适于Kirchhoff叠前深度偏移的地震走时李代数积分算法
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摘要
本文基于拟微分算子理论和李代数积分法,根据程函方程和波场坐标变换,提出一种新的适于横向变速介质Kirchhoff叠前深度偏移的地震波走时算法.该算法与Kirchhoff叠前时间偏移所用李代数时间积分表达相比,差异在于增加了波数一次项,且二次项的系数在求积时亦需进行修正.针对单平方根算子象征、李代数积分、指数映射和走时多项式的求解而言,皆需对以往Kirchhoff叠前时间偏移中所用算法进行深化调整.文中数值算例对比了本文李代数积分表达与时间积分的区别,本算法计算结果与线性横向变速介质中的理论值相当吻合.通过走时多项式中各项对结果的影响分析,可知非对称项使计算精度得到了进一步提高.数值试验表明,本算法对横向变速介质中走时求取是可行的,且不需要存储海量走时表,有利于提高Kirchhof叠前深度偏移的精度和效率.
Based on pseudo difference operator and Lie algebra integral method,we proposed a new travel-time calculation method for pre-stack Kirchhoff depth migration in medium with lateral velocity variation using eikonal equation and coordinate transform of wave-field. Comparing with Lie algebra time integral used in the pre-stack Kirchhoff time migration,our method contains the first-order item of wave-number and the second-order item also needs to be corrected in depth integral.Therefore we improve the performance of previous method in calculations of the symbol of single root square operator,Lie algebra integral,exponential mapping and the coefficients of travel-time.Here we compare the expression of improved Lie algebra integral with time integral,and the calculated travel-time is coincident with the theoretical value in medium with laterally linear velocity variation.Through the effect analysis of some items on the final travel-time,we can conclude that the odd item improved the precision further and our algorithm is suitable for the calculation of travel-time in medium with lateral velocity variation,and what s more,there is no mass storage of travel-time tables,which is very beneficial to improving the precision and efficiency of pre-stack Kirchhoff depth migration.
Based on pseudo difference operator and Lie algebra integral method,we proposed a new travel-time calculation method for pre-stack Kirchhoff depth migration in medium with lateral velocity variation using eikonal equation and coordinate transform of wave-field. Comparing with Lie algebra time integral used in the pre-stack Kirchhoff time migration,our method contains the first-order item of wave-number and the second-order item also needs to be corrected in depth integral.Therefore we improve the performance of previous method in calculations of the symbol of single root square operator,Lie algebra integral,exponential mapping and the coefficients of travel-time.Here we compare the expression of improved Lie algebra integral with time integral,and the calculated travel-time is coincident with the theoretical value in medium with laterally linear velocity variation.Through the effect analysis of some items on the final travel-time,we can conclude that the odd item improved the precision further and our algorithm is suitable for the calculation of travel-time in medium with lateral velocity variation,and what s more,there is no mass storage of travel-time tables,which is very beneficial to improving the precision and efficiency of pre-stack Kirchhoff depth migration.
引文
[1] 刘洪,刘国峰,李博等.基于横向导数的走时计算方法及其在叠前时间偏移中的应用.石油物探,2009,48(1) :3～10 Liu H,Liu G F,Li B,et al.Traveltime calculation by lateral derivative and its application in pre-stack time migration.Geophysical Prospecting for Petroleum(in Chinese),2009,48(1) :3～10
[2] Dix C H.Seismic velocities from surface measurements.Geophysics,1955,20(1) :68～86
[3] Cerveny V.Seismic Ray Theory.Cambridge University Press,2001
[4] Vinje V,Iversen E,Gjoystdal H.Traveltime and amplitude estimation using wavefront construction.Geophysics,1993,58(8) :1157～1166
[5] Vinje V,Iversen E,Astebol K,et al.Estimation of multivalued arrivals in 3D models using wavefront construction-Part Ⅰ.Geophysical Prospecting,1996,44(5) :819～842
