## 2 System Model  
Notation: In this paper, bold letters denote vectors and matrices; \(\omega\) omega denotes the angular frequency of electric current in the transmitter; \(w\) denotes the number of sets of secondary field data we collected in the process of detection; \(\mathbf{H}_{P}\), \(\mathbf{B}_{P}\) and \(\mathbf{B}_{s}\) respectively denote the values of the primary field magnetic field intensity, primary field magnetic induction and secondary field magnetic induction, which contain \(x\), \(y\), and \(z\) components; \(\mathbf{C}\) denotes the collected secondary field data generated by the simulation system; \(\theta\), \(\phi\), and \(\psi\) denote the pitch angle, roll angle and yaw angle for a vertically placed cylinder target, respectively; and \(j\) denotes the imaginary unit. In the detection system, the metal target is located at a certain position in the underground space. When the primary field (generated by the transmitter) acts on the metal target, a secondary field is generated by the eddy current of the metal target. The secondary field data are collected by a detector sequentially placed in different positions on the detection plane (the detector includes a transmitter and a receiver; in this paper, we assume that these two components are coincident). Estimation of the target's parameters can be obtained from these secondary field data. As shown in Figure 1, in EMI detection, the commonly used data collection scheme is 2D scanning [14; 23; 24]. The detector is placed on the scanning line to collect data successively. On each scanning line, the distance between two adjacent sampling points is called the point interval \(p_{s}\), and the distance between the scanning lines is called the line interval \(l_{s}\). The plane where the scanning line is located is called the detection plane. Assuming that there are \(2M+1\) scanning lines, \(2N+1\) sets of data are collected on each scanning line, ultimately producing \(w=(2M+1)(2N+1)\) sets of data collected on the detection plane. Each set of data are represented by \(\mathbf{C}\left(k\right)\) (\(k=1\cdots w\)). Based on these collected data, the initial value estimation algorithm can be used to obtain the approximate estimation of the target state, and, by using these initial values, an accurate estimation of the metal target's parameters can be obtained quickly by the KF and EKF algorithms. In Figure 1, the target position vector is \(\mathbf{r}\), the posture is \(\psi\), \(\theta\), \(\phi\), and the polarizability is \(\mathbf{\beta}=\left[ \begin{smallmatrix}\beta_{x}&\beta_{y}&\beta_{z}\\ \end{smallmatrix} \right]^{T}\). To obtain an estimation of the target's parameters from secondary field data, it is necessary to characterize the EMI response with a suitable forward model. A spherical coordinate system is used in the primary field calculation because calculating the primary field in this coordinate system is simpler than that in a Cartesian coordinate system. As shown in Figure 2, there is a circular coil with a radius of \(a\) in the space, and a harmonic current \(I=I_{0}e^{-j\omega t}\) with an angular frequency of \(\omega\) is applied to the coil. In the spherical coordinate system, the center of the toroidal coil lies at the origin. \(\rho,\gamma,\eta\) are the radial distance, polar angle and azimuthal angle, respectively. The inducted field strength at any position \(P(\rho,\gamma,\eta)\) in space is [25]:  
\[H_{\rho}=j\frac{\mu_{0}k_{r}a^{2}I_{0}\cos\gamma}{2\rho^{2}}\left[1+\frac{1}{ jk_{r}\rho}\right]\exp^{-jk_{r}\rho}, \tag{1}\]  
\[H_{\gamma}=-\frac{\mu_{0}(k_{r}a)^{2}I_{0}\sin\gamma}{4\rho}\left[1+\frac{1}{ jk_{r}\rho}-\frac{1}{(k_{r}\rho)^{2}}\right]\exp^{-jk_{r}\rho}, \tag{2}\]  
\[H_{\eta} =0 \tag{3}\]  
where \(\mu_{0}\) is the vacuum permeability, \(k_{r}=\frac{2\pi}{\lambda}\) is the wave number of electromagnetic waves in the propagation medium, \(\lambda=\frac{2\pi r}{\omega}\) is the wavelength of the electromagnetic waves, and \(v\) is the wave velocity in the medium.