## 4 Measured Data Test  
To further verify the feasibility and effectiveness of the three-dimensional L-BFGS inversion method based on the multiplicative objective function, this method is used to carry out an experimental calculation of the measured data.  
Figure 12 shows the measured magnetic anomaly data of the Tongdao copper-nickel deposit in Hunan Province, China. The map uses the Chinese geodetic coordinate system [20]. N-type faults have developed in this area, and the copper-nickel sulphide filled by cracks is enriched in the magnetite and pyrrhotite enrichment area. Therefore, inversion can be performed to predict the spatial scope of copper-nickel deposits by locating the locations of ultrabasic rocks containing magnetite and pyrrhotite [21]. The measured data size is 141 \(\times\) 125, and the grid spacing is 10 m. The magnetic field strength in this area is 47,290 nT, and the geomagnetic inclination and geomagnetic declination are 39.0\({}^{\circ}\) and \(-\)2.5\({}^{\circ}\), respectively.  
In the three-dimensional L-BFGS inversion using the magnetic method based on the multiplicative objective function, the underground subdivision is divided into 30 \(\times\) 30 \(\times\) 30 cube units, and the size of the grid units is a cube with a side length of 50 m.  
The data fitting difference (\(RMS\)) of the inversion is shown in Figure 13. The magnetic three-dimensional limited-memory BFGS inversion based on the multiplicative objective function is iterated a total of 20 times, and the initial \(RMS\) value converges from 6.5 to 0.98. The traditional magnetic three-dimensional inversion is iterated a total of 19 times, and the initial \(RMS\) value converges from 6.5 to 1.13, meeting the iterative stop condition. The \(RMS\) descent curve of the magnetic method based on the multiplicative objective function is smooth, which proves that the magnetic method based on the multiplicative objective function has good stability and convergence.  
Figure 11: Inversion results of the traditional magnetic three-dimensional inversion of the oblique prism model.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Single Prism** & \
\begin{tabular}{c} **Iterations** \\ **Number/Times** \\ \end{tabular} &
\begin{tabular}{c} **Inversion** \\ **Time/Seconds** \\ \end{tabular} & **Final RMS** \\ \hline Limited-memory BFGS & \(11\) & 26.9 & 0.88 \\ three-dimensional inversion & \(36\) & \(1600.6\) & 0.98 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of the inversion efficiency of the oblique prism.  
The measured inversion results of three-dimensional limited-memory BFGS inversion using the magnetic method based on the multiplicative objective function are shown in Figure 14, and the measured results of traditional three-dimensional inversion using the magnetic method are shown in Figure 15. The measured inversion horizontal section results of the three-dimensional limited-memory BFGS inversion based on the multiplicative objective function show that there are two obvious north-eastward faults, and the spatial distribution of underground faults can be clearly seen according to the vertical sections in both directions. The traditional magnetic three-dimensional inversion also has good results. As seen from Figure 15, the inversion results can basically distinguish two faults. In terms of the computational efficiency, as can be seen from Table 3, the measurement of the three-dimensional limited-memory BFGS inversion using the magnetic method  
Figure 12: Magnetic anomaly.  
Figure 13: Fitting difference curve of the measured data inversion.  
based on the multiplicative objective function takes approximately 35 s. However, the traditional three-dimensional inversion using the magnetic method takes approximately 2600 s. Compared with traditional three-dimensional inversion using the magnetic method based on a multiplicative objective function, three-dimensional limited-memory BFGS inversion using the magnetic method has higher inversion efficiency.