地震偏移的最优可分近似算法实现
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摘要
针对地震偏移算法中单程波算子的特征函数近似,采用最优可分表示法将该特征函数展开为空间变量(g)和水平波数(k)的可分表达式,以此可分近似表达式为基础,运用正反Fourier变换重新构造单程波算子.为了克服特征函数最优可分近似计算在奇点及其领域产生的数值振荡,引入等价黏性技巧以增强算法的数值计算稳定性,并采用分频最优可分及对空间变量(g)线性插值的方法,不仅提高了计算精度,也节约计算机时.文中具体研究了单程波算子的脉冲响应和二维叠前深度偏移.结果表明,在不甚大步长情况下,本文构造的算子具有适应横向强变速的能力,运用Marmousi模型验证了本文方法适合复杂构造的成像.
An optimal separable approximation(OSA) is applied to the characteristic function of one-way wave operator in the seismic migration algorithm.The characteristic function is expressed as products of functions in space variable(g) and functions in wave number variable(k).Based on the optimal separable approximation,the mixed domain algorithm with Fourier and inverse Fourier transformation is used to approximate the one-way wave operator.To overcome numerical instability from approximating characteristics function near the singular point,an equivalent sticking skill is adopted to improve algorithmic stability.Moreover,separate frequency computation and linear interpolation for space variables are used to improve precision and save time.The impulse response of one-way wave operator on simples and the complex lateral variable velocity model shows that the optimal separable approximation can apply to lateral velocity variation in the case of small steps.Results on the Marmousi model test show the new method can also be used to image complex structure.
An optimal separable approximation(OSA) is applied to the characteristic function of one-way wave operator in the seismic migration algorithm.The characteristic function is expressed as products of functions in space variable(g) and functions in wave number variable(k).Based on the optimal separable approximation,the mixed domain algorithm with Fourier and inverse Fourier transformation is used to approximate the one-way wave operator.To overcome numerical instability from approximating characteristics function near the singular point,an equivalent sticking skill is adopted to improve algorithmic stability.Moreover,separate frequency computation and linear interpolation for space variables are used to improve precision and save time.The impulse response of one-way wave operator on simples and the complex lateral variable velocity model shows that the optimal separable approximation can apply to lateral velocity variation in the case of small steps.Results on the Marmousi model test show the new method can also be used to image complex structure.
引文
[1]Samuel H Gray,John Etgen,Joe Dellinger,et al.Seismic migrationproblems and solutions.Geophysics,2001,66(5):1662~1640
[2]马在田.地震成像技术-有限差分法偏移.北京:石油工业出版社,1989Ma Z T.Seismic Imaging Technology:Finite_difference MethodMigration.Beijing:Petroleum Industry Press,1989
[3]Ristow D,Ruhl T.3_D implicit finite_difference migration bymultiway_splitting.Geophysics,1997,62(2):554~567
[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,et al.Split_step Fouriermigration.Geophysics,1990,55(4):410~421
[6]Wu R S,Jin S.Windowed GSP(generalized screen propagators)migration applied to SEG/EAEG salt model data.67th Ann.Internat.Mtg.,Soc.of Expl.Geophys.Expanded Abstracts,1997,1746~1749
[7]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
[8]Liu L N,Zhang J F.3_D wavefield extrapolation with optimum split_step Fourier method.Geophysics,2004,in press
