基于非分裂CFS-PML边界条件的频散介质GPR时域有限元模拟

通过数值模拟研究探地雷达（GPR）高频电磁波在频散介质中的传播规律，对提高实测资料的解释精度具有重要意义.复频移完全匹配层边界条件（CFS-PML）以其优越的吸收特性被广泛用于一阶电磁波动方程的GPR时域有限差分数值模拟中，其实现过程大都涉及电磁场的卷积计算，辅助变量较多，降低计算效率.为此，本文从复拉伸坐标系下的Debye频散介质电磁波动方程出发，通过合理构造辅助微分方程，推导了二阶Debye频散介质电磁波动方程的非分裂CFS-PML边界条件实现公式，避免了电磁波场的分裂和卷积计算.在此基础上，利用Galerkin法和Newmark-β差分法推导了基于非分裂CFS-PML边界条件的GPR有限元方程及其时域差分离散格式.两个GPR模型的模拟结果表明：本文提出的基于辅助微分方程的非分裂CFS-PML边界条件实现方法可有效地吸收大角度入射的低频虚假反射波，提高模拟精度；相比于非频散介质，高频电磁波在频散介质中传播衰减更强、子波持续时间增大、分辨率和传播速度降低、直达波和反射波的主频更小，分析结果有助于提高实测GPR资料的解译精度.

Finite element time domain simulation of ground penetrating radar in dispersive media based on non-splitting CFS-PML boundary condition

It is of great significance to study the propagation law of high frequency electromagnetic wave of ground penetrating radar (GPR) in dispersive media through numerical simulation for improving the interpretation accuracy of observed data. Complex frequency shift perfectly matched layer (CFS-PML) has been widely used in finite difference time domain numerical simulation of GPR based on the first-order electromagnetic wave equation due to their superior absorption characteristics and its implementation methods involve the convolution calculation of electromagnetic fields and introduce many auxiliary variables, which reduce the computational efficiency. Therefore, starting from the Debye dispersive media electromagnetic wave equation in the complex stretch coordinate, this paper derives the non-splitting CFS-PML realization formulas of second order Debye dispersive media electromagnetic wave equation by constructing the auxiliary differential equation reasonably, which avoids the splitting and convolution calculation of electromagnetic wave field. On this basis, Galerkin method and Newmark-β difference method are used to derive time domain finite element equation with non-splitting CFS-PML and its time difference discretization scheme. The simulated results of two GPR models demonstrate that the proposed non-splitting CFS-PML boundary condition implementation method based on auxiliary differential equations can effectively absorb the low frequency spurious reflected waves with large angle incidence and improve the simulation accuracy. Compared with non-dispersive media, the high frequency electromagnetic wave of GPR in dispersive media has stronger attenuation, longer wavelet duration, lower resolution and propagation velocity. These analysis results are helpful to improve the interpretation accuracy of observed GPR data.