عنوان مقاله [English]
Ground-penetrating radar (GPR) is a popular geophysical method for high-resolution imaging of the shallow subsurface structures. Numerical modeling of radar waves plays a significant role in interpretation, processing, and imaging of GPR data. A number of different approaches have been presented for the numerical modeling of GPR data. The most common approach for GPR modeling is the finite-difference method (FDM) because the FDM approach is conceptually simple and easy to program. The difficulties in applying boundary conditions at non-linear boundaries and the lack of sufficient accuracy in complex geometries are the most important drawbacks of FDM.
This paper presents a finite-element method, for simulation of ground-penetrating radar (GPR) in two dimensions in the time-domain. FEM is a powerful and versatile numerical technique for handling problems involving complex geometries and inhomogeneous media. The technique is based on a weak formulation of Maxwell’s equations. In the FEM method, the wavefield is discretized on the elements using Lagrange interpolation, and integration over an element is accomplished based upon the Gaussian-quad integration rule. The major difference between the various numerical methods is in the spatial discretization. In the elemental-based methods, the complex geometry of the problem is divided into several smaller and simpler elements, then the integrals are calculated for each element. These methods have no with any regular or irregular geometry. The responses of the model in the finite-element methods are approximated in nodal points, so nodal polynomials of Lagrange are used for interpolation of the model response. Besides, the systematic generality of the method makes it possible to construct general-purpose computer programs for solving a wide range of problems. In this paper, at first, Maxwell’s equations are discretized, then the boundary condition is applied to minimize artificial reflections from the edges of the computation domain. Although the governing equations and mechanisms are completely different between radar and seismic waves, most of GPR data processing approaches are derived from seismic data processing. Due to similarities in these two techniques, accordingly, we implement the first-order Clayton and Engquist absorbing boundary conditions (firstorder CE-ABC) introduced in the numerical finite-difference modeling of seismic wave propagation. This boundary condition is simple to apply. The presented formulations are in matrix notation that simplifies the implementation of the relations in computer programs, especially in MATLAB application. After spatial discretization with FEM, time discretization is done by Finite-Central Difference (FCD). The time discretization is the most massive and time-consuming step in modeling, which spatial discretization has an important role in this process. The stiffness, mass and damping matrices are sparse and symmetrical in FEM; so if we use the optimized numerical algorithms and storages strategies, computational costs and processing-time can be reduced significantly. To investigate the efficiency of FEM, the computer program has been written in MATLAB and has been tested on two models. The results show that the radar wave simulation via FEM is an accurate and effective approach in complex models.