[6] Vinje V,Iversen E,Astebol K,et al.Estimation of multivalued arrivals in 3D models using wavefront construction-Part Ⅱ:Tracing and interpolation.Geophysical Prospecting,1996,44(5) :843～858
[7] Vidale J.Finite-difference calculation of travel times.Bulletin of the Seismological Society of America,1988,78(6) :2062～2076
[8] Vidale J.Finite-difference calculation of traveltime in three dimensions.Geophysics,1990,55(5) :521～526
[9] Operto M S,Xu S,Gilles L.Can we quantitatively image complex structures with rays?Geophysics,2000,65(4) :1223～1238
[10] Podvin P,Lecomte I.Finite difference computation of traveltimes in very contrasted velocity models:a massively parallel approach and its associated tools.Geophys.J.Int.,1991,105(11) :271～284
[11] Moser T J.Shortest path calculation of seismic rays.Geophysics,1991,56(1) :59～67
[12] Van T J,Symes W W.Upwind finite-difference calculation of traveltimes.Geophysics,1991,56(6) :812～821
[13] Qin F,Olsen K,Luo Y,et al.Finite difference solution of the eikonal equation along expanding wavefronts.Geophysics,1992,57(3) :478～487
[14] Lee K J,Gibson Jr R L.An improved mesh generation scheme for wavefront construction method.Geophysics,2007,22(1) :T1～T8
[15] Chambers K,Kendall J M.A practical implementation of wave front construction for 3-D isotropic media.Geophys.J.Int.,2008,173(3) :1～9
[16] 李振春,刘玉莲,张建磊等.基于矩形网格的有限差分走时计算方法.地震学报,2004,26(6) :644～650 Li Z C,Liu Y L,Zhang J L,et al.Finite-difference calculation of traveltime based on rectangular grid.Acta Seismologica Sinica(in Chinese),2004,26(6) :644～650
[17] 孙章庆,孙建国,韩复兴等.波前快速推进法起伏地表地震波走时计算.勘探地球物理进展,2007,30(5) :392～395 Sun Z Q,Sun J G,Han F X,et al.Seismic traveltimes calculation on relief surface by fast marching method.Progress in Exploration Geophysics(in Chinese),2007,30(5) :392～395
[18] 孙章庆,孙建国,韩复兴.复杂地表条件下基于线性插值和窄带技术的地震波走时计算.地球物理学报,2009,52(11) :2846～2853 Sun Z Q,Sun J G,Han F X.Traveltimes computation using linear interpolation and narrow band technique under complex topographic conditions.Chinese J.Geophys.(in Chinese),2009,52(11) :2846～2853
[19] Feng K.On difference schemes and Symplectic geometry.Proc 1984 Beijing Symp Diff Geometry and Diff Equation.Beijing:Science Press,1985． 42～58
[20] 刘学深,丁培柱.量子系统保结构计算新进展.物理学进展,2004,24(1) :47～89 Liu X S,Ding P Z.New progress of structure-preserving computation for quantum system.Progress in Physics(in Chinese),2004,24(1) :47～89
[21] 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) :1779～1793 Liu H,Yuan J H,Chen J B,et al.Theory of large-step wavefield depth extrapolation.Chinese J.Geophys.(in Chinese),2006,49(6) :1779～1793
[22] 刘洪,王秀闽,曾锐等.单程波算子积分解的象征表示.地球物理学进展,2007,22(2) :463～471 Liu H,Wang X M,Zeng R,et al.Symbol description to integral solution of one-way wave operator.Progress in Geophysics(in Chinese),2007,22(2) :463～471
[23] 刘国峰,刘洪,王秀闽等.Kirchhoff积分时间偏移两种走时计算及并行算法.地球物理学进展,2009,24(1) :131～136 Liu G F,Liu H,Wang X M,et al.Two kinds of traveling time computation and parallel computing methods of Kirchhoff migration.Progress in Geophysics(in Chinese),2009,24(1) :131～136
[24] 王秀闽.积分偏移的计算几何与并行实现[博士论文].北京:中国科学院地质与地球物理研究所,2008 Wang X M.Computational geometry and parallel implementation of integral migration[Ph.D.thesis](in Chinese).Beijing:Institute of Geology and Geophysics,Chinese Academy of Sciences,2008
[25] Liu G F,Liu H,Li B,et al.Getting pre-stack time migration travel times from the single square root operator.Applied Geophysics,2009,6(2) :129～137
[26] 李博,刘国峰,刘洪.地震叠前时间偏移的一种图形处理器提速实现方法.地球物理学报,2009,52(1) :245～252 Li B,Liu G F,Liu H.A method of using GPU to accelerate seismic pre-stack time migration.Chinese J.Geophys.(in Chinese),2009,52(1) :245～252
[27] 刘国峰,刘钦,李博等.油气勘探地震资料处理GPU/CPU协同并行计算.地球物理学进展,2009,24(5) :1671～1678 Liu G F,Liu Q,Li B,et al.GPU/CPU co-processing parallel computation for seismic data processing in oil and gas exploration.Progress in Geophysics(in Chinese),2009,24(5) :1671～1678
