## 3 Numerical Experiments
Meanwhile,  
\[SSIM_{m}=\frac{(2a_{m^{inv}}a_{m^{true}}+c_{1})(2b_{m^{inv}m^{true}}+c_{2})}{ \left(a_{m^{inv}}^{2}+a_{m^{true}}^{2}+c_{1}\right)\left(b_{m^{inv}}^{2}+b_{m^ {true}}^{2}+c_{2}\right)} \tag{23}\]  
where \(a_{m}\), \(b_{m}\), and \(b_{mm}\) represent the mean, variance, and covariance values of the 3D images corresponding to the models; \(c_{1}\) and \(c_{2}\) are constants that maintain stability. In our tests, the smaller the \(RRMS_{m}\) value and the closer the \(SSIM_{m}\) value is to 1, the better the inversion result. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Area** & **Air** & **Layer 1** & **Layer 2** & **A** & **B** & **C** & **D** \\ \hline \(\varepsilon_{r}\) & 1 & 3 & 4 & 8 & 1 & 8 & 1 \\ \(\sigma\) (mS/m) & 0 & 1 & 3 & 10 & 0 & 10 & 0 \\ \hline \end{tabular}
\end{table} 
Table 1: Electrical material properties of true models. The final model reconstruction errors \(RRMS_{m}\) of \(\varepsilon_{r}\) and \(\sigma\) in Table 2 indicate that strategy TV2 exhibited the smallest values, demonstrating that the TV2-regularized inversion obtained the best agreement between the inverted and true models. Meanwhile, the structural similarity indices \(SSIM_{m}\) of \(\varepsilon_{r}\) and \(\sigma\) in Table 2 suggest that strategy TV3 showed the closest values to 1, indicating the highest structural fidelity between the inverted and true models. However, under the same number of iterations, inversion with the TV3 strategy had the longest computation time (approximately 18% longer), indicating that the line  
Figure 4: The 3D images for (**a–d**) the relative permittivity models and (**e–h**) the conductivity models: (**a,e**) inverted models without model constraints; (**b,f**) inverted models with model constraint TV1; (**c,g**) inverted models with composite model constraint TV2; (**d,h**) inverted models with composite model constraint TV3. Parameter settings of strategies TV1–TV3 are shown in Table 2.  
Figure 5: Comparison of TV regularization test in 1D distribution 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 of strategies TV1–TV3 are shown in Table 2.  
search process in inversion was difficult and thus the stability decreased when using this parameter setting in TV3. As shown in Figure 4, compared with the inversion without model constraints, the oscillation artifacts in the inversion under TV model constraints are mitigated. As shown in Figure 4b-d,f-h, when \(\lambda_{\varepsilon}\) and \(\lambda_{\sigma}\) increase, the artifacts in the reconstructed models gradually decrease. However, excessive \(\lambda_{\varepsilon}\) and \(\lambda_{\sigma}\) values (Figure 4d,h) lead to the over-smoothing of block anomalies, suppressing reconstruction artifacts at the cost of losing the true medium characteristics. This artifact suppression also occurred in the conductivity reconstruction of anomaly C in Figure 4g, where the variations in the updating gradients were minor and mistakenly weakened. Thus, a trade-off between anomaly preservation and artifact suppression is important in model-constrained inversion. The 1D profiles in Figure 5 further validate these findings. Both the TV1 and TV2 curves demonstrate improved smoothness compared to the unregularized case (curve \"none\"), while TV3 exhibits excessive smoothing, failing to capture the actual amplitude variations of the model parameters. Figure 6 demonstrates several key findings regarding the inversion convergence. The data misfits in Figure 6a for TV1 and TV2 are consistently lower than in the unregularized case, suggesting that appropriate TV regularization improves inversion convergence.