Numerical simulation of 2D visco-acoustic wave equation with an optimized combined compact difference scheme
Wang Yong1,2, Xu Youde3, Gao Gang1,2, Gui Zhixian1,2, Chen Ying4, Wang Yanan5
1. Key Laboratory of Exploration Technologies for Oil and Gas Resources, Ministry of Education, Yangtze University, Wuhan, Hubei 430100, China; 2. College of Geophysics and Petroleum Resources, Yangtze University, Wuhan, Hubei 430100, China; 3. West Branch, Research Institute of Exploration and Development, Shengli Oilfield Branch Co., SINOPEC, Dongying, Shandong 257001, China; 4. Surveying Service Center, Equipment Service Department, BGP Inc., CNPC, Zhuozhou, Hebei 072751, China; 5. GRI, BGP Inc., CNPC, Zhuozhou, Hebei 072751, China
Abstract：The numerical simulation of seismic wavefield has great significance in geophysical exploration and seismology.In this paper,a combined compact difference scheme is proposed for the numerical simulation of visco-acoustic wave equation.First the second-order discrete scheme of the displacement field is established according to the Taylor series expansion and visco-acoustic wave equation,and a combined compact difference scheme is used to calculate the spatial derivative of the displacement field.Then the accuracy,dispersion and stability of the difference scheme are analyzed.Finally the optimization of the combination compact difference scheme is discussed based on the idea of dispersion-relation-preserving.The following understandings are obtained from our theoretical research:A.3-point 6-order combined compact difference scheme has smaller truncation errors and lower simulation dispersion than the conventional 7-point 6-order center difference and 5-point 6-order compact difference; B.The dispersion relation and the stability of the visco-acoustic wave equation difference scheme are not only related to the space grid size and time step,but also to the media quality factor and the seismic dominant frequency; C.The numerical wavenumber of the optimized combination compact difference scheme is more close to the true wavenumber than that before optimization,which is more beneficial to suppress the numerical dispersion and improve the calculation efficiency.The numerical simulation and wavefield of visco-acoustic wave equation are analyzed on uniform and Marmousi models based on the perfectly matched layer (PML) boundary condition.Numerical simulation results show that the proposed scheme has higher accuracy and computational efficiency in complex media.
Alterman Z,Karal F C.Propagation of seismic wave in layered media by finite difference methods.Bulletin of the Seismological Society of America,1968,58(1):367-398.
[2]
Graves R W.Simulating seismic wave propagation in 3D elastic media using staggered-grid finite diffe-rences.Bulletin of the Seismological Society of America,1996,86(4):1091-1106.
[3]
董良国,马在田,曹景忠等.一阶弹性波方程交错网格高阶差分解法.地球物理学报,2000,43(3):411-419.Dong Liangguo,Ma Zaitian,Cao Jingzhong et al.The staggered-grid high-order difference method of first-order elastic equation.Chinese Journal of Geophysics,2000,43(3):411-419.
[4]
Saenger E H,Gold N and Shapiro S A.Modeling the propagation of elastic waves using a modified finite-difference grid.Wave Motion,2000,31(1):77-92.
[5]
Saenger E H,Bohlen T.Finite-difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid.Geophysics,2004,69(2):583-591.
[6]
唐佳,王凡,刘福烈.三维波动方程正演的三级并行加速.石油地球物理勘探,2016,51(5):1049-1054.Tang Jia,Wang Fang,Liu Fulie.3D wave equation forward modeling based on three-level parallel acce-leration.OGP,2016,51(5):1049-1054.
[7]
唐怀谷,何兵寿.一阶声波方程时间四阶精度差分格式的伪谱法求解.石油地球物理勘探,2017,52(1):71-80.Tang Huaigu,He Bingshou.Pseudo spectrum method of first-order acoustic wave equation finite-difference schemes with fourth-order time difference accuracy.OGP,2017,52(1):71-80.
[8]
董良国,马在田,曹景忠.一阶弹性波方程交错网格高阶差分解法稳定性研究.地球物理学报,2000,43(6):856-864.Dong Liangguo,Ma Zaitian,Cao Jingzhong.A study on stability of the staggered-grid high-order difference method of first-order elastic wave equation.Chinese Journal of Geophysics,2000,43(6):856-864.
