## 2 Mixed L\({}_{\rm p}\) Norm Regularization for 3-D Magnetic Inversion
### Synthetic Data Test  
The synthetic model consists of two cubes. Cube 1 has a side length of 400 m, with its center located at X = 1000 m, Y = 1000 m, and Z = 500 m, and a magnetic susceptibility of 0.04 (SI). Cube 2 has a side length of 200 m, with its center located at X = 2000 m, Y = 2000 m, and Z = 300 m, and a magnetic susceptibility of 0.08 (SI). The geomagnetic field strength is 55,000 nT, with a dip angle of 90\({}^{\circ}\) and a declination of 0\({}^{\circ}\). Both cubes have a magnetic inclination of 90\({}^{\circ}\) and a declination of 0\({}^{\circ}\). The magnetic anomalies generated by the synthetic model at the observation plane of Z = 0 m are shown in Figure 3, which includes 2% Gaussian noise. Figures 4 and 5 present the inversion results using conventional L\({}_{1}\) and L\({}_{2}\) norms, as well as the mixed L\({}_{\rm p}\) norm regularization. It can be observed that the L\({}_{1}\) norm provides stronger boundary constraints compared to the L\({}_{2}\) norm, with the inversion results more effectively restoring the model boundaries both laterally and vertically. The L\({}_{2}\) norm inversion results display divergence in the vertical boundaries of the anomaly body. In contrast, the mixed L\({}_{\rm p}\) norm not only better restores the model boundaries but also yields physical-property values closer to the true values. Among the mixed norms, L\({}_{02}\) provides better results than L\({}_{12}\). The result along profile X = 1000 m at the depth slice of Z = 350 m (Figure 6a) shows that the horizontal boundary delineation of the anomalous body is similar for both conventional norm inversion and mixed L\({}_{\rm p}\) norm regularization. However, the restoration of physical properties is notably superior using the mixed norm, with peak values following the order L\({}_{12}\)\(>\) L\({}_{02}\)\(>\) L\({}_{1}\)\(>\) L\({}_{2}\). The result at the position of X = 1000 m along profile A (as illustrated in Figure 6b) reveals significant differences in the delineation of the vertical boundary of the anomalous body across the various norms. Among the different norms, L\({}_{02}\) provides the most focused result, while L\({}_{1}\) and L\({}_{12}\) produce similar outcomes, and L\({}_{2}\) exhibits the most divergence, with a distinct \"tailing\" effect along the Z direction. The restoration of physical properties follows a similar trend as Figure 6a, with peak values of L\({}_{12}\)\(>\) L\({}_{02}\)\(>\) L\({}_{1}\)\(>\) L\({}_{2}\). Overall, from the perspective of boundary delineation and physical property restoration,  
Figure 3: Magnetic anomalies caused by the synthetic model. The white dashed boxes outline the true cubes, while the black dashed line locates the profile A shown in Figures 4 and 5.  
the mixed L\({}_{\rm p}\) norm regularization inversion yields better results compared to conventional L\({}_{1}\) and L\({}_{2}\) norm regularization, with L\({}_{02}\) outperforming L\({}_{01}\). Figure 4: Magnetic inversion results using L\({}_{1}\) and L\({}_{2}\) norm regularization for the synthetic model. (**a,b**) show the results at depth slice of Z = 350 m and along profile A using the L\({}_{1}\) norm regularization algorithm; (**c,d**) are those using the L\({}_{2}\) norm regularization algorithm. The black boxes outline the true model, and the black dashed lines in (**a–d**) locate the profiles shown in Figure 6.  
Figure 5: Magnetic inversion results using mixed L\({}_{\rm p}\) norm regularization for the synthetic model. (**a,b**) show the results at depth slice of Z = 350 m and along profile A using the L\({}_{02}\) norm regularization algorithm; (**c,d**) are those using the L\({}_{12}\) norm regularization algorithm. The black boxes outline the true model, and the black dashed lines in (**a–d**) locate the profiles shown in Figure 6.  
Indeed, there is a clear mismatch in the depth of anomalous bodies between the true and recovered models.