测井约束地震波形反演的同伦摄动法
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摘要
测井数据和地震数据是地震勘探中两种最重要的资料.测井约束地震波形反演是在非线性波形反演的基础上,利用已知测井资料详细的垂直分辨能力和地震资料均匀密集的水平采样特点,通过迭代反演来求取一个具有较高分辨率的速度参数.本文建立了测井约束反演模型,研究了测井约束下地震波形反演的同伦摄动求解方法.同伦摄动法作为一种新的、求解数学物理中各种非线性问题的有效方法,具有计算速度快、计算精度高的优点.这对于提高反演的精度和效率是十分有益的.为了表征该方法的有效性和稳定性,分别对水平层状介质模型和逆冲断层带模型进行了数值模拟,并与Landweber迭代法相对比,结果表明该算法具有更好的收敛性,能够取得更为满意的反演效果.
The well logging data and seismic data are two most important data in the seismic prospecting.Well log constrained seismic waveform inversion is based on the nonlinear waveform inversion techniques.It makes full use of detailed vertical resolution of well log data and horizontal dense sampling of seismic data.Its core is to obtain an approximate velocity parameter of higher resolution by iterative inversion.This paper established the well log constraint inversion model,and studied homotopy perturbation method for solving this problem.Homotopy perturbation method is a novel and effective method,and can solve various nonlinear equations of various branches in mathematics and physics.This method has the advantages of fast computation and high precision.This improves the accuracy and stability in inversion.In order to show the effectiveness and stability,we carried out numerical simulations of the horizontally stratified medium and the thrust fault belt model,and compared with Landweber iteration.Numerical results show that the new method improves the convergence,and can produce more satisfactory effect.
The well logging data and seismic data are two most important data in the seismic prospecting.Well log constrained seismic waveform inversion is based on the nonlinear waveform inversion techniques.It makes full use of detailed vertical resolution of well log data and horizontal dense sampling of seismic data.Its core is to obtain an approximate velocity parameter of higher resolution by iterative inversion.This paper established the well log constraint inversion model,and studied homotopy perturbation method for solving this problem.Homotopy perturbation method is a novel and effective method,and can solve various nonlinear equations of various branches in mathematics and physics.This method has the advantages of fast computation and high precision.This improves the accuracy and stability in inversion.In order to show the effectiveness and stability,we carried out numerical simulations of the horizontally stratified medium and the thrust fault belt model,and compared with Landweber iteration.Numerical results show that the new method improves the convergence,and can produce more satisfactory effect.
引文
[1]Tarantola A.Inversion of seismic reflection data in theacoustic approximation.Geophysics,1984,49(8):1259-1266.
[2]Mora P.Elastic wave-field inversion of reflection andtransmission data.Geophysics,1988,53(6):750-759.
[3]Pratt R G,Shipp R M.Seismic waveform inversion in thefrequency domain,Part II:Fault delineation in sedimentsusing crosshole data.Geophysics,1999,64(3):902-914.
[4]Hicks G J,Pratt R G.Reflection waveform inversion usinglocal descent methods:Estimating attenuation and velocityover a gas-sand deposit.Geophysics,2001,66(2):598-612.
[5]Ravaut C,Operto S,Improta L,Virieux J,et al.Multiscaleimaging of complex structures from multifold wide-apertureseismic data by frequency-domain full-waveform tomography:Application to a thrust belt.Geophy.J.Int.,2004,159(3):1032-1056.
[6]Gao F,Levander A R,Pratt R G,et al.Waveformtomography at a groundwater contamination site:Vsp-surfacedata set.Geophysics,2006,71(1):H1-H11.
[7]Bleibinhaus F,Hole J A,Ryberg T,et al.Structure of theCalifornia Coast ranges and San Andreas fault at SAFODfrom seismic waveform inversion and reflection imaging.Journal of Geophysical Research,2007,112:B06315,doi:10.1029/2006JB004611.
[8]冯国峰,韩波,刘家琦.二维波动方程约束反演的大范围收敛广义脉冲谱方法.地球物理学报,2003,46(2):265-270.Feng G F,Han B,Liu J Q.Widely convergent generalizedpulse-spectrum methods for 2-D wave Equation Inversion.Chinese J.Geophys(in Chinese),2003,46(2):265-270.
[9]傅红笋,韩波.二维波动方程速度的正则化-同伦-测井约束反演.地球物理学报,2005,48(6):1441-1448.Fu H S,Han B.A regularization homotopy method for theinverse problem of 2-D wave equation and well log constraintinversion.Chinese J.Geophys.(in Chinese),2005,48(6):1441-1448.
