插值小波尺度法探地雷达数值模拟及四阶Runge Kutta辅助微分方程吸收边界条件

冯德山; 杨道学; 王珣

doi:10.7498/aps.65.234102

摘要

应用迭代插值方法构造了插值小波尺度函数，并将该尺度函数的导数用于离散Maxwell方程组的空间微分，使用四阶Runge Kutta（four order Runge Kutta，RK4）算法计算时间导数，导出了插值小波尺度法的探地雷达（ground penetrating radar，GPR）正演公式，与常规的基于中心差分的时域有限差分算法（finite difference time domain，FDTD）相比，插值小波尺度算法提高了GPR波动方程的空间与时间离散精度.首先，采用具有解析解的层状模型，分别将FDTD算法及插值小波尺度法应用于层状模型正演，单道雷达数据与解析解拟合表明：相同的网格剖分方式，插值小波尺度法比FDTD具有更高的精度.然后，将辅助微分方程完全匹配层（auxiliary differential equation perfecting matched layer，ADE-PML）边界条件应用到插值小波尺度法GPR正演中，在均匀介质模型中对比了FDTD-CPML（坐标伸缩完全匹配层），FDTD-RK4ADE-PML、插值小波尺度RK4ADE-PML的反射误差，结果表明：插值小波尺度RK4ADE-PML吸收效果优于另外两种条件下的吸收边界.最后，应用加载UPML（各向异性完全匹配层）的FDTD和RK4ADE-PML的插值小波尺度法开展了二维GPR模型的正演，展示了RK4ADE-PML对倏逝波的良好吸收效果.

关键词:

Abstract

Ground penetrating radar (GPR) forward is one of the geophysical research directions.Through the forward of geological model,the database of radar model can be enriched and the characteristics of typical geological radar echo images can be understood,which in turn can guide the data interpretation of GPR measured profile,thereby improving the GPR data interpretation level.In this article,the interpolating wavelet scale function by using iterative interpolation method is presented,and the derivative of scale function is used in spatial differentiation of discrete Maxwell equations. The forward modeling formula of GPR based on the interpolation wavelet scale method is derived by using fourth-order Runge-Kutta method (RK4) for calculating the higher time derivative.Compared with the conventional finite difference time domain (FDTD) algorithm based on the central difference method,the interpolation wavelet scale algorithm improves the accuracy of GPR wave equation in both space and time discretization.Firstly,the FDTD algorithm and the interpolation wavelet scale method are applied to the forward modeling of a layered model with analytic solution. Single channel radar data and analytical solution fitting indicate that the interpolation wavelet scale method has higher accuracy than FDTD,with the same mesh generation used.Therefore,auxiliary differential equation perfectly matching layer (ADE-PML) boundary condition is used on an interpolation wavelet scale,and the comparisons between reflection errors obtained using CPML (FDTD),RK4ADE-PML (FDTD),and RK4ADE-PML (interpolating wavelet scales) in a homogeneous medium model show that the absorption effect of RK4ADE-PML (interpolating wavelet scales) is better than the other two absorbing boundaries.Finally,interpolation wavelet scale method,with both UPML,FDTD and RK4ADE-PML loaded,is used for two-dimensional GPR forward modeling,showing good absorption effect for evanescent wave.From all the experimental results,the following conclusions are obtained.1) Using the derivative of the interpolating wavelet scale function instead of central difference schemes for the spatial derivative discretization of Maxwell equations and time derivative calculated using the fourth-order Runge Kutta algorithm,the interpolating wavelet scale algorithm has higher accuracy than regular FDTD algorithm due to the improvement in the spatial and time accuracy of GPR wave equation.2) The best absorption layer parameters of interpolating wavelet scale RK4ADE-PML are selected, when the maximum value of the reflection error is the minimum.The maximum reflection error can reach-93 dB,which increases 20 dB compared with that of UMPL boundary in FDTD algorithm.And the higher simulation accuracy of interpolating wavelet scale algorithm than FDTD algorithm is confirmed after calculating single channel radar data.3) Comparing wave field snapshots of GPR forward modeling,radar pictures from wide-angle method and section method indicates that interpolating wavelet scale RK4ADE-PML reduces reflection error of absorption boundary,improves both spatial and time accuracy,is more effective than UPML boundary in eliminating false reflection of large angle incidence, and has better absorption effect for evanescent wave and low-frequency wave.

