none 10.1190/geo2019-0264.1 Society of Exploration Geophysicists Society of Exploration Geophysicists 186 114542249 5981 2025121211125896300 10.1190 2025-12-12T16:13:05Z 2020-03-11T03:54:09Z 2 Geophysics 0016-8033 1942-2156 05 1 2020 85 3 Xingguo Huang University of Bergen 1 , Department of Earth Science, Allegaten 41, 5020 Bergen, . E-mail: xingguo.huang@uib.no ; xingguo.huang19@gmail.com . Norway http://orcid.org/0000-0001-9719-6297 Hui Sun Southwest Jiaotong University 2 , Faculty of Geosciences and Environmental Engineering, Chengdu 610031, . E-mail: sunhui@swjtu.edu.cn (corresponding author). China http://orcid.org/0000-0002-6334-2409 Zhangqing Sun Jilin University 3 , College for Geo-exploration Science and Technology, 938 Ximinzhu Street, Changchun 130026, . E-mail: sun_zhangq@jlu.edu.cn . China Nuno Vieira da Silva 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 . UK 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 . UK ABSTRACT 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. 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