虚谱法交错网格地震波场数值模拟
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摘要
提高空间差分精度、有效压制人为边界反射是波动方程波场模拟的关键。虚谱法利用模型空间的全部信息对波场函数进行傅里叶变换,可以得到精确的波场空间导数,使数值频散效应减弱,进而实现宽频带地震波场模拟。阐述了求解弹性波波动方程的方法原理,讨论了数值模拟中Gibbs效应和边界反射问题的解决方法,即在半网格点处计算空间导数并采用最佳匹配层边界条件。设计了5层水平层状介质模型,讨论了虚谱法的模拟精度和计算效率,试算表明,适当增大差分网格和时间延拓步长不会影响计算精度,但计算效率可以得到大幅度提高。分别采用不同的差分方法对Marmousi2模型和SEG/EAGE模型进行数值模拟,结果表明,虚谱法交错网格模拟结果信噪比高,在同等模拟精度条件下较其它方法具有更高的计算效率。
Enhancing the precision of spatial difference and effectively suppressing artificial boundary reflection is the key of wave equation wavefield simulation.Pseudo spectral method carried out Fourier transform on wavefield function by utilizing all information of model space,which can achieve accurate wavefield spatial deviative and attenuate numerical dispersion effect,and eventually realize wide frequency bandwidth seismic wavefield simulation.The principles for solving the wave equation of elastic wave were elaborated. The method for suppressing boundary reflection and Gibbs phenomena was discussed,that is calculating spatial deviative at point of half-grid and adopting perfectly matched layer boundary condition. A layered medium model with five horizontal layers was designed; the simulation precision and calculation efficiency of pseudo spectral method were discussed.The trial computation shows that properly increasing difference grid and time continuation step size can largely enhance computation efficiency without influencing computation precision.Numerical tests on Marmousi2 and SEG/ EAGE show that simulation results by pseudo spectral method with staggered grid has higher S/N and higher computation efficiency with same simulation precision.
Enhancing the precision of spatial difference and effectively suppressing artificial boundary reflection is the key of wave equation wavefield simulation.Pseudo spectral method carried out Fourier transform on wavefield function by utilizing all information of model space,which can achieve accurate wavefield spatial deviative and attenuate numerical dispersion effect,and eventually realize wide frequency bandwidth seismic wavefield simulation.The principles for solving the wave equation of elastic wave were elaborated. The method for suppressing boundary reflection and Gibbs phenomena was discussed,that is calculating spatial deviative at point of half-grid and adopting perfectly matched layer boundary condition. A layered medium model with five horizontal layers was designed; the simulation precision and calculation efficiency of pseudo spectral method were discussed.The trial computation shows that properly increasing difference grid and time continuation step size can largely enhance computation efficiency without influencing computation precision.Numerical tests on Marmousi2 and SEG/ EAGE show that simulation results by pseudo spectral method with staggered grid has higher S/N and higher computation efficiency with same simulation precision.
引文
[1]Carcione J M,Hermanz G C,Kroode A P E.Seismic modeling[J].Geophysics,2002,67(4):1304-1325
[2]陆基孟.地震勘探原理(下册)[M].东营:石油大学出版社,1993:123-193
[3]贺振华.反射地震资料偏移处理与反演方法[M].重庆:重庆大学出版社,1989:12-15
[4]陈耿毅,余钦范,蔡希玲,等.地震模拟技术新进展——第67届EAGE年会论文综述[J].勘探地球物理进展,2005,28(6):439-448
[5]张永刚.地震波场数值模拟方法[J].石油物探, 2003,42(2):143-148
[6]Altermen Z S,Loewenthal D.Seismic wave in a quarter and three quarter plane[J].Geophysical Journal of the Royal Astronomical Society,1970,20 (1):101-126
[7]Alford R M,Kelly K R,Boore D M.Accuracy of finite difference modeling of the acoustic wave equation [J].Geophysics,1974,39(6):834-842
[8]Virieux J.SH wave propagation in heterogeneous media:velocity-stress finite-difference method[J]. Geophysics.1984,49(11):1933-1957
[9]Levander A R.Fourth-order finite-difference P-SV seismograms[J].Geophysics,1988,53(11):1425- 1436
[10]Crase E.High-order(space and time) finite-difference modeling of elastic wave equation[J].Expanded Abstracts of 60~(th) Annual SEG Mtg,1990,987-991
[11]CarcioneJ M.Staggered mesh for the anisotropic and viscoelastic wave equation[J].Geophysics,1999, 64(6):1863-1866
[12]Magnier S A,Mora P,Tarantola A.Finite difference on minimal grids[J].Geophysics,1994,59(9):1435- 1443
[13]Kosloff D D,Baysalt E.Forward modeling by a Fourier method[J].Geophysics,1982,47(10 ): 1402-1412
