基于快速匹配法的VTI介质走时计算
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摘要
地震波全波形反演是当今地质构造反演的潮流。在层析成像等为其提供初始模型的预处理中,地震波初至走时是一个非常重要的物理量。因而,高效高精度且稳定的走时计算方法对于各向异性建模具有重要的研究意义。为实现高效高精度且稳定的走时计算,首先利用扰动理论及泰勒公式将具有垂直对称轴的横向各向同性(VTI)介质程函方程展开,得到走时解;然后引入各向同性快速推进法(fast marching method,FMM),运用改进后的迎风差分格式求取差分格式黏滞解获取单点走时。结合窄带推进技术,得到了一种新的基于快速匹配法的VTI介质走时计算方法。通过对均匀弱各向异性模型计算结果和解析值的对比,评估了其误差,相对误差稳定于0.5%以下。针对该方法的有效性和稳定性,对层状介质模型和盐丘模型反射波走时进行了试算,取得了较好的效果。理论分析和模型试算表明,该方法对VTI介质走时计算具有较高的精度,能够应用于各向异性层析成像和全波场偏移等研究中。
Full-wave seismic inversion is a trend of geological structure interpretation.Seismic firstarrival travel time is an important parameter in processes of providing initial models like tomography.An efficient,accurate and stable method for travel time computation is very important for anisotropic modeling.Firstly,we use the perturbation theory and Taylor's theorem to expand the eikonal equation of the VTI media,by which we achieve the analytical solution of the homogeneous media(Alkhalifa2012);next,we introduce a fast marching method of isotropic media,and use the upwind difference to obtain the single point travel time.Combining with narrowband propulsion technique,we set a new travel-time computation method in VTI media based on the fast matching method.The calculated result is evaluated by comparing with the analytical values of the weak anisotropic homogenous model.The effectiveness and stability of the method is verified through calculating the travel time of the reflected waves in the layered medium model.The theoretical analysis and model trial show that this method is efficient and precise,and can be applied to anisotropic tomography and full wave field migration.
Full-wave seismic inversion is a trend of geological structure interpretation.Seismic firstarrival travel time is an important parameter in processes of providing initial models like tomography.An efficient,accurate and stable method for travel time computation is very important for anisotropic modeling.Firstly,we use the perturbation theory and Taylor's theorem to expand the eikonal equation of the VTI media,by which we achieve the analytical solution of the homogeneous media(Alkhalifa2012);next,we introduce a fast marching method of isotropic media,and use the upwind difference to obtain the single point travel time.Combining with narrowband propulsion technique,we set a new travel-time computation method in VTI media based on the fast matching method.The calculated result is evaluated by comparing with the analytical values of the weak anisotropic homogenous model.The effectiveness and stability of the method is verified through calculating the travel time of the reflected waves in the layered medium model.The theoretical analysis and model trial show that this method is efficient and precise,and can be applied to anisotropic tomography and full wave field migration.
引文
[1]Um J,Thurber C.A Fast Algorithm for Two-Point Seismic Ray Tracing[J].Bulletin of the Seismological Society of America,1987,77(3):972-986.
[2]Moser T J,Nolet G,Snieder A R.Ray Bending Revisited[J].Bulletin of the Seismological Society of America,1992,82(1):259-288.
[3]Langan R T.Tracing of Rays Through Heterogeneous Media:An Accurate and Efficient Procedure[J].Geophysics,1985,50(9):1456-1465.
[4]Dijkstra E W.A Note on Two Problems in Connection with Graphs[J].Numerische Mathematics,1959,1(1):269-271.
[5]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 Zhangqing,Sun Jianguo,Han Fuxing.Traveltimes Computation Using Linear Interpolation and Narrow Band Technique Under Complex Topographical Conditions[J].Chinese Journal of Geophysics,2009,52(11):2846-2853.
[6]孙章庆,孙建国,岳玉波,等.基于快速推进迎风双线性插值法的三维地震波走时计算[J].地球物理学报,2015,58(6):2011-2023.Sun Zhangqing,Sun Jianguo,Yue Yubo,et al.3DTraveltime Computation Using Fast Marching Upwind Bilinear Interpolation Method[J].Chinese Journal of Geophysics,2015,58(6):2011-2023.
