Direct numerical calculation of ray Jacobian and its application
LIANG Quan1,2,3, MAO Weijian1,2, OUYANG Wei1,2, LI Wuqun1,2, ZHANG Jianlei4
1. Center for Computational & Exploration Geophysics, Institute of Geodesy and Geophysics, Chinese Academy of Science, Wuhan, Hubei 430077, China; 2. State Key Laboratory of Geodesy and Earth's Dynamics, Wuhan, Hubei 430077, China; 3. University of Chinese Academy of Sciences, Beijing 100049, China; 4. Research & Development Center, BGP, CNPC, Zhuozhou, Hebei 072751, China
Abstract：Ray Jacobian plays a fundamental role in solving transport equation,obtaining the geometrical propagating factor of seismic wavefront and the amplitude of the Green’s function.The ray Jacobian is of great importance for true amplitude migration imaging based on the ray theory and is a part of the weight function of the true amplitude imaging condition of generalized Radon transform (GRT) based inverse scattering migration.In the medium with strong velocity variation,the ray Jacobian computed by dynamic ray tracing may be rather problematic for the unstable second spatial derivatives of the velocity field,which results in generating pseudo-caustic points and missing real caustic points.To compute ray Jacobian accurately,we focus on its definition and calculate it through the ratio of the element cross-sectional area of a ray tube to the element initial angular area.Since the direct method has a clear physical meaning,we can get an accurate Jacobian,and real caustic points can be detected effectively.In addition,the singularity in inverse scattering amplitude-preserved migration caused by the caustics can be avoided by adopting a reasonable ray Jacobian smoothing threshold.Application to a salt dome model has proved the method is effective and applicable.
梁全, 毛伟建, 欧阳威, 李武群, 张建磊. 射线雅可比的直接数值计算方法及应用[J]. 石油地球物理勘探, 2020, 55(3): 557-566.
LIANG Quan, MAO Weijian, OUYANG Wei, LI Wuqun, ZHANG Jianlei. Direct numerical calculation of ray Jacobian and its application. Oil Geophysical Prospecting, 2020, 55(3): 557-566.
李振春.地震偏移成像技术研究现状与发展趋势[J].石油地球物理勘探,2014,49(1):1-21.LI Zhenchun.Research status and development trends for seismic migration technology[J].Oil Geophysical Prospecting,2014,49(1):1-21.
[2]
岳玉波,李振春,钱忠平,等.复杂地表条件下保幅高斯束偏移[J].地球物理学报,2012,55(4):1376-1383.YUE Yubo,LI Zhenchun,QIAN Zhongping,et al.Amplitude-preserved Gaussian beam migration under complex topographic conditions[J].Chinese Journal of Geophysics,2012,55(4):1376-1383.
[3]
徐升,Lambaré G.复杂介质下保真振幅Kirchhoff深度偏移[J].地球物理学报,2006,49(5):1431-1444.XU Sheng, Lambaré G.True amplitude Kirchhoff prestack depth migration in complex media[J].Chinese Journal of Geophysics,2006,49(5):1431-1444.
Thierry P,Lambaré G,Podvin P,et al.3-D preserved amplitude prestack depth migration on a workstation[J].Geophysics,1999,64(1):222-229.
[6]
Beylkin G and Burridge R.Linearized inverse scatte-ring problems in acoustic and elasticity[J].Wave Motion,1990,12(1):15-52.
[7]
Li W Q,Mao W J,Li X L,et al.Angle-domain inverse scattering migration/inversion in isotropic media[J].Journal of Applied Geophysics,2018,154:196-209.
[8]
陈云峰,王华忠,任浩然,等.线性反演最小二乘叠前偏移的矩阵形式解析[J].石油地球物理勘探,2014,49(6):1091-1096.CHEN Yunfeng,WANG Huazhong,REN Haoran,et al.Matrix representation of least square prestack migration based on linearization[J].Oil Geophysical Prospecting,2014,49(6):1091-1096.
[9]
李江,李庆春,张向辉.最小二乘偏移方法研究进展综述[J].地球物理学进展,2016,31(4):1601-1607.LI Jiang,LI Qingchun,ZHANG Xianghui.Overview of progress in least squares migration studying[J].Progress in Geophysics,2016,31(4):1601-1607.
[10]
Červený V.Seismic Ray Theory[M].Cambridge University Press,2001.
[11]
Operto M S,Xu S,Lambaré G.Can we quantitatively image complex structures with rays?[J].Geophysics,2000,65(4):1223-1238.
[12]
Keho T H and Beydoun W B.Paraxial ray Kirchhoff migration[J].Geophysics,1988,53(12):1540-1546.
[13]
Cohen J K,Hagin F G,Bleistein N.Three-dimensional Born inversion with an arbitrary reference[J].Geophysics,1986,51(8):1552-1558.
[14]
Forgues E,Lambaré G.Parameterization study for acoustic and elastic ray plus born inversion[J].Journal of Seismic Exploration,1997,6(2-3):253-277.
Lambare G,Virieux J,Madariaga R,et al.Iterative asymptotic inversion in the acoustic approximation[J].Geophysics,1992,57(9):1138.
[18]
Thierry P,Operto S,Lambaré G.Fast 2-D ray+Born migration/inversion in complex media[J].Geophy-sics,1999,64(1):162-181.
