单程波李代数深度积分的精度分析和算法改进
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摘要
本文在利用拟微分算子和象征理论研究单程波传播规律的基础上,详细分析了单平方根算子李代数深度积分存在的问题,利用改进的算法对单平方根算子进行大步长延拓,计算其李代数积分和指数映射,得到地震波走时的解析表达式.对方法改进前后李代数积分象征中系数校正情况和线性横向变速介质中走时的计算误差进行了对比,数值算例表明,改进后的方法收敛更快,减少了计算时间且提高了计算精度.
Based on the research of the propagation rules of one-way seismic wave using pseudo differential operator and the symbol theory, we analyze the existing problems in Lie algebra depth integral of single square root operator in detail. Making use of the improved algorithm to perform large-step extrapolation of single square root operator and calculating Lie algebra integral and exponential mapping, we can finally obtain the analytical expression of seismic travel time. Besides, we compare the coefficient corrections in symbol of Lie algebra integral and calculation error in medium with laterally linear velocity variation before and after algorithm modification. The numerical examples prove that the improved algorithm converges faster, reduces the computing time and improves the numerical precision.
Based on the research of the propagation rules of one-way seismic wave using pseudo differential operator and the symbol theory, we analyze the existing problems in Lie algebra depth integral of single square root operator in detail. Making use of the improved algorithm to perform large-step extrapolation of single square root operator and calculating Lie algebra integral and exponential mapping, we can finally obtain the analytical expression of seismic travel time. Besides, we compare the coefficient corrections in symbol of Lie algebra integral and calculation error in medium with laterally linear velocity variation before and after algorithm modification. The numerical examples prove that the improved algorithm converges faster, reduces the computing time and improves the numerical precision.
引文
[1] 刘洪,李幼铭.油气勘探二次创业的油储地球物理方法研究回顾.石油与天然气地质,2008,29(5) :648～653 Liu H,Li Y M.Review on geophysical methodology of oil reservoirs in the second round of hydrocarbon exploration.Oil & Gas Geology(in Chinese),2008,29(5) :648～653
[2] Cordes H O.The Technique of Pseudo Differential Operators.Cambridge:Cambridge University Press,1995
[3] 陈恕行.拟微分算子理论(第二版).北京:高等教育出版社, 1995 Chen S X.Pseudo Differential Operator(Edition 2) (in Chinese).Beijing:Higher Education Press,1995
[4] Naimark M A,Stern A I.Theory of Group Representations.New York:Springer-Verlag New York Inc.,1982
[5] Feng K.On difference schemes and symplectic geometry.In:Feng K ed.Symplectic Differential Geometry and Differential Equations.Beijing:Science Press,1985
[6] Iserles A,Munthe-Kaas H,Nfrstt S,et al.Lie-group methods.Acta Numerica,2000,9(1) :215～263
[7] 田畴.李群及其在微分方程中的应用.北京:科学出版社,2003 Tian C.Applications of Lie Groups to Differential Equations(in Chinese).Beijing:Science Press,2003
[8] 陈景波,秦孟兆.射线追踪、辛几何算法与波场的数值模拟.计算物理,2001,18(6) :481～486 Chen J B,Qin M Z.Ray tracing,symplectic geometry algorithm and numerical modeling of wave field.Computational Physics(in Chinese),2001,18(6) :481～486
[9] 秦孟兆,陈景波.Maslov渐近理论与辛几何算法.地球物理学报,2000,43(4) :522～533 Qin M Z,Chen J B.Maslov asymptotic theory and symplectic algorithm.Chinese J.Geophys.(in Chinese),2000,43(4) :522～533
[10] 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
[11] Zhang G Q.A new algorithm for finite-difference migration of steep dips.Geophysics,1988,53(2) :167～175
[12] 张关泉.一维波动方程的反演问题.中国科学,1988,31(7) :707～721 Zhang G Q.One-dimensional wave equation inverse problem.Science China(in Chinese),1988,31(7) :707～721
[13] 张关泉.波场分裂、平方根算子与偏移.反射地震学论文集,上海:同济大学出版社,2000 Zhang G Q.Wave field split,square root operator and migration.Memoir of Reflection Seismology(in Chinese).Shanghai:Tongji University Press,2000
[14] Chen J B,Munthe-Kaas H,Qin M Z.Square-conservative schemes for a class of evolution equations using Lie-group methods.SIAM J.Numer.Anal.,2002,39:2164～2178
[15] 刘超颖,王成祥.非稳态相移法叠前深度偏移.地球物理学进展,2004,19(3) :543～546 Liu C Y,Wang C X.Prestack depth migration by nonstationary phase shift method.Progress in Geophysics(in Chinese),2004,19(3) :543～546
