基于波前重建和李代数积分的地震波走时计算
|
摘要
地震波走时计算在数值模拟、层析反演和偏移成像中均有重要意义。将波前重建与李代数积分相结合,提出了一种新的适应横向变速介质的非对称走时算法,称之为wave-front construction-Lie algebra integral(WFC-LAI)算法。本算法利用一次波前重建计算成像射线走时进行坐标变换,将深度域单平方根算子透镜项转化为常数,在射线坐标系下计算李代数积分和指数映射,得到地震波走时的解析表达式。数值试验表明,该方法计算结果与线性横向变速介质中走时的理论值吻合。通过与波前重建结果对比,WFC-LAI算法对于求取横向变速介质中地震波走时是可行的,节省了存储空间,易于并行,有利于提高Kirchhoff积分叠前深度偏移的精度和效率。
Seismic travel-time calculation is significant in numerical modeling,tomography inversion and migration.In this paper,integrated using wave-front construction and Lie algebra integral,we proposed a new method,called wave-front construction-Lie algebra integral(WFC-LAI)method,which is applicable to unsymmetrical travel-time calculation in laterally velocity variant medium.We make coordinate conversion by travel-time of image rays computed by wave-front construction and convert the lens item of single square root operator in depth domain to a constant,then in the ray coordinates we determine analytical expression of paraxial travel-time using Lie algebra integral and exponent mapping.Through numerical experiments we know our result coincides with the theoretical value in the medium with linear laterally variant velocity.Comparing with wave-front construction we can conclude that in laterally velocity variant medium,WFC-LAI method is applicable to travel-time calculation and saves storage space and computing time,which is extremely beneficial to improve the precision and efficiency of Kirchhoff pre-stack depth migration.
Seismic travel-time calculation is significant in numerical modeling,tomography inversion and migration.In this paper,integrated using wave-front construction and Lie algebra integral,we proposed a new method,called wave-front construction-Lie algebra integral(WFC-LAI)method,which is applicable to unsymmetrical travel-time calculation in laterally velocity variant medium.We make coordinate conversion by travel-time of image rays computed by wave-front construction and convert the lens item of single square root operator in depth domain to a constant,then in the ray coordinates we determine analytical expression of paraxial travel-time using Lie algebra integral and exponent mapping.Through numerical experiments we know our result coincides with the theoretical value in the medium with linear laterally variant velocity.Comparing with wave-front construction we can conclude that in laterally velocity variant medium,WFC-LAI method is applicable to travel-time calculation and saves storage space and computing time,which is extremely beneficial to improve the precision and efficiency of Kirchhoff pre-stack depth migration.
引文
[1]Gray S H.Efficient traveltime calculations for Kirch-hoff migration[J].Geophysics,1986,51(8):1685-1688.
[2]Vidale J E.Finite-difference calculation of traveltimes[J].Bull Seis Soc Am,1988,78(6):2062-2076.
[3]Gray S H,May W P.Kirchhoff migration using ei-konal equation traveltimes[J].Geophysics,1994,59(5):810-817.
[4]Vinje V,Iversen E,Gjoystdal H.Traveltime and am-plitude estimation using wavefront construction[J].Geophysics,1993,58(8):1157-1166.
[5]Vinje V,Iversen E,Astebol K,et al.Estimation of multivalued arrivals in3D models using wavefront construction—Part I[J].Geophysical Prospecting,1996,44(5):819-842.
[6]Vinje V,Iversen E,Astebol K,et al.Estimation of mul-tivalued arrivals in3D models using wavefront construction—Part II:tracing and interpolation[J].Geophysicsal Prospecting,1996,44(5):843-858.
[7]Lambare G,Lucio P S,Hanyga A.Two-dimensional multivalued traveltime and amplitude maps by uniform sampling of ray field[J].Geophys J Internat,1996,125(2):584-598.
[8]Lucio P S,Lambare G,Hanyga A.3D multivalued travel time and amplitude maps[J].PAGEOPH,1996,148(3):449-479.
[9]Vinje V,Astebol K,Iversen E,et al.3-Dray modeling by wavefront construction in open models[J].Geophysics,1999,64(6):1912-1919.
[10]Lee K J,Gibson Jr R L.An improved mesh genera-tion scheme for wavefront construction method[J].Geophysics,2007,22(1):1-8.