[10]Han B,Fu H S,Liu L.A homotopy method for well logconstraint waveform Inversion.Geophysics,2007,72(1):R1-R7.
[11]王守东.井间地震资料全变差正则化波形反演.石油物探,2011,50(3):234-240.Wang S D.Total variation regularization waveform inversionof crosswell seismic data.Geophysical Prospecting forPetroleum(in Chinese),2011,50(3):234-240.
[12]He J H.Homotopy perturbation technique.Comput.Meth.Appl.Mech.Eng.,1999,178(3-4):257-262.
[13]He J H.A coupling method of a homotopy technique and aperturbation technique for non-linear problems.Int.J.Non-Linear Mech,2000,35(1):37-43.
[14]Sharma P R,Methi G.Applications of homotopyperturbation method to partial differential equations.AsianJournal of Mathematic and Statistics,2011,4(3):140-150.
[15]Shakeri F,Dehghan M.Inverse problem of diffusion equationby He's homotopy perturbation method.Phys.Scr.,2007,75(4):551-556.
[16]Ganji D D,Sadighi A.Application of He's homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations.Int.J.Nonlinear Sci.Numer.Simul.,2006,7(4):411-418.
[17]Javidi M,Golbabai A.A numerical solution for solvingsystem of Fredholm integral equations by using homotopyperturbation method.Appl.Math.Comput.,2007,189(2):1921-1928.
[18]Abdulaziz O,Hashim I,Chowdhury M S H.Solvingvariational problems by homotopy-perturbation method.Int.J.Numer.Meth.Eng.,2008,75(6):709-721.
[19]Cao L,Han B,Wang W.Homotopy perturbation method fornonlinear ill-posed operator equations.Int.J.Non.Sci.Numer.Simul.,2009,10(10):1319-1322.
[20]崔岩,王彦飞,杨长春.带先验知识的波阻抗反演正则化方法研究.地球物理学报,2009,52(8):2135-2141.Cui Y,Wang Y F,Yang C C.Regularizing method with apriori knowledge for seismic impedance inversion.Chinese J.Geophys.(in Chinese),2009,52(8):2135-2141.
[21]许琨,王妙月.声波方程频率域有限元参数反演.地球物理学报,2001,44(6):852-864.Kun X,Wang M Y.Finite element inversion of thecoefficients of acoustic equation in frequency domain.ChineseJ.Geophys.(in Chinese),2001,44(6):852-864.
[22]周辉,何樵登,徐世浙.人工神经网络非线性地震波形反演.石油物探,1997,36(1):61-70.Zhou H,He Q D,Xu S Z.Nonlinear seismic waveforminversion using the artifiual neural network.GPP(inChinese),1997,36(1):61-70.
[23]Sambridge M,Drijkoningen G.Genetic algorithms in seismicwaveform inversion.Geophy.J.Int.,1992,109(2):323-342.
[24]姚姚.地球物理非线性反演模拟退火法的改进.地球物理学报,1995,38(5):643-650.Yao Y.Improvement on nonlinear geophysical inversionsimulated annealing.Chinese J.Geophys.(in Chinese),1995,38(5):643-650.
[25]马坚伟,朱亚平,杨慧珠.二维地震波形小波多尺度反演.工程数学学报,2004,21(1):109-113.Ma J W,Zhu Y P,Yang H Z.2-dimension seismic waveformwavelet multiscale inversion.J.Engi.Math.(in Chinese),2004,21(3):109-113.
[26]王彦飞.反演问题的计算方法及其应用[当代科学前沿论丛北京].北京:高等教育出版,2007.Wang Y F.Computational Methods for Inverse Problems andTheir Applications(in Chinese).Beijing:Higher EducationPress,2007.
[2]Mora P.Elastic wave-field inversion of reflection andtransmission data.Geophysics,1988,53(6):750-759.
[3]Pratt R G,Shipp R M.Seismic waveform inversion in thefrequency domain,Part II:Fault delineation in sedimentsusing crosshole data.Geophysics,1999,64(3):902-914.
[4]Hicks G J,Pratt R G.Reflection waveform inversion usinglocal descent methods:Estimating attenuation and velocityover a gas-sand deposit.Geophysics,2001,66(2):598-612.
[5]Ravaut C,Operto S,Improta L,Virieux J,et al.Multiscaleimaging of complex structures from multifold wide-apertureseismic data by frequency-domain full-waveform tomography:Application to a thrust belt.Geophy.J.Int.,2004,159(3):1032-1056.
[6]Gao F,Levander A R,Pratt R G,et al.Waveformtomography at a groundwater contamination site:Vsp-surfacedata set.Geophysics,2006,71(1):H1-H11.
