## 3 Results  
The developed numerical algorithms were tested using real-world data from pavement structures obtained experimentally. As part of this research project, approximately 2000 km of highways in the Akmola region were investigated using GPR. During the calculations, the characteristics of reconstructing the dielectric permittivity for various layers of the medium were analyzed. Additionally, the accuracy of the reconstructed parameters was evaluated. A layered medium model representing the road surface is considered:  
* The first layer consists of asphalt (fine- and coarse-grained) with a glass fiber reinforcement interlayer;
* The second layer is a base layer composed of a mixture of fine crushed stone with a grain size of 4 mm and containing 7% cement by weight;
* The third layer is a base layer composed of a mixture of coarse crushed stone with a grain size of 8 mm;
* The fourth layer is the subgrade layer, which is soil (loam). These data were obtained using the CartScan software package, whose algorithm is described in detail in Section 2.1. Table 1 illustrates the properties of the medium validated by experts from the Road Asset Quality Center. Since the formulation of the problem (6)-(8) was derived from the system of [PERSON]'s equations in the wave formulation, the skin effect occurs as a physical phenomenon. Consequently, the signal can attenuate before reaching the subgrade layer. We determine the depth at which the skin effect occurs. To achieve this, we use the following formula [28]:  
\[H_{s}=\sqrt{\frac{2}{\mu\omega_{0}\sigma}}, \tag{31}\]  
where the reference angular frequency \(\omega_{0}\) is equal to \(\sigma/\varepsilon\). To calculate the depth of the skin effect, we consider the dielectric permittivity \(\varepsilon\) and the electrical conductivity \(\sigma\) of the subgrade layer. The result is \(H_{s}=1.27\) m.  
The graphs of the source (8), the medium's response (5), and the solution of the problem (6)-(8) in the wave formulation are presented below. The Gaussian function (9) with the parameter \(\alpha=1.5\times 10^{8}\) was selected to represent the source (Figure 9). In Figures 9 and 10, time \(t\) ranges from 0 to 50 ns, and depth \(z\) ranges from 0 to 1.27 m. Both the solution and the medium's response correspond to the electrical voltage, while the source corresponds to the gradient of the electrical voltage along the \(Oz\)-axis. Electrical voltage is measured in [V/m], and its gradient is measured in [V/m\({}^{2}\)]. In the case of the wave formulation, all layers of the underground medium were successfully reconstructed. The reconstruction was achieved in 28 iterations with a termination parameter \(\delta=10^{-4}\) for the optimization method (19), resulting in an accuracy of 10\(-\)2 (relative error less than 1%), which represents an excellent outcome. The convergence of the optimization method is highly sensitive to the choice of the initial approximation. The initial approximations of the relative dielectric permittivity were set to 1, representing the minimal physically meaningful value. The initial approximation was selected to be constant across the entire domain and as low as possible (see Figure 11, indicated by the \begin{table}
\begin{tabular}{c c c c c} \hline
**Layer Number** & **1-st Layer** & **2-nd Layer** & **3-rd Layer** & **Subgrade Layer** \\ \hline \(\varepsilon_{r}\) & 3.0 & 6.0 & 4.0 & 20.0 \\ \(\sigma\) [S/m] & 0.00512 & 0.00725 & 0.00592 & 0.0132 \\ \(z\) [m] & 0.15 & 0.45 & 0.7 & 2.2 \\ \hline \end{tabular} \(\varepsilon_{r}\) is relative dielectric permittivity; \(\sigma\) is electrical conductivity; \(z\) is depth.  
\end{table} 
Table 1: Model of the road pavement. Figure 9: Source and medium response in the wave formulation. (**a**) Source in the wave formulation; (**b**) medium response in the wave formulation. dashed line).