Abstract:Under the high frequency approximation on wave phenomena and spatial variations of physical parameters of subsurface medium,scattering waves degrade into reflection waves,and the corresponding scattering wave equation degrades into reflection wave equation based on impedance relative perturbation.Reflection wave equations based on the reflectivity is derived through the definition of reflectivity as the directional derivative of the impedance relative perturbation with respect to the propagation direction of incident wave.Using these reflection wave equations and the linear inverse theory,we establish a waveform imaging method based on reflection wave equations for seismic primary reflection data.According to the reflection wave equation based on the impedance relative perturbation and the reflection wave equation based on the reflectivity,we first apply adjoint operators of reflection wave propagator to establish an approximate inversion of impedance relative perturbation and a reflectivity waveform migration,then reapply the least squares inverse operator of refection wave propagator to establish a least squares inversion of impedance relative perturbation and a reflectivity least squares waveform migration.Aiming at the huge amount of calculation and the instability of the least squares inverse operator of reflection wave propagator,we use an iterative solution to the least squares inversion of impedance relative perturbation and the reflectivity least squares waveform migration.The waveform imaging method proposed in this paper can faithfully use the waveform information of seismic data in the sense of wave equation,and it is a real wave equation migration imaging.
Lailly P.The seismic inverse problem as a sequence of before stack migrations.Conference Expanded Abstracts on Inverse Scattering,Theory and Application,Society for Industrial and Applied Mathematics,1983,206-220.
[2]
Tarantola A.Inversion of seismic reflection data in the acoustic approximation.Geophysics,1984,49(8):1259-1266.
[3]
Tarantola A.Linearized inversion of seismic reflection data.Geophysical Prospecting,1984,32(4):998-1015.
[4]
Gray S,Etgen J,Dellinger J et al.Seismic migration problems and solutions.Geophysics,2001,66(5):1622-1640.
[5]
Yilmaz O.Seismic Data Analysis.SEG,2001.
[6]
Etgen J,Gray S,Zhang Y.An overview of depth imaging in exploration geophysics.Geophysics,2009,74(6):WCA5-WCA17.
[7]
Leveille J,Jones I,Zhou Z et al.Subsalt imaging for exploration,production,and development:A review.Geophysics,2011,76(5):WB3-WB20.
[8]
Claerbout J.Toward a unified theory of reflector mapping.Geophysics,1971,36(3):467-481.
[9]
Claerbout J.Fundamentals of Geophysical Data Processing.McGraw-Hill Book Co Inc,1976.
[10]
Claerbout J.Imaging of the Earth's Interior.Blackwell Scientific Publication,1985.
[11]
Loewenthal D,Lu L,Roberson R et al.The wave equation applied to migration.Geophysical Prospecting,1976,24(2):380-399.
[12]
马在田.地震成像技术:有限差分偏移.北京:石油工业出版社,1989.
[13]
Robein E.Seismic imaging:A review of the techniques,their principles,merits and limitations.EAGE,2010.
[14]
Stolt R,Benson A.Seismic Migration:Theory and Practice.Geophysical Press,London,1986.
[15]
Bleistein N,Cohen J,Stockwell J.Mathematics of Multi-dimensional Seismic Imaging,Migration,and Inversion. Springer, New York,2001.
[16]
李振春等编著.地震叠前成像理论与方法.山东东营:中国石油大学出版社,2011.
[17]
李振春.地震偏移成像技术研究现状与发展趋势.石油地球物理勘探,2014,49(1):1-21.Li Zhenchun.Research status and development trends for seismic migration technology.OGP,2014,49(1):1-21.
[18]
Du Q, Zhu Y, Ba J.Polarity reversal correction for elastic reverse time migration.Geophysics,2012,77(2):S31-S41.
Zhang Y,Duan L and Xie Y.A stable and practical implementation of least-squares reverse time migration.Geophysics,2014,80(1):V23-V31.
[21]
Yao G,Jakubowicz H.Least-squares reverse time migration.SEG Technical Program Expanded Abstracts,2012,31:1-5.
[22]
Yao G,Jakubowicz H.Non-linear least-squares reverse time migration.SEG Technical Program Expanded Abstracts,2012,31:1-5.
[23]
Dong S,Cai J,Guo M et al.Least-squares reverse time migration:towards true amplitude imaging and impoving the resolution.SEG Technical Program Expanded Abstracts,2012,31:1-5.
[24]
Gray S.True-amplitude seismic migration:A comparison of three approaches.Geophysics,1997,62(3):926-936.
[25]
郭振波,李振春.最小平方逆时偏移真振幅成像.石油地球物理勘探,2014,49(1):113-120.Guo Zhenbo,Li Zhenchun.True amplitude imaging based on least-squares reverse time migration.OGP,2014,49(1):113-120.
[26]
黄建平,孙陨松,李振春等.一种基于分频编码的最小二乘裂步偏移方法.石油地球物理勘探,2014,49(4):702-707.Huang Jianping,Sun Yunsong,Li Zhenchun et al.Least-squares split step migration based on frequency division encoding.OGP,2014,49(4):702-709.
[27]
周华敏,陈生昌,任浩然等.基于照明补偿的单程波最小二乘偏移.地球物理学报,2014,57(8):2644-2655.Zhou Huamin,Chen Shengchang,Ren Haoran et al.One way wave equation least squares migration based on illumination compensation.Chinese Journal of Geophysics,2014,57(8):2644-2655.
[28]
Zhang Y,Ratcliffe A,Duan L.Velocity and impedance inversion based on true amplitude reverse time migration.SEG Technical Program Expanded Abstracts, 2013,32:949-953.
