基于分数阶时间导数的黏弹性衰减VTI介质中平面波传播

程时俊, 毛伟建, 欧阳威. 2021. 基于分数阶时间导数的黏弹性衰减VTI介质中平面波传播. 地球物理学报, 64(3): 965-981, doi: 10.6038/cjg2021O0231
引用本文: 程时俊, 毛伟建, 欧阳威. 2021. 基于分数阶时间导数的黏弹性衰减VTI介质中平面波传播. 地球物理学报, 64(3): 965-981, doi: 10.6038/cjg2021O0231
CHENG ShiJun, MAO WeiJian, OUYANG Wei. 2021. Plane wave propagation in viscoelastic attenuative VTI media based on fractional time derivatives. Chinese Journal of Geophysics (in Chinese), 64(3): 965-981, doi: 10.6038/cjg2021O0231
Citation: CHENG ShiJun, MAO WeiJian, OUYANG Wei. 2021. Plane wave propagation in viscoelastic attenuative VTI media based on fractional time derivatives. Chinese Journal of Geophysics (in Chinese), 64(3): 965-981, doi: 10.6038/cjg2021O0231

基于分数阶时间导数的黏弹性衰减VTI介质中平面波传播

  • 基金项目:

    国家重点研发计划(2016YFC0601101),国家自然科学基金(41704143),国家科技重大专项项目《新一代地球物理油气勘探软件系统》(2017ZX05018-001)和中国工程科技发展战略湖北研究院咨询研究项目联合资助

详细信息
    作者简介:

    程时俊, 男, 1990年生, 在读博士, 主要从事黏弹性各向异性介质的正演方法研究.E-mail: chengshijun18@whigg.ac.cn

    通讯作者: 毛伟建, 男, 研究员, 博士生导师, 中国科学院精密测量科学与技术创新研究院, 主要从事地震数据处理, 成像和反演研究.E-mail: wjmao@whigg.ac.cn
  • 中图分类号: P631

Plane wave propagation in viscoelastic attenuative VTI media based on fractional time derivatives

More Information
  • 目前在地震勘探频带范围内通常假设品质因子Q与频率无关,且呈衰减各向同性.事实上,相比较速度各向异性,介质的衰减各向异性同样不可忽视.本文将衰减各向异性和速度各向异性二者与常Q模型相结合,建立了黏弹性衰减VTI介质模型,并基于分数阶时间导数理论,给出了对应的本构关系和波动方程.利用均匀平面波分析和Poynting定理,推导出准压缩波qP、准剪切波qSV和纯剪切波SH的复速度、相速度、能量速度以及品质因子的解析表达式.对模型的正确性进行了数值验证,并分析了qP,qSV和SH波在介质中的传播特性.数值试验结果表明:本模型能够实现理想的恒定Q行为,表现了品质因子和速度的各向异性特征,显示出黏弹性增强将导致能量速度和相速度的频散曲线变化剧烈;速度和衰减各向异性参数与传播角度之间的耦合效应对qP,qSV和SH波的速度和能量影响明显;qP,qSV和SH波的频散曲线和波前面随着衰减各向异性强度的改变发生显著变化,其中耦合在一起的qP和qSV波变化趋势相同,而SH波与它们呈现相反的变化规律.本研究为从常Q模型角度分析地震波在衰减各向异性黏弹性介质中的传播特征奠定了理论基础.

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  • 图 1 

    qP, qSV和SH波品质因子关于频率的变化, 介质属性在表 1中定义

    Figure 1. 

    Quality factors of the qP, qSV, and SH waves as a function of the angular frequency for the medium defined in Table 1

    图 2 

    能量速度和相速度关于频率的变化, 其中每个介质的品质因子在表 2中给出, 实线代表能量速度, 虚线代表相速度

    Figure 2. 

    The energy and phase velocities as a function of the angular frequency, where the quality factors of the four materials are given in Table 2, and the solid and dotted line represent the energy and phase velocities, respectively

    图 3 

    在35 Hz频率时qP、qSV和SH波的能量速度(a)和相速度(b)的极坐标表示, 介质属性如表 1中所示

    Figure 3. 

    Polar representations of the energy (a) and phase velocities (b) of the qP, qSV, and SH waves for a angular frequency 35 Hz. The material properties are shown in the Table 1

    图 4 

    qP, qSV和SH波品质因子的极坐标表示, 频率为35 Hz

    Figure 4. 