[16] 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,YuanJ H,ChenJ B,et al.Theory of large-step wave field depth extrapolation.ChineseJ.Geophys.(in Chinese),2006,49(6) :1779～1793
[17] 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
[18] 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
[19] 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 Kirehhoff migration.Progress in Geophysics(in Chinese),2009,24(1) :131～136
[20] 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
[21] Celledoni E,Iserles A.Methods for the approximation of the matrix exponential in a Lie-algebra setting.IMA J.Numer.Anal.,2001,21(2) :463～488
[22] Moler C,Van Loan C F.Nineteen dubious ways to compute the exponential of a matrix.SIAM Review,1978,20(4) :801～836
[23] Liu Y,Wu R S.A comparison between phase screen,finite difference and eigenfunetion expansion calculation for scalar waves in inhomogeneous media.Bulletin of the Seismological Society of America,1994,84(4) :1154～1168
[24] McLachlan R I,Quispel G R W.Splitting methods.Acta Numerica,2002,11(1) :341～434
[25] Magnus W.On the exponential solution of differential equations for a linear operator.Communications on Pure and Applied Mathematics,1954,7(5) :649～673
[26] 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.High-Frequency Approximation and Sympleetic Geometry(in Chinese).Beijing.Science Press,2001． 1～141
[27] Chen J.Specular ray parameter extraction and stationary phase migration.Geophysics,2004,69(1) :249～256
[28] 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.Travel time calculation by lateral derivative and its application in pre-stack time migration.Geophysical Prospecting for Petroleum(in Chinese),2009,48(1) :3～10
[29] 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
[30] 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
[31] 刘洪,袁江华,勾永锋等.地震逆散射波场和算子的谱分解.地球物理学报,2007,50(1) :240～247 Liu H,Yuan J H,Gou Y F,et al.Spectral factorization of wave field and operator in seismic inverse scattering.Chinese J.Geophys.(in Chinese),2007,50(1) :240～247
[32] 刘洪,何利,刘国锋等.地层滤波公式的李代数积分证明和推广.金翔龙,秦蕴珊,朱日祥等.中国地质与地球物理研究进展--庆贺刘光鼎院士80华诞.北京:科学出版社,2008． 412～422 Liu H,He L,Liu G F,et al.Derivation and generalization of the stratigraphic filtering formula via Lie algebra integral(in Chinese).Jin X L,Qin Y S,Zhu R X,et al.The Research Development of Geology and Geophysics in China The Collection in Memory of 80th Anniversary of Academician Liu Guangding.Beijing:Sciences Press,2008． 412～422
[33] 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
[34] 周纳,刘继勇.非对称走时叠前时间偏移方法及应用.油气地球物理,2010,8(2) :48～52 Zhou N,Liu J Y.The method and application of asymmetric travel-time pre-stack time migration.Petroleum Geophysics(in Chinese),2010,8(2) :48～52
[35] 陈新荣.非对称走时叠前时间偏移在青东4东地区的应用.勘探地球物理进展,2010,33(4) :279～283 Chen X R.The application of asymmetric travel-time prestack time migration in area of Qingdong 4 dong.Progress in Exploration Geophysics(in Chinese),2010,33(4) :279～283
[2] Cordes H O.The Technique of Pseudo Differential Operators.Cambridge:Cambridge University Press,1995
[3] 陈恕行.拟微分算子理论(第二版).北京:高等教育出版社, 1995 Chen S X.Pseudo Differential Operator(Edition 2) (in Chinese).Beijing:Higher Education Press,1995
[4] Naimark M A,Stern A I.Theory of Group Representations.New York:Springer-Verlag New York Inc.,1982
[5] Feng K.On difference schemes and symplectic geometry.In:Feng K ed.Symplectic Differential Geometry and Differential Equations.Beijing:Science Press,1985
[6] Iserles A,Munthe-Kaas H,Nfrstt S,et al.Lie-group methods.Acta Numerica,2000,9(1) :215～263
[7] 田畴.李群及其在微分方程中的应用.北京:科学出版社,2003 Tian C.Applications of Lie Groups to Differential Equations(in Chinese).Beijing:Science Press,2003
[8] 陈景波,秦孟兆.射线追踪、辛几何算法与波场的数值模拟.计算物理,2001,18(6) :481～486 Chen J B,Qin M Z.Ray tracing,symplectic geometry algorithm and numerical modeling of wave field.Computational Physics(in Chinese),2001,18(6) :481～486
