基于SPML和海绵边界的伪谱法弹性波模拟复合吸收边界条件
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摘要
弹性波数值模拟中分裂格式的完全匹配层吸收边界(SPML)难以吸收与边界接近平行传播的地震波,并会产生数值噪声,降低了地震波模拟的精度。针对此问题,提出了复合吸收边界条件,将SPML边界条件与海绵吸收边界条件组合成边界的内外层,实现了对与边界接近平行传播地震波的有效吸收。针对数值噪声沿边界传播的特点,对复合边界中的海绵边界吸收系数进行了改进,提高了海绵边界对平行于边界方向传播的地震波的衰减能力,进一步提高了对入射至边界处地震波的吸收效果。数值测试结果显示,不论对简单速度模型还是复杂速度模型,复合吸收边界对地震波的吸收效果优于常规SPML边界的吸收效果,且不会产生明显的数值噪声。
In pseudo-spectral elastic wave modeling,the conventional split perfectly matched layer(SPML)is less effective in attenuating seismic waves that propagate almost parallel to the boundary and generates numerical noise,reducing the numerical simulation accuracy.Here,we propose a composite absorbing boundary(CAB)by adding a sponge-absorbing boundary layer outside the SPML absorbing boundary to attenuate the unabsorbed seismic waves through the SPML boundary.We modified the sponge boundary absorption coefficient in the CAB to avoid numerical noise generation.The numerical experiments show that the CAB has a higher attenuation capability than the conventional SPML for seismic waves traveling parallel to the boundary,and no significant noise is generated around the boundary.Since the proposed CAB adds an absorption layer outside the original PML and does not introduce additional complex mathematical transformations,it can be easily applied to other types of numerical simulations of seismic waves,such as finite difference and finite element methods.
In pseudo-spectral elastic wave modeling,the conventional split perfectly matched layer(SPML)is less effective in attenuating seismic waves that propagate almost parallel to the boundary and generates numerical noise,reducing the numerical simulation accuracy.Here,we propose a composite absorbing boundary(CAB)by adding a sponge-absorbing boundary layer outside the SPML absorbing boundary to attenuate the unabsorbed seismic waves through the SPML boundary.We modified the sponge boundary absorption coefficient in the CAB to avoid numerical noise generation.The numerical experiments show that the CAB has a higher attenuation capability than the conventional SPML for seismic waves traveling parallel to the boundary,and no significant noise is generated around the boundary.Since the proposed CAB adds an absorption layer outside the original PML and does not introduce additional complex mathematical transformations,it can be easily applied to other types of numerical simulations of seismic waves,such as finite difference and finite element methods.
引文
[1]CLAYTON R,ENGQUIST B.Absorbing boundary conditions for acoustic and elastic wave equations[J].Bulletin of the Seismological Society of America,1977,67(6):1529-1540
[2]薛东川,王尚旭,焦淑静.起伏地表复杂介质波动方程有限元数值模拟方法[J].地球物理学进展,2007,22(2):522-529XUE D C,WANG S X,JIAO S J.Wave equation finiteelement modeling including rugged topography and complicated medium[J].Progress in Geophysics,2007,22(2):522-529
[3]李信富,李小凡,张美根.伪谱法弹性波场数值模拟中的边界条件[J].地球物理学进展,2007,22(5):1375-1379LI X F,LI X F,ZHANG M G.Boundary condition for the pseudo-spectral numerical simulation of seismic wave propagation[J].Progress in Geophysics,2007,22(5):1375-1379
[4]CERJAN C,DAN K,KOSLOFF R,et al.A nonreflecting boundary condition for discrete acoustic and elastic wave equations[J].Geophysics,1985,50(4):705-708
