二阶声波方程频域PML边界条件及频域变网格步长并行计算
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摘要
研究了二阶声波方程频域PML边界条件和频域变网格并行计算技术。PML边界是一种较为理想的吸收边界方法,多用在求解时域应力速度方程中,但对于频域声波正演,二阶位移方程更常用。从一阶声波方程PML吸收边界条件导出频域二阶位移方程PML边界条件,模拟算例得到的频率切片、时间切片和地震记录对比都说明该边界条件吸收效果很好。频域单炮正演不同频率间是独立的,据此低频部分采用大网格计算,高频采用小网格,实现变网格步长计算技术,这是较时间域正演的一个优势,在保证模拟质量的同时,减少计算量和内存消耗。
With the development of the computer,full-waveform inversion based on frequency-domain modeling attracts researchers' attentions again,accordingly,PML boundary conditions for second-order acoustic wave equations and variable grid parallel computation in frequency-domain modeling are studied.PML boundary conditions is a relatively ideal absorbing boundary method and is usually used in solving the stress-velocity equations in time domain,however,for frequency acoustic wave modeling,second order displacement equation is more commonly used.We derived frequency domain second-order displacement equation PML boundary conditions from the first order acoustic wave equation PML boundary conditions and present satisfactory results of simulation examples.A single shot frequency domain modeling is implemented by solving a number of linear equations followed by an inverse Fourier transform on the records,so the frequency domain parallel computing can be performed either in different shots or in different frequencies.According to the frequency dependent character,we study the variable grid spacing computing method,using large grids in low frequencies,and small grids in high frequencies,which is one of the advantages compared to time domain modeling,and reduce the number of computation and memory consumption in the premise of ensuring the quality of simulation.
With the development of the computer,full-waveform inversion based on frequency-domain modeling attracts researchers' attentions again,accordingly,PML boundary conditions for second-order acoustic wave equations and variable grid parallel computation in frequency-domain modeling are studied.PML boundary conditions is a relatively ideal absorbing boundary method and is usually used in solving the stress-velocity equations in time domain,however,for frequency acoustic wave modeling,second order displacement equation is more commonly used.We derived frequency domain second-order displacement equation PML boundary conditions from the first order acoustic wave equation PML boundary conditions and present satisfactory results of simulation examples.A single shot frequency domain modeling is implemented by solving a number of linear equations followed by an inverse Fourier transform on the records,so the frequency domain parallel computing can be performed either in different shots or in different frequencies.According to the frequency dependent character,we study the variable grid spacing computing method,using large grids in low frequencies,and small grids in high frequencies,which is one of the advantages compared to time domain modeling,and reduce the number of computation and memory consumption in the premise of ensuring the quality of simulation.
引文
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[2]Plessix R E.Three-dimensional frequency-domain full-waveform inversion with an iterative solver[J].Geo-physics,2009,74(6):149-157.
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[13]刘四新,曾昭发,徐波.地质雷达数值模拟中有损耗介质吸收边界条件的实现[J].吉林大学学报:地球科学版,2005,35(3):378-381.LIU Si-xin,ZENG Zhao-fa,XU Bo.Realization ofabsorbing boundary condition with lossy media forground penetrating radar simulation[J].Journal of Ji-lin University:Earth Science Edition,2005,35(3):378-381.
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[15]张显文,韩立国,黄玲,等.基于递归积分的复频移PML弹性波方程交错网格高阶差分法[J].地球物理学报,2009,52(7):1800-1807.ZHANG Xian-wen,HAN Li-guo,HUANG Ling,etal.A staggered-grid high-order difference method ofcomplex frequency-shifted PML based on recursiveintergration for elastic wave equation[J].ChineseJournal of Geophysics,2009,52(7):1800-1807.
[16]Hu W,Abubakar A,Habashy T M.Application ofthe nearly perfectly matched layer in acoustic wavemodeling[J].Geophysics,2007,72(5):169-175.
[17]王守东.声波方程完全匹配层吸收边界[J].石油地球物理勘探,2003,38(1):31-34.WANG Shou-dong.Acoustic wave equation PML ab-sorbing boundary[J].Oil Geophysical Prospecting,2003,38(1):31-34.
[18]殷文.基于频率域高阶有限差分法的正演模拟及并行算法[J].吉林大学学报:地球科学版,2008,38(1):144-151.YIN Wen.Forward modeling and parallel algorithmbased on high-order finite-difference method in fre-quency domain[J].Journal of Jilin University:EarthScience Edition,2008,38(1):144-151.
[19]Fokkema J T,van den Berg P M.Seismic applica-tions of acoustic reciprocity[M].New York:Elsevi-er Science Pub Co Inc,1993.