[19]
De Hoop M V,Bleistein N.Generalized Radon transform inversions for reflectivity in anisotropic elastic media[J].Inverse problems,1997,13(3):669-690.
[20]
Hill N R.Gaussian beam migration[J].Geophysics,1990,55(11):1416-1428.
[21]
Hill N R.Prestack Gaussian-beam depth migration[J].Geophysics,2001,66(4):1240-1250.
[22]
Gray S H,Bleistein N.True-amplitude Gaussian-beam migration[J].Geophysics,2009,74(2):S11-S23.
[23]
栗学磊,毛伟建.多波多分量高斯束叠前深度偏移[J].地球物理学报,2016,59(8):2989-3005.LI Xuelei and MAO Weijian.Multimode and multi-component Gaussian beam prestack depth migration[J].Chinese Journal of Geophysics,2016,59(8):2989-3005.
[24]
Mao W J and Stuart G W.Rapid multi-wave-type ray tracing in complex 2-D and 3-D isotropic media[J].Geophysics,1997,62(1):298-308.
[25]
Um J and Thurber C.A fast algorithm for two-point ray tracing[J].Bulletin of the Seismological Society of America,1987,77(3):972-986.
[26]
徐涛,徐果明,高尔根,等.三维复杂介质的块状建模和试射射线追踪[J].地球物理学报,2004,47(6):1118-1126.XU Tao,XU Guoming,GAO Ergen,et al.Block mo-deling and shooting ray tracing in complex 3-D media[J].Chinese Journal of Geophysics,2004,47(6):1118-1126.
[27]
Xu Q R and Mao W J.An efficient ray-tracing method and its application to Gaussian beam migration in complex multilayered anisotropic media[J].Geophy-sics,2018,83(5):T281-T289.
[28]
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(增刊1):1-6.BAI Haijun,SUN Zandong,WANG Xuejun.Raytracing in TTI media using wavefront construction[J].Oil Geophysical Prospecting,2011,46(S1):1-6.
[29]
张凯,陈军屹,祝佰航,等.基于程函方程的VTI介质初至波走时层析反演[J].石油地球物理勘探,2018,53(6):1218-1226.ZHANG Kai,CHEN Junyi,ZHU Baihang,et al.First break traveltime tomographic imaging based on the eikonal equation in VTI medium[J].Oil Geophysical Prospecting,2018,53(6):1218-1226.
[30]
李同宇,张建中.地震射线追踪的线性走时扰动插值法[J].石油地球物理勘探,2018,53(6):1165-1174.LI Tongyu and ZHANG Jianzhong.A linear traveltime perturbation interpolation method for seismic ray tracing[J].Oil Geophysical Prospecting,2018,53(6):1165-1174.
[31]
魏脯力,孙建国,孟祥羽.应用网格走时的射线路径计算方法[J].石油地球物理勘探,2018,53(4):722-729.WEI Puli,SUN Jianguo and MENG Xiangyu.A ray path calculation method based on grid traveltime[J].Oil Geophysical Prospecting,2018,53(4):722-729.
[32]
Moser T J.Shortest path calculation of seismic rays[J].Geophysics,1991,56(1):59-67.
[33]
Fisher R,Lees J M.Shortest path ray tracing with sparse graphs[J].Geophysics,1993,58(7):987-996.
[34]
Vidale J E.Finite-difference calculation of travel times[J].Bulletin of the Seismological Society of America,1988,78(6):2062-2076.
[35]
张霖斌,刘迎曦,赵振峰,等.有限差分法射线追踪[J].石油地球物理勘探,1993,28(6):673-677.ZHANG Linbin,LIU Yingyi,ZHAO Zhenfeng,et al.Finite-difference ray tracing[J].Oil Geophysical Prospecting,1993,28(6):673-677.
[36]
Vinje V,Iversen E,Gjøystdal H.Traveltime and am-plitude estimation using wavefront construction[J].Geophysics,1993,58(8):1157-1166.
[37]
Vidale J E,Houston H.Rapid calculation of seismic amplitudes[J].Geophysics,1990,55(11):1504-1507.
[38]
Buske S.Finite-difference solution of the transport equation:First results[J].Pure and Applied Geophy-sics,1996,148(3-4):565-581.
Vanelle C,Gajewski D.Determination of geometrical spreading from traveltimes[J].Journal of Applied Geophysics,2003,54(3-4):391-400.
[41]
Popov M M,Pšeník I,ervený V.Computation of ray amplitudes in inhomogeneous media with curved interfaces[J].Studia Geophysica et Geodaetica,1978,22(3):248-258.
[42]
ervený V,Klimeš L,Pšeník I.Applications of dy-namic ray tracing[J].Physics of the Earth and Planetary Interiors,1988,51(1-3):25-35.
[43]
Chapman C.Fundamentals of Seismic Wave Propagation[M].Cambridge University Press,2004.
[44]
Ouyang W,Mao W J,Li X L.Approximate solution of nonlinear double-scattering inversion for true amplitude imaging[J].Geophysics,2015,80(1):R43-R55.
[45]
Ouyang W,Mao W J,Li W Q,et al.Approximate non-linear multiparameter inversion with single and double scattering seismic wavefields in acoustic media[J].Geophysical Journal International,2017,208(2):767-789(本文编辑:朱汉东)作者简介梁全博士研究生,1988年生;2011年获长江大学勘查技术与工程专业学士学位,2015年获中国石油大学