[7]Bleibinhaus F,Hole J A,Ryberg T,et al.Structure of theCalifornia Coast ranges and San Andreas fault at SAFODfrom seismic waveform inversion and reflection imaging.Journal of Geophysical Research,2007,112:B06315,doi:10.1029/2006JB004611.
[8]冯国峰,韩波,刘家琦.二维波动方程约束反演的大范围收敛广义脉冲谱方法.地球物理学报,2003,46(2):265-270.Feng G F,Han B,Liu J Q.Widely convergent generalizedpulse-spectrum methods for 2-D wave Equation Inversion.Chinese J.Geophys(in Chinese),2003,46(2):265-270.
[9]傅红笋,韩波.二维波动方程速度的正则化-同伦-测井约束反演.地球物理学报,2005,48(6):1441-1448.Fu H S,Han B.A regularization homotopy method for theinverse problem of 2-D wave equation and well log constraintinversion.Chinese J.Geophys.(in Chinese),2005,48(6):1441-1448.
[10]Han B,Fu H S,Liu L.A homotopy method for well logconstraint waveform Inversion.Geophysics,2007,72(1):R1-R7.
[11]王守东.井间地震资料全变差正则化波形反演.石油物探,2011,50(3):234-240.Wang S D.Total variation regularization waveform inversionof crosswell seismic data.Geophysical Prospecting forPetroleum(in Chinese),2011,50(3):234-240.
[12]He J H.Homotopy perturbation technique.Comput.Meth.Appl.Mech.Eng.,1999,178(3-4):257-262.
[13]He J H.A coupling method of a homotopy technique and aperturbation technique for non-linear problems.Int.J.Non-Linear Mech,2000,35(1):37-43.
[14]Sharma P R,Methi G.Applications of homotopyperturbation method to partial differential equations.AsianJournal of Mathematic and Statistics,2011,4(3):140-150.
[15]Shakeri F,Dehghan M.Inverse problem of diffusion equationby He's homotopy perturbation method.Phys.Scr.,2007,75(4):551-556.
[16]Ganji D D,Sadighi A.Application of He's homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations.Int.J.Nonlinear Sci.Numer.Simul.,2006,7(4):411-418.
[17]Javidi M,Golbabai A.A numerical solution for solvingsystem of Fredholm integral equations by using homotopyperturbation method.Appl.Math.Comput.,2007,189(2):1921-1928.
[18]Abdulaziz O,Hashim I,Chowdhury M S H.Solvingvariational problems by homotopy-perturbation method.Int.J.Numer.Meth.Eng.,2008,75(6):709-721.
[19]Cao L,Han B,Wang W.Homotopy perturbation method fornonlinear ill-posed operator equations.Int.J.Non.Sci.Numer.Simul.,2009,10(10):1319-1322.
[20]崔岩,王彦飞,杨长春.带先验知识的波阻抗反演正则化方法研究.地球物理学报,2009,52(8):2135-2141.Cui Y,Wang Y F,Yang C C.Regularizing method with apriori knowledge for seismic impedance inversion.Chinese J.Geophys.(in Chinese),2009,52(8):2135-2141.
[21]许琨,王妙月.声波方程频率域有限元参数反演.地球物理学报,2001,44(6):852-864.Kun X,Wang M Y.Finite element inversion of thecoefficients of acoustic equation in frequency domain.ChineseJ.Geophys.(in Chinese),2001,44(6):852-864.
[22]周辉,何樵登,徐世浙.人工神经网络非线性地震波形反演.石油物探,1997,36(1):61-70.Zhou H,He Q D,Xu S Z.Nonlinear seismic waveforminversion using the artifiual neural network.GPP(inChinese),1997,36(1):61-70.
[23]Sambridge M,Drijkoningen G.Genetic algorithms in seismicwaveform inversion.Geophy.J.Int.,1992,109(2):323-342.
[24]姚姚.地球物理非线性反演模拟退火法的改进.地球物理学报,1995,38(5):643-650.Yao Y.Improvement on nonlinear geophysical inversionsimulated annealing.Chinese J.Geophys.(in Chinese),1995,38(5):643-650.
[25]马坚伟,朱亚平,杨慧珠.二维地震波形小波多尺度反演.工程数学学报,2004,21(1):109-113.Ma J W,Zhu Y P,Yang H Z.2-dimension seismic waveformwavelet multiscale inversion.J.Engi.Math.(in Chinese),2004,21(3):109-113.
[26]王彦飞.反演问题的计算方法及其应用[当代科学前沿论丛北京].北京:高等教育出版,2007.Wang Y F.Computational Methods for Inverse Problems andTheir Applications(in Chinese).Beijing:Higher EducationPress,2007.
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