Abstract：Anumerical method of preconditioned least squares is proposed based on volume integral equation to accurately simulate zero-offset VSP wave field.First,based on the acoustic scattering theory,the one-dimensional acoustic wave equation in frequency domain can be expressed as the Lippman-Schwinger integral equation by the Green function.On this base how to solve the second kind of Fredholm integral equation can be turned to how to sov-le large linear equations through rectangular integral formula,and this lays a foundation for subsequent accurate calculation.Second,the whole wave field can be solved by the least squares method,rather than adding the scattering component to the whole wave field step by step by iteration.After introducing the preconditioned operator,the numerical calculation process is more stable.Finally,taking a stepped model and a real model based on logging data as examples,the VSP wave field can be solved using the new method,and the results were compared with those from the other conventional finite difference methods.The results show that the new method is feasible,effective and accurate in simulating zero-offset VSP wave field.
颜红芹. 基于声波散射理论的零井源距VSP波场数值模拟方法[J]. 石油地球物理勘探, 2020, 55(3): 567-574.
YAN Hongqin. Numerical simulation of zero-offset VSP wave field based on the acoustic scattering theory. Oil Geophysical Prospecting, 2020, 55(3): 567-574.
赵邦六,董世泰,曾忠.井中地震技术的昨天、今天和明天——井中地震技术发展及应用展望[J].石油地球物理勘探,2017,52(5):1112-1123.ZHAO Bangliu,DONG Shitai,ZENG Zhong.Borehole seismic deveplement,status quo and future:Application prospect of borehole seismic[J].Oil Geophysical Prospecting,2017,52(5):1112-1123.
[2]
侯爱源,李庆忠,张文波.复杂区深井VSP层析求速度场的方法[J].石油地球物理勘探,2017,52(6):1150-1155.HOU Aiyuan,LI Qingzhong,ZHANG Wenbo.A method for velocity field with deep-well VSP tomography in complex areas[J].Oil Geophysical Prospecting,2017,52(6):1150-1155.
[3]
高静怀,汪超,赵伟.用于零偏移距VSP资料的自适应波形反演方法研究[J].地球物理学报,2009,52(12):3091-3100.GAO Jinghuai,WANG Chao,ZHAO Wei.On the method of adaptive waveform inversion with zero-offset VSP data[J].Chinese Journal of Geophysics,2009,52(12):3091-3100.
[4]
Egorov A,Pevzner R,Bóna A,et al.Time-lapse full waveform inversion of vertical seismic profile data:Workflow and application to the CO2 CRC Otway project[J].Geophysical Research Letters,2017,44(14):7211-7218.
[5]
韩璇颖,印兴耀,曹丹平,等.基于分段快速模拟退火的零偏VSP全波形反演[J].石油物探,2019,58(1):103-111.HAN Xuanying,YIN Xingyao,CAO Danping,et al.Zero-offset VSP velocity inversion with FWI using segmented fast simulated annealing[J].Geophysical Prospecting for Petroleum,2019,58(1):103-111.
[6]
邹延延,徐义贤,沙椿.VSP正反演综述[J].地球物理学进展,2009,24(1):145-153.ZOU Yanyan,XU Yixian,SHA Chun.Review on the forward modeling and inversion of vertical seismic profile[J].Progress in Geophysics,2009,24(1):145-153.
[7]
杨飞龙,孙渊,李绪宣,等.基于高斯射线束的斜井VSP正演方法[J].地球物理学进展,2014,29(6):2791-2799.YANG Feilong,SUN Yuan,LI Xuxuan,et al.Devia-ted hole VSP forward method based on the Gaussian beam[J].Progress in Geophysics,2014,29(6):2791-2799.
Virieux J.P-SV wave propagation in heterogeneous media:Velocity-stress finite-difference method[J].Geophysics,1986,51(4):889-901.