Keywords:

作者及机构信息

1.
中南大学地球科学与信息物理学院, 长沙 410083;中南大学, 有色金属成矿预测与地质环境监测教育部重点实验室, 长沙 410083

通信作者: 冯德山, fengdeshan@126.com

基金项目: 国家自然科学基金（批准号：41574116）、中南大学创新驱动项目（批准号：2015CX008）、教育部新世纪优秀人才支持计划（批准号：NCET-12-0551）、中南大学教师研究基金（批准号：2014JSJJ001）和中南大学升华育英人才计划资助的课题.

Authors and contacts

1.
School of Geoscience and Info-Physics, Central South University, Changsha 410083, China;

Corresponding author: Feng De-Shan, fengdeshan@126.com

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 41574116), the Innovation Driven of Central South University, China (Grant No. 2015CX008), the Program for New Century Excellent Talents in University of Ministry of Education of China (Grant No. NCET-12-0551), the Research Foundation of Central South University, China (Grant No. 2014JSJJ001), and the Shenghua Yuying project of Central South University, China.

参考文献

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施引文献

[1]	Li J 2014 Ph. D. Dissertation (Changchun:Jilin University) (in Chinese)[李静2014博士学位论文(长春:吉林大学)]
[2]	Feng D S, Chen J W, Wu Q 2014 Chin. J. Geophys. 57 1322 (in Chinese)[冯德山, 陈佳维, 吴奇2014地球 57 1322]
[3]	Irving J, Knight R 2006 Comput. Geosci. 32 1247
[4]	Liu S X, Zeng Z F 2007 Chin. J. Geophys. 50 320 (in Chinese)[刘四新, 曾昭发2007地球 50 320]
[5]	Diamanti N, Giannopoulos A 2009 J. Appl. Geophys. 67 309
[6]	Teixeira F L 2008 IEEE Trans. Antennas and Propag. 56 2150
[7]	Feng D S, Chen C S, Dai Q W 2010 Chin. J. Geophys. 53 2484 (in Chinese)[冯德山, 陈承申, 戴前伟2010地球 53 2484]
[8]	Li J, Zeng Z F, Liu S X 2012 Comput. Geosci. 49 121
[9]	Wei B, Li X Y, Wang F, Ge D B 2009 Acta Phys. Sin. 58 6174 (in Chinese)[魏兵, 李小勇, 王飞, 葛德彪2009 58 6174]
[10]	Di Q Y, Wang M Y 1999 Chin. J. Geophys. 42 818 (in Chinese)[底青云, 王妙月1999地球 42 818]
[11]	Li Z H, Huang Q H, Wang Y B 2009 Chin. J. Geophys. 52 1915 (in Chinese)[李展辉, 黄清华, 王彦宾2009地球 52 1915]
[12]	Zhuan S X, Ma X K 2012 Acta Phys. Sin. 61 110206 (in Chinese)[颛孙旭, 马西奎2012 61 110206]
[13]	Xu L J, Liu S B, Mo J J, Yuan N C 2006 Acta Phys. Sin. 55 3470 (in Chinese)[徐利军, 刘少斌, 莫锦军, 袁乃昌2006 55 3470]
[14]	Vivek K, Mani M 2006 J. Comput. Appl. Math. 230 803
[15]	Pedro P, Margarete O D, Paulo J S G F, Sônia M G, Anamaria G, José R P 2007 IEEE Trans. Magn. 43 1013
[16]	Marta D L L P, Stewart C, Robert P 2012 J. Comput. Phys. 231 6754
[17]	Rodrigo B B, Marco A C S, Raul R E S 2013 Finite Elem. Anal. Des. 75 71
[18]	Martin R, Komatitsch D, Gedney S D, Bruthiaux E 2010 CMES 56 17
[19]	Zhang W, Shen Y 2010 Geophysics 75 141
[20]	Zhao J G 2014 Jilin University 44 675 (in Chinese)[赵建国2014吉林大学学报44 675]
[21]	Li J X 2007 Ph. D. Dissertation (Tianjin:Tianjin University) (in Chinese)[李建雄2007博士学位论文(天津:天津大学)]
[22]	Deslauriers G, Dubuc S 1989 Constr. Approx. 5 49
[23]	Dubuc 1986 Math. Anal. Appl. 114 185
[24]	Satio N, Beylkin G 1993 IEEE Trans. Signal Process. 41 319
[25]	Sweldens W 1996 Appl. Comput. Harmon. Anal. 3 186
[26]	M Sc Hao J L 2011 Ph. D. Dissertation (Zur Erlangung des akademischen Grades eines)
[27]	Ge D B, Yan Y B 2005 Finite-Differeence Time-Domain Method for Electromagnetic Waves (Xi'an:Xidian University Press) p31(in Chinese)[葛德彪, 闫玉波2005电磁波时域有限差分方法(西安:西安电子科技大学出版社)第31页]

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