Second-generation wavelet finite element based on the lifting scheme for GPR simulation  

制基于提升方案的第二代小波有限元探地雷达正演

作　　者：Feng De-Shan Zhang Hua Wang Xun 冯德山;张华;王珣(中南大学地球科学与信息物理学院,湖南长沙410083;有色金属成矿预测教育部重点实验室,湖南长沙410083)

机构地区：[1]School of Geosciences and Info-Physics,Central South University,Changsha 410083,China [2]Key Laboratory of Metallogenic Prediction of Nonferrous Metals,Ministry of Education,Changsha 410083,China

出　　处：《Applied Geophysics》2020年第1期143-153,170,共12页应用地球物理（英文版）

基　　金：supported by the National Natural Science Foundation of China(Nos.41574116 and 41774132);Hunan Provincial Innovation Foundation for Postgraduate(Grant Nos.CX2017B052);the Fundamental Research Funds for the Central Universities of Central South University(Nos.2018zzts693)。

摘　　要：Ground-penetrating radar(GPR)is a highly efficient,fast and non-destructive exploration method for shallow surfaces.High-precision numerical simulation method is employed to improve the interpretation precision of detection.Second-generation wavelet finite element is introduced into the forward modeling of the GPR.As the finite element basis function,the second-generation wavelet scaling function constructed by the scheme is characterized as having multiple scales and resolutions.The function can change the analytical scale arbitrarily according to actual needs.We can adopt a small analysis scale at a large gradient to improve the precision of analysis while adopting a large analytical scale at a small gradient to improve the efficiency of analysis.This approach is beneficial to capture the local mutation characteristics of the solution and improve the resolution without changing mesh subdivision to realize the efficient solution of the forward GPR problem.The algorithm is applied to the numerical simulation of line current radiation source and tunnel non-dense lining model with analytical solutions.Result show that the solution results of the secondgeneration wavelet finite element are in agreement with the analytical solutions and the conventional finite element solutions,thereby verifying the accuracy of the second-generation wavelet finite element algorithm.Furthermore,the second-generation wavelet finite element algorithm can change the analysis scale arbitrarily according to the actual problem without subdividing grids again.The adaptive algorithm is superior to traditional scheme in grid refinement and basis function order increase,which makes this algorithm suitable for solving complex GPR forward-modeling problems with large gradient and singularity.探地雷达是一种高效快速无损的浅地表勘探方法,为了提高探地雷达探测的解释精度,需要开展探地雷达高精度的数值模拟方法研究。因此,本文将第二代小波有限元引入探地雷达的正演模拟中,以提升方案构造的第二代小波尺度函数代替多项式函数作为有限元基函数,具有多尺度、多分辨的特性。该算法可根据实际需要,任意改变分析尺度,在大梯度处采用小的分析尺度以提高分析精度,而在小梯度处采用大的分析尺度以提高分析效率,从而有利于捕捉解的局部突变特征,在不改变网格剖分的前提下提高分辨率,实现GPR正问题的高效求解。将该算法应用于具有解析解的线电流辐射源及隧道衬砌不密实模型的数值模拟中,结果表明:第二代小波有限元的求解结果与解析解及常规有限元算法求解结果都能较好地吻合,验证了第二代小波有限元算法的正确性,而且第二代小波有限元可以根据实际问题的需要任意改变分析尺度,不需要网格单元的重新剖分,是一种优于传统单元网格加密和阶次升高的自适应算法,特别适合于求解大梯度、奇异性的复杂GPR正演模拟问题。

关 键 词：Ground penetrating radar wave equation second-generation wavelet finite element method lifting scheme forward modeling 

分 类 号：P631.3[天文地球—地质矿产勘探]