[5]BERENGER J P.A perfectly matched layer for the absorption of electromagnetic waves[J].Journal of Computational Physics,1994,114(2):185-200
[6]杜增利,徐峰,高宏亮.虚谱法交错网格地震波场数值模拟[J].石油物探,2010,49(5):430-437DU Z L,XU F,GAO H L.Pseudo-spectral method of the staggered grid seismic wave field numerical simulation[J].Geophysical Prospecting for Petroleum,2010,49(5):430-437
[7]陈可洋.完全匹配层吸收边界条件研究[J].石油物探,2010,49(5):472-477CHEN K Y.Study of absorbing condition by perfectly matched layer[J].Geophysical Prospecting for Petroleum,2010,49(5):472-477
[8]HASTINGS F D,SCHNEIDER J B,BROSCHAT S L.Application of the perfectly matched layer(PML)absorbing boundary condition to elastic wave propagation[J].Journal of the Acoustical Society of America,1996,100(100):3061-3069
[9]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):31-34WANG S D.The perfectly matched layer absorbing boundary of acoustic wave equation[J].Oil Geophysical Prospecting,2003,38(1):31-34
[10]CHEW W C,LIU Q H.Perfectly matched layers for elastodynamics:a new absorbing boundary condition[J].Journal of Computational Acoustics,2011,4(4):341-359
[11]赵海波,王秀明,王东,等.完全匹配层吸收边界在孔隙介质弹性波模拟中的应用[J].地球物理学报,2007,50(2):581-591ZHAO H B,WANG X M,WANG D,et al.Application of the boundary absorption using aperfectly matched layer for elastic wave simulation in poroelastic media[J].Chinese Journal of Geophysics,2007,50(2):581-591
[12]高正辉,孙建国,孙章庆,等.基于完全匹配层构建新方法的2.5维声波近似方程数值模拟[J].石油地球物理勘探,2016,51(6):1128-1133GAO Z H,SUN J G,SUN Z Q,et al.A numerical simulation of the 2.5dimensional acoustic wave approximate equation based on the perfectly matched absorbing layer[J].Oil Geophysical Prospecting,2016,51(6):1128-1133
[13]孙林洁,刘春成,张世鑫.PML边界条件下孔隙介质弹性波可变网格正演模拟方法研究[J].石油物探,2015,54(6):652-664SUN L J,LIU C C,ZHANG S X.A variable grid finitedifference scheme with PML boundary condition in elastic wave of porous media[J].Geophysical Prospecting for Petroleum,2015,54(6):652-664
[14]KRISTEK J,MOCZO P,GALIS M.A brief summary of some PML formulations and discretizations for the velocity-stress equation of seismic motion[J].Studia Geophysica et Geodaetica,2009,53(4):459-474
[15]张衡,刘洪,李博,等.VTI介质声波方程非分裂式PML吸收边界条件研究[J].石油物探,2016,55(6):781-792ZHANG H,LIU H,LI B,et al.The research on unsplit PML absorbing boundary conditions of acoustic equation for VTI media[J].Geophysical Prospecting for Petroleum,2016,55(6):781-792
[16]秦臻,卢明辉,郑晓东,等.弹性波正演模拟中改进的非分裂式PML实现方法[J].应用地球物理,2009,6(2):113-121QIN Z,LU M H,ZHENG X D,et al.The implementation of an improved NPML absorbing boundary condition in elastic wave modeling[J].Applied Geophysics,2009,6(2):113-121
[17]秦臻,任培罡,姚姚,等.弹性波正演模拟中PML吸收边界条件的改进[J].地球科学:中国地质大学学报,2009,34(4):658-664QIN Z,REN P G,YAO Y,et al.Improvement of PML absorbing boundary conditions in elastic wave forward modeling[J].Earth Science:Journal of China University of Geosciences,2009,34(4):658-664
[18]单启铜,乐友喜.PML边界条件下二维粘弹性介质波场模拟[J].石油物探,2007,46(2):126-130SHAN Q T,LE Y X.Two-dimensional viscoelastic medium wave field simulation with PML boundary conditions[J].Geophysical Prospecting for Petroleum,2007,46(2):126-130
[19]张衡,刘洪,李博,等.TTI介质声波方程分裂式PML吸收边界条件研究[J].石油物探,2017,56(3):349-361ZHANG H,LIU H,LI B,et al.The research on split PML absorbing boundary conditions of acoustic equation for TTI media[J].Geophysical Prospecting for Petroleum,2017,56(3):349-361