[7]李培明,梅胜全,马青坡.一种改进的双线性插值射线追踪方法[J].石油地球物理勘探,2013,48(4):553-558.Li Peiming,Mei Shengquan,Ma Qingpo.An Improved Bilinear Interpolation Traveltime Ray Tracing Method[J].Oil Geophysical Prospecting,2013,48(4):553-558.
[8]梅胜全,邓飞,钟本善,等.基于改进的双线性走时插值的三维射线追踪[J].物探化探计算技术,2010,32(2);152-157.Mei Shengquan,Deng Fei,Zhong Benshan,et al.The3DRay Tracing Method Base on the Improved Bilinear Traveltime Interpolation[J].Computing Techniques for Geophysical and Geochemical Exploration,2010,32(2):152-157.
[9]Alkhalifah T.Traveltime Computation with the Linearized Eikonal Equation for Anisotropic Media[J].Geophysical Prospecting,2000,50(4):373-382.
[10]Vidale J.Finite-Difference Calculation of Travel Times[J].Bulletin of the Seismological Society of America,1988,78(6):2062-2076.
[11]孙章庆,孙建国,韩复兴.三维起伏地表条件下地震波走时计算的不等距迎风差分法[J].地球物理学报,2012,55(7):2441-2449.Sun Zhangqing,Sun Jianguo,Han Fuxing.Traveltime Computation Using the Upwind Finite Difference Method with Nonuniform Grid Spacing in a3DUndulating Surface Condition[J].Chinese Journal of Geophysics,2012,55(7):2441-2449.
[12]李兴旺,白超英,李晓玲.四面体单元剖分下三维各向异性TI介质中多次波射线追踪[J].石油地球物理勘探,2017,52(1):48-55.Li Xingwang,Bai Chaoying,Li Xiaoling.Multiples Raytracing in 3D Anisotropic TI Media with a Tetrahedron Cell Model[J].Oil Geophysical Prospecting,2017,52(1):48-55.
[13]李晓玲,白超英,胡光义.起伏层状TI介质中多次波射线追踪[J].石油地球物理勘探,2013,48(6):924-931.Li Xiaoling,Bai Chaoying,Hu Guangyi.Multiple Ray Tracing in an Undulated Layered TI Media[J].Oil Geophysical Prospecting,2013,48(6):924-931.
[14]高亮,李幼铭,陈旭荣,等.地震射线辛几何算法初探[J].地球物理学报,2000,43(3):402-410.Gao Liang,Li Youming,Chen Xurong,et al.An Attempt to Seismic Ray Tracing with Symplectic Algorithm[J].Chinese Journal of Geophysics,2000,43(3):402-410.
[15]秦孟兆,陈景波.Maslov渐近理论与辛几何算法[J].地球物理学报,2000,43(4):522-533.Qin Mengzhao,Chen Jingbo.Maslov Asymptotic Theory and Symplectic Algorithm[J].Chinese Journal of Geophysics,2000,43(4):522-533.
[16]Cameron M,Fomel S,Sethian J.Time-to-Depth Conversion and Seismic Velocity Estimation Using Time-Migration Velocity[J].Geophysics,2008,73(5):205-210.
[17]Rawlinson N,Sambridge M.Multiple Reflection and Transmission Phases in Complex Layered Media Using a Multistage Fast Marching Method[J].Geophysics,2004,69(5):1338-1350.
[18]Rawlinson N,Sambridge M,Saygin E.A Dynamic Objective Function Technique for Generating Multiple Solution Models in Seismic Tomography[J].Geophysical Journal International,2008,174(1):295-308.
[19]Hao Q,Alkhalifah T.An Acoustic Eikonal Equation for Attenuating Transversely Isotropic Media with a Vertical Symmetry Axis[J].Geophysics,2017,82(1):C9-C20.
[20]Sethian J A.AFast-Marching Level Set Method for Monotonically Advancing Fronts[J].Proceedings of the National Academy of Sciences of the United States of America,1996,93(4):1591-1595.
[21]Sethian J A.Level Set Methods and Fast Marching Method[M]//Sethian J A.Level Set Methods and Fast Marching Methods.London:Cambridge University Press,1999:400.
