## 3 Numerical Experiments
### Inversion with TV+MSG Regularization  
In this case, we performed and analyzed the dual-parameter FWI results of our proposed TV+MSG composite constraints. Figure 13a,e and Figure 13b,f display the 3D inversion results of the conventional TV-regularized and the MSG-regularized schemes with the optimal parameter settings (TV2 and MSG2 in the above tests). Figure 13c,d,g,h show the reconstructed 3D models with our TV+MSG approach, where the relative permittivity models were constrained with TV (\(\lambda_{\varepsilon}=0.01\)) and the conductivity models were constrained with MSG (\(\lambda_{\sigma}=0.0025\), \(\beta_{\sigma}^{2}=0.5\) or \(0.2\)). To better compare and analyze the boundary reconstruction accuracy of the blocky anomalies, we plotted the 2D slices in the \(x\)-\(y\) plane crossing through the surfaces of these anomalous bodies, as shown in Figures 14 and 15. Moreover, comparisons of the 1D vertical profiles of the models are plotted in Figure 16. The misfits \(\Phi_{d}(\mathbf{m}^{\text{im}\sigma})/\Phi_{d}(\mathbf{m}^{0})\) and model reconstruction errors \(RRMS_{m}\) with different regularizations are shown in Figure 17. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **None** & **TV2** & **MSG2** & **TV+MSG (0.5)** & **TV+MSG (0.2)** \\ \hline \(\lambda_{\varepsilon}\) in TV & / & 0.01 & / & 0.01 & 0.01 \\ \(\lambda_{\sigma}\) in TV & / & 0.005 & / & / & / \\ \(\lambda_{\varepsilon}\) in MSG & / & / & 0.001 & / & / \\ \(\lambda_{\sigma}\) in MSG & / & / & 0.5 & 0.0025 & 0.0025 \\ \(\beta_{\varepsilon}^{2}\) in MSG & / & / & 0.001 & / & / \\ \(\beta_{\sigma}^{2}\) in MSG & / & / & 0.5 & 0.5 & 0.2 \\ Total iterations & 51 & 51 & 51 & 51 & 51 \\ PDE solvers & 110 & 106 & 110 & 106 & 106 \\ Time (min) & 326.769 & 303.523 & 327.447 & 303.438 & 304.488 \\ \(RRMS_{m}\) of \(\varepsilon_{r}\) & 0.853 & 0.827 & 0.843 & 0.827 & 0.828 \\ \(RRMS_{m}\) of \(\sigma\) & 0.970 & 0.970 & 0.964 & 0.967 & 0.959 \\ \(SSIM_{m}\) of \(\varepsilon_{r}\) & 0.742 & 0.882 & 0.797 & 0.890 & 0.889 \\ \(SSIM_{m}\) of \(\sigma\) & 0.624 & 0.781 & 0.686 & 0.672 & 0.723 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of unregularized inversion and TV+MSG-constrained inversion using different regularization factors. Table 4 summarizes the experimental parameters employed in the test cases, including both the constrained and unconstrained approaches. The final model reconstruction errors \(RRMS_{m}\) indicated that strategy TV2 and the composite strategy TV+MSG exhibited the smallest values for \(\varepsilon_{r}\), while strategy MSG2 and the composite strategy TV+MSG exhibited the smallest values for \(\sigma\). The structural similarity indices \(SSIM_{m}\) for \(\varepsilon_{r}\) suggested that the composite strategy TV+MSG achieved the highest structural fidelity between the inverted and true models. Although the \(SSIM_{m}\) for \(\sigma\) in strategy TV2 showed the closest values to 1, the model reconstruction error \(RRMS_{m}\) for \(\sigma\) was the same as in the case without model constraints. Notably, the \(SSIM_{m}\) results for \(\sigma\) in strategy MSG2 and the composite strategy TV+MSG were better than the results obtained without model constraints. These results,  
Figure 14: 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 relative permittivity models: (**a,g**) true models; (**b,h**) inverted models without model constraints; (**c,i**) 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).