## 4 Discussion  
In an exploration scenario with high-contrast subsurface anomalies, the TV and MSG regularization methods have been proven to be suitable to solve such imaging problems. However, the performance of standalone TV regularization or MSG regularization for dual-parameter imaging in our experiments was sensitive to the different parameters of the subsurface media. The simultaneous optimal reconstruction of both the relative permittivity and conductivity parameters remains unachievable due to their differing physical sensitivities in the inversion process. In this study, the analysis and comparison of the inversion results under different regularization conditions yields important insights for the dual-parameter FWI of 3D GPR. Specifically, our results demonstrate that the combination of different regularizations has the potential to provide more reliable and characteristic-dependent reconstruction for dual- or multi-parameter inversion problems. For relative permittivity reconstruction, our analysis indicates that TV regularization provides optimal performance by effectively suppressing artifacts while maintaining an appropriate model resolution. The MSG-constrained permittivity results, even with optimal regularization factors and focusing parameters, show inferior performance compared to TV regularization, with a tendency towards overfocusing. This finding is particularly significant as it confirms that unregularized permittivity inversion already achieves sufficient focusing, suggesting that the primary role of regularization for permittivity should be artifact reduction rather than structural enhancement. For conductivity estimation, however, MSG regularization proves superior to TV constraints. While anomalies with high conductivity in deep regions will lose a certain degree of accuracy, the MSG approach could significantly mitigate the volumetric effects characteristic of near-subsurface conductivity inversion. The stable downward trend in the conductivity model reconstruction error curves provides strong evidence for MSG's optimization effectiveness. This differential performance between parameters highlights the importance of our composite regularization strategy, which allows TV and MSG constraints to be selectively applied according to each parameter's distinct inversion characteristics. Figure 20: Curves of the misfits and model reconstruction errors in the tests of different initial models with respect to the number of iterations: (**a**) data misfits and (**b**) model reconstruction errors \(RRMS_{m}\) for the relative permittivity and conductivity models. “h” and “s” in the legend refer to the homogeneous initial models and smoothed initial models. Parameter settings are shown in Table 5.  
Our primary contribution in this study is the proposal of a theoretical composite regularization strategy that overcomes the sensitivity disparity between conductivity and permittivity when simultaneously inverting the two parameters in joint inversion, delivering enhanced accuracy and stability. The analysis of the TV, MSG, and TV+MSG regularization approaches, which was combined with in-depth parameter selection studies, provides guidance for model-constrained inversion in multi-parameter imaging. It should be noted that our discussion is limited to 3D imaging problems involving high-contrast media with certain background information, leaving the optimal constraint strategies for other cases open for future research. Table 6 lists the main limitations and corresponding alternative methods that deserve attention in the future. Key considerations may include the automatic tuning of the parameters in different regularizations, noise sensitivity characteristics, and the spatial distribution features in more complex geological scenarios. It is also necessary to systematically discuss the sensitivity of the initial models, improve the accuracy of high conductivity in deep regions, consider applications to field data, and reduce the computational cost of 3D FWI, etc.