## 4 Simulation Platform and Algorithm Performance
### Initial Value Estimation Performance  
The performance of the initial value estimation algorithm is discussed from three aspects: horizontal position estimation performance, depth estimation performance, and magnetic polarizability tensor estimation performance. To objectively evaluate the horizontal position estimation performance, the horizontal position estimation errors at different simulation conditions are compared in this paper  
Figure 6: Cylinder in the detection system. Figure 7: The secondary field strength map of a cylindrical target (\(\theta\)\(=90^{\circ}\)). (**a**) the component \(\mathbf{B}_{x}\) of the secondary field in the \(x\)-axis direction; (**b**) the component \(\mathbf{B}_{y}\) of the secondary field in the \(y\)-axis direction; (**c**) the component \(\mathbf{B}_{z}\) of the secondary field in the \(z\)-axis direction; (**d**) total secondary field \(\mathbf{B}_{s}\). The horizontal position estimation error \(\bar{u}\) at each simulation condition is the average value obtained from 100 Monte Carlo experiments, which is:  
\[\bar{u}=\frac{\sum\limits_{i=1}^{100}\sqrt{\left(\hat{X}_{0}\left(i\right)-X_{0} \right)^{2}+\left(\hat{Y}_{0}\left(i\right)-Y_{0}\right)^{2}}}{100}, \tag{41}\]  
where \(\left(\hat{X}_{0}\left(i\right),\hat{Y}_{0}\left(i\right)\right)\) is the horizontal position estimation obtained by the \(i\)th Monte Carlo experiment, and \(\left(X_{0},Y_{0}\right)\) is the true position of the detected target. The results of the horizontal position estimation error for the cylindrical target at different \(SNR\) are shown in Figure 8. As \(SNR\) increases, the error gradually decreases. When \(SNR\geq 10\) dB, the error is less than 1 cm. The horizontal position estimation exhibits an excellent performance. To evaluate the depth estimation performance, the depth estimation errors at different simulation conditions are compared in this paper. The depth estimation error \(\bar{d}\) at each simulation condition is the average value obtained from 100 Monte Carlo experiments:  
\[\bar{d}=\frac{\sum\limits_{i=1}^{100}\left|\hat{Z}_{0}\left(i\right)-Z_{0} \right|}{100}, \tag{42}\]  
where \(\hat{Z}_{0}\left(i\right)\) is the depth obtained by the \(i\)th Monte Carlo experiment and \(Z_{0}\) is the true depth of the target. The depth estimation errors of cylindrical targets at different \(SNR\) are compared in Figure 9. It can be seen from the figure that, as \(SNR\) increases, the depth estimate error gradually decreases. After \(SNR\) reaches a certain threshold, the error gradually stabilizes to approximately 6 cm. The FWHM method can obtain only an approximate estimation of the target depth. Even in the case of no noise, there are errors in depth estimation. The depth estimation errors of cylindrical targets with depths in the range of 35-105 cm are compared in Figure 10. The error increases with increasing depth but remains within 8 cm. In summary, the FWHM method provides a method for depth estimation with acceptable performance. Figure 8: Estimation error of the horizontal position of the cylindrical target at different signal-to-noise ratio (SNR) values. To evaluate the estimation performance of the magnetic polarizability tensor \(\mathbf{M}\), the relative error of \(\mathbf{M}\) under different simulation conditions is compared in this paper. The relative error \(\bar{e}_{r}\) for each simulation condition is the average value obtained from 100 Monte Carlo experiments:  
\[\bar{e}_{r}=\frac{\sum\limits_{i=1}^{100}\left\|\hat{\mathbf{M}}_{0}\left(i\right)- \mathbf{M}\right\|_{2}}{100\left\|\mathbf{M}\right\|_{2}}, \tag{43}\]  
where \(\hat{\mathbf{M}}_{0}\left(i\right)\) is the estimation of \(\mathbf{M}\) obtained by using the least squares method in the \(i\)th Monte Carlo experiment.