## 3 Synthetic Test  
### Single Magnetic Dipole Source Model  
This study employed the single magnetic dipole source model presented in Section 2, \(x_{0}=2.5\) m, \(y_{0}=2.5\) m, \(z_{0}=5\) m, and a magnetic moment of 10 A\(\cdot\)m\({}^{2}\) to assess the capability of the LM-OBF algorithm in estimating the parameters of a magnetic dipole source under strong background noise conditions. The Earth's magnetic inclination is 60\({}^{\circ}\), and the declination is \(-10^{\circ}\). The dipole's inclination is 20\({}^{\circ}\), and the azimuth is 30\({}^{\circ}\). The spatial domain for forward modeling spans a 5 \(\times\) 5 m area with a grid size of 0.5 m. Gaussian random noise ranging from 10% to 97% is introduced, resulting in a SNR varying from 10 to \(-15\) dB to test the LM-OBF algorithm's noise resistance in parameter estimation under different SNR conditions. For the initial values of the LM algorithm, the initial horizontal position is determined by the maximum value of \(E\), the burial depth \(z_{0}\) is set to 1 m to ensure the stability of the forward modeling and optimization process, and the initial values for the dipole's inclination and azimuth are both set to 0. The results of the parameter estimation are shown in Table 2, which displays the differences between each parameter estimated by the LM-OBF method under various SNR conditions and their true values. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**SNR** & \(x_{0}\) & \(y_{0}\) & \(z_{0}\) & \(m\) & \(I_{0}\) & \(A_{0}\) \\ \hline
10 dB & 0.0029 & 0.0016 & 0.0243 & 0.0320 & 0.242 & 0.424 \\
5 dB & 0.0059 & 0.0047 & 0.0305 & 0.0657 & 0.522 & 0.510 \\
0 dB & 0.0126 & 0.0114 & 0.0312 & 0.1205 & \(-\)0.654 & 0.675 \\ \(-\)5 dB & 0.0124 & 0.0153 & 0.0315 & 0.1704 & \(-\)0.676 & \(-\)1.530 \\ \(-\)10 dB & 0.0215 & 0.0182 & 0.0498 & 0.3716 & \(-\)1.085 & \(-\)0.261 \\ \(-\)15 dB & 0.0258 & 0.0415 & 0.0731 & 0.8203 & \(-\)6.328 & \(-\)8.907 \\ \hline \end{tabular}
\end{table} 
Table 2: Parameter optimization errors of a single magnetic dipole source under various SNR conditions. Figure 4: The flowchart of the LM-OBF algorithm. When the SNR is \(-15\) dB, the LM-OBF algorithm effectively recovers the position of the magnetic dipole source, although there are significant errors in estimating the magnitude and direction of the magnetic moment. As the SNR of the data increases, the accuracy of parameter estimation by the LM-OBF algorithm also gradually improves. At an SNR of \(-10\) dB, the algorithm is already capable of accurately recovering the parameters of the magnetic dipole source, with the positional parameter estimation error being less than \(2\%\) and the errors in the magnitude and direction of the magnetic moment being less than \(5\%\). These tests demonstrate the LM-OBF algorithm's strong capabilities in estimating magnetic dipole parameters under low SNR conditions. The data with an SNR of \(-10\) dB is considered an example to demonstrate the effectiveness of the LM-OBF algorithm. Figure 5a represents the original magnetic dipole, Figure 5c shows the distribution of the energy function \(E\) without noise, and Figure 5b,d display the noisy magnetic dipole data and its energy function \(E\) distribution after noise addition, respectively. Figure 5e illustrates the energy function \(E\) estimated using the LM-OBF algorithm for parameter recovery, and Figure 5f shows the residuals of the parameter estimation. Figure 5: (**a**) Magnetic dipole source without noise. (**b**) Magnetic dipole source with \(-10\) dB noise. (**c**) \(E\) of the magnetic dipole source without noise. (**d**) \(E\) of the magnetic dipole source with noise. (**e**) Recovered \(E\) using the results of LM-OBF. (**f**) Residuals. Figure 6a depicts the original magnetic dipole sources, Figure 6c shows the distribution of the energy function E without noise, and Figure 6b,d display the noisy magnetic dipole data and its corresponding energy function \(E\) distribution after noise addition, respectively.