傅里叶有限差分法三维波动方程正演模拟
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摘要
傅里叶有限差分(FFD)法兼有相位屏法和隐式有限差分法二者的优势,能够处理复杂地质构造中的波传播问题,但在三维情形下,算子的双向分裂会引起明显的方位各向异性误差.本文用Fourier变换计算双向分裂过程中的高阶交叉项,消除了方位各向异性误差.该方法充分利用了FFD法在双域实现的算法结构,明显减少了由于引入误差校正所带来的计算量.将该方法应用于修改后的三维French模型的地震正演问题,并将得到的叠后记录、单炮记录同全波有限差分法的模拟结果进行对比,结果证实了该方法对一次反射波具有较高的模拟精度,在内存需求和计算效率方面则具有更大的优势.
Fourier finite-difference(FFD) operator has both of the advantages of Fourier and finite-difference methods,it can handle the wave propagations in complex media with large velocity contrasts and wide propagation angles.In 3-D case,however,two-way splitting may cause artificial azimuthal anisotropy.In this paper,we handle the remnants of the conventional two-way splitting technique by Fourier transforms,and the azimuthal anisotropy is removed.Based on the dual-domain scheme of the FFD method,the proposed algorithm significantly reduces computational cost caused by error correction.Compared with the finite-difference method by the zero-offset records and shot profiles based on 3-D modified French model,the proposed method can properly model the primary reflected waves and is much more attractive in both computational cost and storage demand.
Fourier finite-difference(FFD) operator has both of the advantages of Fourier and finite-difference methods,it can handle the wave propagations in complex media with large velocity contrasts and wide propagation angles.In 3-D case,however,two-way splitting may cause artificial azimuthal anisotropy.In this paper,we handle the remnants of the conventional two-way splitting technique by Fourier transforms,and the azimuthal anisotropy is removed.Based on the dual-domain scheme of the FFD method,the proposed algorithm significantly reduces computational cost caused by error correction.Compared with the finite-difference method by the zero-offset records and shot profiles based on 3-D modified French model,the proposed method can properly model the primary reflected waves and is much more attractive in both computational cost and storage demand.
引文
[1]Graves R W,Clayton R W.Modeling acoustic waves with paraxialextrapolators.Geophysics,1990,55(3):306~319
[2]Li Z.Compensating finite-difference errors in 3D migration andmodeling.Geophysics,1991,56(6):1650~1660
[3]金胜汶,陈必远,马在田.三维波动方程有限差分正演方法.地球物理学报,1994,37(6):804~810Jin S W,Chen B Y,Ma Z T.Three-dimensional wave equationforward modeling by the finite-difference methods.Chinese J.Geophys.(Acta Geophysica Sinica),1994,37(6):804~810
[4]Gazdag J.Wave equation migration with the phase-shift method.Geophysics,1978,43(7):1342~1351
[5]Stoffa P L,Fokkema J T,Freire R M D,et al.Split-step Fouriermigration.Geophysics,1990,55(4):410~421
[6]Wu R S.Wide-angle elastic wave one-way propagator inheterogeneous media and an elastic wave complex-screen method.J.Geophys.Res.,1994,99(B1):751~766
[7]Wu R S,Jin S,Xie X B.Seismic wave propagation and scattering inheterogeneous crustal waveguides using screen propagators:Ⅰ,SHwaves.Bull.Seism.Soc.Am.,2000,90(2):401~413
[8]Xie X B,Wu R S.Modeling elastic wave forward propagation andreflection using the complex-screen method.J.Acous.Soc.Am.,2001,109(6):2629~2635
[9]Le Rousseau J H,de Hoop M V.Modeling and imaging with thescalar generalized-screen algorithms in isotropic media.Geophysics,2001,66(5):1551~1568
