## 2 Materials and Methods  
### LSRTM  
Compared to Kirchhoff migration and one-way wave equation migration, RTM offers the benefits of high imaging accuracy and no limitation of interface inclination [27]. However, there are also deficiencies, such as high dependence on the velocity model, low imaging resolution, and high data requirements. As an improvement, LSRTM updates the coefficient of reflection iteratively through the idea of inversion, which has a higher resolution image than RTM. This can be attributed to the inversion feature of LSRTM. Compared to other migrations, LSRTM is essentially a waveform inversion of the output reflection coefficient with the advantage of linear inversion. Here, we briefly introduce the workflow of LSRTM for electromagnetic wave equations. Firstly, the reverse migration operator is derived based on the Born approximation. The given background electrical medium and the background electrical field of the waves in 2-D TM\({}_{z}\) mode are satisfied [28]:  
\[ \begin{split}\frac{\partial H_{x}}{\partial t}&=-\frac{ 1}{\mu}\frac{\partial E_{z}}{\partial y}\\ \frac{\partial H_{y}}{\partial t}&=\frac{1}{\mu} \frac{\partial E_{z}}{\partial x}\\ \frac{\partial E_{z}}{\partial t}&=\frac{1}{\varepsilon }(\frac{\partial H_{y}}{\partial x}-\frac{\partial H_{x}}{\partial y}-\sigma E _{z}-J_{z})\end{split}  \tag{1}\]  
where, \(H\) is the magnetic field strength, \(E\) is the electric field strength, \(\varepsilon\) is the dielectric permittivity (F/m), \(\sigma\) is the conductivity (S/m), \(\mu\) is the magnetic permeability (H/m), \(t\) is the time (s), and \(J_{z}\) is the source. In this paper, we focus only on dielectric permittivity \(\varepsilon\) and ignore the influence of conductivity \(\sigma\) and magnetic permeability \(\mu\) on the reflection coefficient. When the initial model \(p_{0}(\varepsilon)\) is near the true model \(p(\varepsilon)\), we split the true model \(p(\varepsilon)\) into background model parameter \(p_{0}(\varepsilon)\) and perturbed model parameter \(\Delta p(\varepsilon)\), which can be expressed as \(p(\varepsilon)=p_{0}(\varepsilon)+\Delta p(\varepsilon)\). Similarly, the wavefields \(U(x,t,x_{s})\) can also be split into background wavefields \(U_{0}(x,t,x_{s})\) and perturbed wavefields \(\Delta U(x,t,x_{s})\), which can be expressed as \(U(x,t,x_{s})=U_{0}(x,t,x_{s})+\Delta U(x,t,x_{s})\). When [PERSON] expands Equation (1) at \(p=p_{0}\), using the Born approximation and ignoring higher order terms, the expression can be obtained as follows:  
\[ \begin{split}\mu\frac{\partial\Delta H_{x}}{\partial t}& +\frac{\partial\Delta E_{z}}{\partial y}&=-\Delta\mu \frac{\partial H_{x}}{\partial t}\\ \mu\frac{\partial\Delta H_{y}}{\partial t}&-\frac{ \partial\Delta E_{z}}{\partial x}&=-\Delta\mu\frac{\partial H_{y }}{\partial t}\\ \varepsilon\frac{\partial\Delta E_{z}}{\partial t}& -\frac{\partial\Delta H_{y}}{\partial x}&+\frac{ \partial H_{x}}{\partial y}&+\sigma\Delta E_{z}&=- \Bigl{(}\Delta\varepsilon\frac{\partial E_{z}}{\partial t}+\Delta\sigma E_{ z}\Bigr{)}\end{split}  \tag{2}\]  
When solving perturbed wavefields, information on background wavefields is generally required, so it is necessary to perform two forward simulations, which is also known as the reverse migration process. During the forward process of GPR simulation, the observed data \(d\) and the relation of model parameter \(\mathbf{m}\), where \(\mathbf{m}\) represents \(\Delta\mathbf{p}\), can be expressed as:  
\[L\mathbf{m}=d \tag{3}\]  
where, \(\mathbf{m}\) is referred to as \(\Delta\varepsilon\) when only the dielectric permittivity is considered; \(L\) is the forward operator. We assume that \(d_{obs}\) is the measured data, and \(d\) is the perturbed wavefield. In LSRTM, the reflection coefficient is iteratively updated by inversion, and the objective function is established as follow [29]:  
\[\min_{\mathbf{m}}S(\mathbf{m})=\frac{1}{2}\|L\mathbf{m}-d_{obs}\|_{2}^{2} \tag{4}\]  
where the simulated reflected wavefields must be satisfied with the Taylor expansion of Equation (1).