基于梯度法和L-BFGS算法的探地雷达时间域全波形反演
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摘要
探地雷达全波形反演充分利用雷达波场的运动学和动力学信息,来反演地下介质电导率和介电常数等参数,这里从TM模式下的麦克斯韦方程组出发,利用单轴各向异性(UPML)吸收边界条件,进行雷达波场时域有限差分正演模拟,给出了电导率和介电常数梯度方向的求取方法,并以步长为自变量通过求取目标函数为极值的方式来确定最优步长。从反演结果可以看出,对于单参数反演,无论是电导率还是介电常数,反演结果都十分接近于真实模型;对于双参数同时反演,反演结果的异常体形态接近于真实模型,但是由于电导率和介电常数之间的耦合影响,使在数值上相比单参数反演得到的反演结果较差;考虑到近似Hessian矩阵中的非对角块元素能够反映不同参数之间的相互作用,因此在双参数同时反演时对比了梯度法和L-BFGS算法,结果显示,利用L-BFGS算法可以更好地解决电导率与介电常数之间的耦合影响。
Full waveform inversion of ground-penetrating radar makes full use of the kinematics and dynamics of radar wave field information to carry out the inversion of the parameters of underground medium such as electrical conductivity and dielectric constant,and then judge the condition of the underground medium.In this paper,in order to carry out the time domain finite difference forward modeling of radar wave field,the maxwell's equations of TM as starting model and the better absorption effect of uniaxial anisotropic(UPML)as absorbing boundary condition were used.The calculate method of gradient direction of the electrical conductivity and dielectric constant are given,and the determination of the optimal step length is through the way of calculating the extreme value of the objective function by step length for the independent variables.The inversion process is divided into three parts:first electrical conductivity is assumed to be known and the inversion of dielectric constant is carry out solely;second the dielectric constant is assumed to be known and the inversion of electrical conductivity is carry out solely;and finally carries on the inversion of the double parameters of electrical conductivity and dielectric constant at the same time.As can be seen from the inversion results,for a single parameter inversion,both the electrical conductivity and dielectric constant,the inversion results are very closed to the real model,for the simultaneous inversion of double parameters,the inversion results of abnormal body shape is closed to the real model,but due to the coupling effect between electrical conductivity and dielectric constant,so in numerical value,it can't get accurate inversion results like single parameter inversion.Considering the approximate Hessian matrix of the diagonal block elements can reflect the interaction between different parameters,so it compared the gradient method and L-BFGS algorithm during the simultaneous inversion of the double parameters,the results showed that it can better solve the coupling effect between dielectric constant and electrical conductivity by using L-BFGS algorithm.
Full waveform inversion of ground-penetrating radar makes full use of the kinematics and dynamics of radar wave field information to carry out the inversion of the parameters of underground medium such as electrical conductivity and dielectric constant,and then judge the condition of the underground medium.In this paper,in order to carry out the time domain finite difference forward modeling of radar wave field,the maxwell's equations of TM as starting model and the better absorption effect of uniaxial anisotropic(UPML)as absorbing boundary condition were used.The calculate method of gradient direction of the electrical conductivity and dielectric constant are given,and the determination of the optimal step length is through the way of calculating the extreme value of the objective function by step length for the independent variables.The inversion process is divided into three parts:first electrical conductivity is assumed to be known and the inversion of dielectric constant is carry out solely;second the dielectric constant is assumed to be known and the inversion of electrical conductivity is carry out solely;and finally carries on the inversion of the double parameters of electrical conductivity and dielectric constant at the same time.As can be seen from the inversion results,for a single parameter inversion,both the electrical conductivity and dielectric constant,the inversion results are very closed to the real model,for the simultaneous inversion of double parameters,the inversion results of abnormal body shape is closed to the real model,but due to the coupling effect between electrical conductivity and dielectric constant,so in numerical value,it can't get accurate inversion results like single parameter inversion.Considering the approximate Hessian matrix of the diagonal block elements can reflect the interaction between different parameters,so it compared the gradient method and L-BFGS algorithm during the simultaneous inversion of the double parameters,the results showed that it can better solve the coupling effect between dielectric constant and electrical conductivity by using L-BFGS algorithm.
引文
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[18]刘玉柱,王光银,杨积忠.基于Born敏感核函数的VTI介质多参数全波形反演[J].地球物理学报,2015(04):1305-1316.LIU Y Z,WANG G Y,YANG J Z.Multi-parameter full-waveform inversion for VTI media based on born sensitivity kernels[J].Chinese Journal of Geophysics,2015(04):1305-1316.(In Chinese)
[19]S.OPERTO,Y.GHOLAMI,V.PRIEUX.A guided tour of multiparameter full-waveform inversion with multicomponent data:From theory to practice[J].The Leading Edge,2013,32(9):1040-1054.
[20]GERHARD PRATT,CHANGSOO SHIN,HICKS.Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion[J].Geophysical Journal International,1998,33(2):341-362.
[21]杨积忠,刘玉柱,董良国.基于Born敏感核函数的速度、密度双参数全波形反演[J].地球物理学报,2016(03):1082-1094.YANG J Z,LIU Y Z,DONG L G.Multi-parameter full waveform inversion for velocity and density based on Born sensitivity kernels[J].Chinese Journal of Geophysics,2016(03):1082-1094.(In Chinese)
[22]陈宝林.最优化理论与算法2版[M].北京:清华大学出版社,2005.CHEN B L.Optimization theory and algorithm 2nd ed[M].Beijing:TsingHua University Press,2005.(In Chinese)
[23]A.PICA,J.P.DIET,A.TARANTOLA.Nonlinear inversion of seismic reflection data in a laterally invariant medium[J].Geophysics,1990,55(3):284-292.
