基于Pade逼近的Cole-Cole频散介质GPR有限元正演
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摘要
针对Cole-Cole频散介质中的复介电常数是jω的分数次幂函数,传统的时域有限元法难以离散及计算时间域分数阶导数,本文采用Pade逼近算法将含有时间分数阶导数的Cole-Cole频散介质电磁波方程推导为一组整数阶辅助微分方程,提出了一种适用于Cole-Cole频散介质的GPR有限元正演模拟算法.在复数伸展坐标系下,通过在频率域Cole-Cole频散介质电磁波方程中引入2个中间变量,并将其变换到时间域,从而以变分形式将PML边界条件加载到Cole-Cole频散介质GPR有限元方程组中,并给出了详细的求解公式.在此基础上,编制了基于Pade逼近的Cole-Cole频散介质GPR有限元正演程序,利用该程序对均匀模型进行计算,并与解析解进行对比,验证了本文构建的GPR有限元正演算法的正确性和有效性.设计了一个复杂Cole-Cole频散介质GPR模型,利用本文构建的GPR有限元正演算法进行模拟并与非频散介质模型的模拟结果进行对比,分析了电磁波在Cole-Cole频散介质中传播衰减增强、子波延伸,分辨率降低等传播规律,有助于实测雷达资料更可靠、更准确的解释.模拟结果表明,基于Pade逼近的GPR有限元正演算法可用于复杂Cole-Cole频散介质结构模拟,且具有较高的计算精度.
Since the complex permittivity of Cole-Cole dispersive media is a fractional power function of jω,it is difficult to calculate the time domain fractional derivative by using the finite element time domain(FETD)method.In this paper,a set of auxiliary integer order differential equations are derived from the fractional derivative Maxwell equations in the Cole-Cole dispersive media by the Pade approximation algorithm and a FETD algorithm for simulating ground penetratingradar(GPR)in the Cole-Cole dispersive media is proposed.A perfect matched layer(PML)boundary condition in the variational form is applied to the Cole-Cole dispersive media through introducing two intermediate variables to the Maxwell equations in the frequency domain,which is then transformed into the time domain.We firstly validate the proposed FETD algorithm through simulating the GPR waves propagating in a homogeneous dispersive medium and comparing the results with the analytical solution.Then we design a complex subsurface model of the Cole-Cole dispersive media,compare the simulated GPR profile with that from the counterpart model of non-dispersive media.We find that the medium dispersion can cause larger attenuation of GPR waves and elongate the GPR wavelet and degrade the resolution.We conclude that the proposed FETD algorithm can simulate GPR waves in the Cole-Cole dispersive media with a high accuracy and can aid in the reliable interpretation of the field GPR data.
Since the complex permittivity of Cole-Cole dispersive media is a fractional power function of jω,it is difficult to calculate the time domain fractional derivative by using the finite element time domain(FETD)method.In this paper,a set of auxiliary integer order differential equations are derived from the fractional derivative Maxwell equations in the Cole-Cole dispersive media by the Pade approximation algorithm and a FETD algorithm for simulating ground penetratingradar(GPR)in the Cole-Cole dispersive media is proposed.A perfect matched layer(PML)boundary condition in the variational form is applied to the Cole-Cole dispersive media through introducing two intermediate variables to the Maxwell equations in the frequency domain,which is then transformed into the time domain.We firstly validate the proposed FETD algorithm through simulating the GPR waves propagating in a homogeneous dispersive medium and comparing the results with the analytical solution.Then we design a complex subsurface model of the Cole-Cole dispersive media,compare the simulated GPR profile with that from the counterpart model of non-dispersive media.We find that the medium dispersion can cause larger attenuation of GPR waves and elongate the GPR wavelet and degrade the resolution.We conclude that the proposed FETD algorithm can simulate GPR waves in the Cole-Cole dispersive media with a high accuracy and can aid in the reliable interpretation of the field GPR data.
