二维弹性及粘弹性TTI介质中地震波场数值模拟:四种不同网格高阶有限差分算法研究
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摘要
本文利用交错网格、辅助网格、旋转交错网格、同位网格有限差分方法分别模拟了二维弹性TTI介质和二维黏弹性TTI介质中的地震波传播.在稳定性条件内,选用不同的网格间距及时间间隔,通过波场快照、合成理论地震图较为系统分析对比了这四种不同网格有限差分数值模拟在计算精度、CPU时间、相移、频散、以及保幅方面的优缺点.数值模拟结果表明:1)这四种不同网格有限差分算法都是很好的波场数值模拟算法;2)就CPU计算时间而言,旋转交错网格有限差分算法的计算效率最高;3)从计算精度来看,同位网格有限差分的计算精度最高;4)从振幅保护方面来看,四种网格的保护振幅的能力相当;5)相移方面,当网格间距增大时,交错网格和旋转交错网格有可能出现相移现象;6)频散方面,同位网格的频散现象不明显.
In this paper,we use the staggered grid,auxiliary grid,rotated staggered grid and non-staggered grid finite difference methods to simulate the wavefield in 2D elastic and viscoelastic TTI media,respectively.Under the stability criterion,we choose different spatial and temporal intervals to get snapshots and synthetic seismogram to compare the four algorithms in terms of accuracy,CPU time,phase shift,dispersion and amplitude protection.The numerical results show that: 1) all of the four finite difference algorithms are suitable for seismic wavefield simulation;2) the rotated staggered grid has the fastest CPU time;3) the non-staggered grid has the highest accuracy;4) the abilities of the four girds to protect the amplitude are nearly the same;5) the staggered grid and rotated staggered gird may appear phase shift when the spatial interval becomes larger;6) the non-staggered grid has a low dispersion phenomenon.
In this paper,we use the staggered grid,auxiliary grid,rotated staggered grid and non-staggered grid finite difference methods to simulate the wavefield in 2D elastic and viscoelastic TTI media,respectively.Under the stability criterion,we choose different spatial and temporal intervals to get snapshots and synthetic seismogram to compare the four algorithms in terms of accuracy,CPU time,phase shift,dispersion and amplitude protection.The numerical results show that: 1) all of the four finite difference algorithms are suitable for seismic wavefield simulation;2) the rotated staggered grid has the fastest CPU time;3) the non-staggered grid has the highest accuracy;4) the abilities of the four girds to protect the amplitude are nearly the same;5) the staggered grid and rotated staggered gird may appear phase shift when the spatial interval becomes larger;6) the non-staggered grid has a low dispersion phenomenon.
引文
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[2]Moczo P,LuckáM,Kristek J,et al.3Ddisplacement finitedifferences and a combined memory optimization[J].Bull.Seism.Soc.Am.,1999,89(1):69-79.
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[5]Virieux J.P-SV wave propagation in heterogeneous media:velocity-stress finite-difference method[J].Geophysics,1986,51(4):889-901.
[6]Levander A R.4th-order finite-difference P-SV seismograms[J].Geophysics,1988,53(11):1425-1436.
[7]Graves R W.Simulating seismic wave propagation in 3Delasticmedia using staggered grid finite differences[J].Bull.Seism.Soc.Am.,1996,86(4):1091-1106.
[8]Crase E.High-order(space and time)finite-difference modeling ofthe elastic wave equation[A].60th Ann.Internat.Mtg.Soc.Expl.Geophys.,Expanded Abstracts,1990:987-991.
[9]Igel H,Riollet B,Mora P.Accuracy of staggered 3-D finite-difference grids for anisotropic wave propagation[C].SEGExpanded Abstracts,1992,11:1244-1246.
[10]Igel H,Mora P,Riollet B.Anisotropic wave propagationthrough finite-difference grids[J].Geophysics,1995,60(4):1203-1216.
[11]Saenger E H,Gold N,Shapiro S A.Modeling the propagation ofelastic waves using a modified finite-difference grid[J].WaveMotion,2000,31(1):77-92.
[12]Saenger E H,Bohlen T.Finite-difference modeling of viscoelasticand anisotropic wave propagation using the rotated staggeredgrid[J].Geophysics,2004,69(2):583-591.
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[14]王晓欢.TTI介质中地震波传播特性研究[硕士学位论文].长春:吉林大学,2006.Wang X H.A study on propagation signatures of seismicwave in TTI media[D].Changchun:Jilin University(inChinese),2006.
[15]Bayliss A,Jordan K E,Lemesurier B J,et al.A 4th-orderaccurate finite-difference scheme for the computation ofelastic-waves[J].Bull.Seism.Soc.Am.,1986,76(4):1115-1132.
[16]谢小碧,姚振兴.二维不均匀介质中点源P-SV波响应的有限差分近似算法[J].地球物理学报,1988,31(5):540-555.Xie X B,Yao Z X.P-SV wave responses for a point source intwo-dimensional heterogeneous media:finite-differencemethod[J].Acta Geophysica Sinica(Chinese J.Geophys.)(in Chinese),1988,31(5):540-555.