    Polar representations of the quality factors of the qP, qSV, and SH waves. The angular frequency is 35 Hz

    图 5 

    qP波的相速度(a)、能量速度(b)以及两个速度的差速度(c)关于参数εv和传播角度的变化

    Figure 5. 

    The phase velocity (a), energy velocity (b), and the difference of the two velocities (c) of the qP wave versus the parameter εv and propagation angle

    图 6 

    qSV波的相速度(a)、能量速度(b)以及两个速度的差速度(c)关于参数εv和传播角度的变化

    Figure 6. 

    The phase velocity (a), energy velocity (b), and the difference of the two velocities (c) of the qSV wave versus the parameter εv and propagation angle

    图 7 

    qP波的相速度(a)、能量速度(b)以及两个速度的差速度(c)关于参数δv和传播角度的变化

    Figure 7. 

    The phase velocity (a), energy velocity (b), and the difference of the two velocities (c) of the qP wave versus the parameter δv and propagation angle

    图 8 

    qSV波的相速度(a)、能量速度(b)以及两个速度的差速度(c)关于参数δv和传播角度的变化

    Figure 8. 

    The phase velocity (a), energy velocity (b), and the difference of the two velocities (c) of the qSV wave versus the parameter δv and propagation angle

    图 9 

    SH波的相速度(a)、能量速度(b)以及两个速度的差速度(c)关于参数γv和传播角度的变化

    Figure 9. 

    The phase velocity (a), energy velocity (b), and the difference of the two velocities (c) of the SH wave versus the parameter γv and propagation angle

    图 10 

    在三个不同衰减各向异性强度介质中关于频率变化的能量速度曲线

    Figure 10. 

    The energy velocities curves in the three material as a function of the frequency, where these materials have the different levels of the attenuation anisotropy

    图 11 

    在三个不同衰减各向异性强度介质中能量速度曲线的极坐标表示

    Figure 11. 

    Polar representations for the energy velocities curves in the three material at 5 Hz, where these materials have the different levels of the attenuation anisotropy

    图 12 

    品质因子关于参数εQ和传播角度的变化

    Figure 12. 

    The quality factors versus the parameter εQ and propagation angle

    图 13 

    品质因子关于参数δQ和传播角度的变化

    Figure 13. 

    The quality factors versus the parameter δQ and propagation angle

    图 14 

    SH波的品质因子关于参数γQ和传播角度的变化

    Figure 14. 

    The quality factor of the SH wave versus the parameter γQ and propagation angle

    表 1 

    黏弹性衰减VTI介质属性

    Table 1. 

    Material properties of the viscoelastic attenuative VTI media

    εQ γQ δQ Q33 Q55 εv γv δv VP(m·s-1) VS(m·s-1)
    -0.17 0.15 0.16 45 35 0.58 0.21 0.15 3000 1600
    下载: 导出CSV

    表 2 

    四个介质的品质因子

    Table 2. 

    Quality factors of the four materials

    介质1 介质2 介质3 介质4
    Q33 Q55 Q33 Q55 Q33 Q55 Q33 Q55
    15 5 45 35 200 150 1015 1015
    下载: 导出CSV

    表 3 

    三个介质的衰减各向异性参数

    Table 3. 

    Attenuation anisotropic parameters of the three materials

    介质1 介质2 介质3
    εQ δQ γQ εQ δQ γQ εQ δQ γQ
    -0.47 0.40 0.35 -0.17 0.15 0.16 0 0 0
    下载: 导出CSV
  •  

    Ba J, Lu M H, Hu B, et al. 2008. The skeleton-relaxed model for poroviscoelastic media. Chinese Journal of Geophysics (in Chinese), 51(5): 1527-1537, doi:10.3321/j.issn:0001-5733.2008.05.027.