引文
Alani A M,Aboutalebi M,Kilic G.2013.Applications of ground penetrating radar(GPR)in bridge deck monitoring and assessment.Journal of Applied Geophysics,97:45-54.
Atteia G E,Hussein K F A.2010.Realistic model of dispersive soils using PLRC-FDTD with applications to GPR systems.Progress in Electromagnetics Research B,26:335-359.
Bikowski J,Huisman J A,Vrugt J A,et al.2012.Integrated analysis of waveguide dispersed GPR pulses using deterministic and Bayesian inversion methods.Near Surface Geophysics,10(6):641-652.
Chang C W,Lin C H,Lien H S.2009.Measurement radius of reinforcing steel bar in concrete using digital image GPR.Construction and Building Materials,23(2):1057-1063.
Chew W C,Liu Q H.1996.Perfectly matched layers for elastodynamics:A new absorbing boundary condition.Journal of Computational Acoustics,4(4):341-359.
Chung H,Cho J,Ha S G,et al.2013.Accurate FDTD dispersive modeling for concrete materials.ETRI Journal,35(5):915-918.
Collino F,Tsogka C.2001.Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media.Geophysics,66(1):294-307.
Dai Q W,Wang H H,Feng D S,et al.2012.Finite element numerical simulation for GPR based on quadratic interpolation.Progress in Geophysics(in Chinese),27(2):736-743,doi:10.6038/j.issn.1004-2903.2012.02.041.
Deparis J,Garambois S.2008.On the use of dispersive APVO GPRcurves for thin-bed properties estimation:Theory and application to fracture characterization.Geophysics,74(1):J1-J12.
Di Q Y,Wang M Y.1999.2Dfinite element modeling for radar wave.Chinese Journal of Geophysics(in Chinese),42(6):818-825,doi:10.3321/j.issn.0001-5733.1999.06.012.
Drossaert F H,Giannopoulos A.2007.A nonsplit complex frequencyshifted PML based on recursive integration for FDTD modeling of elastic waves.Geophysics,72(2):T9-T17.
Feng D S,Chen C S,Wang H H.2012.Finite element method GPR forward simulation based on mixed boundary condition.Chinese Journal of Geophysics(in Chinese),55(11):3774-3785,doi:10.6038/j.issn.0001-5733.2012.11.024.
Feng D S,Chen J W,Wu Q.2014.A hybrid ADI-FDTD subgridding scheme for efficient GPR simulation of dispersion media.Chinese Journal of Geophysics(in Chinese),57(4):1322-1334,doi:10.6038/cjg20140429.
Feng D S,Wang X.2016.The GPR simulation of bi-phase random concrete medium using finite element of B-spline wavelet on the interval.Chinese Journal of Geophysics(in Chinese),59(8):3098-3109,doi:10.6038/cjg20160832.
Giannakis I,Giannopoulos A,Davidson N.2012.Incorporating dispersive electrical properties in FDTD GPR models using a general Cole-Cole dispersion function.∥2012 14th International Conference on Ground Penetrating Radar(GPR).Shanghai:IEEE,232-236.
Giannakis I,Giannopoulos A.2014.A novel piecewise linear recursive convolution approach for dispersive media using the finite-difference time-domain method.IEEE Transactions on Antennas and Propagation,62(5):2669-2678.
Giannakis I,Giannopoulos A,Warren C.2016.A realistic FDTDnumerical modeling framework of ground penetrating radar for landmine detection.IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing,9(1):37-51.
González-Drigo R,Pérez-Gracia V,Di Capua D,et al.2008.GPRsurvey applied to modernista buildings in barcelona:The cultural heritage of the college of industrial engineering.Journal of Cultural Heritage,9(2):196-202.
Komatitsch D,Tromp J.2003.A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation.Geophysical Journal International,154(1):146-153.
Li Y F,Li G F,Wang Y.2010.Application of convolution perferctly matched layer in finite element method calculation for 2Dacoustic wave.Acta Acustica(in Chinese),35(6):601-607.