[17]Tsingas C,Vafidis A,Kanasewich E R.Elastic wave propagationin transversely isotropic media using finite differences[J].Geophysical Prospecting,1990,38(8):933-949.
[18]Dai N,Vafidis A,Kanasewich E R.Wave propagation inheterogeneous,porous media:a velocity-stress,finite-difference method[J].Geophysics,1995,60(2):327-340.
[19]Zhang W,Chen X F.Traction image method for irregularfree surface boundaries in finite difference seismic wavesimulation[J].Geophys.J.Int.,2006,167(1):337-353.
[20]祝贺君,张伟,陈晓非.二维各向异性介质中地震波场的高阶同位网格有限差分模拟[J].地球物理学报,2009,52(6):1536-1546.Zhu H J,Zhang W,Chen X F.Two-dimensional seismicwave simulation in anisotropic media by non-staggered finitedifference method[J].Chinese J.Geophys.(in Chinese),2009,52(6):1536-1546.
[21]Zhang W,Shen Y,Zhao L.Three-dimensional anisotropicseismic wave modelling in spherical coordinates by acollocated-grid finite-difference method[J].Geophys.J.Int.,2012,188(3):1359-1381.
[22]严红勇,刘洋.黏弹TTI介质中旋转交错网格高阶有限差分数值模拟[J].地球物理学报,2012,55(4):1354-1365.Yan H Y,Liu Y.Rotated staggered grid high-order finite-difference numerical modeling for wave propagation inviscoelastic TTI media[J].Chinese J.Geophys.(inChinese),2012,55(4):1354-1365.
[23]Zhu J L,Dorman J.Two-dimensional,three-component wavepropagation in a transversely isotropic medium with arbitrary-orientation-finite-element modeling[J].Geophysics,2000,65(3):934-942.
[24]Anderson D L.Elastic wave propagation in layered anisotropicmedia[J].J.Geophys.Res.,1961,66(9):2953-2963.
[25]Love A E H.A Treatise on the Mathematical Theory ofElasticity[M].New York:Dover Publications Inc.,1944.
[26]Carcione J M.Wave propagation in anisotropic linear viscoelasticmedia:theory and simulated wavefield[J].Geophys.J.Int.,1990,101(3):739-750.
[27]Carcione J M.Constitutive model and wave equations for linear,viscoelastic,anisotropic Media[J].Geophysics,1995,60(2):537-548.
[28]裴正林,王尚旭.任意倾斜各向异性介质中弹性波波场交错网格高阶有限差分法模拟[J].地震学报,2005,27(4):441-451.Pei Z L,Wang S X.A staggered-grid high-order finite-difference modeling for elastic wave field in arbitrary tiltanisotropic media[J].Acta Seismologica Sinica(in Chinese),2005,27(4):441-451.
[29]Cerjan C,Kosloff D,Kosloff R,et al.A nonreflecting boundarycondition for discrete acoustic and elastic wave equations[J].Geophysics,1985,50(4):705-708.
[30]唐小平,白超英,刘宽厚.伪谱法分离波动方程弹性波模拟[J].石油地球物理勘探,2012,47(1):19-26.Tang X P,Bai C Y,Liu K H.Elastic wavefield simulationusing separated equations through pseudo-spectral method[J].OGP(in Chinese),2012,47(1):19-26.
[31]Zhou B,Greenhalgh S.Raypath and traveltime computationsfor 2Dtransversely isotropic media with dipping symmetryaxes[J].Exploration Geophysics,2006,37(2):150-159.
[32]Zhou B,Greenhalgh S.On the computation of elastic wavegroup velocities for a General anisotropic medium[J].J.Geophys.Eng.,2004,1(3):205-215.
[2]Moczo P,LuckáM,Kristek J,et al.3Ddisplacement finitedifferences and a combined memory optimization[J].Bull.Seism.Soc.Am.,1999,89(1):69-79.
[3]Madariaga R.Dynamics of an expanding circular fault[J].Bull.Seism.Soc.Am.,1976,66(3):639-666.
[4]Virieux J.SH-wave propagation in heterogeneous media:velocity-stress finite-difference method[J].Geophysics,1984,49(11):1933-1942.
[5]Virieux J.P-SV wave propagation in heterogeneous media:velocity-stress finite-difference method[J].Geophysics,1986,51(4):889-901.
[6]Levander A R.4th-order finite-difference P-SV seismograms[J].Geophysics,1988,53(11):1425-1436.
[7]Graves R W.Simulating seismic wave propagation in 3Delasticmedia using staggered grid finite differences[J].Bull.Seism.Soc.Am.,1996,86(4):1091-1106.
[8]Crase E.High-order(space and time)finite-difference modeling ofthe elastic wave equation[A].60th Ann.Internat.Mtg.Soc.Expl.Geophys.,Expanded Abstracts,1990:987-991.
[9]Igel H,Riollet B,Mora P.Accuracy of staggered 3-D finite-difference grids for anisotropic wave propagation[C].SEGExpanded Abstracts,1992,11:1244-1246.
[10]Igel H,Mora P,Riollet B.Anisotropic wave propagationthrough finite-difference grids[J].Geophysics,1995,60(4):1203-1216.