     

    Blangy J P. 1994. AVO in transversely isotropic media-An overview. Geophysics, 59(5): 775-781. doi: 10.1190/1.1443635

     

    Caputo M. 1967. Linear models of dissipation whose Q is almost frequency independent-Ⅱ. Geophysical Journal of the Royal Astronomical Society, 13(5): 529-539. doi: 10.1111/j.1365-246X.1967.tb02303.x

     

    Carcione J M. 1990. Wave propagation in anisotropic linear viscoelastic media: theory and simulated wavefields. Geophysical Journal International, 101(3): 739-750. doi: 10.1111/j.1365-246X.1990.tb05580.x

     

    Carcione J M. 1992. Anisotropic Q and velocity dispersion of finely layered media. Geophysical Prospecting, 40(7): 761-783. doi: 10.1111/j.1365-2478.1992.tb00551.x

     

    Carcione J M. 1994. Wavefronts in dissipative anisotropic media. Geophysics, 59(4): 644-657. doi: 10.1190/1.1443624

     

    Carcione J M, Cavallini F, Mainardi F, et al. 2002. Time-domain modeling of constant-Q seismic waves using fractional derivatives.//Seismic Waves in Laterally Inhomogeneous Media. Birkhäuser, Basel: Springer, 1719-1736.

     

    Carcione J M. 2009. Theory and modeling of constant-Q P-and S-waves using fractional time derivatives. Geophysics, 74(1): T1-T11. doi: 10.1190/1.3008548

     

    Chen H M. 2017. Study on numerical simulation of wave equations and viscoacoustic full waveform inversion[Ph. D. thesis] (in Chinese). Beijing: China University of Petroleum (Beijing).

     

    Chen W, Holm S. 2004. Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency. The Journal of the Acoustical Society of America, 115(4): 1424-1430. doi: 10.1121/1.1646399

     

    Crampin S, Chesnokov E M, Hipkin R G. 1984. Seismic anisotropy-the state of the art: Ⅱ. Geophysical Journal International, 76(1): 1-16. doi: 10.1111/j.1365-246X.1984.tb05017.x

     

    Hestholm S. 1999. Three-dimensional finite difference viscoelastic wave modelling including surface topography. Geophysical Journal International, 139(3): 852-878. doi: 10.1046/j.1365-246x.1999.00994.x

     

    Kjartansson E. 1979. Constant Q-wave propagation and attenuation. Journal of Geophysical Research: Solid Earth, 84(B9): 4737-4748. doi: 10.1029/JB084iB09p04737

     

    Li Q Q, Zhou H, Zhang Q C, et al. 2016. Efficient reverse time migration based on fractional Laplacian viscoacoustic wave equation. Geophysical Journal International, 204(1): 488-504. doi: 10.1093/gji/ggv456

     

    Li Q Q, Fu L Y, Zhou H, et al. 2019. Effective Q-compensated reverse time migration using new decoupled fractional Laplacian viscoacoustic wave equation. Geophysics, 84(2): S57-S69. doi: 10.1190/geo2017-0748.1

     

    McDonal F J, Angona F A, Mills R L, et al. 1958. Attenuation of shear and compressional waves in Pierre shale. Geophysics, 23(3): 421-439. doi: 10.1190/1.1438489

     

    Picotti S, Carcione J M, Santos J E, et al. 2010. Q-anisotropy in finely-layered media. Geophysical Research Letters, 37(6): L06302, doi:10.1029/2009GL042046.

     

    Picotti S, Carcione J M, Santos J E. 2012. Oscillatory numerical experiments in finely layered anisotropic viscoelastic media. Computers & Geosciences, 43: 83-89.

     

    Prasad M, Nur A. 2003. Velocity and attenuation anisotropy in reservoir rocks.//73rd Ann. Internat Mtg., Soc. Expi. Geophys.. Expanded Abstracts, 1652-1655.

     

    Qiao Z H, Sun C Y, Wu D S. 2019. Theory and modelling of constant-Q viscoelastic anisotropic media using fractional derivative. Geophysical Journal International, 217(2): 798-815. doi: 10.1093/gji/ggz050

     

    Robertsson J O A, Blanch J O, Symes W W. 1994. Viscoelastic finite-difference modeling. Geophysics, 59(9): 1444-1456. doi: 10.1190/1.1443701

     

    Shi X C, Mao W J, Li X L. 2019. Viscoelastic Q-compensated Gaussian beam migration based on vector-wave imaging. Chinese Journal of Geophysics (in Chinese), 62(4): 1480-1491, doi:10.6038/cjg2019L0797.

