## 3 Numerical Experiments
### Inversion with Different Initial Models  
The initial models used in the above tests were the idealized background medium models matching the true models. To further explore the sensitivity of our algorithm to the initial models, we conducted experiments using different initial models, as shown in Figures 18 and 19. The homogeneous initial models in Figure 18a,d were designed as the media in the deepest layers of the true models, and the smoothed initial models in Figure 19a,d were constructed by smoothing the true models. Table 5 lists the corresponding parameters for this experiment. Figure 18b,e and Figure 19b,e show the inversion results without model constraints. Figure 18c,f and Figure 19c,f display the 3D inversion results of our TV+MSG schemes. The misfits \(\Phi_{d}(\mathbf{m}^{inv})/\Phi_{d}(\mathbf{m}^{0})\) and model reconstruction errors \(RRMS_{m}\) are shown in Figure 20.  
Figure 18: Homogeneous initial models and inverted models for (**a**–**c**) the relative permittivity and (**d**–**f**) the conductivity: (**a**) homogeneous initial relative permittivity model; (**b**) inverted relative permittivity models without model constraints; (**c**) inverted relative permittivity models with composite model constraint TV+MSG; (**d**) homogeneous initial conductivity model; (**e**) inverted conductivity models without model constraints; (**f**) inverted conductivity models with composite model constraint TV+MSG. Red circles in (**a**) and white \(\times\)-shaped markers in (**d**), respectively, represent the transmitters and receivers in the data acquisition setup for 3D GPR. Parameter settings are shown in Table 5.  
The final model reconstruction errors \(RRMS_{m}\) and structural similarity indices \(SSIM_{m}\), shown in Table 5, indicated that the initial models had significant influences on the inversion results. Compared with the results given above, the homogeneous initial models generated larger \(RRMS_{m}\) values and smaller \(SSIM_{m}\) values. As seen for the reconstructed models in Figure 18, when the initial guess in the shallow region substantially deviated from the true background media, the inversion process exhibited depth-dependent convergence behavior where shallow medium parameters were preferentially optimized during inversion iterations. When using the homogeneous models as the initial models, the model reconstructions, shown in Figure 19, were superior. As evidenced in Figure 20b, the curves of the model reconstruction errors for conductivity without any model constraint strategy (blue dashed line marked with \"o\") displayed an upward trend. However, the TV+MSG regularized inversion exhibited stable behavior with poor initial models--that is, the artifacts were eliminated and the background media were recovered to some extent, although  
Figure 19: Smoothed initial models and inverted models for (**a–c**) the relative permittivity and (**d–f**) the conductivity: (**a**) smoothed initial relative permittivity model; (**b**) inverted relative permittivity models without model constraints; (**c**) inverted relative permittivity models with composite model constraint TV+MSG; (**d**) smoothed initial conductivity model; (**e**) inverted conductivity models without model constraints; (**f**) inverted conductivity models with composite model constraint TV+MSG. Red circles in (**a**) and white \(\times\)-shaped markers in (**d**), respectively, represent the transmitters and receivers in the data acquisition setup for 3D GPR. Parameter settings are shown in Table 5.