[20]MARTIN R,KOMATITSCH D.An unsplit convolutional perfectly matched layer technique improved at grazing incidence for the viscoelastic wave equation[J].Geophysical Journal International,2009,179(1):333-344
[21]RODEN J A,GEDNEY S D.Convolution PML(CPML):an efficient FDTD implementation of the CFS-PML for arbitrary media[J].Microwave&Optical Technology Letters,2015,27(5):334-339
[22]姜永金,毛钧杰,柴舜连.CFS-PML边界条件在PSTD算法中的实现与性能分析[J].微波学报,2004,20(4):36-39JIANG Y J,MAO J J,CHAI S L.Implementation and analysis of the perfectly matched layer media with CFS for the PSTD method[J].Journal of Microwaves,2004,20(4):36-39
[23]张鲁新,符力耘,裴正林.不分裂卷积完全匹配层与旋转交错网格有限差分在孔隙弹性介质模拟中的应用[J].地球物理学报,2010,53(10):2470-2483ZHANG L X,FU L Y,PEI Z L.Finite difference modeling of Biot’s poroelastic equations with unsplit convolutional PML and rotated staggered grid[J].Chinese Journal of Geophysics,2010,53(10):2470-2483
[24]MARTIN R,KOMATITSCH D,GEDNEY S D.A variational formulation of a stabilized unsplit convolutional perfectly matched layer for the isotropic or anisotropic seismic wave equation[J].Computer Modeling in Engineering&Sciences,2008,37(3):274-304
[25]TOIVANEN J I,STEFANSKI T P,KUSTER N,et al.Comparison of CPML implementations for the GPUaccelerated FDTD solver[J].Progress in Electromagnetics Research M,2011,19:61-75
[26]GAZDAG J.Modeling of the acoustic wave equation with transform methods[J].Geophysics,1981,46(6):854-859
[27]KOSLOFF D.Elastic wave calculation by the Fourier method[J].Bulletin of the Seismological Society of America,1984,74(3):138-142
[28]ZOU Z H,YU W H.Wave field forward modeling and theoretical analysis of weaken in discrete media[J].Applied Geophysics,2006,3(2):75-81
[29]COLLINO F,MONK P B.Optimizing the perfectly matched layer[J].Computer Methods in Applied Mechanics&Engineering,1998,164(1):157-171
[30]COLLINO F,TSOGKA C.Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media[J].Geophysics,2001,66(1):294-307
[31]高刚,贺振华,黄德济,等.完全匹配层人工边界条件中的衰减因子分析[J].石油物探,2011,50(5):430-433GAO G,HE Z H,HUANG D J,et al.Analysis on attenuation factor in the processing of artificial boundary conditions of PML[J].Geophysical Prospecting for Petroleum,2011,50(5):430-433
[32]周凤玺,张家齐,张海威.完全匹配层中衰减函数的参数优化分析[J].石油物探,2016,55(6):793-799ZHOU F X,ZHANG J Q,ZHANG H W.The parameter optimization analysis for the attenuation function in the perfectly matched layer[J].Geophysical Prospecting for Petroleum,2016,55(6):793-799
[33]KOMATITSCH D.A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation[J].Geophysical Journal International,2003,154(1):146-153
[34]WENG C C,WEEDON W H.A 3Dperfectly matched medium from modified maxwell’s equations with stretched coordinates[J].Microwave&Optical Technology Letters,1994,7(13):599-604
[35]KOMATITSCH D.An un-split convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation[J].Geophysics,2008,73(4):T51-T61
[2]薛东川,王尚旭,焦淑静.起伏地表复杂介质波动方程有限元数值模拟方法[J].地球物理学进展,2007,22(2):522-529XUE D C,WANG S X,JIAO S J.Wave equation finiteelement modeling including rugged topography and complicated medium[J].Progress in Geophysics,2007,22(2):522-529
