可变交错网格优化差分系数法地震波正演模拟
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摘要
传统变网格有限差分系数是通过Taylor级数展开得到的,在大的波数范围内,数值频散较严重。为此,提出一种基于最小二乘算法的可变交错网格优化差分系数法,即建立基于频散关系的平方误差函数,并引入约束条件,通过求取条件极值得到可变交错网格优化差分算子。频散分析表明,可变交错网格优化差分系数法能在更大波数范围内满足频散关系。模型正演结果证明,相同的空间差分算子长度,可变交错网格优化差分系数法相比Taylor级数展开法,能有效提高正演模拟的精度。
Traditional variable grid finite-difference coefficients are obtained by Taylor series expansion,and numerical dispersion is severe for a wide range of wave numbers.An optimized difference coefficient method of a variable staggered-grid based on the least squares theory is proposed in this paper.First,a square error function based on the dispersion relation is established.Next,with the constraint condition,the optimized difference operators for the variable staggered-grid are obtained by solving the conditional extremum,which can effectively suppress the numerical dispersion.Dispersion analyses show that the optimal variable staggered-grid finite-difference scheme can satisfy the dispersion relation for a wider wave number range.Numerical examples demonstrate that,for the same spatial difference operator length,the proposed optimized finite-difference scheme offers more accurate seismic wave field forward modelling than the Taylor expansion method.
Traditional variable grid finite-difference coefficients are obtained by Taylor series expansion,and numerical dispersion is severe for a wide range of wave numbers.An optimized difference coefficient method of a variable staggered-grid based on the least squares theory is proposed in this paper.First,a square error function based on the dispersion relation is established.Next,with the constraint condition,the optimized difference operators for the variable staggered-grid are obtained by solving the conditional extremum,which can effectively suppress the numerical dispersion.Dispersion analyses show that the optimal variable staggered-grid finite-difference scheme can satisfy the dispersion relation for a wider wave number range.Numerical examples demonstrate that,for the same spatial difference operator length,the proposed optimized finite-difference scheme offers more accurate seismic wave field forward modelling than the Taylor expansion method.
引文
[1]ALTERMAN 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
[2]VIRIEUX J.P-SV wave propagation in heterogeneous media:velocity-stress finite difference method[J].Geophysics,1986,51(4):889-901
[3]MOCZO P.Finite-difference technique for SH-waves in2-D media using irregular grids:application to the seismic response problem[J].Geophysical Journal International,1989,99(2):321-329
[4]JASTRAM C,BEHLE A.Acoustic modeling on a grid of vertically varying spacing[J].Geophysical Prospecting,1992,40(1):157-169
[5]JASTRAM C,TESSMER E.Elastic modeling on a grid with vertically varying spacing[J].Geophysical Prospecting,1994,42(2):357-370
[6]李胜军,孙成禹,倪长宽,等.声波方程有限差分数值模拟的变网格步长算法[J].工程地球物理学报,2007,4(3):207-212LI S J,SUN C Y,NI C K,et al.Acoustic equation numerical modeling on a grid of varying spacing[J].Chinese Journal of Engineering Geophysics,2007,4(3):207-212
[7]李振春,李庆洋,黄建平,等.一种稳定的高精度双变网格正演模拟与逆时偏移方法[J].石油物探,2014,53(2):127-136LI Z C,LI Q Y,HUANG J P,et al.A stable and highprecision dual-variable grid forward modeling and reverse time migration method[J].Geophysical Prospecting for Petroleum,2014,53(2):127-136
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[9]孟凡顺,李洋森,王璐,等.基于可变交错网格的地震波数值模拟[J].地球物理学进展,2013,28(6):2936-2943MENG F S,LI Y S,WANG L,et al.Based on variable and staggered grids seismic numerical simulation[J].Progress in Geophysics,2013,28(6):2936-2943
[10]KANG T S.Finite-difference seismic simulation combining discontinuous grids with locally variable timesteps[J].Bulletin of the Seismological Society of America,2004,94(1):207-219
[11]LI H,ZHANG W,ZHANG Z,et al.Elastic wave finitedifference simulation using discontinuous curvilinear grid with non-uniform time step:two-dimensional case[J].Geophysical Journal International,2015,202(1):102-118
[12]孙林洁,刘春成,张世鑫.PML边界条件下孔隙介质弹性波可变网格正演模拟方法研究[J].石油物探,2015,54(6):652-664SUN L J,LIU C C,ZHANG S X.A variable grid finitedifference scheme with PML boundary condition in elastic wave of porous media[J].Geophysical Prospecting for Petroleum,2015,54(6):652-664