[11]Chambers K,Kendall J M.A practical implemen-tation of wave front construction for3-D istropic media[J].Geophysical Journal International,2008,173(3):1-9.
[12]Operto MS,Xu S,Gilles L.Can we quantitatively image complex structures with rays[J].Geophysics,2000,65(4):1223-1238.
[13]Podvin P,Lecomte I.Finite difference computation of traveltimes in very contrasted velocity models:a massively parallel approach and its associated tools[J].Geophy J Int,1991,105(11):271-284.
[14]Moser T J.Shortest path calculation of seismic rays[J].Geophysics,1991,56(1):59-67.
[15]Van T J,Symes W W.Upwind finite-difference cal-culation of traveltimes[J].Geophysics,1991,56(6):812-821.
[16]Qin F,Olsen K,Luo Y,et al.Finite-difference so-lution of the eikonal equation along expanding wavefronts[J].Geophysics,1992,57(3):478-487.
[17]赵连锋,朱介寿,曹俊兴,等.有序波前重建法的射线追踪[J].地球物理学报,2003,46(3):415-420.ZHAO Lian-feng,ZHUJie-shou,CAO Jun-xing,et al.Ray tracing using ordinal wavefront reconstraction method[J].Chinese Journal of Geophysics,2003,46(3):415-420.
[18]孙建国,何洋.基于波前构建的射线追踪:一种Java实现[J].吉林大学学报:地球科学版,2007,37(4):814-820.SUNJian-guo,HE Yang.Ray tracing based on wave-front construction:a Java implementation[J].Journal of Jilin University:Earth Science Edition,2007,37(4):814-820.
[19]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[J].勘探地球物理进展,2007,30(5):392-395.SUN Zhang-qing,SUNJian-guo,HAN Fu-xing,et al.Seismic traveltime calculation on relief surface byfast-marching method[J].Progress in Exploration Geophysics,2007,30(5):392-395.
[20]郭继茹,冯,王俊祥,等.最佳路径射线追踪算法研究[J].吉林大学学报:地球科学版,2008,38(增刊):73-75.GUO Ji-ru,FENG Xuan,WANG Jun-xiang,et al.Study of shortest path method of ray tracing algorithm[J].Journal of Jilin University:Earth Science Edition,2008,38(Sup.):73-75.
[21]韩复兴,孙建国,杨昊.基于二维三次卷积插值算法的波前构建射线追踪[J].吉林大学学报:地球科学版,2008,38(2):336-340.HAN Fu-xing,SUN Jian-guo,YANG Hao.Ray tracing of wave-front construction by bicubic convolution interpolation[J].Journal of Jilin University:Earth Science Edition,2008,38(2):336-340.
[22]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[J].地球物理学报,2009,52(11):2846-2853.SUN Zhang-qing,SUN Jian-guo,HAN Fu-xing.Traveltimes computation using linear interpolation and narrow band technique under complex topographic conditions[J].Chinese Journal of Geophysics,2009,52(11):2846-2853.
[23]李波涛,杨长春,陈雨红,等.基于波前面三角形网格剖分的波前重建法三维射线追踪[J].地球物理学进展,2009,24(2):507-512.LI Bo-tao,YANG Chang-chun,CHEN Yu-hong,et al.3D wavefront construction method ray tracing based on triangular grid layout on wavefront[J].Progress in Geophysics,2009,24(2):507-512.
[24]杨昊,孙建国,韩复兴.波前扩展有限差分地震波走时算法的C++语言描述[J].吉林大学学报:地球科学版,2007,37(3):615-619.YANG Hao,SUNJian-guo,HAN Fu-xing.An C++language programimplement of travel-time calculation of expanding wavefronts finite-difference method[J].Journal of Jilin University:Earth Science Edition,2007,37(3):615-619.
[25]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[J].地震学报,2004,26(6):644-650.LI Zhen-chun,LI U Yu-lian,ZHANG Jian-lei,et al.Finite-difference calculation of traveltime based on rectangular grid[J].Acta Seismologica Sinica,2004,26(6):644-650.
[26]Feng K.On difference schemes and symplectic geo-metry[C]//Feng K.Proc1984Beijing Symp Diff Geometry and Diff Equations.Beijing:Science Press,1985:42-58.
