## 3 Numerical Experiments
First, we analyze the sensitivity of the regularization factors \(\lambda_{\varepsilon}\) and \(\lambda_{\sigma}\) in the MSG-constrained FWI. Under fixed focusing parameters (\(\beta_{\varepsilon}^{2}=\beta_{\sigma}^{2}=0.5\)), Figures 7a-d and 8a-d show the 3D relative permittivity and conductivity reconstructed models with four regularization factor settings (MSG1-MSG4). The MSG-constrained inversion results showed a progressive improvement over the unconstrained case (Figures 7h and 8h), with artifact reduction and the enhanced focusing of block anomalies as the regularization factors (\(\lambda_{\varepsilon}\) and \(\lambda_{\sigma}\)) increased. In particular, the values of shallow block-shaped anomalous bodies in the conductivity models were closer to those of the true models. However, a larger \(\lambda_{\varepsilon}\) and \(\lambda_{\sigma}\) (MSG3/MSG4) led to overfocused anomalies with size underestimation and then weakened the updating of deep anomalies due to the incorrect reconstruction of the shallow media. Detailed comparisons can be found in Figure 4:5 The MSG3 and MSG4 strategies exhibited sharp boundaries at a depth of 0.5 m, while they failed to capture the actual amplitude in depth. Figure 10a also demonstrates the poor data fitting and convergence of MSG3 and MSG4. As shown in Figure 10b, while the relative permittivity model errors of MSG3 and MSG4 (blue and purple dashed lines with markers * and \(\triangle\)) decreased rapidly during the initial iterations, excessive focusing on shallow anomalies impeded error reduction in later stages. The reconstruction errors of the conductivity models, shown in Figure 10b, exhibited similar early-stage behavior but demonstrated particularly poor late-iteration performance, ultimately showing increasing error trends. The difficulties of updating the media in depth also led to greater calculation times, as shown in Figure 4:6 These results demonstrate that regularization parameter selection critically impacts MSG-constrained FWI's performance. In this test, MSG2 performed the best overall in terms of the boundary clarity of the reconstructed models, the convergence of the misfit, and the reduction of the model reconstruction errors. Figure 4:7 Curves of the misfits and model reconstruction errors in the TV regularization test with different \(\beta_{\varepsilon}^{2}\) and \(\beta_{\sigma}^{2}\) with respect to the number of iterations: (**a**) data misfits and (**b**) model reconstruction errors \(RRMS_{m}\) for the relative permittivity and conductivity models. Parameter settings of strategies MSG2 and MSG5–MSG7 are shown in Figure 4:8  
Then, while keeping the regularization factors constant (\(\lambda_{\varepsilon}=\lambda_{\sigma}=0.001\) as in MSG2), as shown in Figure 7b,e-g and Figure 8b,e-g, we evaluated how varying the focusing parameters (MSG2/MSG5-MSG7) influenced the reconstructions of both model parameters. The comparative results demonstrate that the improvement persists in the tested range of focusing parameters with the MSG framework. Smaller values of focusing factors \(\beta_{\varepsilon}^{2}\) and \(\beta_{\sigma}^{2}\) produce sharper shallow-anomaly boundaries with fewer oscillation artifacts and smoother backgrounds. For example, when \(\beta_{\varepsilon}^{2}=\beta_{\sigma}^{2}=0.1\) (Figures 7e and 8e), the results demonstrate minimal artifacts in background regions and superior shallow-anomaly resolution, although the block-shaped anomaly C in the deep part is more difficult to distinguish. As evidenced by the 1D model distributions in Figure 11, increasing focusing factors (see the results for MSG5, MSG6, MSG2, and MSG7) correspond to weaker focusing capabilities. For instance, the purple curves (\(\beta_{\sigma}^{2}=0.8\)) exhibit the smoothest trends compared with other results. Notably, the differences caused by the focusing factors \(\beta_{\varepsilon}^{2}\) and \(\beta_{\sigma}^{2}\) are not significant compared those due to the variation of the regularization factors \(\lambda_{\varepsilon}\) and \(\lambda_{\sigma}\).