[13]马光克,李洋森,孙万元,等.可变网格高阶有限差分法逆时偏移研究[J].石油物探,2016,55(5):719-736MA G K,LI Y S,SUN W Y,et al.Acoustic pre-stack reverse time migration using variable grid finite-difference method[J].Geophysical Prospecting for Petroleum,2016,55(5):719-736
[14]ALFORD R M,KEILY K R,BORE D M.Accuracy of finite difference modeling of the acoustic wave equation[J].Geophysics,1974,39(6):834-842
[15]董良国,李培明.地震波传播数值模拟中的频散问题[J].天然气工业,2004,24(6):53-56DONG L G,LI P M.Dispersion problem in seismic wave propagation numerical modeling[J].Natural Gas Industry,2004,24(6):53-56
[16]DU Q Z,LI B,HOU B.Numerical modeling of seismic wavefields in transversely isotropic media with a compact staggered-grid finite difference scheme[J].Applied Geophysics,2009,6(1):42-49
[17]TAM C K W,WEBB J C.Dispersion-relation-preserving finite difference schemes for computational acoustics[J].Journal of Computational Physics,1993,107(2):262-281
[18]LIU Y,SEN M K.Scalar wave equation modeling with time-space domain dispersion-relation-based staggeredgrid finite-difference schemes[J].Bulletin of the Seismological Society of America,2011,101(101):141-159
[19]杜启振,白清云,李宾.横向各向同性介质优化差分系数法地震波场数值模拟[J].石油地球物理勘探,2010,45(2):170-176DU Q Z,BAI Q Y,LI B.Optimized difference coefficient method seismic wave field numerical simulation in vertical transverse isotropic medium[J].Oil Geophysical Prospecting,2010,45(2):170-176
[20]梁文全,王彦飞,杨长春.基于优化方法的时间-空间域隐格式有限差分算子确定方法[J].石油物探,2015,54(3):254-259LIANG W Q,WANG Y F,YANG C C.Determination on the implicit finite-difference operator based on optimization method in time-space domain[J].Geophysical Prospecting for Petroleum,2015,54(3):254-259
[21]YANG L,YAN H,LIU H.Least squares staggeredgrid finite-difference for elastic wave modelling[J].Exploration Geophysics,2014,45(4):255-260
[22]YAN H Y,YANG L,LIU H.Acoustic reverse-time migration using optimal staggered-grid finite-difference operator based on least squares[J].Acta Geophysica,2015,63(3):715-734
[23]IGEL H,MORA P,RIOLLET B.Anisotropic wave propagation through finite-difference grids[J].Geophysics,1995,60(4):1203-1216
[24]雍鹏,黄建平,李振春,等.一种时空域优化的高精度交错网格差分算子与正演模拟[J].地球物理学报,2016,59(11):4223-4233YONG P,HUANG J P,LI Z C,et al.Optimized staggered-grid finite-difference method in time-space domain based on exact time evolution schemes[J].Chinese Journal of Geophysics,2016,59(11):4223-4233
[25]梁文全,王彦飞,杨长春.声波方程数值模拟矩形网格有限差分系数确定法[J].石油地球物理勘探,2017,52(1):56-62LIANG W Q,WANG Y F,YANG C C.Acoustic wave equation modeling with rectangle grid finite difference operator and its linear time space domain solution[J].Oil Geophysical Prospecting,2017,52(1):56-62
[26]KINDELAN M,KAMEL A,SGUAZZERO P.On the construction and efficiency of staggered numerical differentiators for the wave equation[J].Geophysics,1990,55(1):107-110
[27]ZHANG J,YAO Z.Optimized explicit finite-difference schemes for spatial derivatives using maximum norm[J].Journal of Computational Physics,2013,250(10):511-526
[28]YANG L,YAN H,LIU H.Optimal implicit staggeredgrid finite-difference schemes based on the sampling approximation method for seismic modeling[J].Geophysical Prospecting,2016,64(3):595-610
[2]VIRIEUX J.P-SV wave propagation in heterogeneous media:velocity-stress finite difference method[J].Geophysics,1986,51(4):889-901
[3]MOCZO P.Finite-difference technique for SH-waves in2-D media using irregular grids:application to the seismic response problem[J].Geophysical Journal International,1989,99(2):321-329
[4]JASTRAM C,BEHLE A.Acoustic modeling on a grid of vertically varying spacing[J].Geophysical Prospecting,1992,40(1):157-169
[5]JASTRAM C,TESSMER E.Elastic modeling on a grid with vertically varying spacing[J].Geophysical Prospecting,1994,42(2):357-370
[6]李胜军,孙成禹,倪长宽,等.声波方程有限差分数值模拟的变网格步长算法[J].工程地球物理学报,2007,4(3):207-212LI S J,SUN C Y,NI C K,et al.Acoustic equation numerical modeling on a grid of varying spacing[J].Chinese Journal of Engineering Geophysics,2007,4(3):207-212
[7]李振春,李庆洋,黄建平,等.一种稳定的高精度双变网格正演模拟与逆时偏移方法[J].石油物探,2014,53(2):127-136LI Z C,LI Q Y,HUANG J P,et al.A stable and highprecision dual-variable grid forward modeling and reverse time migration method[J].Geophysical Prospecting for Petroleum,2014,53(2):127-136
