0}(\lambda r)\lambda d\lambda \tag{8}\]  
Figure 1: Schematic diagram of the working mode of the transient electromagnetic method.  
The frequency domain response is converted into the time domain perpendicular magnetic field component by a cosine transform:  
\[B_{z}(t)=\frac{\mu_{0}}{2\pi}\int_{0}^{\infty}\text{Re}\bigg{[}\int_{0}^{\infty} \lambda J_{0}(\lambda r)\frac{r_{TE}(\lambda,\omega)}{i\omega}d\lambda\bigg{]} \cos(\omega t)d\omega \tag{9}\]