[27]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[J].石油物探,2009,48(1):3-10.LI U Hong,LI U Guo-feng,LI Bo,et al.Traveltime calculation by lateral derivative and its application in pre-stack time migration[J].Geophysical Prospecting for Petroleum,2009,48(1):3-10.
[28]刘洪,袁江华,陈景波,等.大步长波场深度延拓的理论[J].地球物理学报,2006,49(6):1779-1793.LI U Hong,YUAN Jiang-hua,CHEN Jing-bo,et al.Theory of large-step wavefield depth extrapolation[J].Chinese Journal of Geophysics,2006,49(6):1779-1793.
[29]刘洪,王秀闽,曾锐,等.单程波算子积分解的象征表示[J].地球物理学进展,2007,22(2):463-471.LI U Hong,WANG Xiu-min,ZENG Rui,et al.Symbol description to integral solution of one-way wave operator[J].Progress in Geophysics,2007,22(2):463-471.
[30]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[J].地球物理学进展,2009,24(1):131-136.LI U Guo-feng,LI U Hong,WANG Xiu-min,et al.Two kinds of traveling time computation and parallel computing methods of Kirchhoff migration[J].Progress in Geophysics,2009,24(1):131-136.
[31]王秀闽.积分偏移的计算几何与并行实现[D].北京:中国科学院地质与地球物理研究所,2008.WANG Xiu-min.Computational geometry and parallel implementation of integral migration[D].Beijing:Institute of Geology and Geophysics,Chinese Academy of Sciences,2008.
[32]Liu Guo-feng,Liu Hong,Li Bo,et al.Getting pre-stack time migration travel times from the single square root operator[J].Applied Geophysics,2009,6(2):129-137.
[33]Magnus W.On the exponential solution of diffe-rential equations for a linear operator[J].Communications on Pure and Applied Mathematics,1954,7(5):649-673.
[34]Iserles A,Munthe-Kaas H,Nfrstt S,et al.Lie-group methods[J].Acta Numerica,2000,9(1):215-263.
[35]Chen J B,Munthe-Kass H,Qin M Z.Square-con-servative schemes for a class of evolution equations using Lie-group methods[J].SIAMJ Nnmer Anal,2002,39(6):2164-2178.
[36]Moler C,Van Loan C F.Ninteen dubious ways to compute the exponential of a matrix[J].SIAM Review,1978,20(4):801-836.
[37]Liu Y,Wu R S.Acomparison between phase screen,finite difference and eigenfunction expansion calculation for scalar waves in inhomogeneous media[J].Bulletin of the Seismological Society of America,1994,84(4):1154-1168.
[38]Celledini E,Iserles A.Methods for the approxima-tion of the matrix exponential in a Lie-algebraic setting[J].I MA Journal of Numerical Analysis,2001,21(2):463-488.
[39]Nolet G.地震层析成像[M].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[M].Translated by WANG Chun-yong,WU Ning-yuan,LI You-ming,et al.Beijing:Academic Book Press,1989.
[40]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[J].地球物理学报,2009,52(1):245-252.LI Bo,LI U Guo-feng,LI U Hong.A method of using GPU to accelerate seismic pre-stack time migration[J].Chinese Journal of Geophysics,2009,52(1):245-252.
[41]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[J].地球物理学进展,2009,24(5):1671-1678.LI U Guo-feng,LI U Qin,LI Bo,et al.GPU/CPUco-processing parallel computation for seismic data processing in oil and gas exploration[J].Progress in Geophysics,2009,24(5):1671-1678.
[2]Vidale J E.Finite-difference calculation of traveltimes[J].Bull Seis Soc Am,1988,78(6):2062-2076.
[3]Gray S H,May W P.Kirchhoff migration using ei-konal equation traveltimes[J].Geophysics,1994,59(5):810-817.
[4]Vinje V,Iversen E,Gjoystdal H.Traveltime and am-plitude estimation using wavefront construction[J].Geophysics,1993,58(8):1157-1166.
[5]Vinje V,Iversen E,Astebol K,et al.Estimation of multivalued arrivals in3D models using wavefront construction—Part I[J].Geophysical Prospecting,1996,44(5):819-842.
[6]Vinje V,Iversen E,Astebol K,et al.Estimation of mul-tivalued arrivals in3D models using wavefront construction—Part II:tracing and interpolation[J].Geophysicsal Prospecting,1996,44(5):843-858.
