{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,12]],"date-time":"2025-12-12T17:04:56Z","timestamp":1765559096168,"version":"3.48.0"},"reference-count":49,"publisher":"Society of Exploration Geophysicists","issue":"6","content-domain":{"domain":["library.seg.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2021,11,1]]},"abstract":"<jats:title>ABSTRACT<\/jats:title>\n                  <jats:p>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.<\/jats:p>","DOI":"10.1190\/geo2020-0747.1","type":"journal-article","created":{"date-parts":[[2021,7,19]],"date-time":"2021-07-19T13:12:00Z","timestamp":1626700320000},"page":"T543-T551","update-policy":"https:\/\/doi.org\/10.1190\/crossmark-policy","source":"Crossref","is-referenced-by-count":3,"title":["Homotopy solutions of the acoustic eikonal equation for strongly attenuating transversely isotropic media with a vertical symmetry axis"],"prefix":"10.1190","volume":"86","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9719-6297","authenticated-orcid":false,"given":"Xingguo","family":"Huang","sequence":"first","affiliation":[{"name":"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 ."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stewart","family":"Greenhalgh","sequence":"additional","affiliation":[{"name":"Swiss Federal Institute of Technology (ETH) Zurich, Institute of Geophysics 2 , Department of Earth Sciences, Zurich 8092, Switzerland. E-mail: gstewart@retired.ethz.ch ."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8051-755X","authenticated-orcid":false,"given":"Yun","family":"Long","sequence":"additional","affiliation":[{"name":"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 ."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jun","family":"Lin","sequence":"additional","affiliation":[{"name":"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 ."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2835-2844","authenticated-orcid":false,"given":"Xu","family":"Liu","sequence":"additional","affiliation":[{"name":"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 ."}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"186","published-online":{"date-parts":[[2021,9,23]]},"reference":[{"key":"2025121212002279300_r1","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1190\/1.1444361","article-title":"Acoustic approximations for processing in transversely isotropic media","volume":"63","author":"Alkhalifah","year":"1998","journal-title":"Geophysics"},{"key":"2025121212002279300_r2","doi-asserted-by":"crossref","first-page":"1239","DOI":"10.1190\/1.1444815","article-title":"An acoustic wave equation for anisotropic media","volume":"65","author":"Alkhalifah","year":"2000","journal-title":"Geophysics"},{"issue":"2","key":"2025121212002279300_r3","doi-asserted-by":"crossref","first-page":"U13","DOI":"10.1190\/1.3553015","article-title":"Scanning anisotropy parameters in complex media","volume":"76","author":"Alkhalifah","year":"2011","journal-title":"Geophysics"},{"issue":"3","key":"2025121212002279300_r4","doi-asserted-by":"crossref","first-page":"WA31","DOI":"10.1190\/1.3555040","article-title":"Traveltime approximations for transversely isotropic media with an inhomogeneous background","volume":"76","author":"Alkhalifah","year":"2011","journal-title":"Geophysics"},{"key":"2025121212002279300_r5","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1111\/j.1365-2478.2012.01067.x","article-title":"Traveltime approximations for inhomogeneous transversely isotropic media with a horizontal symmetry axis","volume":"61","author":"Alkhalifah","year":"2013","journal-title":"Geophysical Prospecting"},{"key":"2025121212002279300_r6","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511580994","volume-title":"Viscoelastic waves in layered media","author":"Borcherdt","year":"2009"},{"key":"2025121212002279300_r7","volume-title":"Wave fields in real media: Theory and numerical simulation of wave propagation in anisotropic, anelastic, porous and electromagnetic media","author":"Carcione","year":"2015","edition":"3rd ed."