Li Z H,Huang Q H,Wang Y B.2009.A 3-D staggered grid PSTDmethod for borehole radar simulations in dispersive media.ChineseJournalof Geophysics(in Chinese),52(7):1915-1922,doi:10.3969/j.issn.0001-5733.2009.07.027.
Liu L B,Qian R Y.2015.Ground penetrating radar:A critical tool in near-surface geophysics.Chinese Journal of Geophysics(in Chinese),58(8):2606-2617,doi:10.6038/cjg20150802.
Liu Q H,Fan G X.1999.Simulations of GPR in dispersive media using a frequency-dependent PSTD algorithm.IEEE Transactions on Geoscience and Remote Sensing,37(5):2317-2324.
Liu S X,Zeng Z F.2007.Numerical simulation for Ground Penetrating Radar wave propagation in the dispersive medium.Chinese Journal of Geophysics(in Chinese),50(1):320-326,doi:10.3321/j.issn.0001-5733.2007.01.040.
Liu Y S,Teng J W,Liu S L,et al.2013.Explicit finite element method with triangle meshes stored by sparse format and its perfectly matched layers absorbing boundary condition.Chinese Journal of Geophysics(in Chinese),56(9):3085-3099,doi:10.6038/cjg20130921.
Lu T,Cai W,Zhang P W.2005.Discontinuous Galerkin timedomain method for GPR simulation in dispersive media.IEEETransactions on Geoscience and Remote Sensing,43(1):72-80.
Mangel A R,Moysey S M J,Van Der Kruk J.2015.Resolving precipitation induced water content profiles by inversion of dispersive GPR data:A numerical study.Journal of Hydrology,525:496-505.
Mustafa S,Abbosh A M,Nguyen P T.2014.Modeling human head tissues using fourth-order Debye model in convolutionbased three-dimensional finite-difference time-domain.IEEETransactions on Antennas and Propagation,62(3):1354-1361.
Newmark N M.1959.A method of computation for structural dynamics.Journal of the Engineering Mechanics Division,85(3):67-94.
Rekanos I T,Papadopoulos T G.2010.An auxiliary differential equation method for FDTD modeling of wave propagation in Cole-Cole dispersive media.IEEE Transactions on Antennas and Propagation,58(11):3666-3674.
Rekanos I T.2012.FDTD schemes for wave propagation in DavidsonCole dispersive media using auxiliary differential equations.IEEETransactions on Antennas and Propagation,60(3):1467-1478.
Shahmansouri A,Rashidian B.2012.GPU implementation of splitfield finite-difference time-domain method for Drude-Lorentz dispersive media.Progress in Electromagnetics Research,125:55-77.
Shibayama J,Ando R,Nomura A,et al.2009.Simple trapezoidal recursive convolution technique for the frequency-dependent FDTD analysis of a Drude-Lorentz model.IEEE Photonics Technology Letters,21(2):100-102.
Solla M,Lorenzo H,Rial F I,et al.2011.GPR evaluation of the
Roman masonry arch bridge of Lugo(Spain).NDT&E International,44(1):8-12.
Teixeira F L.2008.Time-domain finite-difference and finite-element methods for Maxwell equations in complex media.IEEETransactions on Antennas and Propagation,56(8):2150-2166.
Wang H H,Dai Q W.2014.Finite element method GPR forward simulation in dispersive medium.Journal of Central South University(Science and Technology)(in Chinese),45(3):790-797.
Wang H H,Dai Q W,Feng D S.2014.Element-free method forward modeling for GPR based on an improved sarma-type absorbing boundary.Journal of Environmental&Engineering Geophysics,19(4):277-285.
Zhao J G,Shi R Q.2013.Perfectly matched layer-absorbing boundary condition for finite-element time-domain modeling of elastic wave equations.Applied Geophysics,10(3):323-336.
