2.5维地震波场褶积微分算子法数值模拟
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摘要
早期的褶积微分算子都是基于正反傅立叶变换而实现的,其精度比四阶有限差分的精度稍高,本文将计算数学中的Forsyte广义正交多项式微分算子与褶积算子相结合,构建了一个新的快速、高精度褶积微分算子,其计算结果非常接近实验函数微分的精确值,精度与16阶有限差分的精度相当,远优于错格伪谱法的精确度.另外,2.5维数值模拟比二维模拟可以更真实地模拟三维介质的某个剖面的波场,并且2.5维地震波模拟的计算量比三维模拟的计算量及计算耗时要大大减少.本文利用基于Forsyte广义正交多项式褶积微分算子法计算2.5维非均匀介质地震波场,模拟结果表明,该算法的计算速度快,计算精度高,能够直观、高效地反映复杂介质中波场的传播规律,并且2.5维波场数值模拟具有更高的计算效率,是一种非常值得深入研究并广泛应用的方法.
Early convolutional differentiators were all based on the Fourier transformation,their precision was a little higer than that of the four-order finite difference.For improving the precision and efficiency of seismic modeling,the authors develop a new modeling approach(Convolutional Forsyte Polynomial Differentiator Method) by using optimized convolutional operators for spatial differentiation and staggered-grid finite-difference for time differentiation in wave equation computation.The solution of this new method is much close to the exact value,and the precision is nearly equal to that of 16-order finite difference,and higher than that of the pseudospectral method.2.5D modelling is more efficient than 3D modelling,and more effective to model the 3D seismic field than 2D modelling.This paper applies the convolutional Forsyte polynomial differentiator method to model 2.5D seismic waves field.The numerical results show that the algorithm can bring reliable outcomes with high precision and fast speed,and 2.5D modelling is much efficient and economical,and deserves more attentation,research and extensive application.
Early convolutional differentiators were all based on the Fourier transformation,their precision was a little higer than that of the four-order finite difference.For improving the precision and efficiency of seismic modeling,the authors develop a new modeling approach(Convolutional Forsyte Polynomial Differentiator Method) by using optimized convolutional operators for spatial differentiation and staggered-grid finite-difference for time differentiation in wave equation computation.The solution of this new method is much close to the exact value,and the precision is nearly equal to that of 16-order finite difference,and higher than that of the pseudospectral method.2.5D modelling is more efficient than 3D modelling,and more effective to model the 3D seismic field than 2D modelling.This paper applies the convolutional Forsyte polynomial differentiator method to model 2.5D seismic waves field.The numerical results show that the algorithm can bring reliable outcomes with high precision and fast speed,and 2.5D modelling is much efficient and economical,and deserves more attentation,research and extensive application.
引文
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[16]张中杰,滕吉文,杨顶辉.声波与弹性波场数值模拟中的褶积微分算子法[J].地震学报,1996,18(1):63~69.Zhang Z J,Teng J W,Yang D H.Acoustic and elastic wavemodeling by a convolutional differentiator[J].Acta Seis-mologica Sinica,1996,18(1):63~69.
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[18]戴志阳,孙建国,查显杰.地震波混合阶褶积算法模拟[J].物探化探计算技术,2005,27(2):111~114.Dai Z Y,Sun J G,Li X J.Seismic wave modeling with hybridorder convolution differentiator[J].Computing techniques forgeophysical and geochemical exploration,2005,27(2):111~114.
[19]谢靖.物探数据处理的数学方法-下册[M].北京:地质出版社,1981,97~104.Xie J.Numerical method for geophysical data process[M].Beijing:Geological Publishing House,1981,97~104.
[20]Gazdag J,Modelling of the acoustic wave equation withtransform method[J].Geophysics,1981,46:854.
[21]Cerjan C,Kosloff D,et al.A nonreflecting boundary condi-tion for discrete acoustic and elastic wave equation[J].Geo-physics,1985,50:705~708.
[22]Kosloff R.Absorbing boundaries for wave propagation prob-lems[J].J.Comput.Phys.,1986,63:363~376.
[2]John W,Stockwell J.2.5-D wave equations and high-frequen-cy asymptotics[J].Geophysics,1995,60:556~562.
[3]Song Z M,Paul R.Williamson.Frequency-domain acoustic-wave modeling and inversion of crosshole data:part I—2.5-Dmodeling method propagation[J].Geophysics,1995,60:784~795.
[4]Takashi F,Hiroshi T.2.5-D modelling of elastic waves usingthe pseudospectral method[J].Geophys,J.Int.1996,124:820~832.