[28] 齐民友,徐超江,王维克.现代偏微分方程引论.武汉:武汉大学出版社,2005 Qi M Y,Xu C J,Wang W K.Introduction of Modern Partial Differential Equation(in Chinese).Wuhan:Wuhan University Press,2005
[29] Iserles A,Munthe K H,Nfrstt S,et al.Lie-group methods.Acta Numerica,2000,9(1) :215～263
[30] Celledini E,Iserles A.Methods for the approximation of the matrix exponential in a Lie-algebraic setting.IMA J.Numer.Anal.,2001,21(2) :463～488
[31] Chen J B,Munthe-Kass H,Qin M Z.Square-conservative schemes for a class of evolution equations using Lie_group methods.SIAMJ.Numer.Anal.,2002,39(6) :2164～2178
[32] Magnus W.On the exponential solution of differential equations for a linear operator.Communications on Pure and Applied Mathematics,1954,7(5) :649～673
[33] 李世雄.波动方程的高频近似与辛几何.北京:科学出版社,2001． 1～141 Li S X.The High-Frequency Approximation and Symplectic Geometry(in Chinese).Beijing:Science Press,2001． 1～141
[34] Chen J.Specular ray parameter extraction and stationary phase migration.Geophysics,2004,69(1) :249～256
[35] McLachlan R I,Quispel G W.Splitting methods.Acta Numerica,2002,11(1) :341～434
[36] Moler C,Van Loan C F.Nineteen dubious ways to compute the exponential of a matrix.SIAM Review,1978,20(4) :801～836
[37] Liu Y,Wu R S.A comparison between phase screen,finite difference and eigenfunction expansion calculation for scalar waves in inhomogeneous media.Bulletin of the Seismological Society of America,1994,84(4) :1154～1168
[38] Nolet G.地震层析成像.王椿镛,吴宁远,李幼铭等译.北京:学术书刊出版社,1989 Nolet G.Seismic Tomography(in Chinese).Translated by Wang C Y,Wu N Y,Li Y M,et al.Beijing:Academic Book Press,1989
[2] Dix C H.Seismic velocities from surface measurements.Geophysics,1955,20(1) :68～86
[3] Cerveny V.Seismic Ray Theory.Cambridge University Press,2001
[4] Vinje V,Iversen E,Gjoystdal H.Traveltime and amplitude estimation using wavefront construction.Geophysics,1993,58(8) :1157～1166
[5] Vinje V,Iversen E,Astebol K,et al.Estimation of multivalued arrivals in 3D models using wavefront construction-Part Ⅰ.Geophysical Prospecting,1996,44(5) :819～842
[6] Vinje V,Iversen E,Astebol K,et al.Estimation of multivalued arrivals in 3D models using wavefront construction-Part Ⅱ:Tracing and interpolation.Geophysical Prospecting,1996,44(5) :843～858
[7] Vidale J.Finite-difference calculation of travel times.Bulletin of the Seismological Society of America,1988,78(6) :2062～2076
[8] Vidale J.Finite-difference calculation of traveltime in three dimensions.Geophysics,1990,55(5) :521～526
[9] Operto M S,Xu S,Gilles L.Can we quantitatively image complex structures with rays?Geophysics,2000,65(4) :1223～1238
[10] Podvin P,Lecomte I.Finite difference computation of traveltimes in very contrasted velocity models:a massively parallel approach and its associated tools.Geophys.J.Int.,1991,105(11) :271～284
[11] Moser T J.Shortest path calculation of seismic rays.Geophysics,1991,56(1) :59～67
[12] Van T J,Symes W W.Upwind finite-difference calculation of traveltimes.Geophysics,1991,56(6) :812～821
[13] Qin F,Olsen K,Luo Y,et al.Finite difference solution of the eikonal equation along expanding wavefronts.Geophysics,1992,57(3) :478～487
[14] Lee K J,Gibson Jr R L.An improved mesh generation scheme for wavefront construction method.Geophysics,2007,22(1) :T1～T8
[15] Chambers K,Kendall J M.A practical implementation of wave front construction for 3-D isotropic media.Geophys.J.Int.,2008,173(3) :1～9
[16] 李振春,刘玉莲,张建磊等.基于矩形网格的有限差分走时计算方法.地震学报,2004,26(6) :644～650 Li Z C,Liu Y L,Zhang J L,et al.Finite-difference calculation of traveltime based on rectangular grid.Acta Seismologica Sinica(in Chinese),2004,26(6) :644～650
[17] 孙章庆,孙建国,韩复兴等.波前快速推进法起伏地表地震波走时计算.勘探地球物理进展,2007,30(5) :392～395 Sun Z Q,Sun J G,Han F X,et al.Seismic traveltimes calculation on relief surface by fast marching method.Progress in Exploration Geophysics(in Chinese),2007,30(5) :392～395
