Abstract:The decoupling of the wave equation refers to the decoupling of an elastic wave equation into wave equations that can describe the independent propagation of various wave patterns, which plays an important role in numerical simulation of seismic waves, seismic migration, and multi-component seismology. In most anisotropic media, the qP wave and qS wave are generally coupled for propagation without exact decoupling nature, but it is found that the ellipsoidal anisotropic (EA) media are an exception. First, on the basis of the exact dispersion relation equation of the elastic wave in homogeneous EA media, three decoupled dispersion equations are decomposed from this equation by the factorization method and then transformed into the completely and exactly decoupled wave equations of the qP wave, qSV wave, and SH wave in homogeneous EA media by inverse Fourier transform. The theoretical formulas and numerical examples indicate that the qP wave, qSV wave, and SH wave can be completely and exactly decoupled and propagate independently in homogeneous EA media by decoupled wave equations. The wavefront of the qP wave and SH wave is ellipsoidal, and the wavefront of the qSV wave is spherical, independent of anisotropic parameters. The three complete decoupled wave equations are suitable not only for weak anisotropic EA media but also for strong anisotropic EA media.
Alkhalifah T. An acoustic wave equation for anisotropic media[J]. Geophysics, 2000, 65(4):1239-1250.
[3]
Alkhalifah T. An acoustic wave equation for orthor-hombic anisotropy[J]. Geophysics, 2003, 68(4):1169-1172.
[4]
Grechka V, Zhang L, Rector J W III. Shear waves in acoustic anisotropic media[J]. Geophysics, 2004, 69(2):576-582.
[5]
Zhou H, Zhang G, Bloor R. An anisotropic acoustic wave equation for modeling and migration in 2D TTI media[C]. SEG Technical Program Expanded Abstracts, 2006, 25:194-198.
Fowler P J, Du X, Fletcher R P. Coupled equations for reverse time migration in transversely isotropic media[J]. Geophysics, 2010, 75(1):S11-S22.
[8]
Duveneck E, Bakker P M. Stable P-wave modeling for reverse time migration in tilted TI media[J]. Geo-physics, 2011, 76(2):S65-S75.
[9]
Zhang Y, Zhang H, Zhang G. A stable TTI reverse time migration and its implementation[J]. Geophy-sics, 2011, 76(3):WA3-WA11.
[10]
Cheng J B, Kang W. Simulating propagation of separated wave modes in general anisotropic media, Part I:qP-wave propagators[J]. Geophysics, 2014, 79(1):C1-C18.
[11]
Cheng J B, Kang W. Simulating propagation of separated wave modes in general anisotropic media, Part II:qS-wave propagators[J]. Geophysics, 2016, 81(2):C39-C52.
[12]
郭成锋, 杜启振, 张明强, 等. 改进的TTI介质纯P波方程正演模拟与逆时偏移[J]. 地球物理学报, 2017, 60(1):258-270.GUO Chengfeng, DU Qizhen, ZHANG Mingqiang, et al. Numerical simulation and reverse time migration using an improved pure P-wave equation in tilted transversely isotropic media[J]. Chinese Journal of Geophysics, 2017, 60(1):258-270.
[13]
慕鑫茹, 黄建平, 李振春, 等. 基于最佳平方逼近的TTI介质解耦qP波与qSV波逆时偏移[J]. 石油地球物理勘探, 2019, 54(6):1280-1292.MU Xinru, HUANG Jianping, LI Zhenchun, et al. Reverse time migration of decoupled qP- and qSV-waves in TTI media with the optimal quadratic approximation[J]. Oil Geophysical Prospecting, 2019, 54(6):1280-1292.
[14]
谷一鹏, 印兴耀, 梁锴, 等. VTI介质弹性波相速度扩展各向异性线性近似[J]. 石油地球物理勘探, 2020, 55(3):635-642.GU Yipeng, YIN Xingyao, LIANG Kai, et al. Exten-ded anisotropic linear approximation for elastic phase velocity in VTI media[J]. Oil Geophysical Prospecting, 2020, 55(3):635-642.
[15]
孙上饶, 梁锴, 印兴耀, 等. 三维TTI介质弹性波相、群速度的近似配方表征法[J]. 石油地球物理勘探, 2021, 56(3):496-504, 518.SUN Shangrao, LIANG Kai, YIN Xingyao, et al. Approximate 3D phase and group velocities for elastic wave in TTI media based on an approximate match method[J]. Oil Geophysical Prospecting, 2021, 56(3):496-504, 518.
[16]
单俊臻, 吴国忱, 龚诚诚. HTI介质方位观测PP波反射系数一阶扰动近似[J]. 石油地球物理勘探, 2019, 54(2):371-379.SHAN Junzhen, WU Guochen, GONG Chengcheng. First-order perturbation approximation of PP wave reflection coefficient for HTI medium based on azimuthal geometry[J]. Oil Geophysical Prospecting, 2019, 54(2):371-379.
何兵寿, 武雪峤, 高琨鹏. TI介质中qP波方程逆时偏移技术的研究现状与展望[J]. 石油物探, 2021, 60(1):34-45.HE Bingshou, WU Xueqiao, GAO Kunpeng. Research status and prospect of qP wave reverse time migration in TI media[J]. Geophysical Prospecting for Petroleum, 2021, 60(1):34-45.
[19]
何兵寿, 高琨鹏, 徐国浩. 各向异性介质中弹性波逆时偏移技术的研究现状与展望[J]. 石油物探, 2021, 60(2):210-223.HE Bingshou, GAO Kunpeng, XU Guohao. Elastic wave reverse time migration in anisotropic media:State of the art and perspectives[J]. Geophysical Prospecting for Petroleum, 2021, 60(2):210-223.
[20]
Bakulin A, Grechka V, Tsvankin L. Estimation of fracture parameters from reflection seismic data, Part 1:HTI model due to a single fracture set[J]. Geophy-sics, 2000, 65(6):1788-1802.
[21]
Byun B. Seismic parameters for media with elliptical velocity dependencies[J]. Geophysics, 1982, 47(12):1621-1626.
[22]
Helbig K. Elliptical anisotropy:Its significance and meaning[J]. Geophysics, 1983, 48(7):825-832.
[23]
Thomsen L. Weak elastic anisotropy[J]. Geophy-sics, 1986, 51(10):1954-1966.
[24]
李磊, 郝重涛. 横向各向同性介质和斜方介质各向异性参数的约束条件[J]. 地球物理学报, 2011, 54(11):2819-2830.LI Lei, HAO Zhongtao. Constraints on anisotropic parameters in transversely isotropic media and the extensions to orthorhombic media[J]. Chinese Journal of Geophysics, 2011, 54(11):2819-2830.
[25]
梁锴, 曹丹平, 印兴耀, 等. 倾斜椭球各向异性介质弹性波传播特征[J]. 石油地球物理勘探, 2018, 53(6):1142-1151.LIANG Kai, CAO Danping, YIN Xingyao, et al. Elastic wave propagation features in tilted ellipsoidal anisotropic media[J]. Oil Geophysical Prospecting, 2018, 53(6):1142-1151.