[22]Alkhalifah T,Fomel S.Implementing the Fast Marching Eikonal Solver:Spherical Versus Cartesian Coordinates[J].Geophysical Prospecting,2010,49(2):165-178.
[23]Popovici A M,Sethian J A.3-D Imaging Using Higher Order Fast Marching Traveltimes[J].Geophysics,2002,67(10):604-609.
[24]Rawlinson N,Sambridge M.Wave Front Evolution in Strongly Heterogeneous Layered Media Using the Fast Marching Method[J].Geophysical Journal of the Royal Astronomical Society,2004,156(3):631-647.
[25]Rawlinson N,Sambridge M,Rawlinson N,et al.The Fast Marching Method:An Effective Tool for Tomographic Imaging and Tracking Multiple Phases in Complex Layered Media[J].Exploration Geophysics,2005,36(4):341-350.
[26]De Kool M,Rawlinson N,Sambridge M.A Practical Grid-Based Method for Tracking Multiple Refraction and Reflection Phases in ThreeDimensional Heterogeneous Media[J].Geophysical Journal of the Royal Astronomical Society,2006,167(1):253-270.
[27]张文生,何樵登,朱建伟,等.横向各向同性介质中群速度的计算[J].物探化探计算技术,1997,19(2):97-102.Zhang Wensheng,He Qiaodeng,Zhu Jianwei,et al.Calculation of Group Velocity in Transversely Isotropic Media[J].Geophysical and Geochemical Exploration Technology,1997,19(2):97-102.
[28]董良国,魏建新.横向各向同性介质中弹性波的物理模拟[J].石油物探,1999,38(1):76-84.Dong Liangguo,Wei Jianxin.Physical Simulation of Elastic Waves in Transversely Isotropic Media[J].Petroleum Geophysical Exploration,1999,38(1):76-84.
[29]孟庆生,何樵登,王德利.均匀横向各向同性介质中P波及SV波的射线规律[J].吉林大学学报(地球科学版),2002,32(4):378-381.Meng Qingsheng,He Qiaodeng,Wang Deli.Ray Law of P Wave and SV Wave in Uniform Transversely Isotropic Medium[J].Journal of Jilin University(Earth Science Edition),2002,32(4):378-381.
[30]张建中.三维TI介质中P波NMO速度及VSP走时联合反演[D].北京:中国地震局地质研究所,2005.Zhang Jianzhong.P-Wave NMO Velocity and VSPTravel Time Joint Inversion in 3D TI Media[D].Beijing:Institute of Geology,China Earthquake Administration,2005.
[31]郝奇.VTI介质速度和各向异性参数建模研究[D].长春:吉林大学,2010.Hao Qi.Modeling of Velocity and Anisotropy Parameters of VTI Media[D].Changchun:Jilin University,2010.
[32]白海军,孙赞东,王学军.基于波前构建法的TTI介质射线追踪[J].石油地球物理勘探,2011,46(a01):1-6.Bai Haijun,Sun Zandong,Wang Xuejun.TTI Media Ray Tracing Based on Wavefront Construction[J].Oil Geophysical Prospecting,2011,46(a01):1-6.
[33]Alkhalifah T.Acoustic Approximations for Seismic Processing in Transversely Isotropic Media[J].Geophysics,1998,63(2):623-631.doi:10.1190/1.1444361.
[34]Bender C M,Orszag S A.Advanced Mathematical Methods for Scientists and Engineers[M].London:McGraw-Hill,1999.
[35]Alkhalifah T.Scanning Anisotropy Parameters in Complex Media[J].Geophysics,2011,76(2):U13-U22.doi:10.1190/1.3553015.
[2]Moser T J,Nolet G,Snieder A R.Ray Bending Revisited[J].Bulletin of the Seismological Society of America,1992,82(1):259-288.
[3]Langan R T.Tracing of Rays Through Heterogeneous Media:An Accurate and Efficient Procedure[J].Geophysics,1985,50(9):1456-1465.
[4]Dijkstra E W.A Note on Two Problems in Connection with Graphs[J].Numerische Mathematics,1959,1(1):269-271.
