用于波场成像的谱法LU分解
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摘要
地震波场模拟和偏移成像等有限差分隐格式算法中的重要环节 ,是实现亥姆霍兹算子表示矩阵H的快速求逆运算 .在螺旋边界条件下 ,H具有Toeplitz结构的正定厄密矩阵 ,其快速求逆可由谱法LU分解实现 .本文分析了谱法LU分解对提高计算速度的原理及特点 ,并着重讨论了在不同类型的介质模型中 ,采用谱法分解矩阵H时带来的数值误差、误差的分布及其对波场计算的影响 .研究结果表明 ,对均匀介质而言 ,矩阵H各列具有相同的非零元素分布 ,谱法LU分解的误差在吸收边界条件下 ,不影响波场模拟和成像计算 ;但对于非均匀介质模型 ,矩阵H各列具有不同的非零元素分布 ,谱法LU分解的误差随介质不均匀性程度的增大而增大 ,势必影响非均匀介质中波场计算 .在波场模拟和成像等有限差分隐格式算法中 ,采用谱法LU分解完成矩阵求逆时 ,必须考虑到并尽量减少该方法的误差对波场计算的影响 .
The inversion of Helmholtz matrix, a finite-difference representation of Helmholtz operator, plays a critical role in the numerical methods of seismic modeling and 3D depth migration. The Helmholtz matrix has a Toeplitz structure with a helical boundary condition. It is easy to be inversed by an efficient LU decomposition method, a spectral factorization method (SF method). In this paper we discuss the efficiency of SF method in LU decomposition, the errors of SF method in decomposing the matrix H of several different velocity models, errors distribution and their effects on seismic imaging. The result in this paper indicates that for constant velocity model all the arrays of matrix H have same nonzero values, and the errors brought by SF method will not affect the computation of wave fields if an absorb boundary condition is used; for non-constant velocity model, each array in matrix H has different nonzero values, and the error increases with variability degree of the velocity model, which are fatal to wave fields calculation. Therefore, once a SF method is used to calculate the wave field in seismic modeling and imaging, the errors given by this method should be considered to ensure a correct wave fields, which will be discussed in another paper.
The inversion of Helmholtz matrix, a finite-difference representation of Helmholtz operator, plays a critical role in the numerical methods of seismic modeling and 3D depth migration. The Helmholtz matrix has a Toeplitz structure with a helical boundary condition. It is easy to be inversed by an efficient LU decomposition method, a spectral factorization method (SF method). In this paper we discuss the efficiency of SF method in LU decomposition, the errors of SF method in decomposing the matrix H of several different velocity models, errors distribution and their effects on seismic imaging. The result in this paper indicates that for constant velocity model all the arrays of matrix H have same nonzero values, and the errors brought by SF method will not affect the computation of wave fields if an absorb boundary condition is used; for non-constant velocity model, each array in matrix H has different nonzero values, and the error increases with variability degree of the velocity model, which are fatal to wave fields calculation. Therefore, once a SF method is used to calculate the wave field in seismic modeling and imaging, the errors given by this method should be considered to ensure a correct wave fields, which will be discussed in another paper.
引文
[1] KummerB ,BehleA ,DorauF .Hybridmodelingofelastic wavepropagationintwo dimensionallaterallyinhomogeneousmedia.Geophysics,1987,52(2):765～771
[2] KuoJT ,YengYC ,PecholcsP ,etal.Anoteontheinfluenceoftheelasticityofwavepropagationinanacoustic elasticmedia.Comp.andMathAppl.,1985,11:887～896
[3] MuftiIR .Seismicmodelingintheimplicitmode.Geophysicalprospecting,198433:619～656
[4] 罗明秋,刘 洪,李幼铭.地震波传播的哈密顿表述及辛几何算法.地球物理学报,2001,44(1):120～128LUOMingqiu,LIUHong,LIYouming.Hamiltoniandescriptionofseismicwavepropagationandsymplecticmethod.ChineseJ .Geophys.,2001,44(1):120～128
[5] 罗明秋,刘 洪,李幼铭.基于螺旋线上谱因式分解的地震波场隐式辛算法.地球物理学报,2000,43(3):379～388LUOMingqiu,LIUHong,LIYouming.Seismicwavemodelingwithimplicitsymplecticmethodbasedonspectralfactorizationonhelix.ChineseJ.Geophys.,2000,43(3):379～388
[6] ClaerboutJF .ImagingtheEarth’sInterior.BlackwellScientificPublication,Inc.1985
[7] HolbergO .Towardsoptimumone waywavepropagation.GeophysicsProspect,1988,36(3):99～114
