基于三角形剖分的复杂GPR模型有限元法正演模拟

Finite element method forward simulation for complex geoelectricity GPR model based on triangle mesh dissection

针对基于矩形网格剖分的时域有限差分法(FDTD)和有限单元法(FEM),对于物性参数分布复杂或几何特征不规则的模型适应性差的问题,从雷达波所满足的Maxwell方程出发,推导探地雷达(GPR)有限元波动方程,通过采用三角形网格剖分和线性插值基函数,在满足时间步长与空间网格差分稳定性前提下,应用Galerkin有限单元法求解GPR波波动方程;同时为消除FEM进行GPR正演模拟时来自截断边界处的超强反射,采用透射边界条件把GPR波在截断边界处的反射波透射出去,进而压制来自截断边界处的反射波。然后,编制GPR有限元正演模拟的Matlab程序。应用该程序分别对起伏分界面、"V"字形2个复杂地电模型进行FEM正演模拟,得到基于三角形网格剖分的FEM正演模拟GPR剖面图,并把该正演模拟剖面图与常规的基于矩形剖分的FEM正演模拟剖面图进行对比,结果表明:基于三角形剖分的FEM对于复杂GPR模型的物性参数分界面拟合更好,其模拟所得的正演剖面与实际模型更相符,具有更高的模拟精度,更有利于指导雷达剖面的数据解译。

For the problem that finite difference time domain method （FDTD） and finite element method （FEM） based on the rectangular mesh dissection have poor adaptability to the patterns that physical parameter distributions are complex or geometric features are irregular, ground penetrating radar （GPR） finite element wave equation was deduced; from the Maxwell equation which met with the radar wave, under the premise of meeting the time step and space grid difference stability, triangular mesh dissection and linear interpolation function and Galerkin finite element method were adopted to solve GPR wave undulation equation. Meanwhile, in order to eliminate the super strong reflection on the interceptive boundary when the FEM was conducting GPR forward simulations, the transmission absorbed boundary condition was adapted to transmit reflected waves that in the interceptive boundary of GPR wave, and then suppressed the reflected wave from the interceptive boundary. Afterwards, the Matlab program of GPR finite element forward simulations was established, and this program conducted FEM forward simulations for two complex geoelectric model of curl interface under-ground and ＂V＂ shape respectively. The GPR profiles of FEM forward simulations was obtained based on triangular mesh dissection. The profiles based on this forward simulations and the FEM forward simulations profiles based on conventional rectangular dissection were compared. The results show that the FEM based on triangular dissection fits better to the physical parameter boundary surface of the complex GPR model, the forward simulations profiles are more consistent with actual models, and it has higher simulations accuracy and is more advantageous to the guidance of the radar profile data interpretation.