## 3 Example of Theoretical Model Synthesis
### Examples of Oblique Prismatic Synthesis  
As shown in Figure 7, there is a slanted prism composed of multiple single prisms in the ground, and they are buried 500 m below the surface of the earth. The inversion area is set to 10 km \(\times\) 10 km \(\times\) 10 km, and it is divided into 9261 cube units. The magnetic susceptibility of the anomalous body is 0.25 SI, the magnetic susceptibility of the surrounding rock is 0.001 SI, and the geomagnetic field intensity is 5000 nT. The inclination and declination of the geomagnetic field are 90\({}^{\circ}\) and 0\({}^{\circ}\), and the inclination and declination are 90\({}^{\circ}\) and 0\({}^{\circ}\), respectively. The measurement points are evenly arranged on the ground surface directly above the abnormal body. A total of 400 (20 \(\times\) 20) measurement points are arranged, and the measurement points are represented by black circles, as shown in Figure 8. The grid distribution of the abnormal body is the same as that of the single abnormal body. Thus, according to Equation (12), \(\Delta\)V is 12.5 \(\times\) 10\({}^{8}\), and the constant \(\delta\) is also 0.00179. To make the inversion closer to the actual situation, a 5% Gaussian random error is added to the synthetic data to synthesize observation data \(d\), and the magnetic susceptibility value equal to the average magnetic susceptibility of the surrounding rock is selected as the initial inversion model. The data fitting difference (_RMS_) of the inversion is shown in Figure 9. The three-dimensional limited-memory BFGS (L-BFGS) inversion based on the multiplicative objective function is iterated a total of 11 times, and the initial value of _RMS_ converges from 6.71 to 0.88. The traditional three-dimensional inversion is iterated a total of 36 times, and the initial _RMS_ value converges from 6.71 to 0.98. The iteration stop condition is met. 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. \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 & & & \\ three-dimensional inversion & 8 & 3.7 & 0.84 \\ \hline Traditional magnetic & & & \\ three-dimensional inversion & 26 & 1800.2 & 0.92 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the inversion efficiency of a single prism. Figure 6: Inversion results of the traditional three-dimensional magnetic inversion single prism. The horizontal slice diagram and vertical slice diagram of the oblique prism obtained by means of magnetic three-dimensional limited-memory BFGS inversion based on the multiplicative objective function are shown in Figure 10. The horizontal slice diagram and vertical slice diagram of the oblique prism inversion results obtained by means of the traditional magnetic method for three-dimensional inversion are shown in Figure 11. The black boxes in both figures represent the locations of anomalous bodies. It can be seen from the figure that the positioning of the abnormal body with high magnetic permeability in the three-dimensional limited-memory BFGS inversion results based on the multiplicative objective function is more accurate, and the shape of the inversion results is closer to the real abnormal body. The true value of the abnormal body is close to the true magnetic susceptibility of the inversion model of 0.25 SI, which proves that the three-dimensional limited-memory BFGS inversion method based on the multiplicative objective function can be used to obtain the optimal solution. Figure 8: Schematic diagram of the observation system. Figure 7: Oblique prism anomaly. invert the abnormal body accurately. For the skin effect, this is related to the difference in the longitudinal resolution of the magnetic method itself.