戴前伟,王洪华,冯德山等.2012.基于双二次插值的探地雷达有限元数值模拟.地球物理学进展,27(2):736-743,doi:10.6038/j.issn.1004-2903.2012.02.041.
底青云,王妙月.1999.雷达波有限元仿真模拟.地球物理学报,42(6):818-825,doi:10.3321/j.issn.0001-5733.1999.06.012.
冯德山,陈承申,王洪华.2012.基于混合边界条件的有限单元法GPR正演模拟.地球物理学报,55(11):3774-3785,doi:10.6038/j.issn.0001-5733.2012.11.024.
冯德山,陈佳维,吴奇.2014.混合ADI-FDTD亚网格技术在探地雷达频散媒质中的高效正演.地球物理学报,57(4):1322-1334,doi:10.6038/cjg20140429.
冯德山,王珣.2016.区间B样条小波有限元GPR模拟双相随机混凝土介质.地球物理学报,59(8):3098-3109,doi:10.6038/cjg20160832.
李义丰,李国峰,王云.2010.卷积完全匹配层在两维声波有限元计算中的应用.声学学报,35(6):601-607.
李展辉,黄清华,王彦宾.2009.三维错格时域伪谱法在频散介质井中雷达模拟中的应用.地球物理学报,52(7):1915-1922,doi:10.3969/j.issn.0001-5733.2009.07.027.
刘澜波,钱荣毅.2015.探地雷达:浅表地球物理科学技术中的重要工具.地球物理学报,58(8):2606-2617,doi:10.6038/cjg20150802.
刘四新,曾昭发.2007.频散介质中地质雷达波传播的数值模拟.地球物理学报,50(1):320-326,doi:10.3321/j.issn.0001-5733.2007.01.040.
刘有山,滕吉文,刘少林等.2013.稀疏存储的显式有限元三角网格地震波数值模拟及其PML吸收边界条件.地球物理学报,56(9):3085-3099,doi:10.6038/cjg20130921.
王洪华,戴前伟.2014.频散介质中探地雷达有限元法正演模拟.中南大学学报(自然科学版),45(3):790-797.
Atteia G E,Hussein K F A.2010.Realistic model of dispersive soils using PLRC-FDTD with applications to GPR systems.Progress in Electromagnetics Research B,26:335-359.
Bikowski J,Huisman J A,Vrugt J A,et al.2012.Integrated analysis of waveguide dispersed GPR pulses using deterministic and Bayesian inversion methods.Near Surface Geophysics,10(6):641-652.
Chang C W,Lin C H,Lien H S.2009.Measurement radius of reinforcing steel bar in concrete using digital image GPR.Construction and Building Materials,23(2):1057-1063.
Chew W C,Liu Q H.1996.Perfectly matched layers for elastodynamics:A new absorbing boundary condition.Journal of Computational Acoustics,4(4):341-359.
Chung H,Cho J,Ha S G,et al.2013.Accurate FDTD dispersive modeling for concrete materials.ETRI Journal,35(5):915-918.
Collino F,Tsogka C.2001.Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media.Geophysics,66(1):294-307.
Dai Q W,Wang H H,Feng D S,et al.2012.Finite element numerical simulation for GPR based on quadratic interpolation.Progress in Geophysics(in Chinese),27(2):736-743,doi:10.6038/j.issn.1004-2903.2012.02.041.
Deparis J,Garambois S.2008.On the use of dispersive APVO GPRcurves for thin-bed properties estimation:Theory and application to fracture characterization.Geophysics,74(1):J1-J12.
Di Q Y,Wang M Y.1999.2Dfinite element modeling for radar wave.Chinese Journal of Geophysics(in Chinese),42(6):818-825,doi:10.3321/j.issn.0001-5733.1999.06.012.
Drossaert F H,Giannopoulos A.2007.A nonsplit complex frequencyshifted PML based on recursive integration for FDTD modeling of elastic waves.Geophysics,72(2):T9-T17.
