## 3 Example of Theoretical Model Synthesis
### Example of Single Prism Synthesis  
As shown in Figure 2, a single prism with a size of 4 km \(\times\) 4 km \(\times\) 4 km is placed underground, and the top is buried at a depth of 2 km. 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 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 3. According to Equation (12), \(\Delta\)V is 12.5 \(\times\) 10\({}^{8}\) and the constant \(\delta\) is 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 initial model of inversion is set to the same value as the surrounding rock, and all subsequent theoretical model calculations are the same. The data fitting difference (\(RMS\)) of the inversion is shown in Figure 4. The three-dimensional limited-memory BFGS (L-BFGS) inversion based on the multiplicative objective function is iterated eight times, and the initial \(RMS\) value converges from 9.33 to 0.89. The traditional three-dimensional inversion is iterated 26 times, and the initial \(RMS\) value converges from 9.33 to 1.11, 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. The horizontal slice diagram and vertical slice diagram of a single prism obtained by means of magnetic three-dimensional limited-memory BFGS inversion based on the multiplicative objective function are shown in Figure 5. The horizontal slice diagram and vertical slice diagram of the single-prism inversion results of the traditional three-dimensional magnetic inversion are shown in Figure 6. The black boxes in both figures represent the locations of anomalous bodies. It can be seen from the figure that the positioning of the high-magnetic-permeability anomaly 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 anomaly. 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 invert the abnormal body more accurately. Although the traditional  
Figure 3: Schematic diagram of the observation system. Figure 2: Single prism anomaly. three-dimensional magnetic inversion results show that the shape recovery of the abnormal body is also good, the true recovery of the abnormal body is poor and only recovers to approximately 0.15 SI. As shown in Table 1, compared with the traditional magnetic three-dimensional inversion, the _RMS_ of the magnetic three-dimensional limited-memory BFGS inversion based on the multiplicative objective function drops below 1, and both have better inversion results. However, the time required for the magnetic three-dimensional limited-memory BFGS inversion based on the multiplicative objective function is only approximately 3.7 s, while the logarithmic obstacle method takes more than 1800 s. The inversion efficiency is thus greatly improved.