[7]Lambare G,Lucio P S,Hanyga A.Two-dimensional multivalued traveltime and amplitude maps by uniform sampling of ray field[J].Geophys J Internat,1996,125(2):584-598.
[8]Lucio P S,Lambare G,Hanyga A.3D multivalued travel time and amplitude maps[J].PAGEOPH,1996,148(3):449-479.
[9]Vinje V,Astebol K,Iversen E,et al.3-Dray modeling by wavefront construction in open models[J].Geophysics,1999,64(6):1912-1919.
[10]Lee K J,Gibson Jr R L.An improved mesh genera-tion scheme for wavefront construction method[J].Geophysics,2007,22(1):1-8.
[11]Chambers K,Kendall J M.A practical implemen-tation of wave front construction for3-D istropic media[J].Geophysical Journal International,2008,173(3):1-9.
[12]Operto MS,Xu S,Gilles L.Can we quantitatively image complex structures with rays[J].Geophysics,2000,65(4):1223-1238.
[13]Podvin P,Lecomte I.Finite difference computation of traveltimes in very contrasted velocity models:a massively parallel approach and its associated tools[J].Geophy J Int,1991,105(11):271-284.
[14]Moser T J.Shortest path calculation of seismic rays[J].Geophysics,1991,56(1):59-67.
[15]Van T J,Symes W W.Upwind finite-difference cal-culation of traveltimes[J].Geophysics,1991,56(6):812-821.
[16]Qin F,Olsen K,Luo Y,et al.Finite-difference so-lution of the eikonal equation along expanding wavefronts[J].Geophysics,1992,57(3):478-487.
[17]赵连锋,朱介寿,曹俊兴,等.有序波前重建法的射线追踪[J].地球物理学报,2003,46(3):415-420.ZHAO Lian-feng,ZHUJie-shou,CAO Jun-xing,et al.Ray tracing using ordinal wavefront reconstraction method[J].Chinese Journal of Geophysics,2003,46(3):415-420.
[18]孙建国,何洋.基于波前构建的射线追踪:一种Java实现[J].吉林大学学报:地球科学版,2007,37(4):814-820.SUNJian-guo,HE Yang.Ray tracing based on wave-front construction:a Java implementation[J].Journal of Jilin University:Earth Science Edition,2007,37(4):814-820.
[19]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[J].勘探地球物理进展,2007,30(5):392-395.SUN Zhang-qing,SUNJian-guo,HAN Fu-xing,et al.Seismic traveltime calculation on relief surface byfast-marching method[J].Progress in Exploration Geophysics,2007,30(5):392-395.
[20]郭继茹,冯,王俊祥,等.最佳路径射线追踪算法研究[J].吉林大学学报:地球科学版,2008,38(增刊):73-75.GUO Ji-ru,FENG Xuan,WANG Jun-xiang,et al.Study of shortest path method of ray tracing algorithm[J].Journal of Jilin University:Earth Science Edition,2008,38(Sup.):73-75.
[21]韩复兴,孙建国,杨昊.基于二维三次卷积插值算法的波前构建射线追踪[J].吉林大学学报:地球科学版,2008,38(2):336-340.HAN Fu-xing,SUN Jian-guo,YANG Hao.Ray tracing of wave-front construction by bicubic convolution interpolation[J].Journal of Jilin University:Earth Science Edition,2008,38(2):336-340.
[22]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[J].地球物理学报,2009,52(11):2846-2853.SUN Zhang-qing,SUN Jian-guo,HAN Fu-xing.Traveltimes computation using linear interpolation and narrow band technique under complex topographic conditions[J].Chinese Journal of Geophysics,2009,52(11):2846-2853.
[23]李波涛,杨长春,陈雨红,等.基于波前面三角形网格剖分的波前重建法三维射线追踪[J].地球物理学进展,2009,24(2):507-512.LI Bo-tao,YANG Chang-chun,CHEN Yu-hong,et al.3D wavefront construction method ray tracing based on triangular grid layout on wavefront[J].Progress in Geophysics,2009,24(2):507-512.