Feng D S,Chen C S,Wang H H.2012.Finite element method GPR forward simulation based on mixed boundary condition.Chinese Journal of Geophysics(in Chinese),55(11):3774-3785,doi:10.6038/j.issn.0001-5733.2012.11.024.
Feng D S,Chen J W,Wu Q.2014.A hybrid ADI-FDTD subgridding scheme for efficient GPR simulation of dispersion media.Chinese Journal of Geophysics(in Chinese),57(4):1322-1334,doi:10.6038/cjg20140429.
Feng D S,Wang X.2016.The GPR simulation of bi-phase random concrete medium using finite element of B-spline wavelet on the interval.Chinese Journal of Geophysics(in Chinese),59(8):3098-3109,doi:10.6038/cjg20160832.
Giannakis I,Giannopoulos A,Davidson N.2012.Incorporating dispersive electrical properties in FDTD GPR models using a general Cole-Cole dispersion function.∥2012 14th International Conference on Ground Penetrating Radar(GPR).Shanghai:IEEE,232-236.
Giannakis I,Giannopoulos A.2014.A novel piecewise linear recursive convolution approach for dispersive media using the finite-difference time-domain method.IEEE Transactions on Antennas and Propagation,62(5):2669-2678.
Giannakis I,Giannopoulos A,Warren C.2016.A realistic FDTDnumerical modeling framework of ground penetrating radar for landmine detection.IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing,9(1):37-51.
González-Drigo R,Pérez-Gracia V,Di Capua D,et al.2008.GPRsurvey applied to modernista buildings in barcelona:The cultural heritage of the college of industrial engineering.Journal of Cultural Heritage,9(2):196-202.
Komatitsch D,Tromp J.2003.A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation.Geophysical Journal International,154(1):146-153.
Li Y F,Li G F,Wang Y.2010.Application of convolution perferctly matched layer in finite element method calculation for 2Dacoustic wave.Acta Acustica(in Chinese),35(6):601-607.
Li Z H,Huang Q H,Wang Y B.2009.A 3-D staggered grid PSTDmethod for borehole radar simulations in dispersive media.ChineseJournalof Geophysics(in Chinese),52(7):1915-1922,doi:10.3969/j.issn.0001-5733.2009.07.027.
Liu L B,Qian R Y.2015.Ground penetrating radar:A critical tool in near-surface geophysics.Chinese Journal of Geophysics(in Chinese),58(8):2606-2617,doi:10.6038/cjg20150802.
Liu Q H,Fan G X.1999.Simulations of GPR in dispersive media using a frequency-dependent PSTD algorithm.IEEE Transactions on Geoscience and Remote Sensing,37(5):2317-2324.
Liu S X,Zeng Z F.2007.Numerical simulation for Ground Penetrating Radar wave propagation in the dispersive medium.Chinese Journal of Geophysics(in Chinese),50(1):320-326,doi:10.3321/j.issn.0001-5733.2007.01.040.
Liu Y S,Teng J W,Liu S L,et al.2013.Explicit finite element method with triangle meshes stored by sparse format and its perfectly matched layers absorbing boundary condition.Chinese Journal of Geophysics(in Chinese),56(9):3085-3099,doi:10.6038/cjg20130921.
Lu T,Cai W,Zhang P W.2005.Discontinuous Galerkin timedomain method for GPR simulation in dispersive media.IEEETransactions on Geoscience and Remote Sensing,43(1):72-80.
Mangel A R,Moysey S M J,Van Der Kruk J.2015.Resolving precipitation induced water content profiles by inversion of dispersive GPR data:A numerical study.Journal of Hydrology,525:496-505.
Mustafa S,Abbosh A M,Nguyen P T.2014.Modeling human head tissues using fourth-order Debye model in convolutionbased three-dimensional finite-difference time-domain.IEEETransactions on Antennas and Propagation,62(3):1354-1361.
Newmark N M.1959.A method of computation for structural dynamics.Journal of the Engineering Mechanics Division,85(3):67-94.