[8] HaleD .3DdepthmigrationviaMcClellantransformations.Geophysics,1991,56(11):1778～1785
[9] HaleD .Stableexplicitdepthextrapolationofseismicwavefields.Geophysics,1991,56(11):1770～1777
[10] EtgenJT .Stabilityofexplicitdepthextrapolationthroughlaterallyvaryingmedia.SEG 64thInternationalMeeting,ExpandedAbstracts,1994,1266～1269
[11] GravesRW ,ClaytonRW .Modelingacousticwaveswithparaxialextrapolators.Geophysics,1990,55(3):306～319
[12] ZhouH ,McMechanGA .One pass3Dseismicextrapolationwiththe45°waveequation.Geophysics,1997,62(6):1817～1824
[13] GuanquanZhang,YuZhang,HongboZhou.Helicalfinite differenceschemesfor3Ddepthmigration.SEG 70 thInternationalMeeting,ExpandedAbstracts,2000.862～865
[14] RistowD ,R¨uhlT .3Dimplicitfinitedifferencemigrationbymultiway splitting.SEG 65 thInternationalMeeting,ExpandedAbstracts,1995.1220～1223
[15] ClaerboutJF .Multidimensionalrecursivefiltersviaahelixwithapplicationtovelocityestimationand3Dmigration,the68thAnn.SEGmeeting,1998.1995～1998
[16] ClaerboutJF .Multidimensionalrecursivefiltersviaahelix.Geophysics,1998,63:1532～1541
[17] 石根华.数值流形方法与非连续变形分析.北京:清华大学出版社,1997.146～167SHIGenhua.NumericalManifoldMethodandDiscontinuousDeformationanalysis.Beijing:TsinghuaUniversityPress,1997.146～167
[18] LUOMingqiu,GAOHongwei,LIUHong,etal.AcomparisonontheLUdecompositionofLaplacianmatrixwithspectralanddirecteliminationmethods.SEG 70 thInternationalMeeting,ExpandedAbstracts,2000.2349～2353
[19] RickettJ.Implicit3Ddepthmigrationbywavefieldextrapolationwithhelicalboundarycondition.the68thAnn.SEGmeeting,1998.1124～1127
[2] KuoJT ,YengYC ,PecholcsP ,etal.Anoteontheinfluenceoftheelasticityofwavepropagationinanacoustic elasticmedia.Comp.andMathAppl.,1985,11:887～896
[3] MuftiIR .Seismicmodelingintheimplicitmode.Geophysicalprospecting,198433:619～656
[4] 罗明秋,刘 洪,李幼铭.地震波传播的哈密顿表述及辛几何算法.地球物理学报,2001,44(1):120～128LUOMingqiu,LIUHong,LIYouming.Hamiltoniandescriptionofseismicwavepropagationandsymplecticmethod.ChineseJ .Geophys.,2001,44(1):120～128
[5] 罗明秋,刘 洪,李幼铭.基于螺旋线上谱因式分解的地震波场隐式辛算法.地球物理学报,2000,43(3):379～388LUOMingqiu,LIUHong,LIYouming.Seismicwavemodelingwithimplicitsymplecticmethodbasedonspectralfactorizationonhelix.ChineseJ.Geophys.,2000,43(3):379～388
[6] ClaerboutJF .ImagingtheEarth’sInterior.BlackwellScientificPublication,Inc.1985
[7] HolbergO .Towardsoptimumone waywavepropagation.GeophysicsProspect,1988,36(3):99～114
[8] HaleD .3DdepthmigrationviaMcClellantransformations.Geophysics,1991,56(11):1778～1785
[9] HaleD .Stableexplicitdepthextrapolationofseismicwavefields.Geophysics,1991,56(11):1770～1777
[10] EtgenJT .Stabilityofexplicitdepthextrapolationthroughlaterallyvaryingmedia.SEG 64thInternationalMeeting,ExpandedAbstracts,1994,1266～1269
[11] GravesRW ,ClaytonRW .Modelingacousticwaveswithparaxialextrapolators.Geophysics,1990,55(3):306～319
[12] ZhouH ,McMechanGA .One pass3Dseismicextrapolationwiththe45°waveequation.Geophysics,1997,62(6):1817～1824
[13] GuanquanZhang,YuZhang,HongboZhou.Helicalfinite differenceschemesfor3Ddepthmigration.SEG 70 thInternationalMeeting,ExpandedAbstracts,2000.862～865
[14] RistowD ,R¨uhlT .3Dimplicitfinitedifferencemigrationbymultiway splitting.SEG 65 thInternationalMeeting,ExpandedAbstracts,1995.1220～1223
[15] ClaerboutJF .Multidimensionalrecursivefiltersviaahelixwithapplicationtovelocityestimationand3Dmigration,the68thAnn.SEGmeeting,1998.1995～1998
[16] ClaerboutJF .Multidimensionalrecursivefiltersviaahelix.Geophysics,1998,63:1532～1541
[17] 石根华.数值流形方法与非连续变形分析.北京:清华大学出版社,1997.146～167SHIGenhua.NumericalManifoldMethodandDiscontinuousDeformationanalysis.Beijing:TsinghuaUniversityPress,1997.146～167
[18] LUOMingqiu,GAOHongwei,LIUHong,etal.AcomparisonontheLUdecompositionofLaplacianmatrixwithspectralanddirecteliminationmethods.SEG 70 thInternationalMeeting,ExpandedAbstracts,2000.2349～2353
[19] RickettJ.Implicit3Ddepthmigrationbywavefieldextrapolationwithhelicalboundarycondition.the68thAnn.SEGmeeting,1998.1124～1127
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