[11]Saenger E H,Gold N,Shapiro S A.Modeling the propagation ofelastic waves using a modified finite-difference grid[J].WaveMotion,2000,31(1):77-92.
[12]Saenger E H,Bohlen T.Finite-difference modeling of viscoelasticand anisotropic wave propagation using the rotated staggeredgrid[J].Geophysics,2004,69(2):583-591.
[13]Hustedt B,Operto S,Virieux J.Mixed-grid and staggered-grid finite-difference methods for frequency-domain acousticwave modelling[J].Geophys.J.Int.,2004,157(3):1269-1296.
[14]王晓欢.TTI介质中地震波传播特性研究[硕士学位论文].长春:吉林大学,2006.Wang X H.A study on propagation signatures of seismicwave in TTI media[D].Changchun:Jilin University(inChinese),2006.
[15]Bayliss A,Jordan K E,Lemesurier B J,et al.A 4th-orderaccurate finite-difference scheme for the computation ofelastic-waves[J].Bull.Seism.Soc.Am.,1986,76(4):1115-1132.
[16]谢小碧,姚振兴.二维不均匀介质中点源P-SV波响应的有限差分近似算法[J].地球物理学报,1988,31(5):540-555.Xie X B,Yao Z X.P-SV wave responses for a point source intwo-dimensional heterogeneous media:finite-differencemethod[J].Acta Geophysica Sinica(Chinese J.Geophys.)(in Chinese),1988,31(5):540-555.
[17]Tsingas C,Vafidis A,Kanasewich E R.Elastic wave propagationin transversely isotropic media using finite differences[J].Geophysical Prospecting,1990,38(8):933-949.
[18]Dai N,Vafidis A,Kanasewich E R.Wave propagation inheterogeneous,porous media:a velocity-stress,finite-difference method[J].Geophysics,1995,60(2):327-340.
[19]Zhang W,Chen X F.Traction image method for irregularfree surface boundaries in finite difference seismic wavesimulation[J].Geophys.J.Int.,2006,167(1):337-353.
[20]祝贺君,张伟,陈晓非.二维各向异性介质中地震波场的高阶同位网格有限差分模拟[J].地球物理学报,2009,52(6):1536-1546.Zhu H J,Zhang W,Chen X F.Two-dimensional seismicwave simulation in anisotropic media by non-staggered finitedifference method[J].Chinese J.Geophys.(in Chinese),2009,52(6):1536-1546.
[21]Zhang W,Shen Y,Zhao L.Three-dimensional anisotropicseismic wave modelling in spherical coordinates by acollocated-grid finite-difference method[J].Geophys.J.Int.,2012,188(3):1359-1381.
[22]严红勇,刘洋.黏弹TTI介质中旋转交错网格高阶有限差分数值模拟[J].地球物理学报,2012,55(4):1354-1365.Yan H Y,Liu Y.Rotated staggered grid high-order finite-difference numerical modeling for wave propagation inviscoelastic TTI media[J].Chinese J.Geophys.(inChinese),2012,55(4):1354-1365.
[23]Zhu J L,Dorman J.Two-dimensional,three-component wavepropagation in a transversely isotropic medium with arbitrary-orientation-finite-element modeling[J].Geophysics,2000,65(3):934-942.
[24]Anderson D L.Elastic wave propagation in layered anisotropicmedia[J].J.Geophys.Res.,1961,66(9):2953-2963.
[25]Love A E H.A Treatise on the Mathematical Theory ofElasticity[M].New York:Dover Publications Inc.,1944.
[26]Carcione J M.Wave propagation in anisotropic linear viscoelasticmedia:theory and simulated wavefield[J].Geophys.J.Int.,1990,101(3):739-750.
[27]Carcione J M.Constitutive model and wave equations for linear,viscoelastic,anisotropic Media[J].Geophysics,1995,60(2):537-548.
[28]裴正林,王尚旭.任意倾斜各向异性介质中弹性波波场交错网格高阶有限差分法模拟[J].地震学报,2005,27(4):441-451.Pei Z L,Wang S X.A staggered-grid high-order finite-difference modeling for elastic wave field in arbitrary tiltanisotropic media[J].Acta Seismologica Sinica(in Chinese),2005,27(4):441-451.
[29]Cerjan C,Kosloff D,Kosloff R,et al.A nonreflecting boundarycondition for discrete acoustic and elastic wave equations[J].Geophysics,1985,50(4):705-708.
[30]唐小平,白超英,刘宽厚.伪谱法分离波动方程弹性波模拟[J].石油地球物理勘探,2012,47(1):19-26.Tang X P,Bai C Y,Liu K H.Elastic wavefield simulationusing separated equations through pseudo-spectral method[J].OGP(in Chinese),2012,47(1):19-26.
[31]Zhou B,Greenhalgh S.Raypath and traveltime computationsfor 2Dtransversely isotropic media with dipping symmetryaxes[J].Exploration Geophysics,2006,37(2):150-159.
[32]Zhou B,Greenhalgh S.On the computation of elastic wavegroup velocities for a General anisotropic medium[J].J.Geophys.Eng.,2004,1(3):205-215.
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