[2] SEBASTIAN BUSCH,JAN VAN DER KRUK,JUTTA BIKOWSKI,et al.Quantitative conductivity and permittivity estimation using full-waveform inversion of on-ground GPR data[J].Geophysics,2012,77(6):H79-H91.
[3] F LAVOUE,R BROSSIER,L METIVIER,et al.Two-dimensional permittivity and conductivity imaging by full waveform inversion of multioffset GPR data:a frequency-domain quasi-Newton approach[J].Geophysical Journal International,2014,197(1):248-268.
[4] ALBERT TARANTOLA.Inversion of seismic reflection data in the acoustic approximation[J].Geophysics,1984,49(8):1259-1266.
[5] PETER MORA.Nonlinear two‐dimensional elastic inversion of multioffset seismic data[J].Geophysics,1987,52(9):1211-1228.
[6] E.CRASE,A.PICA,M.NOBLE,et al.Robust elastic nonlinear waveform inversion:Application to real data[J].Geophysics,1990,55(5):527-538.
[7] RGERHARD PRATT,M.H.WORTHINGTON.Inverse theory applied to multi-source cross-hole tomography,part I:acoustic wave-equation method[J].Geophysical Prospecting,1990,38(3):287-310.
[8] C SHIN,YH CHA.Waveform inversion in the Laplace domain[J].Geophysical Journal International,2008,173(3):922-931.
[9] JACQUES R.ERNST,ALAN G.GREEN,HANSRUEDI MAURER.Application of a new 2Dtimedomain full-waveform inversion scheme to crosshole radar data[J].Geophysics,2007,72(5):J53-J64.
[10]SEIICHIRO KURODA,MUTSUO TAKEUCHI,HEE JOON KIM.Full-waveform inversion algorithm for interpreting crosshole radar data:a theoretical approach[J].Geosciences Journal,2007,11(3):211-217.
[11]MELES,GIOVANNI ANGELO,VAN DER KRUK,et al.A new vector waveform inversion algorithm for simultaneous updating of conductivity and permittivity parameters from combination crosshole/borehole-to-surface GPR data[J].IEEE Transactions on Geoscience and Remote Sensing,2010,48(9):3391-3407.
[12]KALOGEROPOULOS A,KRUK J V D,HUGENSCHMIDT J,et al.Chlorides and moisture assessment in concrete by GPR full waveform inversion[J].Near Surface Geophysics,2011,9(1790):277-285.
[13]王兆磊,周辉,李国发.用地质雷达数据资料反演二维地下介质的方法[J].地球物理学报,2007(03):897-904.WANG Z L,ZHOU H,LI G F.Inversion of ground-penetrating radar data for 2Delectric parameters[J].Chinese Journal of Geophysics,2007(03):897-904.(In Chinese)
[14]LAMBOT S,SLOB E C,VAN D B I,et al.Estimating soil electric properties from monostatic ground penetrating radar signal inversion in the frequency domain[J].Water Resources Research,2004,40(4):1035-1042.
[15]KLOTZSCHE A,JAN V D K,MELES G,et al.Crosshole GPR full-waveforminversionof waveguides acting as preferential flow paths within aquifer systems[J].Geophysics,2012,77(4):57.
[16]R.STREICH,M.BECKEN.Sensitivity of controlled-source electromagnetic fields in planarly layered media[J].Geophysical Journal International,2011,187(2):705-728.
[17]葛德彪,闫玉波.电磁波时域有限差分方法3版[M].西安:西安电子科技大学出版社,2011.GE D B,YAN Y B.Finite-difference time-domain method for electromagnetic waves 3rd ed[M].Xi'an:Xi'an University Press,2011.(In Chinese)
[18]刘玉柱,王光银,杨积忠.基于Born敏感核函数的VTI介质多参数全波形反演[J].地球物理学报,2015(04):1305-1316.LIU Y Z,WANG G Y,YANG J Z.Multi-parameter full-waveform inversion for VTI media based on born sensitivity kernels[J].Chinese Journal of Geophysics,2015(04):1305-1316.(In Chinese)
[19]S.OPERTO,Y.GHOLAMI,V.PRIEUX.A guided tour of multiparameter full-waveform inversion with multicomponent data:From theory to practice[J].The Leading Edge,2013,32(9):1040-1054.
[20]GERHARD PRATT,CHANGSOO SHIN,HICKS.Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion[J].Geophysical Journal International,1998,33(2):341-362.
[21]杨积忠,刘玉柱,董良国.基于Born敏感核函数的速度、密度双参数全波形反演[J].地球物理学报,2016(03):1082-1094.YANG J Z,LIU Y Z,DONG L G.Multi-parameter full waveform inversion for velocity and density based on Born sensitivity kernels[J].Chinese Journal of Geophysics,2016(03):1082-1094.(In Chinese)
[22]陈宝林.最优化理论与算法2版[M].北京:清华大学出版社,2005.CHEN B L.Optimization theory and algorithm 2nd ed[M].Beijing:TsingHua University Press,2005.(In Chinese)
[23]A.PICA,J.P.DIET,A.TARANTOLA.Nonlinear inversion of seismic reflection data in a laterally invariant medium[J].Geophysics,1990,55(3):284-292.
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