Analysis of quantitative relations between different exact adjoint operator pairs in acoustic least-square migration
WANG Jiansen1, REN Yuxiao2, CHEN Lei2, YAN Dong3, YANG Chuangen3, XU Xinji2
1. School of Qilu Transportation, Shandong University, Jinan, Shandong 250061, China; 2. Geotechnical and Structural Engineering Research Center, Shandong University, Jinan, Shandong 250061, China; 3. Huaneng Tibet Hydropower Safety Engineering Technology Research Center, Linzhi, Tibet 860000, China
Abstract：Least-square migration (LSM) is frequently mentioned in high-resolution imaging, whose successful application depends on the adjoint characteristic of forward-migration operator pairs. Normally, a forward-migration operator pair can be designed according to the Born approximation theory or/and reverse time migration (RTM) process. Its adjoint characteristic can be affected by the discretization and numerical implementation methods of wave equations, and the dot-product test can be used for the numerical test of this characteristic. However, the relations and the diffe-rences between the imaging results of different adjoint operator pairs are not clear. Considering this, three pairs of exact adjoint operators are derived by starting from the matrix expression of the second-order acoustic wave equation, two of which are constructed only on the basis of the Born approximation theory and the RTM process separately, and the third one is based on the self-adjoint discretization of the acoustic wave equation. They are named as Born-AdjBorn, DeRTM-RTM, and self-adjoint Born-RTM operator pairs, respectively, and the corresponding LSM processes are called LSBM, LSRTM, and self-adjoint LSBRTM, respectively. The matrix analysis and mathematical derivation indicate that a series of quantitative relations exist between the imaging results using the three operator pairs, and all these quantitative relations are validated by numerical experiments.
TARANTOLA A. Inversion of seismic reflection data in the acoustic approximation[J]. Geophysics, 1984, 49(8):1259-1266.
[2]
SCHUSTER G T. Least-squares cross-well migration[C]. SEG Technical Program Expanded Abstracts, 1993, 12:110-113.
[3]
NEMETH T, WU C, SCHUSTER G T. Least-squares migration of incomplete reflection data[J]. Geophysics, 1999, 64(1):208-221.
[4]
CLAPP M L, CLAPP R G, BIONDI B L. Regularized least-squares inversion for 3-D subsalt imaging[C]. SEG Technical Program Expanded Abstracts, 2005, 24:1814-1817.
[5]
WANG J, KUEHL H, SACCHI M D. High-resolution wave-equation AVA imaging:Algorithm and tests with a data set from the Western Canadian Sedimentary Basin[J]. Geophysics, 2005, 70(5):S91-S99.
[6]
DONG S, CAI J, GUO M, et al. Least-squares reverse time migration:Towards true amplitude imaging and improving the resolution[C]. SEG Technical Program Expanded Abstracts, 2012, 31:1-5.
[7]
DUTTA G, SCHUSTER G T. Attenuation compensation for least-squares reverse time migration using the viscoacoustic-wave equation[J]. Geophysics, 2014, 79(6):S251-S262.
[8]
ZENG C, DONG S, WANG B. Least-squares reverse time migration:Inversion-based imaging toward true reflectivity[J]. The Leading Edge, 2014, 33(9):962-964, 966, 968.
[9]
GUO P, MCMECHAN G. Adjoint-based least-squares reverse time migration for viscoelastic media[C]. SEG Technical Program Expanded Abstracts, 2017, 36:4358-4363.
[10]
李庆洋, 黄建平, 李振春, 等. 优化的多震源最小二乘逆时偏移[J]. 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[J]. Oil Geophysical Prospecting, 2016, 51(2):334-341.
[11]
段心标, 王华忠, 邓光校. 应用反射理论的最小二乘逆时偏移成像[J]. 石油地球物理勘探, 2020, 55(6):1305-1311.DUAN Xinbiao, WANG Huazhong, DENG Guang-xiao. Least-squares reverse-time migration based on reflection theory[J]. Oil Geophysical Prospecting, 2020, 55(6):1305-1311.
[12]
BAYSAL E, KOSLOFF D D, SHERWOOD J W C. Reverse time migration[J]. Geophysics, 1983, 48(11):1514-1524.
[13]
Whitmore N D. Iterative depth migration by backward time propagation[C]. SEG Technical Program Expanded Abstracts, 1983, 2:382-385.
[14]
LI S C, REN Y X, LIU L B, et al. Reverse time migration of seismic forward-prospecting data in tunnels based on beamforming methods[J]. Rock Mechanics and Rock Engineering, 2019, 52(9):3261-3278.
[15]
FENG Z G, SCHUSTER G T. Elastic least-squares reverse time migration[J]. Geophysics, 2017, 82(2):S143-S157.
[16]
方修政, 钮凤林, 吴迪. 基于逆散射成像条件的最小二乘逆时偏移[J]. 地球物理学报, 2018, 61(9):3770-3782.FANG Xiuzheng, NIU Fenglin, WU Di. Least-squares reverse-time migration enhanced with the inverse scattering imaging condition[J]. Chinese Journal of Geophysics, 2018, 61(9):3770-3782.
[17]
黄建平, 曹晓莉, 李振春, 等. 最小二乘逆时偏移在近地表高精度成像中的应用[J]. 石油地球物理勘探, 2014, 49(1):107-112.HUANG Jianping, CAO Xiaoli, LI Zhenchun, et al. Least square reverse time migration in high resolution imaging of near surface[J]. Oil Geophysical Prospecting, 2014, 49(1):107-112.