[18] 孙章庆,孙建国,韩复兴.复杂地表条件下基于线性插值和窄带技术的地震波走时计算.地球物理学报,2009,52(11) :2846～2853 Sun Z Q,Sun J G,Han F X.Traveltimes computation using linear interpolation and narrow band technique under complex topographic conditions.Chinese J.Geophys.(in Chinese),2009,52(11) :2846～2853
[19] Feng K.On difference schemes and Symplectic geometry.Proc 1984 Beijing Symp Diff Geometry and Diff Equation.Beijing:Science Press,1985． 42～58
[20] 刘学深,丁培柱.量子系统保结构计算新进展.物理学进展,2004,24(1) :47～89 Liu X S,Ding P Z.New progress of structure-preserving computation for quantum system.Progress in Physics(in Chinese),2004,24(1) :47～89
[21] 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) :1779～1793 Liu H,Yuan J H,Chen J B,et al.Theory of large-step wavefield depth extrapolation.Chinese J.Geophys.(in Chinese),2006,49(6) :1779～1793
[22] 刘洪,王秀闽,曾锐等.单程波算子积分解的象征表示.地球物理学进展,2007,22(2) :463～471 Liu H,Wang X M,Zeng R,et al.Symbol description to integral solution of one-way wave operator.Progress in Geophysics(in Chinese),2007,22(2) :463～471
[23] 刘国峰,刘洪,王秀闽等.Kirchhoff积分时间偏移两种走时计算及并行算法.地球物理学进展,2009,24(1) :131～136 Liu G F,Liu H,Wang X M,et al.Two kinds of traveling time computation and parallel computing methods of Kirchhoff migration.Progress in Geophysics(in Chinese),2009,24(1) :131～136
[24] 王秀闽.积分偏移的计算几何与并行实现[博士论文].北京:中国科学院地质与地球物理研究所,2008 Wang X M.Computational geometry and parallel implementation of integral migration[Ph.D.thesis](in Chinese).Beijing:Institute of Geology and Geophysics,Chinese Academy of Sciences,2008
[25] Liu G F,Liu H,Li B,et al.Getting pre-stack time migration travel times from the single square root operator.Applied Geophysics,2009,6(2) :129～137
[26] 李博,刘国峰,刘洪.地震叠前时间偏移的一种图形处理器提速实现方法.地球物理学报,2009,52(1) :245～252 Li B,Liu G F,Liu H.A method of using GPU to accelerate seismic pre-stack time migration.Chinese J.Geophys.(in Chinese),2009,52(1) :245～252
[27] 刘国峰,刘钦,李博等.油气勘探地震资料处理GPU/CPU协同并行计算.地球物理学进展,2009,24(5) :1671～1678 Liu G F,Liu Q,Li B,et al.GPU/CPU co-processing parallel computation for seismic data processing in oil and gas exploration.Progress in Geophysics(in Chinese),2009,24(5) :1671～1678
[28] 齐民友,徐超江,王维克.现代偏微分方程引论.武汉:武汉大学出版社,2005 Qi M Y,Xu C J,Wang W K.Introduction of Modern Partial Differential Equation(in Chinese).Wuhan:Wuhan University Press,2005
[29] Iserles A,Munthe K H,Nfrstt S,et al.Lie-group methods.Acta Numerica,2000,9(1) :215～263
[30] Celledini E,Iserles A.Methods for the approximation of the matrix exponential in a Lie-algebraic setting.IMA J.Numer.Anal.,2001,21(2) :463～488
[31] Chen J B,Munthe-Kass H,Qin M Z.Square-conservative schemes for a class of evolution equations using Lie_group methods.SIAMJ.Numer.Anal.,2002,39(6) :2164～2178
[32] Magnus W.On the exponential solution of differential equations for a linear operator.Communications on Pure and Applied Mathematics,1954,7(5) :649～673
[33] 李世雄.波动方程的高频近似与辛几何.北京:科学出版社,2001． 1～141 Li S X.The High-Frequency Approximation and Symplectic Geometry(in Chinese).Beijing:Science Press,2001． 1～141
[34] Chen J.Specular ray parameter extraction and stationary phase migration.Geophysics,2004,69(1) :249～256
[35] McLachlan R I,Quispel G W.Splitting methods.Acta Numerica,2002,11(1) :341～434
[36] Moler C,Van Loan C F.Nineteen dubious ways to compute the exponential of a matrix.SIAM Review,1978,20(4) :801～836
[37] Liu Y,Wu R S.A comparison between phase screen,finite difference and eigenfunction expansion calculation for scalar waves in inhomogeneous media.Bulletin of the Seismological Society of America,1994,84(4) :1154～1168
[38] Nolet G.地震层析成像.王椿镛,吴宁远,李幼铭等译.北京:学术书刊出版社,1989 Nolet G.Seismic Tomography(in Chinese).Translated by Wang C Y,Wu N Y,Li Y M,et al.Beijing:Academic Book Press,1989
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