基于UPML吸收边界条件的GPR有限元数值模拟

摘要

To decrease the influence of truncation boundaries on the GPR numerical simulation precision availably and improve the precision of complex GPR geo-electric model numerical simulation, the FEM, combined with the UPML boundary condition that is good for absorbing broadband, was used to simulate the propagation of GPR wave in complex GPR geo-electric model. Firstly, starting from wave equation satisfying the GPR, Galerkin method was employed to deduce the two-dimension GPR FEM equation; then, based on the two frequency-domain curl equations which conformed in the UPML field, the GPR time-domain FEM wave equation and its solution were deduced by Fourier transform in details, and then the corresponding GPR simulation algorithm was set up. Finally, the program that was compiled by above algorithms was used to forward model the two complex GPR geo-electric models, and two profile maps of the forward models were obtained. The simulated results demonstrate that UPML boundary condition can fully absorb the super-strong reflection wave on the truncation boundaries, and greatly weaken the strong reflection of truncating boundaries. FEM can simulate complex GPR geo-electric model high accurately. Using the FEM algorithm based on UPML boundary condition can simulate the complex GPR geo-electric models quickly and efficiently.%为了进一步减小截断边界对探地雷达(GPR)数值模拟精度的影响、提高复杂GPR地电模型数值模拟的精度，采用具有良好宽频带吸收特性的单轴各向异性完全匹配层(UPML)吸收边界条件与有限单元法(FEM)相结合的算法来模拟GPR波在复杂地电模型中的传播。从GPR满足的波动方程出发，利用伽辽金法导出二维GPR有限元方程。然后，对UPML区域满足的两个频率域旋度方程采用傅里叶变换，推导GPR时间域有限元波动方程及其求解方法，并建立相应的GPR数值模拟算法。应用以上算法编制的程序对两个复杂GPR地电模型进行了正演模拟，得到了相应的雷达剖面图。模拟结果表明：UPML吸收边界条件能够对截断边界处的强烈反射波进行充分的吸收，大大减弱了截断边界处的强反射；FEM能够对复杂的GPR地电模型进行高精度的模拟。基于UPML吸收边界条件的FEM算法能够对复杂的GPR地电模型进行快速、高效的模拟。