none 10.1190/geo2020-0747.1 Society of Exploration Geophysicists Society of Exploration Geophysicists 186 128783732 5981 2025121212002279300 10.1190 2025-12-12T17:00:30Z 2021-07-19T13:12:00Z 3 Geophysics 0016-8033 1942-2156 11 01 2021 86 6 Xingguo Huang Jilin University, College of Instrumentation and Electrical Engineering 1 , Changchun 130026, China. E-mail: xingguohuang@jlu.edu.cn ; longy@jlu.edu.cn (corresponding author); lin_jun@jlu.edu.cn . http://orcid.org/0000-0001-9719-6297 Stewart Greenhalgh Swiss Federal Institute of Technology (ETH) Zurich, Institute of Geophysics 2 , Department of Earth Sciences, Zurich 8092, Switzerland. E-mail: gstewart@retired.ethz.ch . Yun Long Jilin University, College of Instrumentation and Electrical Engineering 1 , Changchun 130026, China. E-mail: xingguohuang@jlu.edu.cn ; longy@jlu.edu.cn (corresponding author); lin_jun@jlu.edu.cn . http://orcid.org/0000-0001-8051-755X Jun Lin Jilin University, College of Instrumentation and Electrical Engineering 1 , Changchun 130026, China. E-mail: xingguohuang@jlu.edu.cn ; longy@jlu.edu.cn (corresponding author); lin_jun@jlu.edu.cn . Xu Liu King Fahd University of Petroleum and Minerals (KFUPM) 3 , College of Petroleum Engineering and Geosciences (CPG), Dhahran 30261, Saudi Arabia. E-mail: liu.xu@kfupm.edu.sa . http://orcid.org/0000-0003-2835-2844 ABSTRACT Many applications in seismology involve the modeling of seismic-wave traveltimes in anisotropic media. We have developed homotopy solutions of the acoustic eikonal equation for P-wave traveltimes in attenuating transversely isotropic media with a vertical symmetry axis. Instead of the commonly used perturbation theory, we use the homotopy analysis method to express the traveltimes by a Taylor series expansion over powers of an embedding parameter. For the derivation, we first perform homotopy analysis of the eikonal equation and derive the linearized ordinary differential equations for the coefficients of the Taylor series expansion. Then, we obtain the homotopy solutions for the traveltimes by solving the linearized ordinary differential equations. Results of our investigation with approximate formulas demonstrate that the analytical expressions are efficient methods for the computation of traveltimes from the eikonal equation. In addition, these formulas are also effective methods for benchmarking approximate numerical solutions in strongly attenuating anisotropic media. 09 23 2021 11 01 2021 07 19 2021 T543 T551 1 10.1190/crossmark-policy library.seg.org true 2020-10-04 2021-06-16 2021-06-22 2021-09-23 10.1190/geo2020-0747.1 https://pubs.geoscienceworld.org/geophysics/article/86/6/T543/607947/Homotopy-solutions-of-the-acoustic-eikonal https://pubs.geoscienceworld.org/seg/geophysics/article-pdf/86/6/T543/5459453/geo-2020-0747.1.pdf https://pubs.geoscienceworld.org/seg/geophysics/article-pdf/86/6/T543/5459453/geo-2020-0747.1.pdf https://pubs.geoscienceworld.org/geophysics/article/86/6/t543/607947/homotopy-solutions-of-the-acoustic-eikonal Geophysics Alkhalifah 63 623 1998 10.1190/1.1444361 Acoustic approximations for processing in transversely isotropic media Geophysics Alkhalifah 65 1239 2000 10.1190/1.1444815 An acoustic wave equation for anisotropic media Geophysics Alkhalifah 76 2 U13 2011 10.1190/1.3553015 Scanning anisotropy parameters in complex media Geophysics Alkhalifah 76 3 WA31 2011 10.1190/1.3555040 Traveltime approximations for transversely isotropic media with an inhomogeneous background Geophysical Prospecting Alkhalifah 61 495 2013 10.1111/j.1365-2478.2012.01067.x Traveltime approximations for inhomogeneous transversely isotropic media with a horizontal symmetry axis Viscoelastic waves in layered media Borcherdt 2009 10.1017/CBO9780511580994 Wave fields in real media: Theory and numerical simulation of wave propagation in anisotropic, anelastic, porous and electromagnetic media Carcione 3rd ed. 2015 Seismic ray theory Červený 2001 10.1017/CBO9780511529399 Geophysical Journal International Červený 178 939 2009 10.1111/j.1365-246X.2009.04218.x Perturbation Hamiltonians in heterogeneous anisotropic weakly dissipative media Geophysical Journal International Chapman 109 1 1992 10.1111/j.1365-246X.1992.tb00075.x Traveltime tomography in anisotropic media — Part 1: Theory SIAM Review Chapman 41 417 1999 10.1137/S0036144599352058 On the theory of complex rays Geophysics da Silva 81 4 U25 2016 10.1190/geo2015-0349.1 A new parameter set for anisotropic multiparameter full-waveform inversion and application to a North Sea data set Geophysics Gajewski 57 27 1992 10.1190/1.1443186 Vector wavefields for weakly attenuating anisotropic media by the ray method Geophysical Prospecting Golikov 60 433 2012 10.1111/j.1365-2478.2011.01003.x Traveltime parameters in a tilted elliptical anisotropic medium Geophysics Golikov 77 6 C43 2012 10.1190/geo2011-0457.1 Traveltime parameters in tilted