{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T10:19:56Z","timestamp":1769595596239,"version":"3.49.0"},"reference-count":58,"publisher":"Society of Exploration Geophysicists","issue":"3","funder":[{"DOI":"10.13039\/501100001809","name":"National Science Foundation of China","doi-asserted-by":"crossref","award":["41804100"],"award-info":[{"award-number":["41804100"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Science Foundation of China","doi-asserted-by":"crossref","award":["41804100"],"award-info":[{"award-number":["41804100"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["library.seg.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2020,5,1]]},"abstract":"<jats:title>ABSTRACT<\/jats:title>\n                  <jats:p>The complex traveltime solutions of the complex eikonal equation are the basis of inhomogeneous plane-wave seismic imaging methods, such as Gaussian beam migration and tomography. We have developed analytic approximations for the complex traveltime in transversely isotropic media with a titled symmetry axis, which is defined by a Taylor series expansion over the anisotropy parameters. The formulation for the complex traveltime is developed using perturbation theory and the complex point-source method. The real part of the complex traveltime describes the wavefront, and the imaginary part of the complex traveltime describes the decay of the amplitude of waves away from the central ray. We derive the linearized ordinary differential equations for the coefficients of the Taylor-series expansion using perturbation theory. The analytical solutions for the complex traveltimes are determined by applying the complex point-source method to the background traveltime formula and subsequently obtaining the coefficients from the linearized ordinary differential equations. We investigate the influence of the anisotropy parameters and of the initial width of the ray tube on the accuracy of the computed traveltimes. The analytical formulas, as outlined, are efficient methods for the computation of complex traveltimes from the complex eikonal equation. In addition, those formulas are also effective methods for benchmarking approximated solutions.<\/jats:p>","DOI":"10.1190\/geo2019-0264.1","type":"journal-article","created":{"date-parts":[[2020,3,11]],"date-time":"2020-03-11T03:54:09Z","timestamp":1583898849000},"page":"T191-T207","update-policy":"https:\/\/doi.org\/10.1190\/crossmark-policy","source":"Crossref","is-referenced-by-count":2,"title":["A complex point-source solution of the acoustic eikonal equation for Gaussian beams in transversely isotropic media"],"prefix":"10.1190","volume":"85","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9719-6297","authenticated-orcid":false,"given":"Xingguo","family":"Huang","sequence":"first","affiliation":[{"name":"University of Bergen 1 , Department of Earth Science, Allegaten 41, 5020 Bergen, . E-mail: xingguo.huang@uib.no ; xingguo.huang19@gmail.com .","place":["Norway"]}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6334-2409","authenticated-orcid":false,"given":"Hui","family":"Sun","sequence":"additional","affiliation":[{"name":"Southwest Jiaotong University 2 , Faculty of Geosciences and Environmental Engineering, Chengdu 610031, . E-mail: sunhui@swjtu.edu.cn (corresponding author).","place":["China"]}]},{"given":"Zhangqing","family":"Sun","sequence":"additional","affiliation":[{"name":"Jilin University 3 , College for Geo-exploration Science and Technology, 938 Ximinzhu Street, Changchun 130026, . E-mail: sun_zhangq@jlu.edu.cn .","place":["China"]}]},{"given":"Nuno","family":"Vieira da Silva","sequence":"additional","affiliation":[{"name":"Imperial College London 4 Formerly , South Kensington Campus, Department of Earth Science and Engineering, London SW7 2AZ, , presently , Tarland Road, Westhill, Aberdeen AB32 6JZ. E-mail: nuno.vdasilva@gmail.com .","place":["UK"]},{"name":"Total E&P UK 4 Formerly , South Kensington Campus, Department of Earth Science and Engineering, London SW7 2AZ, , presently , Tarland Road, Westhill, Aberdeen AB32 6JZ. E-mail: nuno.vdasilva@gmail.com .","place":["UK"]}]}],"member":"186","published-online":{"date-parts":[[2020,4,14]]},"reference":[{"key":"2025121211125896300_r1","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","ISSN":"https:\/\/id.crossref.org\/issn\/0016-8033","issn-type":"print"},{"issue":"2","key":"2025121211125896300_r2","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","ISSN":"https:\/\/id.crossref.org\/issn\/0016-8033","issn-type":"print"},{"issue":"3","key":"2025121211125896300_r3","doi-asserted-by":"crossref","first-page":"WA31","DOI":"10.1190\/1.3555040","article-title":"Traveltime approximations for transversely isotropic media with an 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