[10]
陈可洋.标量声波波动方程高阶交错网格有限差分法[J].中国海上油气,2009,21(4):232-236.CHENKeyang.High-order staggered-grid finite diffe-rence scheme for scalar acoustic wave equation[J].China Offshore Oil and Gas,2009,21(4):232-236.
[11]
蔡晓慧,刘洋,王建民,等.基于自适应优化有限差分方法的全波VSP逆时偏移[J].地球物理学报,2015,58(9):3317-3334.CAI Xiaohui,LIU Yang,WANG Jianmin,et al.Full-wavefield VSP reverse-time migration based on the adaptive optimal finite-difference scheme[J].Chinese Journal of Geophysics,2015,58(9):3317-3334.
[12]
Fokkema J T,van den Berg P M.Seismic Applica-tions of Acoustic Reciprocity[M].Elsevier,Amsterdam,1993.
[13]
Lam C,Bakrac S,van den Berg P M,et al.On the background model for non-linear inversion of seismic data[C].SEG Technical Program Expanded Abstracts,2006,25:2012-2016.
[14]
Yang J,Abubakar A,van den Berg P M,et al.A CG-FFT approach to the solution of a stress-velocity formulation of three-dimensional elastic scattering problems[J].Journal of Computational Physics,2008,227(24):10018-10039.
[15]
Haffinger P,Gisolf A,van den Berg P M.Towards high resolution quantitative subsurface models by full waveform inversion[J].Geophysical Journal International,2013,193(2):788-797.
[16]
Abubakar A,van den Berg P M.Iterative forward and inverse algorithms based on domain integral equations for three-dimensional electric and magnetic objects[J].Journal of Computational Physics,2004,195(1):236-262.
[17]
李杰,缪竟鸿.对比源反演算法在二维弹性波成像中的应用[J].系统工程与电子技术,2012,34(8):1560-1564.LI Jie,MIAO Jinghong.Application of the contrast source inversion algorithm on 2-D elastic wave imaging[J].Systems Engineering and Electronics,2012,34(8):1560-1564.
[18]
Kleinman R E,van den Berg P M.Iterative methods for solving integral equations[J].Radio Science,1991,26(1):175-181.
[19]
秦博.多维第二类Fredholm积分方程高精度数值算法研究[D].四川成都:电子科技大学,2015.QIN Bo.Research on High Precision Numerical Algorithm for Multidimensional Fredholm Integral Equation of the Second Kind[D].University of Electronic Science and Technology of China,Chengdu,Sichuan,2015.
[20]
Fokkema J T,Van Den Berg P M.Stretched back-grounds for acoustic scattering models[J].Journal of Computational Physics,2012,231(4):1728-1742.
[21]
陆晓.一维Lippmann-Schwinger方程及其应用[J].大学物理,2001,20(2):14-16.LU Xiao.One-dimensional Lippmann-Schwinger equation and its applications[J].College Physics,2001,20(2):14-16.
[22]
张明.应用数值分析[M].北京:石油工业出版社,2012.ZHANG Ming.Applied Numerical Analysis[M].Petroleum Industry Press,Beijing,2012.
[23]
Saad Y.Iterative Methods for Sparse Linear Systems[M].SIAM,Philadelphia,2003.
[24]
杨庆节,刘财,耿美霞,等.交错网格任意阶导数有限差分格式及差分系数推导[J].吉林大学学报(地球科学版),2014,44(1):375-385.YANG Qingjie,LIU Cai,GENG Meixia,et al.Staggered grid finite difference scheme and coefficients deduction of any number of derivatives[J].Journal of Jilin University(Earth Science Edition),2014,44(1):375-385.
[25]
Backus G E.Long-wave elastic anisotropy produced by horizontal layering[J].Journal of Geophysical Research,1962,67(11):4427-4440.
[26]
曹丹平.基于Backus等效平均的测井资料尺度粗化方法研究[J].石油物探,2015,54(1):105-111.CAO Danping.The upscaling method of the well logging data based on Backus equivalence average me-thod[J].Geophysical Prospecting for Petroleum,2015,54(1):105-111.