[14]Fornberg B.The pseudo-spectral method:accurate representation in elastic wave calculations[J].Geophysics, 1988,53(5):625-637
[15]Reshef M,Kosloff D,Edwards M,et al.Three-dimensional acoustic modeling by the Fourier method [J].Geophysics,1988,53(9):1175-1183
[16]Reshef M,Kosloff D,Edwards M,et al.Three-dimensional elastic modeling by the Fourier method [J].Geophysics,1988,53(9):1184-1193
[17]傅旦丹,何樵登.正交各向异性介质地震弹性波场的伪谱法正演模拟[J].石油物探,2001,40(3):8-14
[18]牟永光,裴正林.三维复杂介质地震数值模拟[M].北京:石油工业出版社,2005:36
[19]庄东海,王超,肖斌.虚谱法三维地震波动方程正演模拟[J].江汉石油学院学报,1998,20(4):30-33
[20]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
[21]王守东.声波方程完美匹配层吸收边界[J].石油地球物理勘探,2003,38(1):31-34
[22]Sherif R E,Geldart L P.Exploration seismology [M].New York,USA:Cambridge University Press, 1995:389-392
[23]Cerjan C,Kosloff D,Kosloff R,et al.A nonreflecting boundary condition for discrete acoustic and elastic wave equation[J].Geophysics,1985,50(9):705-708
[24]Correa J P,Spiegelmant M,Carbotte S,et al.Centered and staggered Fourier derivatives and Hilbert transforms[J].Geophysics,2002,67(5):1558-1563
[25]董良国,马在田,曹景忠.一阶弹性波方程交错网格高阶差分解法稳定性研究[J].地球物理学报,2000, 43(6):856-859
[26]Martin G S,Wiley R,Marfurt K J.Marmousi2:an elastic upgrade for Marmousi[J].The Leading Edge, 2006,26(2):156-166
[27]Aminzadeh F,Burkhard N,Long J,et al.Three dimensional SEG/EAEG models - an update[J].The Leading Edge,1996,15(2):131-134
[28]刘文革,贺振华,黄德济,等.高精度傅里叶有限差分法模型正演[J].石油地球物理勘探,2007,42(6): 629-633
[29]Ristow D.Fourier finite-difference migration[J].Geophysics, 1994,59(12):1882-1893
[2]陆基孟.地震勘探原理(下册)[M].东营:石油大学出版社,1993:123-193
[3]贺振华.反射地震资料偏移处理与反演方法[M].重庆:重庆大学出版社,1989:12-15
[4]陈耿毅,余钦范,蔡希玲,等.地震模拟技术新进展——第67届EAGE年会论文综述[J].勘探地球物理进展,2005,28(6):439-448
[5]张永刚.地震波场数值模拟方法[J].石油物探, 2003,42(2):143-148
[6]Altermen Z S,Loewenthal D.Seismic wave in a quarter and three quarter plane[J].Geophysical Journal of the Royal Astronomical Society,1970,20 (1):101-126
[7]Alford R M,Kelly K R,Boore D M.Accuracy of finite difference modeling of the acoustic wave equation [J].Geophysics,1974,39(6):834-842
[8]Virieux J.SH wave propagation in heterogeneous media:velocity-stress finite-difference method[J]. Geophysics.1984,49(11):1933-1957
[9]Levander A R.Fourth-order finite-difference P-SV seismograms[J].Geophysics,1988,53(11):1425- 1436
[10]Crase E.High-order(space and time) finite-difference modeling of elastic wave equation[J].Expanded Abstracts of 60~(th) Annual SEG Mtg,1990,987-991
[11]CarcioneJ M.Staggered mesh for the anisotropic and viscoelastic wave equation[J].Geophysics,1999, 64(6):1863-1866
[12]Magnier S A,Mora P,Tarantola A.Finite difference on minimal grids[J].Geophysics,1994,59(9):1435- 1443
[13]Kosloff D D,Baysalt E.Forward modeling by a Fourier method[J].Geophysics,1982,47(10 ): 1402-1412
[14]Fornberg B.The pseudo-spectral method:accurate representation in elastic wave calculations[J].Geophysics, 1988,53(5):625-637
[15]Reshef M,Kosloff D,Edwards M,et al.Three-dimensional acoustic modeling by the Fourier method [J].Geophysics,1988,53(9):1175-1183
[16]Reshef M,Kosloff D,Edwards M,et al.Three-dimensional elastic modeling by the Fourier method [J].Geophysics,1988,53(9):1184-1193
[17]傅旦丹,何樵登.正交各向异性介质地震弹性波场的伪谱法正演模拟[J].石油物探,2001,40(3):8-14
[18]牟永光,裴正林.三维复杂介质地震数值模拟[M].北京:石油工业出版社,2005:36
[19]庄东海,王超,肖斌.虚谱法三维地震波动方程正演模拟[J].江汉石油学院学报,1998,20(4):30-33
[20]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
[21]王守东.声波方程完美匹配层吸收边界[J].石油地球物理勘探,2003,38(1):31-34
[22]Sherif R E,Geldart L P.Exploration seismology [M].New York,USA:Cambridge University Press, 1995:389-392
[23]Cerjan C,Kosloff D,Kosloff R,et al.A nonreflecting boundary condition for discrete acoustic and elastic wave equation[J].Geophysics,1985,50(9):705-708
[24]Correa J P,Spiegelmant M,Carbotte S,et al.Centered and staggered Fourier derivatives and Hilbert transforms[J].Geophysics,2002,67(5):1558-1563
[25]董良国,马在田,曹景忠.一阶弹性波方程交错网格高阶差分解法稳定性研究[J].地球物理学报,2000, 43(6):856-859
[26]Martin G S,Wiley R,Marfurt K J.Marmousi2:an elastic upgrade for Marmousi[J].The Leading Edge, 2006,26(2):156-166
[27]Aminzadeh F,Burkhard N,Long J,et al.Three dimensional SEG/EAEG models - an update[J].The Leading Edge,1996,15(2):131-134
[28]刘文革,贺振华,黄德济,等.高精度傅里叶有限差分法模型正演[J].石油地球物理勘探,2007,42(6): 629-633
[29]Ristow D.Fourier finite-difference migration[J].Geophysics, 1994,59(12):1882-1893
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