     

    Thomsen L. 1986. Weak elastic anisotropy. Geophysics, 51(10): 1954-1966. doi: 10.1190/1.1442051

     

    Tsvankin I. 1997. Anisotropic parameters and P-wave velocity for orthorhombic media. Geophysics, 62(4): 1292-1309. doi: 10.1190/1.1444231

     

    Wang N, Zhou H, Chen H M, et al. 2018. A constant fractional-order viscoelastic wave equation and its numerical simulation scheme. Geophysics, 83(1): T39-T48. doi: 10.1190/geo2016-0609.1

     

    Wang N, Zhu T Y, Zhou H, et al. 2020. Fractional Laplacians viscoacoustic wavefield modeling with k-space-based time-stepping error compensating scheme. Geophysics, 85(1): T1-T13. doi: 10.1190/geo2019-0151.1

     

    Wu Y, Fu L Y, Chen G X. 2017. Forward modeling and reverse time migration of viscoacoustic media using decoupled fractional Laplacians. Chinese Journal of Geophysics (in Chinese), 60(4): 1527-1537, doi:10.6038/cjg20170425.

     

    Yan H Y, Liu Y. 2012. Rotated staggered grid high-order finite-difference numerical modeling for wave propagation in viscoelastic TTI media. Chinese Journal of Geophysics (in Chinese), 55(4): 1354-1365, doi:10.6038/j.issn.0001-5733.2012.04.031.

     

    Yang R H, Mao W J, Chang X. 2015. An efficient seismic modeling in viscoelastic isotropic media. Geophysics, 80(1): T63-T81. doi: 10.1190/geo2013-0439.1

     

    Zhao H B, Chen B J, Li K Z, et al. 2011. VSP record numerical modeling in viscoelastic media and its application in the study of Q-value estimation method. Chinese Journal of Geophysics (in Chinese), 54(2): 329-335, doi:10.3969/j.issn.0001-5733.2011.02.008.

     

    Zhu T Y, Carcione J M, Harris J M. 2013. Approximating constant-Q seismic propagation in the time domain. Geophysical Prospecting, 61(5): 931-940. doi: 10.1111/1365-2478.12044

     

    Zhu T Y, Harris J M. 2014. Modeling acoustic wave propagation in heterogeneous attenuating media using decoupled fractional Laplacians. Geophysics, 79(3): T105-T116. doi: 10.1190/geo2013-0245.1

     

    Zhu T Y, Carcione J M. 2014. Theory and modelling of constant-Q P-and S-waves using fractional spatial derivatives. Geophysical Journal International, 196(3): 1787-1795. doi: 10.1093/gji/ggt483

     

    Zhu Y P, Tsvankin I. 2006. Plane-wave propagation in attenuative transversely isotropic media. Geophysics, 71(2): T17-T30. doi: 10.1190/1.2187792

     

    Zhu Y P, Tsvankin I. 2007. Plane-wave attenuation anisotropy in orthorhombic media. Geophysics, 72(1): D9-D19. doi: 10.1190/1.2387137

     

    巴晶, 卢明辉, 胡彬等. 2008. 黏弹双相介质中的松弛骨架模型. 地球物理学报, 51(5): 1527-1537, doi:10.3321/j.issn:0001-5733.2008.05.027. http://www.geophy.cn//CN/abstract/abstract1347.shtml

     

    陈汉明. 2017. 波动方程数值模拟与粘滞波形反演方法研究[博士论文]. 北京: 中国石油大学(北京).

     

    石星辰, 毛伟建, 栗学磊. 2019. 矢量黏弹性衰减补偿高斯束偏移. 地球物理学报, 62(4): 1480-1491, doi:10.6038/cjg2019L0797. http://www.geophy.cn//CN/abstract/abstract14953.shtml

     

    吴玉, 符力耘, 陈高祥. 2017. 基于分数阶拉普拉斯算子解耦的黏声介质地震正演模拟与逆时偏移. 地球物理学报, 60(4): 1527-1537, doi:10.6038/cjg20170425. http://www.geophy.cn//CN/abstract/abstract13619.shtml

     

    严红勇, 刘洋. 2012. 黏弹TTI介质中旋转交错网格高阶有限差分数值模拟. 地球物理学报, 55(4): 1354-1365, doi:10.6038/j.issn.0001-5733.2012.04.031. http://www.geophy.cn//CN/abstract/abstract8630.shtml

     

    赵海波, 陈百军, 李奎周等. 2011. 黏弹性介质VSP记录模拟及在估算Q值研究中应用. 地球物理学报, 54(2): 329-335, doi:10.3969/j.issn.0001-5733.2011.02.008. http://www.geophy.cn//CN/abstract/abstract7748.shtml

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出版历程
收稿日期:  2020-06-17
修回日期:  2020-11-04
上线日期:  2021-03-10

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