[5]Cao S,Greenhalgh S.2.5-D modeling of seismic wave propa-gation:Boundary condition,stability criterion,and efficiency[J].Geophysics,1998,63:2082~2090
[6]Hiroshi Takenaka,et al.Quasi-cylindrical 2.5D wave model-ing for large-scale seismic surveys[J].Geophys.Res.Lett.,2003,30(21):2086.
[7]Stig-Kyrre Foss,Bjorn Ursin.2.5-D modeling,inversion andangle migration in anisotropic elastic media[J].Geophysicalprospecting,2004,52:65~84..
[8]Amléia Novais,Lúcio T.Santos.2.5D finite-difference solu-tion of the acoustic wave equation[J].Geophysical Prospec-ting,2005,53:523~531.
[9]Sinclair C,Greenhalgh S,Zhou B.2.5-D Modelling of ElasticWaves in Anisotropic Media using the Spectral-Element Meth-od-a Preliminary Investigation[J].AESC2006,Melbourne,Australia.
[10]范祯祥,郑仙种.变背景速度2.5维反散射技术在地震勘探中的应用问题[J].地球物理学报,1995,38:112~122.Fan Z X,Zheng X Z.The application problem of 2.5 dimen-sional Born inverse scattering method with variable back-ground velocity in seismic prospecting[J].Chinese J.Geo-phys.(in chinese),1995,38:112~122.
[11]牛滨华,孙春岩.裂隙各向异性介质2.5维弹性波场数值模拟[J].地球科学,1995,20:107~111.Niu B H,Sun C Y.Numerical modeling with 2.5 wave fieldin anisotropic media containing cracks[J].Earth Science-Jour-nal of China University of Geosciences,1995,20:107~111.
[12]熊彬,罗延钟.电导率分块均匀的瞬变电磁2.5维有限元数值模拟[J].地球物理学报,2006,49(2):590~597.Xiong B,Luo Y Z.Finite element modeling of 2.5-D TEMwith block homogeneous conductivity[J].Chinese J.Geo-phys.(in chinese),2006,49(2):590~597.
[13]JoséM.Carcione,Gérard C.Herman,et al.Y2K ReviewArticle:Seismic modeling[J].Geophysics,2002,67(4):1304~1325.
[14]郑洪伟,李廷栋,高锐,等.数值模拟在地球动力学中的研究进展[J].地球物理学进展,2006,21(2):360~369.Zheng H W,Li T D,Gao R,et al.,The advance of numeri-cal simulation in geodynamics[J].Progress in Geophysics(inChinese),2006,21(2):360~369.
[15]Zhou B,Greenhalgh S A.Seismic scalar wave equation mod-eling by a convolutional differentiator[J].Bulletin of the Seis-mological Society of America,1992,82(1):289~303.
[16]张中杰,滕吉文,杨顶辉.声波与弹性波场数值模拟中的褶积微分算子法[J].地震学报,1996,18(1):63~69.Zhang Z J,Teng J W,Yang D H.Acoustic and elastic wavemodeling by a convolutional differentiator[J].Acta Seis-mologica Sinica,1996,18(1):63~69.
[17]戴志阳,孙建国,查显杰.地震波场模拟中的褶积微分算子法[J].吉林大学学报(地球科学版),2005,35(4):520~524.Dai Z Y,Sun J G,Li X J.Seismic wave field modeling withconvolution differentiator algorithm[J].Journal of JiLin Uni-versity(Earth Science Edition),2005,35(4):520~524.
[18]戴志阳,孙建国,查显杰.地震波混合阶褶积算法模拟[J].物探化探计算技术,2005,27(2):111~114.Dai Z Y,Sun J G,Li X J.Seismic wave modeling with hybridorder convolution differentiator[J].Computing techniques forgeophysical and geochemical exploration,2005,27(2):111~114.
[19]谢靖.物探数据处理的数学方法-下册[M].北京:地质出版社,1981,97~104.Xie J.Numerical method for geophysical data process[M].Beijing:Geological Publishing House,1981,97~104.
[20]Gazdag J,Modelling of the acoustic wave equation withtransform method[J].Geophysics,1981,46:854.
[21]Cerjan C,Kosloff D,et al.A nonreflecting boundary condi-tion for discrete acoustic and elastic wave equation[J].Geo-physics,1985,50:705~708.
[22]Kosloff R.Absorbing boundaries for wave propagation prob-lems[J].J.Comput.Phys.,1986,63:363~376.
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