[5]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 Zhangqing,Sun Jianguo,Han Fuxing.Traveltimes Computation Using Linear Interpolation and Narrow Band Technique Under Complex Topographical Conditions[J].Chinese Journal of Geophysics,2009,52(11):2846-2853.
[6]孙章庆,孙建国,岳玉波,等.基于快速推进迎风双线性插值法的三维地震波走时计算[J].地球物理学报,2015,58(6):2011-2023.Sun Zhangqing,Sun Jianguo,Yue Yubo,et al.3DTraveltime Computation Using Fast Marching Upwind Bilinear Interpolation Method[J].Chinese Journal of Geophysics,2015,58(6):2011-2023.
[7]李培明,梅胜全,马青坡.一种改进的双线性插值射线追踪方法[J].石油地球物理勘探,2013,48(4):553-558.Li Peiming,Mei Shengquan,Ma Qingpo.An Improved Bilinear Interpolation Traveltime Ray Tracing Method[J].Oil Geophysical Prospecting,2013,48(4):553-558.
[8]梅胜全,邓飞,钟本善,等.基于改进的双线性走时插值的三维射线追踪[J].物探化探计算技术,2010,32(2);152-157.Mei Shengquan,Deng Fei,Zhong Benshan,et al.The3DRay Tracing Method Base on the Improved Bilinear Traveltime Interpolation[J].Computing Techniques for Geophysical and Geochemical Exploration,2010,32(2):152-157.
[9]Alkhalifah T.Traveltime Computation with the Linearized Eikonal Equation for Anisotropic Media[J].Geophysical Prospecting,2000,50(4):373-382.
[10]Vidale J.Finite-Difference Calculation of Travel Times[J].Bulletin of the Seismological Society of America,1988,78(6):2062-2076.
[11]孙章庆,孙建国,韩复兴.三维起伏地表条件下地震波走时计算的不等距迎风差分法[J].地球物理学报,2012,55(7):2441-2449.Sun Zhangqing,Sun Jianguo,Han Fuxing.Traveltime Computation Using the Upwind Finite Difference Method with Nonuniform Grid Spacing in a3DUndulating Surface Condition[J].Chinese Journal of Geophysics,2012,55(7):2441-2449.
[12]李兴旺,白超英,李晓玲.四面体单元剖分下三维各向异性TI介质中多次波射线追踪[J].石油地球物理勘探,2017,52(1):48-55.Li Xingwang,Bai Chaoying,Li Xiaoling.Multiples Raytracing in 3D Anisotropic TI Media with a Tetrahedron Cell Model[J].Oil Geophysical Prospecting,2017,52(1):48-55.
[13]李晓玲,白超英,胡光义.起伏层状TI介质中多次波射线追踪[J].石油地球物理勘探,2013,48(6):924-931.Li Xiaoling,Bai Chaoying,Hu Guangyi.Multiple Ray Tracing in an Undulated Layered TI Media[J].Oil Geophysical Prospecting,2013,48(6):924-931.
[14]高亮,李幼铭,陈旭荣,等.地震射线辛几何算法初探[J].地球物理学报,2000,43(3):402-410.Gao Liang,Li Youming,Chen Xurong,et al.An Attempt to Seismic Ray Tracing with Symplectic Algorithm[J].Chinese Journal of Geophysics,2000,43(3):402-410.
[15]秦孟兆,陈景波.Maslov渐近理论与辛几何算法[J].地球物理学报,2000,43(4):522-533.Qin Mengzhao,Chen Jingbo.Maslov Asymptotic Theory and Symplectic Algorithm[J].Chinese Journal of Geophysics,2000,43(4):522-533.
[16]Cameron M,Fomel S,Sethian J.Time-to-Depth Conversion and Seismic Velocity Estimation Using Time-Migration Velocity[J].Geophysics,2008,73(5):205-210.
[17]Rawlinson N,Sambridge M.Multiple Reflection and Transmission Phases in Complex Layered Media Using a Multistage Fast Marching Method[J].Geophysics,2004,69(5):1338-1350.
[18]Rawlinson N,Sambridge M,Saygin E.A Dynamic Objective Function Technique for Generating Multiple Solution Models in Seismic Tomography[J].Geophysical Journal International,2008,174(1):295-308.
