## 3 Numerical Experiments
However, excessive regularization (TV3) slowed data fitting, with its final misfit being approximately 15 times higher than in the other cases, indicating that improper regularization impedes inversion convergence. The relative permittivity errors (solid line) in Figure 6b follow a similar trend: TV1 and TV2 achieved faster error reductions than unregularized inversion, while TV3 lagged significantly. The conductivity errors (dashed line) in Figure 6b show less regularity, but it could also be concluded that over-regularization degrades conductivity recovery. When comprehensively evaluating the inversion results in Figures 4-6 and Table 2, TV2 exhibits optimal overall performance compared to the other cases in terms of the clarity of the high-contrast subsurface anomalies, inversion convergence, the reduction of reconstruction errors, improvements in structural fidelity, and calculation times. However, improper regularization factor selection will lead to problematic oversmoothing that suppresses correct gradient changes and severely impedes convergence (15 times higher final misfit). In addition, while TV regularization improved the reconstruction results of the relative permittivity models, its effectiveness for conductivity remains limited. This difference arises because, in FWI for GPR, conductivity models mainly exhibit large volumetric  
Figure 6: Curves of the misfits and model reconstruction errors in the TV regularization test with respect to iterations: (**a**) data misfits and (**b**) model reconstruction errors \(RRMS_{m}\) for the relative permittivity and conductivity models. Parameter settings of strategies TV1–TV3 are shown in Table 2.  
Figure 8: The 3D images for the conductivity models: (**a**–**g**) inverted models with model constraints MSG1–MSG7; (**h**) inverted models without model constraints. Parameter settings of strategies MSG1–MSG7 are shown in Table 3.  
Figure 9: Comparison of MSG regularization tests with different \(\lambda_{\varepsilon}\) and \(\lambda_{\sigma}\) 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 of strategies MSG1–MSG4 are shown in Figure 4:0  
Figure 4:1 Comparison of MSG regularization tests with different \(\beta_{2}^{2}\) and \(\beta_{\sigma}^{2}\) 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 of strategies MSG2 and MSG5–MSG7 are shown in Figure 4:2  
Figure 4:3 Curves of the misfits and model reconstruction errors in the MSG regularization test with different \(\lambda_{\epsilon}\) and \(\lambda_{\sigma}\) 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 MSG1–MSG4 are shown in Figure 4:4  
Relevant parameters corresponding to the inversion comparisons listed in Table 3 indicate that strategy MSG2 exhibited the smallest values for the final model reconstruction errors \(RRMS_{m}\) of \(\varepsilon_{r}\) and \(\sigma\), indicating that MSG2-regularized inversion obtained the best agreement between the inverted and true models. Moreover, strategies MSG3 and MSG4 showed closer values to 1 for the structural similarity indices \(SSIM_{m}\) of \(\varepsilon_{r}\) and \(\sigma\), indicating higher structural fidelity between the inverted and true models. Meanwhile, the calculation time for inversion with the MSG3 or MSG4 strategies was approximately 50% longer than in the other strategies. Overall, the differences in the parameters in Table 3 across MSG1-MSG2 and MSG5-MSG7 were similar.