},{"key":"2025121212002279300_r8","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511529399","volume-title":"Seismic ray theory","author":"\u010cerven\u00fd","year":"2001"},{"key":"2025121212002279300_r9","doi-asserted-by":"crossref","first-page":"939","DOI":"10.1111\/j.1365-246X.2009.04218.x","article-title":"Perturbation Hamiltonians in heterogeneous anisotropic weakly dissipative media","volume":"178","author":"\u010cerven\u00fd","year":"2009","journal-title":"Geophysical Journal International"},{"key":"2025121212002279300_r10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/j.1365-246X.1992.tb00075.x","article-title":"Traveltime tomography in anisotropic media \u2014 Part 1: Theory","volume":"109","author":"Chapman","year":"1992","journal-title":"Geophysical Journal International"},{"key":"2025121212002279300_r11","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1137\/S0036144599352058","article-title":"On the theory of complex rays","volume":"41","author":"Chapman","year":"1999","journal-title":"SIAM Review"},{"issue":"4","key":"2025121212002279300_r12","doi-asserted-by":"crossref","first-page":"U25","DOI":"10.1190\/geo2015-0349.1","article-title":"A new parameter set for anisotropic multiparameter full-waveform inversion and application to a North Sea data set","volume":"81","author":"da Silva","year":"2016","journal-title":"Geophysics"},{"key":"2025121212002279300_r13","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1190\/1.1443186","article-title":"Vector wavefields for weakly attenuating anisotropic media by the ray method","volume":"57","author":"Gajewski","year":"1992","journal-title":"Geophysics"},{"key":"2025121212002279300_r15","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1111\/j.1365-2478.2011.01003.x","article-title":"Traveltime parameters in a tilted elliptical anisotropic medium","volume":"60","author":"Golikov","year":"2012","journal-title":"Geophysical Prospecting"},{"issue":"6","key":"2025121212002279300_r14","doi-asserted-by":"crossref","first-page":"C43","DOI":"10.1190\/geo2011-0457.1","article-title":"Traveltime parameters in tilted transversely isotropic media","volume":"77","author":"Golikov","year":"2012","journal-title":"Geophysics"},{"key":"2025121212002279300_r16","doi-asserted-by":"crossref","first-page":"810","DOI":"10.1190\/1.1443639","article-title":"Kirchhoff migration using eikonal equation traveltimes","volume":"59","author":"Gray","year":"1994","journal-title":"Geophysics"},{"key":"2025121212002279300_r17","doi-asserted-by":"crossref","first-page":"679","DOI":"10.1007\/PL00001114","article-title":"Ray tracing in elastic and viscoelastic media","volume":"157","author":"Hanyga","year":"2000","journal-title":"Pure and Applied Geophysics"},{"issue":"1","key":"2025121212002279300_r18","doi-asserted-by":"crossref","first-page":"C9","DOI":"10.1190\/geo2016-0160.1","article-title":"An acoustic eikonal equation for attenuating transversely isotropic media with a vertical symmetry axis","volume":"82","author":"Hao","year":"2017","journal-title":"Geophysics"},{"issue":"4","key":"2025121212002279300_r19","doi-asserted-by":"crossref","first-page":"WA67","DOI":"10.1190\/geo2016-0632.1","article-title":"An acoustic eikonal equation for attenuating orthorhombic media","volume":"82","author":"Hao","year":"2017","journal-title":"Geophysics"},{"issue":"5","key":"2025121212002279300_r20","doi-asserted-by":"crossref","first-page":"C151","DOI":"10.1190\/geo2015-0352.1","article-title":"The offset-midpoint traveltime pyramid of P-waves in homogeneous orthorhombic media","volume":"81","author":"Hao","year":"2016","journal-title":"Geophysics"},{"doi-asserted-by":"crossref","unstructured":"Hao\n              Q.\n            \n            \n              Waheed\n              U.\n            \n            \n              Alkhalifah\n              T.\n            \n          , 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.","key":"2025121212002279300_r21","DOI":"10.3997\/2214-4609.201801124"},{"key":"2025121212002279300_r50","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1007\/s11075-013-9781-0","article-title":"Usage of the homotopy analysis method for solving the nonlinear and linear integral equations of the second kind","volume":"67","author":"Hetmaniok","year":"2014","journal-title":"Numerical Algorithms"},{"key":"2025121212002279300_r22","doi-asserted-by":"crossref","first-page":"1240","DOI":"10.1190\/1.1487071","article-title":"Prestack Gaussian-beam depth migration","volume":"66","author":"Hill","year":"2001","journal-title":"Geophysics"},{"key":"2025121212002279300_r23","doi-asserted-by":"crossref","first-page":"1648","DOI":"10.1093\/gji\/ggy524","article-title":"Traveltime approximation for strongly anisotropic media using the homotopy analysis