[9]刘洪,刘礼农,杨辉.波场延拓算子的最佳乘积基函数展开方法研究.见:张中杰,高锐,吕庆田等编.中国大陆地球深部结构与动力学研究.北京:科学出版社,2004Liu H,Liu L N,Yang H.The optical products base function ofwavefield extrapolation operators.In:Zhang Z J,Gao R,Lu Q T,etal.eds.Research of deep earth structure and dynamics of Chinesemainland.Beijing:Science Press,2004
[10]Chen Jingbo,Liu Hong.Optimization approximation with separablevariables for the one_way wave operator.Geophysical ResearchLetters,2004,31(6):L06613
[11]宋健.高维函数和流形在低维可视空间中的最优表达.科学通报,2001,46(12):977~984Song J.The optimization expression of functions and manifolds in highdimensions by ones in low dimensions.Chinese Science Bull.(inChinese),2001,46(12):977~984
[12]徐树方.矩阵计算的理论和方法.北京:北京大学出版社,1995Xu S F.Theory and Method on Matrix Analysis.Beijing:PekingUniversity Press,1995
[13]Claerbout J F.Imaging the Earth’s Interior.New York:BlackwellScientific Publication,1985
[14]杨辉,刘洪,李幼铭.单程波方程偏移算子高阶辛格式.地球物理学报,2003,46(4):533~538Yang H,Liu H,Li Y M.High order symplectic scheme for one_waywave equation operator.Chinese J.Geophy.(in Chinese),2003,46(4):533~538
[15]刘礼农,刘洪,李幼铭.SEG/EAGE盐丘和推覆体模型的波动方程三维叠前深度偏移成像.地球物理学报,2004,47(2):312~320Liu L N,Liu H,Li Y M.Wave_equation 3_D prestack depthmigration for the SEG/EAGE salt and overthrust model.Chinese J.Geophys.(in Chinese),2004,47(2):312~320
[2]马在田.地震成像技术-有限差分法偏移.北京:石油工业出版社,1989Ma Z T.Seismic Imaging Technology:Finite_difference MethodMigration.Beijing:Petroleum Industry Press,1989
[3]Ristow D,Ruhl T.3_D implicit finite_difference migration bymultiway_splitting.Geophysics,1997,62(2):554~567
[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,et al.Split_step Fouriermigration.Geophysics,1990,55(4):410~421
[6]Wu R S,Jin S.Windowed GSP(generalized screen propagators)migration applied to SEG/EAEG salt model data.67th Ann.Internat.Mtg.,Soc.of Expl.Geophys.Expanded Abstracts,1997,1746~1749
[7]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
[8]Liu L N,Zhang J F.3_D wavefield extrapolation with optimum split_step Fourier method.Geophysics,2004,in press
[9]刘洪,刘礼农,杨辉.波场延拓算子的最佳乘积基函数展开方法研究.见:张中杰,高锐,吕庆田等编.中国大陆地球深部结构与动力学研究.北京:科学出版社,2004Liu H,Liu L N,Yang H.The optical products base function ofwavefield extrapolation operators.In:Zhang Z J,Gao R,Lu Q T,etal.eds.Research of deep earth structure and dynamics of Chinesemainland.Beijing:Science Press,2004
[10]Chen Jingbo,Liu Hong.Optimization approximation with separablevariables for the one_way wave operator.Geophysical ResearchLetters,2004,31(6):L06613
[11]宋健.高维函数和流形在低维可视空间中的最优表达.科学通报,2001,46(12):977~984Song J.The optimization expression of functions and manifolds in highdimensions by ones in low dimensions.Chinese Science Bull.(inChinese),2001,46(12):977~984
[12]徐树方.矩阵计算的理论和方法.北京:北京大学出版社,1995Xu S F.Theory and Method on Matrix Analysis.Beijing:PekingUniversity Press,1995
[13]Claerbout J F.Imaging the Earth’s Interior.New York:BlackwellScientific Publication,1985
[14]杨辉,刘洪,李幼铭.单程波方程偏移算子高阶辛格式.地球物理学报,2003,46(4):533~538Yang H,Liu H,Li Y M.High order symplectic scheme for one_waywave equation operator.Chinese J.Geophy.(in Chinese),2003,46(4):533~538
[15]刘礼农,刘洪,李幼铭.SEG/EAGE盐丘和推覆体模型的波动方程三维叠前深度偏移成像.地球物理学报,2004,47(2):312~320Liu L N,Liu H,Li Y M.Wave_equation 3_D prestack depthmigration for the SEG/EAGE salt and overthrust model.Chinese J.Geophys.(in Chinese),2004,47(2):312~320
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