基于三角形网格频域有限元的Debye频散介质探地雷达数值模拟

Numerical Simulation of Debye Dispersive Ground Penetrating Radar Based on Triangular Grid Frequency Domain Finite Element

摘要: 通过数值模拟研究探地雷达(GPR)高频电磁波在频散介质规律，对指导实测雷达资料的处理与解释具有重要意义。频率域数值模拟通过计算一系列频率的响应特征并利用傅里叶变换转换到时间域，具有计算稳定，精度高的优点，而在GPR数值模拟中得到广泛应用。本文发展了基于三角形网格频率有限元法的Debye频散介质GPR数值模拟算法。从Debye频散介质模型满足的频率域电磁波动方程出发，在复拉伸坐标系下通过合理构造合理辅助微分方程，推导了完全匹配层(PML)边界条件的二阶电磁波方程。在此基础上，利用Galerkin法推导频率域三角形网格的有限元方程。数值模拟的结果表明：本文构建的PML边界条件可较好地吸收边界截断边界处的超强反射波；相较于非频散介质，GPR高频电磁波Debye频散介质中传播时能量衰减更快，子波持续时间更长，传播速度和分辨率降低，分析结果有利于GPR数据解释和反演。

Abstract: Numerical investigation into the propagation characteristics of high-frequency electromagnetic waves generated by GPR in dispersive media is of critical importance for guiding the processing and interpretation of field-measured radar data. Numerical simulation in the frequency domain, which computes electromagnetic response characteristics across discrete frequencies and transforms them into the time domain via Fourier transform, offers enhanced computational stability and high accuracy, thereby enabling its widespread application in GPR modeling. This study develops a GPR numerical simulation algorithm based on the frequency-domain finite element method with triangular meshes for Debye dispersive media. Starting from the frequency-domain electromagnetic wave equations governing the Debye dispersive medium model, the second-order electromagnetic wave equations with PML boundary conditions were systematically derived in the complex-stretched coordinate system through the appropriate construction of auxiliary differential equations. Building upon this foundation, the finite element equations for the frequency-domain triangular meshes were rigorously derived using the Galerkin method. The numerical simulation results demonstrate that the PML boundary conditions constructed in this study effectively absorb strong spurious reflections at truncated boundaries. Compared to non-dispersive media, high-frequency GPR electromagnetic waves propagating in Debye dispersive media exhibit faster energy attenuation, prolonged wavelet duration, and reduced propagation velocity and resolution. These analytical findings provide valuable insights for GPR data interpretation and inversion.