## 3 Materials and Methods
### Electromagnetic and Magnetic Field Response Modeling
#### 3.1.1 Electromagnetic Forward Modeling  
The time-domain electromagnetic (TEM) method is widely used to detect conductive anomalies in the subsurface. This method operates by generating primary magnetic fields that induce eddy currents within conductive targets, which in turn produce secondary magnetic fields that decay over time. Under far-field conditions--where the distance between the sensor and the target significantly exceeds the target's dimensions--the secondary field response can be approximated by a 3D orthogonal magnetic dipole model, which simplifies the complex response of the target into three independent magnetic dipole components oriented along orthogonal axes, represented by the intrinsic magnetic dipole strength \(L_{1}\), \(L_{2}\), and \(L_{3}\)[7; 32]. The induced magnetic moment \(\vec{m}(r_{T})\) at the target location \(r_{T}\) can be expressed as follows:  
\[\vec{m}(r_{T})=\frac{2\pi}{\mu_{0}}M\cdot\vec{B}_{T}(r_{T}), \tag{1}\]  
where \(\vec{B}_{T}\) is the primary magnetic field at the target location, \(\mu_{0}\) denotes the permeability of free space, and \(M\) is the magnetic polarizability tensor (MPT) that describes the target's anisotropic magnetization response in its local coordinate system. The MPT \(M\) is represented as follows:  
\[M=U \begin{bmatrix}L_{1}&0&0\\ 0&L_{2}&0\\ 0&0&L_{3}\end{bmatrix} U^{T}, \tag{2}\]  
where \(L_{1}\), \(L_{2}\), and \(L_{3}\) are the principal magnetic polarizability components along each orthogonal axis. Due to the axial symmetry of the problem, the rotation matrix \(U\) aligns the local coordinate system of the target with the global coordinate system and is defined as follows:  
\[U= \begin{bmatrix}\cos\alpha\cos\beta&\sin\alpha\cos\beta&-\sin\beta\\ -\sin\alpha&\cos\alpha&0\\ \cos\alpha\sin\beta&\sin\alpha\sin\beta&\cos\beta\end{bmatrix} , \tag{3}\]  
where \(\alpha\) and \(\beta\) are the azimuth and dip angles, respectively, which specify the target's orientation in space. To fully illustrate the relationship between the local coordinate system of the object and the global coordinate system, we provide a schematic diagram of the dipole model, as shown in Figure 3.  
The total secondary field response \(F\) of the three orthogonal dipole components can be expressed as the sum of the responses generated by each component. Based on the forward model, we define the response as follows:  
\[F(r_{T},r_{R})=\sum_{i=1}^{3}F_{i}(r_{T},r_{R})=\sum_{i=1}^{3}\frac{\mu_{0}}{4 \pi|r_{R}|^{3}}m_{i}\cdot\left[\frac{3r_{R}r_{R}^{T}}{|r_{R}|^{2}}-\textbf{I} \right], \tag{4}\]  
where  
* \(r_{T}\) and \(r_{R}\) denote the positions of the target and receiver, respectively. * \(F_{i}(r_{T},r_{R})\) represents the secondary field response from each orthogonal dipole component (i.e., \(i=1,2,3\)). * \(\mu_{0}\) is the permeability of free space. * \(m_{i}\) denotes the intrinsic dipole strengths in the \(i\)-th direction, specifically \(L_{1}\), \(L_{2}\), and \(L_{3}\) corresponding to the three orthogonal dipole components. * \(r_{R}r_{R}^{T}\) is the outer product of the receiver position vector \(\textbf{r}_{R}\), forming a matrix. * **I** is the identity matrix, ensuring the response considers both radial and angular components of the field generated by each dipole orientation. This formulation captures the combined secondary field response as the sum of contributions from each of the three orthogonal dipole components, characterized by the target parameters \((x,y,z,L_{1},L_{2},L_{3},\alpha,\beta)\). Here, \(\alpha\) and \(\beta\) represent the rotation and inclination angles of the dipole orientations relative to the coordinate system, which affects the directional response of each component. This unified expression allows the model to calculate the total secondary field response at the receiver position, given the dipole parameters and orientation angles.