## 2 Materials and Methods
### Formulation of the Direct and Inverse Problems of GPR  
For a more rigorous description of signal propagation in an inhomogeneous medium, the forward and inverse problems of GPR are formulated. The forward problem simulates the behavior of the electromagnetic field given known parameters, while the inverse problem focuses on recovering unknown parameters of the subsurface medium, which is crucial for the subsequent interpretation of GPR data.  
The physical formulation of a GPR problem is as follows: an external current source \(j^{xx}\) is activated on the asphalt surface, which has a bell-shaped temporal profile \(r(t)\). Simultaneously, the electromagnetic field (the medium's response) is measured on the asphalt surface over a duration of approximately 50 nanoseconds. The measurement duration depends on the frequency of the electromagnetic pulse; for instance, it is approximately 50 ns for 400 MHz and 16 ns for 1000 MHz. Based on these measurements, it is required to determine the properties of the medium (dielectric permittivity \(\varepsilon\), electrical conductivity \(\sigma\)) at depths ranging from zero to 2.2 m.  
Consider a problem in which the parameters \(\varepsilon\) and \(\sigma\) depend on depth \(z\), and the external current source is modeled as a cable of considerable length, positioned centrally and aligned along the roadway (along the \(Oz\)-axis). In this scenario, neglecting the influence of the cable's ends and the lateral boundaries of the road, we arrive at the following Cauchy problem for the \(E_{2}\) component of the electric field vector [22]:  
\[\varepsilon\frac{\partial^{2}}{\partial t^{2}}E_{2}+\sigma\frac{\partial}{ \partial t}E_{2}=\frac{1}{\mu}\Delta_{xx}E_{2}-w(x)v(z)r^{\prime}(t), \tag{1}\]  
\[E_{2}|_{t=0}=0,\quad\quad\frac{\partial}{\partial t}E_{2}|_{t=0}=0, \tag{2}\]  
Here, \(\varepsilon=\varepsilon_{0}\varepsilon_{r}\), \(\varepsilon_{0}=8.854\cdot 10^{-12}\) [F/m], \(\mu=\mu_{0}\mu_{r}\), \(\mu_{0}=1.257\cdot 10^{-6}\) [H/m], \(w(x)\), and \(v(z)\) are functions describing the spatial extent of the source.  
To study the direct problem (1)-(2), we choose a vertically heterogeneous medium model. Based on the standard road model, we consider a four-layer medium. At the media boundaries, we adopt the average geophysical characteristics of the layers. Thus, the horizontal dotted lines \(z^{1}=0.15\) m, \(z^{2}=0.45\) m, \(z^{3}=0.7\) m, and \(z^{4}=2.2\) m correspond to the boundaries between homogeneous layers (these values are derived from pavement standards). In each layer, the values of \(\varepsilon_{r}\) and \(\sigma\) are constant.  
For subsequent calculations, we consider a vertically inhomogeneous medium model in which the geophysical properties are averaged at the layer boundaries, which is consistent with practical observations.  
We formulate a one-dimensional approximation of problem (1)-(2). We apply the Fourier transform operator with respect to the variable \(x\) and rewrite (1)-(2) as [22]:  
\[\varepsilon U_{tt}+\sigma U_{t}=\frac{1}{\mu}\Big{(}U_{zz}-\lambda^{2}U\Big{)} -W_{\lambda}v(z)r^{\prime}(t), \tag{3}\]  
\[U|_{t=0}=0,\quad\quad U_{t}|_{t=0}=0, \tag{4}\]  
where \(U=F_{x}[E_{2}(x,z,t)]\); \(W_{\lambda}=F_{x}[w(x)]\), and \(\lambda\) is the Fourier transform parameter in \(x\).  
Statement of the Direct Problem: Given the piecewise constant functions \(\varepsilon(z)\) and \(\sigma(z)\), as well as the positive constant \(\mu\), determine the function \(U\) as the solution to the generalized Cauchy problem defined by Equations (3) and (4).  
Additional Information: Suppose additional information is available regarding the solution of the direct problem (3)-(4) at \(z=0\), representing the medium's response:  
\[U^{\lambda}(0,t)=f(t). \tag{5}\]  
Formulation of the Inverse Problem: Given \(\sigma(z),\ \mu\) and the additional information (5) (with \(\lambda=\lambda_{0}\) fixed), determine \(\varepsilon(z)\) and \(U^{\lambda_{0}}\) from Equations (3) and (4).