[3]李信富,李小凡,张美根.伪谱法弹性波场数值模拟中的边界条件[J].地球物理学进展,2007,22(5):1375-1379LI X F,LI X F,ZHANG M G.Boundary condition for the pseudo-spectral numerical simulation of seismic wave propagation[J].Progress in Geophysics,2007,22(5):1375-1379
[4]CERJAN C,DAN K,KOSLOFF R,et al.A nonreflecting boundary condition for discrete acoustic and elastic wave equations[J].Geophysics,1985,50(4):705-708
[5]BERENGER J P.A perfectly matched layer for the absorption of electromagnetic waves[J].Journal of Computational Physics,1994,114(2):185-200
[6]杜增利,徐峰,高宏亮.虚谱法交错网格地震波场数值模拟[J].石油物探,2010,49(5):430-437DU Z L,XU F,GAO H L.Pseudo-spectral method of the staggered grid seismic wave field numerical simulation[J].Geophysical Prospecting for Petroleum,2010,49(5):430-437
[7]陈可洋.完全匹配层吸收边界条件研究[J].石油物探,2010,49(5):472-477CHEN K Y.Study of absorbing condition by perfectly matched layer[J].Geophysical Prospecting for Petroleum,2010,49(5):472-477
[8]HASTINGS F D,SCHNEIDER J B,BROSCHAT S L.Application of the perfectly matched layer(PML)absorbing boundary condition to elastic wave propagation[J].Journal of the Acoustical Society of America,1996,100(100):3061-3069
[9]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):31-34WANG S D.The perfectly matched layer absorbing boundary of acoustic wave equation[J].Oil Geophysical Prospecting,2003,38(1):31-34
[10]CHEW W C,LIU Q H.Perfectly matched layers for elastodynamics:a new absorbing boundary condition[J].Journal of Computational Acoustics,2011,4(4):341-359
[11]赵海波,王秀明,王东,等.完全匹配层吸收边界在孔隙介质弹性波模拟中的应用[J].地球物理学报,2007,50(2):581-591ZHAO H B,WANG X M,WANG D,et al.Application of the boundary absorption using aperfectly matched layer for elastic wave simulation in poroelastic media[J].Chinese Journal of Geophysics,2007,50(2):581-591
[12]高正辉,孙建国,孙章庆,等.基于完全匹配层构建新方法的2.5维声波近似方程数值模拟[J].石油地球物理勘探,2016,51(6):1128-1133GAO Z H,SUN J G,SUN Z Q,et al.A numerical simulation of the 2.5dimensional acoustic wave approximate equation based on the perfectly matched absorbing layer[J].Oil Geophysical Prospecting,2016,51(6):1128-1133
[13]孙林洁,刘春成,张世鑫.PML边界条件下孔隙介质弹性波可变网格正演模拟方法研究[J].石油物探,2015,54(6):652-664SUN L J,LIU C C,ZHANG S X.A variable grid finitedifference scheme with PML boundary condition in elastic wave of porous media[J].Geophysical Prospecting for Petroleum,2015,54(6):652-664
[14]KRISTEK J,MOCZO P,GALIS M.A brief summary of some PML formulations and discretizations for the velocity-stress equation of seismic motion[J].Studia Geophysica et Geodaetica,2009,53(4):459-474
[15]张衡,刘洪,李博,等.VTI介质声波方程非分裂式PML吸收边界条件研究[J].石油物探,2016,55(6):781-792ZHANG H,LIU H,LI B,et al.The research on unsplit PML absorbing boundary conditions of acoustic equation for VTI media[J].Geophysical Prospecting for Petroleum,2016,55(6):781-792
[16]秦臻,卢明辉,郑晓东,等.弹性波正演模拟中改进的非分裂式PML实现方法[J].应用地球物理,2009,6(2):113-121QIN Z,LU M H,ZHENG X D,et al.The implementation of an improved NPML absorbing boundary condition in elastic wave modeling[J].Applied Geophysics,2009,6(2):113-121
[17]秦臻,任培罡,姚姚,等.弹性波正演模拟中PML吸收边界条件的改进[J].地球科学:中国地质大学学报,2009,34(4):658-664QIN Z,REN P G,YAO Y,et al.Improvement of PML absorbing boundary conditions in elastic wave forward modeling[J].Earth Science:Journal of China University of Geosciences,2009,34(4):658-664
[18]单启铜,乐友喜.PML边界条件下二维粘弹性介质波场模拟[J].石油物探,2007,46(2):126-130SHAN Q T,LE Y X.Two-dimensional viscoelastic medium wave field simulation with PML boundary conditions[J].Geophysical Prospecting for Petroleum,2007,46(2):126-130