[29]
Duan L,Zhang Y and Peng L.Band-limited impedance perturbation inversion using cross-correlative least-squares RTM.SEG Technical Program Expanded Abstracts,2014,33:3720-3725.
[30]
Berkhout A.Seismic Migration:Imaging of Acoustic Energy by Wavefield Extrapolation,Part A:Theoretical Aspects(2nd Ed).Elsevier,1982.
[31]
Gazdag J.Wave equation migration with the phase-shift method.Geophysics,1978, 43(7):1342-1351.
Chen J,Liu H.Two kinds of separable approxima-tions for the one-way wave operator.Geophysics,2006,71(1):T1-T5.
[36]
陈生昌,马在田,Wu Rushan.波动方程偏移成像阴影的照明补偿.地球物理学报,2007,50(3):844-850.Chen Shengchang,Ma Zaitian,Wu Rushan.Illumination compensation for wave equation migration shadow.Chinese Journal of Geophysics,2007,50(3):844-850.
[37]
刘少勇,韩冰凯,顾汉明等.带限局部平面波传播算子与偏移方法.石油地球物理勘探,2017,52(5):948-955.Liu Shaoyong,Han Bingkai,Gu Hanming et al.Band-limited local plane wave propagator and migration.OGP,2017,52(5):948-955.
[38]
Hemon C.Wave equations and models.Geophysical Prospecting,1978,26(4):790-821.
[39]
McMechan G.Migration by extrapolation of time-dependent boundary values.Geophysical Prospecting,1983,31(3):413-420.
[40]
Baysal E,Kosloff D and Sherwood J.Reverse time migration.Geophysics,1983,48(11):1514-1524.
[41]
Whitmore D.Iterative depth migration by backward time propagation.SEG Technical Program Expanded Abstracts,1983,2:382-385.
[42]
French W.Two-dimensional and three-dimensional migration of model-experiment reflection.Geophy-sics,1974,39(1):265-277.
[43]
Schneider W.Integral formulation for migration in two and three dimensions.Geophysics,1978,43(1):49-76.
[44]
Sun J.Limited aperture migration.Geophysics,2000,65(2):584-595.
[45]
Cerveny V,Popov M and Psencilk I.Computation of wave fields in inhomogeneous media Gaussian beam approach.Geophysical Journal of the Royal Astronomical Society,1982,70(1):109-128.
[46]
Costa C,Raz S and Kosloff D.Gaussian beam migration.SEG Technical Program Expanded Abstracts,1989,8:1169-1171.
[47]
Hill N.Prestack Gaussian-beam depth migration.Geo-physics,2001,66(4):1240-1250.
[48]
Tang Y,Biondi B.Least-squares migration/inversion of blended data.SEG Technical Program Expanded Abstracts,2009,28:2859-2863.
[49]
Ren H,Wu R and Wang H.Wave equation least square imaging using the local angular Hessian for amplitude correction.Geophysical Prospecting,2011,59(4):651-661.
[50]
黄建平,李振春,孔雪等.碳酸盐岩裂缝型储层最小二乘偏移成像方法研究.地球物理学报,2013,56(5):1716-1725.Huang Jianping,Li Zhenchun,Kong Xue et al.A study of least squares migration method for fractured type carbonate reservoir.Chinese Journal of Geophy-sics,2013,56(5):1716-1725.
[51]
李庆洋,黄建平,李振春等.优化的多震源最小二乘逆时偏移.石油地球物理勘探,2016,51(2):334-341.Li Qingyang,Huang Jianping,Li Zhenchun et al.Optimized multi-source least-squares reverse time migration.OGP,2016,51(2):334-341.
[52]
李娜.基于Huber范数的多震源最小二乘逆时偏移.石油地球物理勘探,2017,52(5):941-947.Li Na.Multi-source least-squares reverse time migration based on Huber norm.OGP,2017,52(5):941-947.
[53]
Beylkin G.Imaging of discontinuities in the inverse scattering problem by the inversion of a causal Radon transform.Journal Mathematics Physics,1985,26(1):99-108.
[54]
Bleistein N.On the imaging of reflectors in the earth.Geophysics,1987,52(7):931-942.
[55]
Zhang Y,Zhang G,Bleistein N.Theory of true amplitude one-way wave equation and true amplitude common-shot migration.SEG Technical Program Expanded Abstracts,2003,22:925-928.
[56]
Xu S,Zhang Y,Tang B.3D angle gathers from reverse time migration.Geophysics,2011,76(2):S77-S92.
Devaney J.Mathematical Foundations of Imaging,Tomography and Wavefield Inversion.Cambridge University Press,2012.
[59]
Goodman J.Introduction to Fourier Optics(3rd Ed).Roberts and Company Publishers Inc,2005.
[60]
陈生昌,陈国新.时间二阶积分波场的全波形反演.地球物理学报,2016,59(10):3765-3776.Chen Shengchang,Chen Guoxin.Full waveform inversion of the second-order time integral wavefield.Chinese Journal of Geophysics,2016,59(10):3765-3776.
[61]
Sava P, Fomel S.Angle-domain common-image ga-thers by wavefield continuation methods.Geophy-sics,2003,68(3):1065-1074.
[62]
Kirsch A.An Introduction to the Mathematical Theory of Inverse Problems(2nd Ed).Springer,2011.
[63]
陈生昌,周华敏.再论地震数据偏移成像.地球物理学报,2016,59(2):643-65.Chen Shengchang,Zhou Huamin.Re-exploration to migration of seismic data.Chinese Journal of Geophysics,2016,59(2):643-654.