[19]Hao Q,Alkhalifah T.An Acoustic Eikonal Equation for Attenuating Transversely Isotropic Media with a Vertical Symmetry Axis[J].Geophysics,2017,82(1):C9-C20.
[20]Sethian J A.AFast-Marching Level Set Method for Monotonically Advancing Fronts[J].Proceedings of the National Academy of Sciences of the United States of America,1996,93(4):1591-1595.
[21]Sethian J A.Level Set Methods and Fast Marching Method[M]//Sethian J A.Level Set Methods and Fast Marching Methods.London:Cambridge University Press,1999:400.
[22]Alkhalifah T,Fomel S.Implementing the Fast Marching Eikonal Solver:Spherical Versus Cartesian Coordinates[J].Geophysical Prospecting,2010,49(2):165-178.
[23]Popovici A M,Sethian J A.3-D Imaging Using Higher Order Fast Marching Traveltimes[J].Geophysics,2002,67(10):604-609.
[24]Rawlinson N,Sambridge M.Wave Front Evolution in Strongly Heterogeneous Layered Media Using the Fast Marching Method[J].Geophysical Journal of the Royal Astronomical Society,2004,156(3):631-647.
[25]Rawlinson N,Sambridge M,Rawlinson N,et al.The Fast Marching Method:An Effective Tool for Tomographic Imaging and Tracking Multiple Phases in Complex Layered Media[J].Exploration Geophysics,2005,36(4):341-350.
[26]De Kool M,Rawlinson N,Sambridge M.A Practical Grid-Based Method for Tracking Multiple Refraction and Reflection Phases in ThreeDimensional Heterogeneous Media[J].Geophysical Journal of the Royal Astronomical Society,2006,167(1):253-270.
[27]张文生,何樵登,朱建伟,等.横向各向同性介质中群速度的计算[J].物探化探计算技术,1997,19(2):97-102.Zhang Wensheng,He Qiaodeng,Zhu Jianwei,et al.Calculation of Group Velocity in Transversely Isotropic Media[J].Geophysical and Geochemical Exploration Technology,1997,19(2):97-102.
[28]董良国,魏建新.横向各向同性介质中弹性波的物理模拟[J].石油物探,1999,38(1):76-84.Dong Liangguo,Wei Jianxin.Physical Simulation of Elastic Waves in Transversely Isotropic Media[J].Petroleum Geophysical Exploration,1999,38(1):76-84.
[29]孟庆生,何樵登,王德利.均匀横向各向同性介质中P波及SV波的射线规律[J].吉林大学学报(地球科学版),2002,32(4):378-381.Meng Qingsheng,He Qiaodeng,Wang Deli.Ray Law of P Wave and SV Wave in Uniform Transversely Isotropic Medium[J].Journal of Jilin University(Earth Science Edition),2002,32(4):378-381.
[30]张建中.三维TI介质中P波NMO速度及VSP走时联合反演[D].北京:中国地震局地质研究所,2005.Zhang Jianzhong.P-Wave NMO Velocity and VSPTravel Time Joint Inversion in 3D TI Media[D].Beijing:Institute of Geology,China Earthquake Administration,2005.
[31]郝奇.VTI介质速度和各向异性参数建模研究[D].长春:吉林大学,2010.Hao Qi.Modeling of Velocity and Anisotropy Parameters of VTI Media[D].Changchun:Jilin University,2010.
[32]白海军,孙赞东,王学军.基于波前构建法的TTI介质射线追踪[J].石油地球物理勘探,2011,46(a01):1-6.Bai Haijun,Sun Zandong,Wang Xuejun.TTI Media Ray Tracing Based on Wavefront Construction[J].Oil Geophysical Prospecting,2011,46(a01):1-6.
[33]Alkhalifah T.Acoustic Approximations for Seismic Processing in Transversely Isotropic Media[J].Geophysics,1998,63(2):623-631.doi:10.1190/1.1444361.
[34]Bender C M,Orszag S A.Advanced Mathematical Methods for Scientists and Engineers[M].London:McGraw-Hill,1999.
[35]Alkhalifah T.Scanning Anisotropy Parameters in Complex Media[J].Geophysics,2011,76(2):U13-U22.doi:10.1190/1.3553015.