[9]
杜启振,郭成锋,公绪飞.VTI介质纯P波混合法正演模拟及稳定性分析.地球物理学报,2015,58(4):1290-1304.Du Qizhen,Guo Chengfeng,Gong Xufei.Hybrid PS/FD numerical simulation and stability analysis of pure P-wave propagation in VTI media.Chinese Journal of Geophysics,2015,58(4):1290-1304.
[10]
汪勇,段焱文,王婷等.优化近似解析离散化方法的二维弹性波波场分离模拟.石油地球物理勘探,2017,52(3):458-467.Wang Yong,Duan Yanwen,Wang Ting et al.Numerical simulation of elastic wave separation in 2D isotropic medium with the optimal nearly-analytic discretization.OGP,2017,52(3):458-467.
[11]
王璐琛,常旭,王一博.TTI介质的交错网格伪P波正演方法.地球物理学报,2016,59(3):1046-1058.Wang Luchen,Chang Xu,Wang Yibo.Forward mode-ling of pseudo P waves in TTI medium using staggered grid.Chinese Journal of Geophysics,2016,59(3):1046-1058.
[12]
Pan G,Abubakar A and Habashy M T.An effective perfectly matched layer design for acoustic fourth-order frequency-domain finite-difference scheme.Geophysical Journal International,2011,188(1):211-222.
[13]
Liu S L,Yang D H,Lan C et al.Modified symplectic schemes with nearly-analytic discrete operators for acoustic wave simulations.Computer Physics Communications,2017,213:52-63.
[14]
Jing H,Chen Y S,Yang D H et al.Dispersion-relation preserving stereo-modeling method beyond Nyquist frequency for acoustic wave equation.Geophysics,2017,82(1):T1-T15.
[15]
汪勇,段焱文,安一凡等.扩展的近似解析离散化方法及弹性波方程数值模拟.石油地球物理勘探,2017,52(5):928-940,955.Wang Yong,Duan Yanwen,An Yifan et al.Expanded approximate analytic discretization and elastic wave numerical simulation.OGP,2017,52(5):928-940,955.
[16]
Dennis S C R,Hundson J D.Compact h4 finite-diffe-rence approximations to operators of Navier-Stokes type.Journal of Computational Physics,1989,85(2):390-416.
[17]
Lele S K.Compact finite difference scheme with spectral-like resolution.Journal of Computational Physics,1992,103(1):16-42.
[18]
Adams N A,Shariff K.A high-resolution hybrid compact-ENO scheme for Shock-Turbulence interaction problems.Journal of Computational Physics,1996,127(1):27-51.
[19]
Chu P C,Fan C.A three point combined compact difference scheme.Journal of Computational Physics,1998,140(2):370-399.
[20]
Yang D H,Wang S Q,Zhang Z J et al.n-times ab-sorbing boundary conditions for compact finite difference modeling of acoustic and elastic wave propagation in the 2-D TI Medium.Bulletin of the Seismological Society of America,2003,93(6):2389-2401.
[21]
Du Q,Li B and Hou B.Numerical modeling of seismic wavefields in transversely isotropic media with a compact staggered-grid finite difference scheme.Applied Geophysics,2009,6(1):42-49.
[22]
Liu Y,Sen M K.A practical implicit finite-difference method:Examples from seismic modeling.Journal of Geophysics and Engineering,2009,6(3):231-249.
[23]
Liu Y,Sen M K.An implicit staggered-grid finite-difference method for seismic modeling.Geophysical Journal International,2009,179(1):459-474.
[24]
杨宽德,宋国杰,李静爽.Biot流动和喷射流动耦合作用下波传播的FCT紧致差分模拟.地球物理学报,2011,54(5):1348-1357.Yang Kuande,Song Guojie,Li Jingshuang.FCT compact difference simulation of wave propagation based on the Biot and squirt-flow coupling interaction.Chinese Journal of Geophysics,2011,54(5):1348-1357.
[25]
李晓波,董良国.斑状饱和介质的黏弹特性表征与地震波模拟.石油物探,2014,53(3):272-279.Li Xiaobo,Dong Liangguo.Viscoelastic representation of patchy saturation media and its seismic wave simulation.GPP,2014,53(3):272-279.