method","volume":"216","author":"Huang","year":"2019","journal-title":"Geophysical Journal International"},{"key":"2025121212002279300_r24","doi-asserted-by":"crossref","first-page":"907","DOI":"10.1093\/gji\/ggy159","article-title":"On the solution of the complex eikonal equation in acoustic VTI media: A perturbation plus optimization scheme","volume":"214","author":"Huang","year":"2018","journal-title":"Geophysical Journal International"},{"issue":"3","key":"2025121212002279300_r25","doi-asserted-by":"crossref","first-page":"C159","DOI":"10.1190\/geo2017-0552.1","article-title":"3D mapping of kinematic attributes in anisotropic media","volume":"84","author":"Ivanov","year":"2019","journal-title":"Geophysics"},{"key":"2025121212002279300_r26","doi-asserted-by":"crossref","first-page":"751","DOI":"10.1111\/j.1365-246X.2011.05054.x","article-title":"Perturbation expansion of complex-valued traveltime along real-valued reference rays","volume":"186","author":"Klime\u0161","year":"2011","journal-title":"Geophysics Journal International"},{"key":"2025121212002279300_r27","first-page":"1","volume-title":"Theory and applications of complex rays","author":"Kravtsov","year":"1999"},{"key":"2025121212002279300_r28","doi-asserted-by":"crossref","first-page":"30","DOI":"10.5957\/jsr.1992.36.1.30","article-title":"Application of process analysis method to the solution of 2-D nonlinear progressive gravity waves","volume":"36","author":"Liao","year":"1992","journal-title":"Journal of Ship Research"},{"key":"2025121212002279300_r29","doi-asserted-by":"crossref","first-page":"970","DOI":"10.1115\/1.2894068","article-title":"A second-order approximate analytical solution of a simple pendulum by the process analysis method","volume":"59","author":"Liao","year":"1992","journal-title":"Journal of Applied Mechanics"},{"key":"2025121212002279300_r30","doi-asserted-by":"crossref","DOI":"10.1201\/9780203491164","volume-title":"Beyond perturbation: Introduction to the homotopy analysis method","author":"Liao","year":"2003"},{"key":"2025121212002279300_r31","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-25132-0","volume-title":"Homotopy analysis method in nonlinear differential equations","author":"Liao","year":"2012"},{"key":"2025121212002279300_r32","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1017\/jfm.2016.162","article-title":"On the steady-state nearly resonant waves","volume":"794","author":"Liao","year":"2016","journal-title":"Journal of Fluid Mechanics"},{"unstructured":"Liao\n              S. J.\n            \n          , 1992c, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. thesis, Shanghai Jiao Tong University.","key":"2025121212002279300_r33"},{"key":"2025121212002279300_r34","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1017\/S0022112099004292","article-title":"A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate","volume":"385","author":"Liao","year":"1999","journal-title":"Journal of Fluid Mechanics"},{"key":"2025121212002279300_r35","doi-asserted-by":"crossref","first-page":"1173","DOI":"10.1016\/S0020-7462(02)00062-8","article-title":"An analytic approximate technique for free oscillations of positively damped systems with algebraically decaying amplitude","volume":"38","author":"Liao","year":"2003","journal-title":"International Journal of Non-Linear Mechanics"},{"key":"2025121212002279300_r36","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/S0020-7462(02)00174-9","article-title":"An analytic approximate approach for free oscillations of self-excited systems","volume":"39","author":"Liao","year":"2004","journal-title":"International Journal of Non-Linear Mechanics"},{"key":"2025121212002279300_r37","doi-asserted-by":"crossref","first-page":"1274","DOI":"10.1016\/j.cnsns.2010.06.026","article-title":"On the homotopy multiple-variable method and its applications in the interactions of nonlinear gravity waves","volume":"16","author":"Liao","year":"2011","journal-title":"Communications in Nonlinear Science and Numerical Simulation"},{"key":"2025121212002279300_r38","doi-asserted-by":"crossref","first-page":"914","DOI":"10.1115\/1.2791935","article-title":"Application of homotopy analysis method in nonlinear oscillations","volume":"65","author":"Liao","year":"1998","journal-title":"Journal of Applied Mechanics"},{"key":"2025121212002279300_r39","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1111\/j.1365-2478.2009.00863.x","article-title":"Generalized