## 2 System Model
Then, the spherical coordinates can be converted to rectangular coordinates according to Formula (4):  
\[\left\{ \begin{array}{l}H_{x}=H_{\rho}\sin\gamma\cos\eta+H_{\gamma}\cos\gamma \cos\eta-H_{\eta}\sin\eta,\\ H_{y}=H_{\rho}\sin\gamma\sin\eta+H_{\gamma}\cos\gamma\sin\eta+H_{\eta}\cos\eta, \\ H_{z}=H_{\rho}\cos\gamma-H_{\gamma}\sin\gamma.\end{array} \right. \tag{4}\]  
The principle of electromagnetic induction in underground metal detection is illustrated by Figure 3, the detector position is \(\mathbf{r}_{d}\), and the target position is \(\mathbf{r}\). When the length of the detected target is much shorter than the distance between the detector and the detected target, the detected target can be idealized as a magnetic dipole, which satisfies the requirements of engineering applications. When the primary field acts on the metal target, an eddy current is generated inside the metal. The magnetic field sensor can be used to obtain the secondary field \(\mathbf{B}_{s}\) generated by the eddy current [26]:  
\[\mathbf{B}_{s}=\frac{\mu_{0}}{4\pi}\left[\frac{3\left(\mathbf{m}\cdot\mathbf{r}\right)\mathbf{ r}}{r^{5}}-\frac{\mathbf{m}}{r^{3}}\right], \tag{5}\]  
where \(\mathbf{m}\) is the induced dipole moment at the metal target position, \(\mu_{0}\) is the permeability of vacuum, \(r\) is the distance from the detector center to the metal target center, and \(\mathbf{r}\) is the vector of the metal  
Figure 1: Process of the metal detection system. target center pointing to the detector center. The value of the induced dipole moment is related to the detected target's posture and polarizability and the primary field strength [26]:  
\[\mathbf{m}=\mathbf{U}\mathbf{\beta}\mathbf{U}^{T}\mathbf{H}_{P}, \tag{6}\]  
where \(\mathbf{U}^{T}\) represents the transpose of the matrix \(\mathbf{U}\) and \(\mathbf{U}\) is the transformation matrix between the observation coordinate system and the metal target coordinate system, also known as the Euler rotation matrix [27]. \(\mathbf{H}_{P}=\left[ \begin{smallmatrix}H_{x}&H_{y}&H_{z}\\ \end{smallmatrix} \right]^{T}\) is the primary field strength calculated according to Formulas (1)-(3) at the detected target position. \(\mathbf{\beta}\), the polarizability of the metal target, is an intrinsic feature of the target and is independent of the position and posture:  
\[\mathbf{\beta}=\left[ \begin{array}{ccc}\beta_{x}&0&0\\ 0&\beta_{y}&0\\ 0&0&\beta_{z}\\ \end{array} \right]. \tag{7}\]  
Formulas (5) and (6) imply that the secondary field has a nonlinear relationship with the position and posture angle of the detected target and scales linearly with the polarizability. Therefore, a nonlinear equation needs to be solved to obtain the position and posture angle, which is a relatively complicated undertaking. In [15], Formula (6) is written as follows:  
\[\mathbf{m}=\mathbf{M}\mathbf{H}_{P}, \tag{8}\]  
where \(\mathbf{M}\) is a \(3\times 3\) magnetic polarizability tensor, which is related to the polarizability and posture of the metal target. \(\mathbf{M}\) fully reflects the dipole response of the metal target and it can be proven that \(\mathbf{M}\) is a \(3\times 3\) symmetric matrix [28]; thus, there are only six different elements in \(\mathbf{M}\). \(\mathbf{M}\) can be expressed as follows:  
\[\mathbf{M}=\left[ \begin{array}{ccc}M_{11}&M_{12}&M_{13}\\ M_{12}&M_{22}&M_{23}\\ M_{13}&M_{23}&M_{33}\\ \end{array} \right]. \tag{9}\]  
Magnetic polarizability tensor \(\mathbf{M}\) is related only to the posture and the polarizability of the target; therefore, it is constant during the detection process. \(\mathbf{M}\) can be directly obtained from the secondary field data without obtaining the target's polarizability and three posture angles. Because the secondary field strength and \(\mathbf{M}\) are linearly related, the use of \(\mathbf{M}\) reduces the number of nonlinear parameters in the EMI model and greatly improves the computational efficiency in the calculation process.