[24]杨昊,孙建国,韩复兴.波前扩展有限差分地震波走时算法的C++语言描述[J].吉林大学学报:地球科学版,2007,37(3):615-619.YANG Hao,SUNJian-guo,HAN Fu-xing.An C++language programimplement of travel-time calculation of expanding wavefronts finite-difference method[J].Journal of Jilin University:Earth Science Edition,2007,37(3):615-619.
[25]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[J].地震学报,2004,26(6):644-650.LI Zhen-chun,LI U Yu-lian,ZHANG Jian-lei,et al.Finite-difference calculation of traveltime based on rectangular grid[J].Acta Seismologica Sinica,2004,26(6):644-650.
[26]Feng K.On difference schemes and symplectic geo-metry[C]//Feng K.Proc1984Beijing Symp Diff Geometry and Diff Equations.Beijing:Science Press,1985:42-58.
[27]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[J].石油物探,2009,48(1):3-10.LI U Hong,LI U Guo-feng,LI Bo,et al.Traveltime calculation by lateral derivative and its application in pre-stack time migration[J].Geophysical Prospecting for Petroleum,2009,48(1):3-10.
[28]刘洪,袁江华,陈景波,等.大步长波场深度延拓的理论[J].地球物理学报,2006,49(6):1779-1793.LI U Hong,YUAN Jiang-hua,CHEN Jing-bo,et al.Theory of large-step wavefield depth extrapolation[J].Chinese Journal of Geophysics,2006,49(6):1779-1793.
[29]刘洪,王秀闽,曾锐,等.单程波算子积分解的象征表示[J].地球物理学进展,2007,22(2):463-471.LI U Hong,WANG Xiu-min,ZENG Rui,et al.Symbol description to integral solution of one-way wave operator[J].Progress in Geophysics,2007,22(2):463-471.
[30]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[J].地球物理学进展,2009,24(1):131-136.LI U Guo-feng,LI U Hong,WANG Xiu-min,et al.Two kinds of traveling time computation and parallel computing methods of Kirchhoff migration[J].Progress in Geophysics,2009,24(1):131-136.
[31]王秀闽.积分偏移的计算几何与并行实现[D].北京:中国科学院地质与地球物理研究所,2008.WANG Xiu-min.Computational geometry and parallel implementation of integral migration[D].Beijing:Institute of Geology and Geophysics,Chinese Academy of Sciences,2008.
[32]Liu Guo-feng,Liu Hong,Li Bo,et al.Getting pre-stack time migration travel times from the single square root operator[J].Applied Geophysics,2009,6(2):129-137.
[33]Magnus W.On the exponential solution of diffe-rential equations for a linear operator[J].Communications on Pure and Applied Mathematics,1954,7(5):649-673.
[34]Iserles A,Munthe-Kaas H,Nfrstt S,et al.Lie-group methods[J].Acta Numerica,2000,9(1):215-263.
[35]Chen J B,Munthe-Kass H,Qin M Z.Square-con-servative schemes for a class of evolution equations using Lie-group methods[J].SIAMJ Nnmer Anal,2002,39(6):2164-2178.
[36]Moler C,Van Loan C F.Ninteen dubious ways to compute the exponential of a matrix[J].SIAM Review,1978,20(4):801-836.
[37]Liu Y,Wu R S.Acomparison between phase screen,finite difference and eigenfunction expansion calculation for scalar waves in inhomogeneous media[J].Bulletin of the Seismological Society of America,1994,84(4):1154-1168.
[38]Celledini E,Iserles A.Methods for the approxima-tion of the matrix exponential in a Lie-algebraic setting[J].I MA Journal of Numerical Analysis,2001,21(2):463-488.
[39]Nolet G.地震层析成像[M].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[M].Translated by WANG Chun-yong,WU Ning-yuan,LI You-ming,et al.Beijing:Academic Book Press,1989.
[40]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[J].地球物理学报,2009,52(1):245-252.LI Bo,LI U Guo-feng,LI U Hong.A method of using GPU to accelerate seismic pre-stack time migration[J].Chinese Journal of Geophysics,2009,52(1):245-252.
[41]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[J].地球物理学进展,2009,24(5):1671-1678.LI U Guo-feng,LI U Qin,LI Bo,et al.GPU/CPUco-processing parallel computation for seismic data processing in oil and gas exploration[J].Progress in Geophysics,2009,24(5):1671-1678.
版权所有：© 2023 中国地质图书馆 中国地质调查局地学文献中心