Rekanos I T,Papadopoulos T G.2010.An auxiliary differential equation method for FDTD modeling of wave propagation in Cole-Cole dispersive media.IEEE Transactions on Antennas and Propagation,58(11):3666-3674.
Rekanos I T.2012.FDTD schemes for wave propagation in DavidsonCole dispersive media using auxiliary differential equations.IEEETransactions on Antennas and Propagation,60(3):1467-1478.
Shahmansouri A,Rashidian B.2012.GPU implementation of splitfield finite-difference time-domain method for Drude-Lorentz dispersive media.Progress in Electromagnetics Research,125:55-77.
Shibayama J,Ando R,Nomura A,et al.2009.Simple trapezoidal recursive convolution technique for the frequency-dependent FDTD analysis of a Drude-Lorentz model.IEEE Photonics Technology Letters,21(2):100-102.
Solla M,Lorenzo H,Rial F I,et al.2011.GPR evaluation of the
Roman masonry arch bridge of Lugo(Spain).NDT&E International,44(1):8-12.
Teixeira F L.2008.Time-domain finite-difference and finite-element methods for Maxwell equations in complex media.IEEETransactions on Antennas and Propagation,56(8):2150-2166.
Wang H H,Dai Q W.2014.Finite element method GPR forward simulation in dispersive medium.Journal of Central South University(Science and Technology)(in Chinese),45(3):790-797.
Wang H H,Dai Q W,Feng D S.2014.Element-free method forward modeling for GPR based on an improved sarma-type absorbing boundary.Journal of Environmental&Engineering Geophysics,19(4):277-285.
Zhao J G,Shi R Q.2013.Perfectly matched layer-absorbing boundary condition for finite-element time-domain modeling of elastic wave equations.Applied Geophysics,10(3):323-336.
戴前伟,王洪华,冯德山等.2012.基于双二次插值的探地雷达有限元数值模拟.地球物理学进展,27(2):736-743,doi:10.6038/j.issn.1004-2903.2012.02.041.
底青云,王妙月.1999.雷达波有限元仿真模拟.地球物理学报,42(6):818-825,doi:10.3321/j.issn.0001-5733.1999.06.012.
冯德山,陈承申,王洪华.2012.基于混合边界条件的有限单元法GPR正演模拟.地球物理学报,55(11):3774-3785,doi:10.6038/j.issn.0001-5733.2012.11.024.
冯德山,陈佳维,吴奇.2014.混合ADI-FDTD亚网格技术在探地雷达频散媒质中的高效正演.地球物理学报,57(4):1322-1334,doi:10.6038/cjg20140429.
冯德山,王珣.2016.区间B样条小波有限元GPR模拟双相随机混凝土介质.地球物理学报,59(8):3098-3109,doi:10.6038/cjg20160832.
李义丰,李国峰,王云.2010.卷积完全匹配层在两维声波有限元计算中的应用.声学学报,35(6):601-607.
李展辉,黄清华,王彦宾.2009.三维错格时域伪谱法在频散介质井中雷达模拟中的应用.地球物理学报,52(7):1915-1922,doi:10.3969/j.issn.0001-5733.2009.07.027.
刘澜波,钱荣毅.2015.探地雷达:浅表地球物理科学技术中的重要工具.地球物理学报,58(8):2606-2617,doi:10.6038/cjg20150802.
刘四新,曾昭发.2007.频散介质中地质雷达波传播的数值模拟.地球物理学报,50(1):320-326,doi:10.3321/j.issn.0001-5733.2007.01.040.
刘有山,滕吉文,刘少林等.2013.稀疏存储的显式有限元三角网格地震波数值模拟及其PML吸收边界条件.地球物理学报,56(9):3085-3099,doi:10.6038/cjg20130921.
王洪华,戴前伟.2014.频散介质中探地雷达有限元法正演模拟.中南大学学报(自然科学版),45(3):790-797.