[20]Chew W C,Weedon W H.A 3D perfectly matchedmedium from modified Maxwell’s equations withstretched coordinates[J].IEEE Transactions on An-tennas and Propagation,1994,43:599-604.
[2]Plessix R E.Three-dimensional frequency-domain full-waveform inversion with an iterative solver[J].Geo-physics,2009,74(6):149-157.
[3]Lysmer,Drake.A finite-element method for seismolo-gy[C]//Bolt B A.Methods in computational physics:Volume 11:Seismology.New York:Academic PressInc,1972:181-216.
[4]Jo C H,Shin C,Suh J H.An optimal 9-point,finite-difference,frequency-space,2-D scalar wave extrapola-tor[J].Geophysics,1996,61(2):529-537.
[5]Shin C,Sohnt H.A frequency-space 2-D scalar waveextrapolator using extended 25-point finite-differenceoperator[J].Geophysics,1998,63(1):289-296.
[6]Operto S,Virieux J,Amestoy P,et al.3D finite-difference frequency-domain modeling of visco-acous-tic wave propagation using a massively parallel directsolver:a feasibility study[J].Geophysics,2007,72(5):195-211.
[7]Operto S,Virieu J,Ribodetti A,et al.Finite-differ-ence frequency-domain modeling of viscoacoustic wavepropagation in 2D tilted transversely isotropic(TTI)media[J].Geophysics,2009,74(5):75-95.
[8]Liao J P,Wang H Z,Ma Z T.2-D elastic wave mod-eling with frequency-space 25-point finite-differenceoperators[J].Applied Geophysics,2009,6(3):259-266.
[9]马召贵,王尚旭,宋建勇.频率域波动方程正演中的多网格迭代算法[J].石油地球物理勘探,2010,45(1):1-5.MA Zhao-gui,WANG Shang-xu,SONG Jian-yong.Multi-grid iterative algorithm in frequency domainwave equation modeling[J].Oil Geophysical Prospec-ting,2010,45(1):1-5.
[10]Engquist B,Majda A.Absorbing boundary condi-tions for numerical simulation of waves[J].Mathe-matics of Computation,1977,31:629-651.
[11]Cerjan C,Kosloff D,Kosloff R,et al.A nonreflect-ing boundary condition for discrete acoustic and elas-tic wave equations[J].Geophysics,1985,50(4),705-708.
[12]Bérenger J P.Aperfectly matched layer for the ab-sorption of electromagnetic waves[J].Journal ofComputational Physics,1994,114:185-200.
[13]刘四新,曾昭发,徐波.地质雷达数值模拟中有损耗介质吸收边界条件的实现[J].吉林大学学报:地球科学版,2005,35(3):378-381.LIU Si-xin,ZENG Zhao-fa,XU Bo.Realization ofabsorbing boundary condition with lossy media forground penetrating radar simulation[J].Journal of Ji-lin University:Earth Science Edition,2005,35(3):378-381.
[14]Gao H,Zhang J.Implementation of perfectly matchedlayers in an arbitrary geometrical boundary for elasticwave modeling[J].Geophysical Journal Internation-al,2008,174(3):1029-1036.
[15]张显文,韩立国,黄玲,等.基于递归积分的复频移PML弹性波方程交错网格高阶差分法[J].地球物理学报,2009,52(7):1800-1807.ZHANG Xian-wen,HAN Li-guo,HUANG Ling,etal.A staggered-grid high-order difference method ofcomplex frequency-shifted PML based on recursiveintergration for elastic wave equation[J].ChineseJournal of Geophysics,2009,52(7):1800-1807.
[16]Hu W,Abubakar A,Habashy T M.Application ofthe nearly perfectly matched layer in acoustic wavemodeling[J].Geophysics,2007,72(5):169-175.
[17]王守东.声波方程完全匹配层吸收边界[J].石油地球物理勘探,2003,38(1):31-34.WANG Shou-dong.Acoustic wave equation PML ab-sorbing boundary[J].Oil Geophysical Prospecting,2003,38(1):31-34.
[18]殷文.基于频率域高阶有限差分法的正演模拟及并行算法[J].吉林大学学报:地球科学版,2008,38(1):144-151.YIN Wen.Forward modeling and parallel algorithmbased on high-order finite-difference method in fre-quency domain[J].Journal of Jilin University:EarthScience Edition,2008,38(1):144-151.
[19]Fokkema J T,van den Berg P M.Seismic applica-tions of acoustic reciprocity[M].New York:Elsevi-er Science Pub Co Inc,1993.
[20]Chew W C,Weedon W H.A 3D perfectly matchedmedium from modified Maxwell’s equations withstretched coordinates[J].IEEE Transactions on An-tennas and Propagation,1994,43:599-604.
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