[10]He Z H,Xiong G J,Zhang Y X.Nonzero offset seismic forwardmodeling by one-way acoustic wave-equation.68th Ann.Internat.Mtg.,Soc.Expl.Geophys.,Expanded Abstracts,1998.1921~1924
[11]熊高君,贺振华,张琳等.共炮记录正演模拟检波点下延记录原理.石油物探,1999,38(2):43~49Xiong G J,He Z H,Zhang L,et al.Geophone record forwardmodeling from common-shot record.Geophysical Prospecting forPetroleum(in Chinese),1999,38(2):43~49
[12]李冰,刘洪,李幼铭.波场延拓短算子构造方法.地球物理学报,2003,46(2):246~251Li B,Liu H,Li Y M.A construction method of the short operator forwavefield extrapolation.Chinese J.Geophys.(in Chinese),2003,46(2):246~251
[13]Claerbout J F.Imaging the Earth’s Interior.Blackwell ScientificPublications Inc.,1985
[14]马在田.高阶有限差分偏移.石油地球物理勘探,1982,17(1):6~15Ma Z T.Finite-difference migration with higher-order approximation.Oil Geophysical Prospecting(in Chinese),1982,17(1):6~15
[15]Huang L,Fehler M.Accuracy analysis of the split-step Fourierpropagator:implications for seismic modeling and migration.Bull.Seism.Soc.Am.,1998,88(1):18~29
[16]Ristow D,Rühl T.Fourier finite-difference migration.Geophysics,1994,59(12):1882~1893
[17]Biondi B.Stable wide-angle Fourier finite-difference downwardextrapolation of 3-D wavefields.Geophysics,2002,67(3):872~882
[18]Brown D L.Applications of operator separation in reflectionseismology.Geophysics,1983,48(3):288~294
[19]Wang Y.ADI plus interpolation:Accurate finite-difference solution to3D paraxial wave equation.Geophysical Prospecting,2001,49(5):547~556
[20]Ristow D,Rühl T.3-D implicit finite-difference migration bymultiway splitting.Geophysics,1997,62(2):554~567
[21]张文生,张关泉.螺旋坐标下的因子分解合成炮叠前深度偏移.地球物理学报,2003,46(4):520~525Zhang W S,Zhang G Q.Factorization synthesized-shot prestack depthmigration in the helical coordinate system.Chinese J.Geophys.(inChinese),2003,46(4):520~525
[22]Zhang J H,Wang W M,Fu L Y,et al.3D Fourier finite-differencemigration by ADI plus interpolation.Geophysical Prospecting,2007,(Accepted)
[23]Fu L Y.Broadband constant-coefficient propagators.GeophysicalProspecting,2005,53(3):299~310
[24]刘礼农,崔凤林,张剑锋.三维复杂构造中地震波模拟的单程波方法.地球物理学报,2004,47(3):514~520Liu L N,Cui F L,Zhang J F.Seismic modeling with one-way waveequation in 3-D complex structures.Chinese J.Geophys.(inChinese),2004,47(3):514~520
[25]谢桂生,刘洪,李幼铭等.界面起伏条件下反射/透射算子+单程波方程的地震波模拟方法.地球物理学报,2005,48(5):1172~1178Xie G S,Liu H,Li Y M,et al.Seismic modeling by reflection/transmission operator and one-way wave equation under the conditionof fluctuating reflectors.Chinese J.Geophys.(in Chinese),2005,48(5):1172~1178
[26]Mufti I R.Large-scale three-dimensional seismic models and theirinterpretive significance.Geophysics,1990,55(9):1166~1182
[27]刘洪,袁江华,陈景波等.大步长波场深度延拓的理论.地球物理学报,2006,49(6):1779~1793Liu H,Yuan J H,Chen J B,et al.Theory of large-step wavefielddepth extrapolation.Chinese J.Geophys.(in Chinese),2006,49(6):1779~1793
[2]Li Z.Compensating finite-difference errors in 3D migration andmodeling.Geophysics,1991,56(6):1650~1660
[3]金胜汶,陈必远,马在田.三维波动方程有限差分正演方法.地球物理学报,1994,37(6):804~810Jin S W,Chen B Y,Ma Z T.Three-dimensional wave equationforward modeling by the finite-difference methods.Chinese J.Geophys.(Acta Geophysica Sinica),1994,37(6):804~810
[4]Gazdag J.Wave equation migration with the phase-shift method.Geophysics,1978,43(7):1342~1351
[5]Stoffa P L,Fokkema J T,Freire R M D,et al.Split-step Fouriermigration.Geophysics,1990,55(4):410~421
[6]Wu R S.Wide-angle elastic wave one-way propagator inheterogeneous media and an elastic wave complex-screen method.J.Geophys.Res.,1994,99(B1):751~766