anisotropy parameters and approximations of attenuations and velocities in viscoelastic media of arbitrary anisotropy type \u2014 Theoretical and experimental aspects","volume":"58","author":"Rasolofosaon","year":"2010","journal-title":"Geophysical Prospecting"},{"issue":"4","key":"2025121212002279300_r40","doi-asserted-by":"crossref","first-page":"C37","DOI":"10.1190\/geo2011-0158.1","article-title":"A new traveltime approximation for TI media","volume":"77","author":"Stovas","year":"2012","journal-title":"Geophysics"},{"issue":"6","key":"2025121212002279300_r41","doi-asserted-by":"crossref","first-page":"C309","DOI":"10.1190\/geo2016-0097.1","article-title":"Application of perturbation theory to a P-wave eikonal equation in orthorhombic media eikonal equation in orthorhombic media","volume":"81","author":"Stovas","year":"2016","journal-title":"Geophysics"},{"key":"2025121212002279300_r42","doi-asserted-by":"crossref","first-page":"1954","DOI":"10.1190\/1.1442051","article-title":"Weak elastic anisotropy","volume":"51","author":"Thomsen","year":"1986","journal-title":"Geophysics"},{"key":"2025121212002279300_r43","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1023\/A:1023359401107","article-title":"Complex rays and wave packets for decaying signals in inhomogeneous, anisotropic and anelastic media","volume":"41","author":"Thomson","year":"1997","journal-title":"Studia Geophysica et Geodaetica"},{"issue":"5","key":"2025121212002279300_r44","doi-asserted-by":"crossref","first-page":"WB203","DOI":"10.1190\/1.3173154","article-title":"Weak anisotropy-attenuation parameters","volume":"74","author":"Vavry\u010duk","year":"2009","journal-title":"Geophysics"},{"issue":"5","key":"2025121212002279300_r45","doi-asserted-by":"crossref","first-page":"WC103","DOI":"10.1190\/geo2012-0413.1","article-title":"Diffraction traveltime approximation for TI media with an inhomogeneous background","volume":"78","author":"Waheed","year":"2013","journal-title":"Geophysics"},{"key":"2025121212002279300_r46","doi-asserted-by":"crossref","first-page":"1218","DOI":"10.1111\/1365-2478.12480","article-title":"Perturbation-based moveout approximations in anisotropic media","volume":"65","author":"Xu","year":"2017","journal-title":"Geophysical Prospecting"},{"key":"2025121212002279300_r47","doi-asserted-by":"crossref","first-page":"7187","DOI":"10.1029\/97JB03536","article-title":"Three-dimensional seismic refraction tomography: A comparison of two methods applied to data from the Faeroe Basin","volume":"103","author":"Zelt","year":"1998","journal-title":"Journal of Geophysical Research: Solid Earth"},{"key":"2025121212002279300_r48","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1111\/j.1365-246X.1994.tb00927.x","article-title":"Complex rays in elastic and anelastic media","volume":"119","author":"Zhu","year":"1994","journal-title":"Geophysical Journal International"}],"container-title":["Geophysics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.geoscienceworld.org\/seg\/geophysics\/article-pdf\/86\/6\/T543\/5459453\/geo-2020-0747.1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.geoscienceworld.org\/seg\/geophysics\/article-pdf\/86\/6\/T543\/5459453\/geo-2020-0747.1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,12]],"date-time":"2025-12-12T17:00:30Z","timestamp":1765558830000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.geoscienceworld.org\/geophysics\/article\/86\/6\/T543\/607947\/Homotopy-solutions-of-the-acoustic-eikonal"},"secondary":[{"URL":"https:\/\/pubs.geoscienceworld.org\/geophysics\/article\/86\/6\/t543\/607947\/homotopy-solutions-of-the-acoustic-eikonal","label":"geoscienceworld"}]},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,19]]},"references-count":49,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2021,11,1]]}},"URL":"https:\/\/doi.org\/10.1190\/geo2020-0747.1","relation":{},"ISSN":["0016-8033","1942-2156"],"issn-type":[{"type":"print","value":"0016-8033"},{"type":"electronic","value":"1942-2156"}],"subject":[],"published":{"date-parts":[[2021,7,19]]},"assertion":[{"value":"2020-10-04","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2021-06-16","order":1,"name":"revised","label":"Revised","group":{"name":"publication_history","label":"Publication History"}},{"value":"2021-06-22","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2021-09-23","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}