[18]
TARANTOLA A. Linearized inversion of seismic reflection data[J]. Geophysical Prospecting, 1984, 32(6):998-1015.
[19]
CLAERBOUT J F, ABMA R. Earth Soundings Ana-lysis:Processing versus Inversion[M]. Blackwell Scientific Publications, Oxford, 1991.
[20]
BEDNAR J. Conference on Inverse Scattering:Theory and Application[M]. Society for Industrial and Applied Mathematics, Philaelphia, 1983.
[21]
KEYS R G, WEGLEIN A B. Generalized linear inversion and the first Born theory for acoustic media[J]. Journal of Mathematical Physics, 1983, 24(6):1444-1449.
[22]
XU L N, SACCHI M D. Preconditioned acoustic least-squares two-way wave-equation migration with exact adjoint operator[J]. Geophysics, 2018, 83(1):S1-S13.
[23]
ZHANG Q, ZHOU H, CHEN H M, et al. Least-squares reverse time migration with and without source wavelet estimation[J]. Journal of Applied Geo-physics, 2016, 134:1-10.
[24]
SUN J Z, FDMEL S, HU J. Least-squares reverse-time migration using one-step two-way wave extrapolation by non-stationary phase shift[C]. SEG Technical Program Expanded Abstracts, 2014, 33:3967-3973.
[25]
HOU J, SYMES W W. An approximate inverse to the extended Born modeling operator[J]. Geophy-sics, 2015, 80(6):R331-R349.
[26]
BORZÌ A, SCHULZ V. Computational Optimization of Systems Governed by Partial Differential Equations[M]. Society for Industrial and Applied Mathema-tics, Philadelphia, 2011.
[27]
CHEN K, SACCHI M D. Elastic least-squares reverse time migration via linearized elastic full-waveform inversion with pseudo-Hessian preconditioning elastic LSRTM[J]. Geophysics, 2017, 82(5):S341-S358.
[28]
JI J. An exact adjoint operation pair in time extrapolation and its application in least-squares reverse-time migration[J]. Geophysics, 2009, 74(5):H27-H33.
[29]
YAO G, JAKUBOWICZ H. Least-squares reverse-time migration in a matrix-based formulation[J]. Geo-physical Prospecting, 2016, 64(3):611-621.
[30]
DAI W, SCHUSTER G T. Plane-wave least-squares reverse-time migration[J]. Geophysics, 2013, 78(4):S165-S177.
[31]
REN H R, WANG H Z, CHEN S C. Least-squares reverse time migration in frequency domain using the adjoint-state method[J]. Journal of Geophysics and Engineering, 2013, 10(3):035002.
[32]
REN Z M, LIU Y, SEN M K. Least-squares reverse time migration in elastic media[J]. Geophysical Journal International, 2016, 208(2):1103-1125.
[33]
李振春, 黄金强, 黄建平, 等. 基于平面波加速的VTI介质最小二乘逆时偏移[J]. 地球物理学报, 2017, 60(1):240-257.LI Zhenchun, HUANG Jinqiang, HUANG Jianping, et al.Fast least-squares reverse time migration based on plane-wave encoding for VTI media[J].Chinese Journal of Geophysics, 2017, 60(1):240-257.
[34]
郭旭, 黄建平, 李振春, 等. 基于一阶速度-应力方程的VTI介质最小二乘逆时偏移[J]. 地球物理学报, 2019, 62(6):2188-2202.GUO Xu, HUANG Jinqiang, LI Zhenchun, et al. Least-squares reverse time migration based on first-order velocity-stress wave equation in VTI media[J]. Chinese Journal of Geophysics, 2019, 62(6):2188-2202.
[35]
陈汉明, 周辉, 田玉昆. 分数阶拉普拉斯算子黏滞声波方程的最小二乘逆时偏移[J]. 石油地球物理勘探, 2020, 55(3):616-626.CHEN Hanming, ZHOU Hui, TIAN Yukun. Least-squares reverse-time migration based on a fractional Laplacian viscoacoustic wave equation[J]. Oil Geophysical Prospecting, 2020, 55(3):616-626.
[36]
XUE Z G, CHEN Y, FOMEL S, et al. Seismic imaging of incomplete data and simultaneous-source data using least-squares reverse time migration with shaping regularization[J]. Geophysics, 2016, 81(1):S11-S20.
[37]
ZHANG D L, SCHUSTER G T. Least-squares reverse time migration of multiples[J]. Geophysics, 2014, 79(1):S11-S21.
[38]
TAN S R, HUANG L J. Least-squares reverse-time migration with a wavefield-separation imaging condition and updated source wavefields[J]. Geophysics, 2014, 79(5):S195-S205.
[39]
LIU Y S, TENG J W, XU T, et al. An efficient step-length formula for correlative least-squares reverse time migration[J]. Geophysics, 2016, 81(4):S221-S238.
[40]
ZHANG Y, DUAN L, XIE Y. A stable and practical implementation of least-squares reverse time migration[J]. Geophysics, 2015, 80(1):V23-V31.
[41]
ZENG C, DONG S Q, WANG B. A guide to least-squares reverse time migration for subsalt imaging:Challenges and solutions[J]. Interpretation, 2017, 5(3):SN1-SN11.
