最优系数有限单元法色散介质GPR频率域正演模拟  

Frequency domain forward modeling of GPR in dispersive media with optimal coefficient finite element method

作　　者：王珣[1] 朱磊 冯德山[1] 许德如 丁思元 刘硕 WANG Xun;ZHU Lei;FENG DeShan;XU DeRu;DING SiYuan;LIU Shuo(School of Geosciences and Info-Physics,Central South University,Changsha 410083,China;School of Earth Sciences,East China University of Technology,Nanchang 330013,China)

机构地区：[1]中南大学地球科学与信息物理学院,长沙410083 [2]东华理工大学地球科学学院,南昌330013

出　　处：《地球物理学报》2023年第12期5173-5186,共14页Chinese Journal of Geophysics

基　　金：国家自然科学基金资助项目(42104143,42074161);湖南省自然科学基金项目(2021JJ30806,2022JJ40584);湖南省研究生科研创新项目(CX20220215,CX20220188);中南大学研究生科研创新项目(2021zzts0270)资助。

摘　　要：工程实际勘探对象如土壤、岩石等多为色散介质,雷达波在其中传播时易发生衰减与畸变,应用常规有限单元法(Finite Element Method,FEM)方法进行数值模拟时,存在数值频散现象.为此,作者以色散介质为研究对象,开展最优系数有限单元法探地雷达(Ground Penetrating Radar,GPR)频率域正演.首先,分析了有限元质量、刚度矩阵的约束条件对有限元求解精度的影响,基于归一化相速度与1的误差最小策略,利用最小二乘法,仅需三个优化参数求取最优的有限元刚度矩阵与质量矩阵.四种不同方法的频散曲线分析及精度对比实验结果表明,优化矩阵在单位波长仅需4.8个网格点下便可达到误差小于0.2%的精度;而一致、集中和折衷矩阵不仅需要更多的网格点,且误差较大.然后,将精确完全匹配层(Exact Perfectly Matched Layer,EPML)吸收边界条件引入最优系数频域有限单元(Finite Element Frequency Domain,FEFD)算法中,简化了吸收参数优化过程,取5层即可达到常规完全匹配层(Perfectly Matched Layer,PML)的10层的吸收效果,能够有效提升正演效率.并将基于EPML的最优系数有限单元法算法引入到城市道路病害模型正演中,实验表明:本文算法能有效压制频散并实现实际色散介质高精度模拟,模拟结果更接近波在地下介质中的实际传播特性.The actual exploration objects in engineering,such as soil and rock,are mostly dispersive media,in which the radar waves tend to attenuate and distort when propagating.Numerical dispersion exists when using conventional finite element method(FEM)method for numerical simulation.This motivates the authors to focus on the dispersive media and develop a frequency domain simulation method for ground penetrating radar(GPR)with the optimal coefficient FEM.Firstly,we analyze the influence of the constraint conditions of the mass matrix and stiffness matrix in FEM on the computational accuracy.And the optimal stiffness matrix and mass matrix are obtained with only three optimization parameters using the least square method according to minimizing the errors between the normalized phase velocity and 1.The results of dispersion curve analysis and accuracy comparison of four different methods show that the proposed optimized matrix can achieve the accuracy of less than 0.2%with only 4.8 grid points per unit wavelength.While the consistent,lumped,and eclectic matrices need more grid points and have larger errors.Then,to simplify the process of absorption parameter optimization,we introduce the exact perfectly matched layer(EPML)absorption boundary conditions into the optimal coefficient finite element frequency domain(FEFD)algorithm.The tests show that the absorption effect of 10-layers conventional perfectly matched layer(PML)can be achieved by taking 5-layers EPML,which means that the EPML can effectively improve the efficiency of simulation.Moreover,our EPML-based optimal coefficient FEM is employed into the forward modeling of an urban road disease model.The experimental results show that the proposed algorithm can effectively suppress the dispersion and realize the high-precision simulation of the actual dispersion medium,and the simulation results are closer to the actual wave propagation characteristics in the underground medium.

关 键 词：探地雷达 数值模拟 色散介质 频率域有限元 数值频散 EPML 

分 类 号：P631[天文地球—地质矿产勘探]