## 5 Conclusions  
In this study, a discrete numerical algorithm was developed and implemented to solve the practical problem of reconstructing a geophysical section aimed at determining the physical characteristics of the pavement, such as the dielectric permittivity of the layers. The relevance of this work stems from the necessity to develop highly accurate and reproducible methods for monitoring road infrastructure conditions.  
A non-destructive subsurface radar sounding method employing an OKO-2 series GPR was utilized in this research. Primary signal processing was performed in the CartScan software environment, where noise removal, trace alignment, and AGC were applied. The obtained radar images served as the basis for subsequent numerical modeling based on [PERSON]'s equations.  
To determine the geoelectric properties of the medium, algorithms for numerical solutions of direct and adjoint problems with conservative properties were developed. This enabled the implementation of an optimization approach to solving the inverse problem based on minimizing the quadratic functional between the theoretical and measured responses of the medium.  
The proposed algorithm outperforms traditional engineering methods in terms of accuracy, interpretability, and noise tolerance. In contrast to machine learning approaches, it does not require large training datasets and preserves the physical meaning of the parameters. This method is non-destructive and obviates the need for core drilling, thereby significantly reducing costs in terms of both time and resources during the inspection of road infrastructure.  
Through experimental studies conducted with the CartScan software, which enables the determination of layer-specific dielectric permittivity, this data was utilized to test the proposed algorithm. Based on numerical calculations, as shown in the upper part of Figure 11, the lowest value of the dielectric permittivity (\(\epsilon_{r}=1\)) was chosen as the initial approximation. A discrete algorithm was developed for solving the optimization problem of reconstructing the dielectric permittivity. The algorithm achieved an accuracy of \(10^{-2}\) within 28 iterations, as shown in the upper part of Figure 14 and defined by the error formula (32).  
Starting from the initial approximation, assumed to be a constant \(\epsilon_{r}=1\) at all points, the algorithm successfully reconstructed a vertically inhomogeneous four-layer medium. This demonstrates that the developed algorithm can be effectively applied in practice at any point along the route to assess the condition of the road surface.  
Future work may focus on extending the method to two-dimensional and three-dimensional models, as well as integrating it with neural network approaches to automate the interpretation of GPR data.  
**Author Contributions:** Conceptualization, K.I.; Methodology, K.I. and A.K.; Software, A.T.; Formal analysis, N.G.; Investigation, K.I. and A.K.; Writing--original draft, N.U.; Writing--review & editing, A.T. and K.S.; Visualization, A.T. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research is funded by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP19680361).  
**Data Availability Statement:** The data that support the findings of this study are available from the corresponding author upon reasonable request.  
**Conflicts of Interest:** The authors declare no conflict of interest.