transversely isotropic media Geophysics Gray 59 810 1994 10.1190/1.1443639 Kirchhoff migration using eikonal equation traveltimes Pure and Applied Geophysics Hanyga 157 679 2000 10.1007/PL00001114 Ray tracing in elastic and viscoelastic media Geophysics Hao 82 1 C9 2017 10.1190/geo2016-0160.1 An acoustic eikonal equation for attenuating transversely isotropic media with a vertical symmetry axis Geophysics Hao 82 4 WA67 2017 10.1190/geo2016-0632.1 An acoustic eikonal equation for attenuating orthorhombic media Geophysics Hao 81 5 C151 2016 10.1190/geo2015-0352.1 The offset-midpoint traveltime pyramid of P-waves in homogeneous orthorhombic media 10.3997/2214-4609.201801124 Hao Q. Waheed U. Alkhalifah T. , 2018, A fast sweeping scheme for P-wave traveltimess in attenuating VTI media: 80th Annual International Conference and Exhibition, EAGE, Extended Abstracts, doi: http://dx.doi.org/10.3997/2214-4609.201801124. Numerical Algorithms Hetmaniok 67 163 2014 10.1007/s11075-013-9781-0 Usage of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind Geophysics Hill 66 1240 2001 10.1190/1.1487071 Prestack Gaussian-beam depth migration Geophysical Journal International Huang 216 1648 2019 10.1093/gji/ggy524 Traveltime approximation for strongly anisotropic media using the homotopy analysis method Geophysical Journal International Huang 214 907 2018 10.1093/gji/ggy159 On the solution of the complex eikonal equation in acoustic VTI media: A perturbation plus optimization scheme Geophysics Ivanov 84 3 C159 2019 10.1190/geo2017-0552.1 3D mapping of kinematic attributes in anisotropic media Geophysics Journal International Klimeš 186 751 2011 10.1111/j.1365-246X.2011.05054.x Perturbation expansion of complex-valued traveltime along real-valued reference rays Theory and applications of complex rays Kravtsov 1 1999 Journal of Ship Research Liao 36 30 1992 10.5957/jsr.1992.36.1.30 Application of process analysis method to the solution of 2-D nonlinear progressive gravity waves Journal of Applied Mechanics Liao 59 970 1992 10.1115/1.2894068 A second-order approximate analytical solution of a simple pendulum by the process analysis method Beyond perturbation: Introduction to the homotopy analysis method Liao 2003 10.1201/9780203491164 Homotopy analysis method in nonlinear differential equations Liao 2012 10.1007/978-3-642-25132-0 Journal of Fluid Mechanics Liao 794 175 2016 10.1017/jfm.2016.162 On the steady-state nearly resonant waves Liao S. J. , 1992c, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. thesis, Shanghai Jiao Tong University. Journal of Fluid Mechanics Liao 385 101 1999 10.1017/S0022112099004292 A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate International Journal of Non-Linear Mechanics Liao 38 1173 2003 10.1016/S0020-7462(02)00062-8 An analytic approximate technique for free oscillations of positively damped systems with algebraically decaying amplitude International Journal of Non-Linear Mechanics Liao 39 271 2004 10.1016/S0020-7462(02)00174-9 An analytic approximate approach for free oscillations of self-excited systems Communications in Nonlinear Science and Numerical Simulation Liao 16 1274 2011 10.1016/j.cnsns.2010.06.026 On the homotopy multiple-variable method and its applications in the interactions of nonlinear gravity waves Journal of Applied Mechanics Liao 65 914 1998 10.1115/1.2791935 Application of homotopy analysis method in nonlinear oscillations Geophysical Prospecting Rasolofosaon 58 637 2010 10.1111/j.1365-2478.2009.00863.x Generalized anisotropy parameters and approximations of attenuations and velocities in viscoelastic media of arbitrary anisotropy type — Theoretical and experimental aspects Geophysics Stovas 77 4 C37 2012 10.1190/geo2011-0158.1 A new traveltime approximation for TI media Geophysics Stovas 81 6 C309 2016 10.1190/geo2016-0097.1 Application of perturbation theory to a P-wave eikonal equation in orthorhombic media eikonal equation in orthorhombic media Geophysics Thomsen 51 1954 1986 10.1190/1.1442051 Weak elastic anisotropy Studia Geophysica et Geodaetica Thomson 41 345 1997 10.1023/A:1023359401107 Complex rays and wave packets for decaying signals in inhomogeneous, anisotropic and anelastic media Geophysics Vavryčuk 74 5 WB203 2009 10.1190/1.3173154 Weak anisotropy-attenuation parameters Geophysics Waheed 78 5 WC103 2013 10.1190/geo2012-0413.1 Diffraction traveltime approximation for TI media with an inhomogeneous background Geophysical Prospecting Xu 65 1218 2017 10.1111/1365-2478.12480 Perturbation-based moveout approximations in anisotropic media Journal of Geophysical Research: Solid Earth Zelt 103 7187 1998 10.1029/97JB03536 Three-dimensional seismic refraction tomography: A comparison of two methods applied to data from the Faeroe Basin Geophysical Journal International Zhu 119 269 1994 10.1111/j.1365-246X.1994.tb00927.x Complex rays in elastic and anelastic media