## 3 Numerical Experiments
### Inversion with TV+MSG Regularization
Parameter settings are shown in Table 4.  
Figure 13: The 3D images for (**a–d**) the relative permittivity models and (**e–h**) the conductivity models: (**a,e**) of inverted models with model constraint TV2; (**b,f**) of inverted models with model constraint MSG2; (**c,g**) of inverted models with composite model constraint TV+MSG (0.5); (**d,h**) of inverted models with composite model constraint TV+MSG (0.2). Parameter settings are shown in Table 4.  
shown in Table 4, indicate that TV regularization and MSG regularization are more effective in models of relative permittivity and conductivity, respectively. Figure 16: Comparison of TV+MSG regularization test in 1D distributions for (**a–d**) the relative permittivity models and (**e–h**) the conductivity models at different positions: (**a,e**) \(x\) = 3.84 m, \(y\) = 3.84 m; (**b,f**) \(x\) = 2.40 m, \(y\) = 2.40 m; (**c,g**) \(x\) = 2.40 m, \(y\) = 3.84 m; (**d,h**) \(x\) = 3.84 m, \(y\) = 2.40 m. Parameter settings are shown in Table 4.  
Figure 15: The 2D slices crossing through the surfaces of the block-shaped anomalous bodies at depths (**a–f**) of \(z\) = 0.48 m and (**g–l**) of \(z\) = 2.64 m for the conductivity models: (**a,g**) true models; (**b,h**) inverted models without model constraints; (**c,l**) inverted models with model constraint TV2; (**d,j**) inverted models with model constraint MSG2; (**e,k**) inverted models with composite model constraint TV+MSG (0.5); (**f,l**) inverted models with composite model constraint TV+MSG (0.2). Parameter settings are shown in Table 4.  
Regarding the reconstructions of the relative permittivity models, as evidenced in Figure 13, the inversion using the composite constraint strategy (Figure 13c,d) achieved similar model reconstruction effects with fewer artifacts compared to the standalone TV2 (Figure 13a), while retaining the correct updates of the deep block-shaped anomalous bodies compared with the standalone MSG2 (Figure 13b). To clearly compare the reconstruction effects in terms of blocky anomalies' boundaries, in Figure 14, we plot 2D images of the relative permittivity models at the positions of the anomalies' upper surfaces (\(z=0.48\) m and \(z=2.64\) m). The values and boundaries of the inverted models in Figure 14e,k and Figure 14f,l (with model constraint TV+MSG) were more similar to those shown in Figure 14c,i (with model constraint TV2). These models exhibited closer fitting to the boundaries of the true blocky anomalous bodies shown in Figure 14a,g than those in Figure 14b,h (without model constraints) and Figure 14d,j (with model constraint MSG2). In addition, the 1D profiles in Figure 16a-d show that the TV+MSG or standalone TV2 constraint strategies achieved model recovery for the relative permittivity at the positions of the background medium, with smoother values compared with the standalone MSG2 approach. In conductivity inversion, the reconstructed models constrained by the TV+MSG regularization strategy (Figure 13g,h) displayed more focused inversion effects with clearer boundaries and more accurate value recovery for the shallow block anomalies (A and B) than that using standalone MSG2, as shown in Figure 13f. The 2D images of the conductivity models at the positions of the anomalies' (A and B) upper surfaces (z = 0.48 m) in Figure 15e,f (with model constraint TV+MSG) also show more focused and clear boundaries than in other inverted images. Moreover, the 1D profiles at positions within \(z=0\) m to \(z=1.5\) m of the conductivity models in Figure 16e-h demonstrate that TV2 regularization oversmoothed the anomalous regions, whereas the MSG2 strategy improved the blocky anomaly recovery, and the TV+MSG approach further sharpened the conductivity values' focusing. This is because the relative permittivity recovery under the composite regularization strategy is closer to the true model than in the standalone MSG model-constrained inversion results.