[19]张衡,刘洪,李博,等.TTI介质声波方程分裂式PML吸收边界条件研究[J].石油物探,2017,56(3):349-361ZHANG H,LIU H,LI B,et al.The research on split PML absorbing boundary conditions of acoustic equation for TTI media[J].Geophysical Prospecting for Petroleum,2017,56(3):349-361
[20]MARTIN R,KOMATITSCH D.An unsplit convolutional perfectly matched layer technique improved at grazing incidence for the viscoelastic wave equation[J].Geophysical Journal International,2009,179(1):333-344
[21]RODEN J A,GEDNEY S D.Convolution PML(CPML):an efficient FDTD implementation of the CFS-PML for arbitrary media[J].Microwave&Optical Technology Letters,2015,27(5):334-339
[22]姜永金,毛钧杰,柴舜连.CFS-PML边界条件在PSTD算法中的实现与性能分析[J].微波学报,2004,20(4):36-39JIANG Y J,MAO J J,CHAI S L.Implementation and analysis of the perfectly matched layer media with CFS for the PSTD method[J].Journal of Microwaves,2004,20(4):36-39
[23]张鲁新,符力耘,裴正林.不分裂卷积完全匹配层与旋转交错网格有限差分在孔隙弹性介质模拟中的应用[J].地球物理学报,2010,53(10):2470-2483ZHANG L X,FU L Y,PEI Z L.Finite difference modeling of Biot’s poroelastic equations with unsplit convolutional PML and rotated staggered grid[J].Chinese Journal of Geophysics,2010,53(10):2470-2483
[24]MARTIN R,KOMATITSCH D,GEDNEY S D.A variational formulation of a stabilized unsplit convolutional perfectly matched layer for the isotropic or anisotropic seismic wave equation[J].Computer Modeling in Engineering&Sciences,2008,37(3):274-304
[25]TOIVANEN J I,STEFANSKI T P,KUSTER N,et al.Comparison of CPML implementations for the GPUaccelerated FDTD solver[J].Progress in Electromagnetics Research M,2011,19:61-75
[26]GAZDAG J.Modeling of the acoustic wave equation with transform methods[J].Geophysics,1981,46(6):854-859
[27]KOSLOFF D.Elastic wave calculation by the Fourier method[J].Bulletin of the Seismological Society of America,1984,74(3):138-142
[28]ZOU Z H,YU W H.Wave field forward modeling and theoretical analysis of weaken in discrete media[J].Applied Geophysics,2006,3(2):75-81
[29]COLLINO F,MONK P B.Optimizing the perfectly matched layer[J].Computer Methods in Applied Mechanics&Engineering,1998,164(1):157-171
[30]COLLINO F,TSOGKA C.Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media[J].Geophysics,2001,66(1):294-307
[31]高刚,贺振华,黄德济,等.完全匹配层人工边界条件中的衰减因子分析[J].石油物探,2011,50(5):430-433GAO G,HE Z H,HUANG D J,et al.Analysis on attenuation factor in the processing of artificial boundary conditions of PML[J].Geophysical Prospecting for Petroleum,2011,50(5):430-433
[32]周凤玺,张家齐,张海威.完全匹配层中衰减函数的参数优化分析[J].石油物探,2016,55(6):793-799ZHOU F X,ZHANG J Q,ZHANG H W.The parameter optimization analysis for the attenuation function in the perfectly matched layer[J].Geophysical Prospecting for Petroleum,2016,55(6):793-799
[33]KOMATITSCH D.A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation[J].Geophysical Journal International,2003,154(1):146-153
[34]WENG C C,WEEDON W H.A 3Dperfectly matched medium from modified maxwell’s equations with stretched coordinates[J].Microwave&Optical Technology Letters,1994,7(13):599-604
[35]KOMATITSCH D.An un-split convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation[J].Geophysics,2008,73(4):T51-T61
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