[8]孙成禹,李胜军,倪长宽,等.波动方程变网格步长有限差分数值模拟[J].石油物探,2008,47(2):123-128SUN C Y,LI S J,NI C K,et al.Wave equation numerical modeling by finite difference method with varying grid spacing[J].Geophysical Prospecting for Petroleum,2008,47(2):123-128
[9]孟凡顺,李洋森,王璐,等.基于可变交错网格的地震波数值模拟[J].地球物理学进展,2013,28(6):2936-2943MENG F S,LI Y S,WANG L,et al.Based on variable and staggered grids seismic numerical simulation[J].Progress in Geophysics,2013,28(6):2936-2943
[10]KANG T S.Finite-difference seismic simulation combining discontinuous grids with locally variable timesteps[J].Bulletin of the Seismological Society of America,2004,94(1):207-219
[11]LI H,ZHANG W,ZHANG Z,et al.Elastic wave finitedifference simulation using discontinuous curvilinear grid with non-uniform time step:two-dimensional case[J].Geophysical Journal International,2015,202(1):102-118
[12]孙林洁,刘春成,张世鑫.PML边界条件下孔隙介质弹性波可变网格正演模拟方法研究[J].石油物探,2015,54(6):652-664SUN L J,LIU C C,ZHANG S X.A variable grid finitedifference scheme with PML boundary condition in elastic wave of porous media[J].Geophysical Prospecting for Petroleum,2015,54(6):652-664
[13]马光克,李洋森,孙万元,等.可变网格高阶有限差分法逆时偏移研究[J].石油物探,2016,55(5):719-736MA G K,LI Y S,SUN W Y,et al.Acoustic pre-stack reverse time migration using variable grid finite-difference method[J].Geophysical Prospecting for Petroleum,2016,55(5):719-736
[14]ALFORD R M,KEILY K R,BORE D M.Accuracy of finite difference modeling of the acoustic wave equation[J].Geophysics,1974,39(6):834-842
[15]董良国,李培明.地震波传播数值模拟中的频散问题[J].天然气工业,2004,24(6):53-56DONG L G,LI P M.Dispersion problem in seismic wave propagation numerical modeling[J].Natural Gas Industry,2004,24(6):53-56
[16]DU Q Z,LI B,HOU B.Numerical modeling of seismic wavefields in transversely isotropic media with a compact staggered-grid finite difference scheme[J].Applied Geophysics,2009,6(1):42-49
[17]TAM C K W,WEBB J C.Dispersion-relation-preserving finite difference schemes for computational acoustics[J].Journal of Computational Physics,1993,107(2):262-281
[18]LIU Y,SEN M K.Scalar wave equation modeling with time-space domain dispersion-relation-based staggeredgrid finite-difference schemes[J].Bulletin of the Seismological Society of America,2011,101(101):141-159
[19]杜启振,白清云,李宾.横向各向同性介质优化差分系数法地震波场数值模拟[J].石油地球物理勘探,2010,45(2):170-176DU Q Z,BAI Q Y,LI B.Optimized difference coefficient method seismic wave field numerical simulation in vertical transverse isotropic medium[J].Oil Geophysical Prospecting,2010,45(2):170-176
[20]梁文全,王彦飞,杨长春.基于优化方法的时间-空间域隐格式有限差分算子确定方法[J].石油物探,2015,54(3):254-259LIANG W Q,WANG Y F,YANG C C.Determination on the implicit finite-difference operator based on optimization method in time-space domain[J].Geophysical Prospecting for Petroleum,2015,54(3):254-259
[21]YANG L,YAN H,LIU H.Least squares staggeredgrid finite-difference for elastic wave modelling[J].Exploration Geophysics,2014,45(4):255-260
[22]YAN H Y,YANG L,LIU H.Acoustic reverse-time migration using optimal staggered-grid finite-difference operator based on least squares[J].Acta Geophysica,2015,63(3):715-734
[23]IGEL H,MORA P,RIOLLET B.Anisotropic wave propagation through finite-difference grids[J].Geophysics,1995,60(4):1203-1216
[24]雍鹏,黄建平,李振春,等.一种时空域优化的高精度交错网格差分算子与正演模拟[J].地球物理学报,2016,59(11):4223-4233YONG P,HUANG J P,LI Z C,et al.Optimized staggered-grid finite-difference method in time-space domain based on exact time evolution schemes[J].Chinese Journal of Geophysics,2016,59(11):4223-4233
[25]梁文全,王彦飞,杨长春.声波方程数值模拟矩形网格有限差分系数确定法[J].石油地球物理勘探,2017,52(1):56-62LIANG W Q,WANG Y F,YANG C C.Acoustic wave equation modeling with rectangle grid finite difference operator and its linear time space domain solution[J].Oil Geophysical Prospecting,2017,52(1):56-62
[26]KINDELAN M,KAMEL A,SGUAZZERO P.On the construction and efficiency of staggered numerical differentiators for the wave equation[J].Geophysics,1990,55(1):107-110
[27]ZHANG J,YAO Z.Optimized explicit finite-difference schemes for spatial derivatives using maximum norm[J].Journal of Computational Physics,2013,250(10):511-526
[28]YANG L,YAN H,LIU H.Optimal implicit staggeredgrid finite-difference schemes based on the sampling approximation method for seismic modeling[J].Geophysical Prospecting,2016,64(3):595-610