[9] 秦孟兆,陈景波.Maslov渐近理论与辛几何算法.地球物理学报,2000,43(4) :522～533 Qin M Z,Chen J B.Maslov asymptotic theory and symplectic algorithm.Chinese J.Geophys.(in Chinese),2000,43(4) :522～533
[10] 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
[11] Zhang G Q.A new algorithm for finite-difference migration of steep dips.Geophysics,1988,53(2) :167～175
[12] 张关泉.一维波动方程的反演问题.中国科学,1988,31(7) :707～721 Zhang G Q.One-dimensional wave equation inverse problem.Science China(in Chinese),1988,31(7) :707～721
[13] 张关泉.波场分裂、平方根算子与偏移.反射地震学论文集,上海:同济大学出版社,2000 Zhang G Q.Wave field split,square root operator and migration.Memoir of Reflection Seismology(in Chinese).Shanghai:Tongji University Press,2000
[14] Chen J B,Munthe-Kaas H,Qin M Z.Square-conservative schemes for a class of evolution equations using Lie-group methods.SIAM J.Numer.Anal.,2002,39:2164～2178
[15] 刘超颖,王成祥.非稳态相移法叠前深度偏移.地球物理学进展,2004,19(3) :543～546 Liu C Y,Wang C X.Prestack depth migration by nonstationary phase shift method.Progress in Geophysics(in Chinese),2004,19(3) :543～546
[16] 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,YuanJ H,ChenJ B,et al.Theory of large-step wave field depth extrapolation.ChineseJ.Geophys.(in Chinese),2006,49(6) :1779～1793
[17] 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
[18] 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
[19] 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 Kirehhoff migration.Progress in Geophysics(in Chinese),2009,24(1) :131～136
[20] 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
[21] Celledoni E,Iserles A.Methods for the approximation of the matrix exponential in a Lie-algebra setting.IMA J.Numer.Anal.,2001,21(2) :463～488
[22] Moler C,Van Loan C F.Nineteen dubious ways to compute the exponential of a matrix.SIAM Review,1978,20(4) :801～836
[23] Liu Y,Wu R S.A comparison between phase screen,finite difference and eigenfunetion expansion calculation for scalar waves in inhomogeneous media.Bulletin of the Seismological Society of America,1994,84(4) :1154～1168
[24] McLachlan R I,Quispel G R W.Splitting methods.Acta Numerica,2002,11(1) :341～434
[25] Magnus W.On the exponential solution of differential equations for a linear operator.Communications on Pure and Applied Mathematics,1954,7(5) :649～673
[26] 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.High-Frequency Approximation and Sympleetic Geometry(in Chinese).Beijing.Science Press,2001． 1～141
[27] Chen J.Specular ray parameter extraction and stationary phase migration.Geophysics,2004,69(1) :249～256
[28] 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.Travel time calculation by lateral derivative and its application in pre-stack time migration.Geophysical Prospecting for Petroleum(in Chinese),2009,48(1) :3～10
[29] 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
[30] 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
[31] 刘洪,袁江华,勾永锋等.地震逆散射波场和算子的谱分解.地球物理学报,2007,50(1) :240～247 Liu H,Yuan J H,Gou Y F,et al.Spectral factorization of wave field and operator in seismic inverse scattering.Chinese J.Geophys.(in Chinese),2007,50(1) :240～247
[32] 刘洪,何利,刘国锋等.地层滤波公式的李代数积分证明和推广.金翔龙,秦蕴珊,朱日祥等.中国地质与地球物理研究进展--庆贺刘光鼎院士80华诞.北京:科学出版社,2008． 412～422 Liu H,He L,Liu G F,et al.Derivation and generalization of the stratigraphic filtering formula via Lie algebra integral(in Chinese).Jin X L,Qin Y S,Zhu R X,et al.The Research Development of Geology and Geophysics in China The Collection in Memory of 80th Anniversary of Academician Liu Guangding.Beijing:Sciences Press,2008． 412～422
[33] 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
[34] 周纳,刘继勇.非对称走时叠前时间偏移方法及应用.油气地球物理,2010,8(2) :48～52 Zhou N,Liu J Y.The method and application of asymmetric travel-time pre-stack time migration.Petroleum Geophysics(in Chinese),2010,8(2) :48～52
[35] 陈新荣.非对称走时叠前时间偏移在青东4东地区的应用.勘探地球物理进展,2010,33(4) :279～283 Chen X R.The application of asymmetric travel-time prestack time migration in area of Qingdong 4 dong.Progress in Exploration Geophysics(in Chinese),2010,33(4) :279～283
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