## 4 Discussion  
The results obtained demonstrate the high accuracy of reconstructing the dielectric permittivity of pavement layers, as evidenced by the low level of discrepancy between the exact and reconstructed values. In particular, the average relative error of \(\varepsilon_{r}\) reconstruction was less than 1%, and the algorithm converged in 28 iterations, as shown in the upper part  
Figure 14: Exact and approximate values of dielectric permittivity and medium response in the wave formulation at \(n=28\). Figure 13: Exact and approximate values of dielectric permittivity and medium response in the wave formulation at \(n=20\). of Figure 14. These results demonstrate the stability of the proposed discrete method for interpreting radar images. The developed algorithm is based on solving the adjoint problem and minimizing the quadratic functional of the discrepancy between the measured and modeled environmental responses. An important advantage of the method is the preservation of physical laws, particularly the law of conservation of energy, through the use of conservative finite-difference schemes. This ensures physical consistency and numerical stability when modeling the propagation of electromagnetic waves in a vertically inhomogeneous medium. Based on this review, the following methods can be identified for solving the interpretation problem (determining the geoelectric section):  
* engineering and technical methods [25];
* selection methods and regularization methods [20];
* machine learning methods [12; 29; 30];
* optimization methods [19; 21]. In comparison with the aforementioned methods, the optimization method demonstrates superior practicality for solving the problem. As noted earlier, it is theoretically justified [22], analyzed for convergence of the iterative steepest descent method [33], and its convergence, along with conservation and uniformity properties, is numerically validated using experimental data (see the upper part of Figure 14). A comparison with existing approaches, presented in Table 3, shows that, in contrast to traditional engineering methods based on heuristic approximations and regularization schemes (e.g., SVD or [PERSON]'s method), the proposed method achieves higher accuracy. Compared to machine learning methods, the developed approach eliminates the need for labeled training datasets, making it more practical in real-world data acquisition scenarios. The formulation of the problem for the geoelectric equation considered here is physically justified in [20; 22]. The rationale behind these explanations is as follows. Assuming that the source is a long cable located on the ground surface and that the dielectric permittivity depends solely on depth (which is valid in this case, as the road surface consists of layers), the system of [PERSON]'s equations can be simplified. Specifically, the equations depend only on the horizontal component of the electric field, while the magnetic field components depend on two variables. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Criteria** & **Engineering** & **Regularization** & **Machine Learning** & **Proposed Method** \\  & **Methods**[25] & **Methods**[20] & [12; 29; 30] & [19; 21] \\ \hline Physical validity & Low & Moderate & Low & High \\ Average & & & & \\ reconstruction error & \(\sim 15\)–\(20\%\) & \(\sim 10\)–\(15\%\) & \(\sim 5\)–\(10\%\) & \(<\)\(1\%\) \\ (\(\epsilon_{r}\)) & & & & \\ Computation time & \(<\)1 & \(\sim 3\)–\(5\) & \(\sim 1800\)–\(3600\) & \(\sim 60\)–\(120\) \\ per profile (seconds) & & & & \\ Data volume & Minimal & Moderate & High & Moderate \\ Customization & Limited & Implementation- & Model-dependent & Flexible (thickness, \(\epsilon_{r}\), steps) \\ flexibility & & & & Yes (initial \\ Need for a priori & & & & \\ data & & & & \\ \multirow{2}{*}{Environmental} & \multirow{2}{*}{Simplified} & \multirow{2}{*}{Linear} & \multirow{2}{*}{Arbitrary} & \multirow{2}{*}{Piecewise constant} \\  & & & & \\ \end{tabular}
\end{table}