[7]Wu R S,Jin S,Xie X B.Seismic wave propagation and scattering inheterogeneous crustal waveguides using screen propagators:Ⅰ,SHwaves.Bull.Seism.Soc.Am.,2000,90(2):401~413
[8]Xie X B,Wu R S.Modeling elastic wave forward propagation andreflection using the complex-screen method.J.Acous.Soc.Am.,2001,109(6):2629~2635
[9]Le Rousseau J H,de Hoop M V.Modeling and imaging with thescalar generalized-screen algorithms in isotropic media.Geophysics,2001,66(5):1551~1568
[10]He Z H,Xiong G J,Zhang Y X.Nonzero offset seismic forwardmodeling by one-way acoustic wave-equation.68th Ann.Internat.Mtg.,Soc.Expl.Geophys.,Expanded Abstracts,1998.1921~1924
[11]熊高君,贺振华,张琳等.共炮记录正演模拟检波点下延记录原理.石油物探,1999,38(2):43~49Xiong G J,He Z H,Zhang L,et al.Geophone record forwardmodeling from common-shot record.Geophysical Prospecting forPetroleum(in Chinese),1999,38(2):43~49
[12]李冰,刘洪,李幼铭.波场延拓短算子构造方法.地球物理学报,2003,46(2):246~251Li B,Liu H,Li Y M.A construction method of the short operator forwavefield extrapolation.Chinese J.Geophys.(in Chinese),2003,46(2):246~251
[13]Claerbout J F.Imaging the Earth’s Interior.Blackwell ScientificPublications Inc.,1985
[14]马在田.高阶有限差分偏移.石油地球物理勘探,1982,17(1):6~15Ma Z T.Finite-difference migration with higher-order approximation.Oil Geophysical Prospecting(in Chinese),1982,17(1):6~15
[15]Huang L,Fehler M.Accuracy analysis of the split-step Fourierpropagator:implications for seismic modeling and migration.Bull.Seism.Soc.Am.,1998,88(1):18~29
[16]Ristow D,Rühl T.Fourier finite-difference migration.Geophysics,1994,59(12):1882~1893
[17]Biondi B.Stable wide-angle Fourier finite-difference downwardextrapolation of 3-D wavefields.Geophysics,2002,67(3):872~882
[18]Brown D L.Applications of operator separation in reflectionseismology.Geophysics,1983,48(3):288~294
[19]Wang Y.ADI plus interpolation:Accurate finite-difference solution to3D paraxial wave equation.Geophysical Prospecting,2001,49(5):547~556
[20]Ristow D,Rühl T.3-D implicit finite-difference migration bymultiway splitting.Geophysics,1997,62(2):554~567
[21]张文生,张关泉.螺旋坐标下的因子分解合成炮叠前深度偏移.地球物理学报,2003,46(4):520~525Zhang W S,Zhang G Q.Factorization synthesized-shot prestack depthmigration in the helical coordinate system.Chinese J.Geophys.(inChinese),2003,46(4):520~525
[22]Zhang J H,Wang W M,Fu L Y,et al.3D Fourier finite-differencemigration by ADI plus interpolation.Geophysical Prospecting,2007,(Accepted)
[23]Fu L Y.Broadband constant-coefficient propagators.GeophysicalProspecting,2005,53(3):299~310
[24]刘礼农,崔凤林,张剑锋.三维复杂构造中地震波模拟的单程波方法.地球物理学报,2004,47(3):514~520Liu L N,Cui F L,Zhang J F.Seismic modeling with one-way waveequation in 3-D complex structures.Chinese J.Geophys.(inChinese),2004,47(3):514~520
[25]谢桂生,刘洪,李幼铭等.界面起伏条件下反射/透射算子+单程波方程的地震波模拟方法.地球物理学报,2005,48(5):1172~1178Xie G S,Liu H,Li Y M,et al.Seismic modeling by reflection/transmission operator and one-way wave equation under the conditionof fluctuating reflectors.Chinese J.Geophys.(in Chinese),2005,48(5):1172~1178
[26]Mufti I R.Large-scale three-dimensional seismic models and theirinterpretive significance.Geophysics,1990,55(9):1166~1182
[27]刘洪,袁江华,陈景波等.大步长波场深度延拓的理论.地球物理学报,2006,49(6):1779~1793Liu H,Yuan J H,Chen J B,et al.Theory of large-step wavefielddepth extrapolation.Chinese J.Geophys.(in Chinese),2006,49(6):1779~1793
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