[26]
姚振岸,孙成禹,唐杰等.基于不同震源机制的黏弹各向异性微地震波场模拟.石油地球物理勘探,2017,52(1):63-70.Yao Zhen'an,Sun Chengyu,Tang Jie et al.Micro-seismic forward modeling in viscoelastic anisotropic media based on different focal mechanisms.OGP,2017,52(1):63-70.
[27]
汪勇,段焱文,王婷等.近似解析离散化方法的黏弹声波方程数值模拟及波场特征分析.石油物探,2017,56(3):362-372.Wang Yong,Duan Yanwen,Wang Ting et al.Numerical simulation and the wave field characteristics analysis of viscoelastic acoustic wave equation based on the nearly-analytic discrete method.GPP,2017,56(3):362-372.
[28]
马灵伟,顾汉明,李宗杰等.应用正演模拟分析近地表黏弹性对深层缝洞储层地震波衰减及成像的影响.石油地球物理勘探,2014,49(4):694-701.Ma Lingwei,Gu Hanming,Li Zongjie et al.Nearsurface visco-elastic medium influences on wave attenuation and imaging of deep fractured-vuggy reservoir with forward modeling.OGP,2014,49(4):694-701.
[29]
罗文山,陈汉明,王成祥等.时间域黏滞波动方程及其数值模拟新方法.石油地球物理勘探,2016,51(4):707-713.Luo Wenshan,Chen Hanming,Wang Chengxiang et al.A novel time-domain viscoacoustic wave equation and its numerical simulation.OGP,2016,51(4):707-713.
[30]
姚振岸,孙成禹,谢俊法等.黏弹TTI介质旋转交错网格微地震波场模拟.石油地球物理勘探,2017,52(02):253-263.Yao Zhen'an,Sun Chengyu,Xie Junfa et al.Micro-seismic forward modeling in visco-elastic TTI media based on rotated staggered grid finite-difference me-thod.OGP,2017,52(2):253-263.
[31]
孙成禹,肖云飞,印兴耀等.黏弹介质波动方程有限差分解的稳定性研究.地震学报,2010,32(2):147-156.Sun Chengyu,Xiao Yunfei,Yin Xingyao et al.Study on the stability of finite difference solution of visco-elastic wave equations.Acta Seismologica Sinica,2010,32(2):147-156.
[32]
Tam C K W,Webb J C.Dispersion-relation-preserving finite difference schemes for computational acoustics.Journal of Computational Physics,1993,107(2):262-281.
[33]
Liu Y.Globally optimal finite-difference schemes based on least squares.Geophysics,2013,78(4):T113-T132.
[34]
Yu C H,Wang D,He Z et al.An optimized dispersion-relation-preserving combined compact difference scheme to solve advection equations.Journal of Computational Physics,2015,300(1):92-115
[35]
王守东.声波方程完全匹配层吸收边界.石油地球物理勘探,2003,38(1):31-34.Wang Shoudong.Absorbing boundary condition for acoustic wave equation by perfectly matched layer.OGP,2003,38(1):31-34.
[36]
刘有山,刘少林,张美根等.一种改进的二阶弹性波动方程的最佳匹配层吸收边界条件.地球物理学进展,2012,27(5):2113-2122.Liu Youshan,Liu Shaolin,Zhang Meigen et al.An improved perfectly matched layer absorbing boundary condition for second order elastic wave equation.Progress in Geophysics,2012,27(5):2113-2122.
[37]
高刚,贺振华,黄德济等.完全匹配层人工边界条件中的衰减因子分析.石油物探,2011,50(5):430-433.Gao Gang,He Zhenhua,Huang Deji et al.Analysis on attenuation factor in the processing of artificial boundary conditions of PML.GPP,2011,50(5):430-433.
[38]
Zhang J,Yao Z.Optimized finite-difference operator for broadband seismic wave modeling.Geophysics,2013,78(1):A13-A18.
[39]
Zhang J,Yao Z.Optimized explicit finite-difference schemes for spatial derivatives using maximum norm.Journal of Computational Physics,2013,250(10):511-526.
