Abstract
In an anisotropic model, traveltime can be determined approximately by numerical solution of the eikonal equation in terms of an anellipticity parameter η, using perturbation theory. However, its accuracy decreases under the effect of strong anisotropy at larger offsets. It becomes invalid for determining normal moveout velocity and anellipticity parameter in seismic processing. We propose a new approach using Levin T-transformation to transform the expanded traveltime in the transversely isotropic medium with vertical axis of symmetry (VTI) into rational form. The objective of this study is to provide a new traveltime approximation that is more accurate at larger offsets. In this study, we derive Levin algorithm and determine the optimal value of Levin parameters, which is a key step in achieving better accuracy. In a numerical experiment, we compare the accuracy between Levin T-transformation and second sequence of Shanks transformation in a homogeneous VTI medium. We also implement both approximations in a velocity analysis and stacking traces using synthetic common midpoint gathers on a multilayer earth model. The proposed method shows a superiority in accuracy to existing methods over a range of offsets with offset-to-depth ratio up to 6 and anellipticity parameter 0–0.5.
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04 April 2022
An Erratum to this paper has been published: https://doi.org/10.1007/s11200-022-0901-y
References
Aitken A.C., 1926. On Bernoulli’s numerical solution of algebraic equations. Proceedings of the Royal Society of Edinburg, 46, 289–305, DOI: https://doi.org/10.1017/S0370164600022070
Alshuhail A.A. and Verschuur D.J., 2019. Robust estimation of VTI models via Joint Migration Inversion: Including multiples in anisotropic parameter estimation. Geophysics, 84, C57–C74, DOI: https://doi.org/10.1190/geo2017-0856.1.
Alkhalifah T. and Tsvankin I., 1995. Velocity analysis for transversely isotropic medium. Geophysics, 60, 1550–1566, DOI: https://doi.org/10.1190/1.1443888.
Alkhalifah T., 1997. Velocity analysis using nonhyperbolic moveout in transversely isotropic medium. Geophysics, 62, 1839–1854, DOI: https://doi.org/10.1190/1.1444285
Alkhalifah T., 1998. Acoustic approximations for processing in transversely isotropic medium. Geophysics, 63, 623–631, DOI: https://doi.org/10.1190/1.1444361.
Alkhalifah T., 2000. The offset-midpoint traveltime pyramid in transversely isotropic medium. Geophysics, 65, 1316–1325, DOI: https://doi.org/10.1190/1.1444823.
Alkhalifah T., 2002. Traveltime computation with the linearized eikonal equation for anisotropic medium. Geophys. Prospect., 50, 373–382, DOI: https://doi.org/10.1046/j.1365-2478.2002.00322.x.
Alkhalifah T., 2011a. Traveltime approximations for transversely isotropic medium with an inhomogeneous background. Geophysics, 76, 31–42, DOI: https://doi.org/10.1190/1.3555040.
Alkhalifah T., 2011b. Scanning anisotropy parameter in complex medium. Geophysics, 76, U13–U22, DOI: https://doi.org/10.1190/1.3553015.
Alkhalifah T., 2012. Traveltime approximations for inhomogeneous transversely isotropic medium with a horizontal symmetry axis. Geophys. Prospect., 61, 495–503, DOI: https://doi.org/10.1111/J.1365-2478.2012.01067.x.
Bender C.M. and Orszag S.A., 1978. Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, New York, 368–379.
Brezinski C., 1980. A general extrapolation algorithm. Numerische Mathematik, 35, 175–187, DOI: https://doi.org/10.1007/bf01396314.
Červený V., 1972. Seismic rays and ray intensities in inhomogeneous anisotropic medium. Geophys. J. R. Astron. Soc., 29, 1–13, DOI: https://doi.org/10.1111/j.1365-246X.1972.tb06147.x.
Červený V., 2001. Seismic Ray Theory. Cambridge University Press, Cambridge, U.K.
Daley P.F., Krebes E.S. and Lines L.R., 2010. A comparison of exact, approximate, and linearized ray tracing methods in transversely isotropic medium. CREWES Research Report, 22, https://www.crewes.org/ForOurSponsors/ResearchReports/2010/CRR201015.pdf.
Dix C.H., 1955. Seismic velocities from surface measurements. Geophysics, 20, 68–86, DOI: https://doi.org/10.1190/1.1442693.
Djebbi R., Plessix R.E. and Alkhalifah T., 2016. Analysis of the traveltime sensitivity kernels for an acoustic transversely isotropic medium with a vertical axis of symmetry. Geophys. Prospect., 65, 22–34, DOI: https://doi.org/10.1111/1365-2478.12361.
Ettrich N. and Gajewski D., 1998. Traveltime computation by perturbation with FD-eikonal solvers in isotropic and weakly anisotropic medium. Geophysics, 63, 1066–1078, DOI: https://doi.org/10.1190/1.1444385.
Fomel S., 2009. Velocity analysis using AB semblance. Geophys. Prospect., 57, 311–321, DOI: https://doi.org/10.1111/j.1365-2478.2008.00741.x.
Fowler P.J., 1994. Finite-difference solutions of the 3-D eikonal equation in spherical coordinates. SEG Technical Program Expanded Abstracts 1994, 1394–1397, DOI: https://doi.org/10.1190/1.1822792.
Gjøystdal H., Reinhardsen J.E. and Åstebol K., 1985. Computer representation of complex 3-D geological structures using a new “solid modeling” technique. Geophys. Prospect., 33, 195–211.
Grechka V. and Tsvankin I., 1998. Feasibility of nonhyperbolic moveout inversion in transversely isotropic medium. Geophysics, 63, 957–969, DOI: https://doi.org/10.1190/1.1444407.
Hao Q. and Alkhalifah T., 2017. An acoustic eikonal equation for attenuating transversely isotropic medium with a vertical symmetry axis. Geophysics, 82, C9–C20, DOI: https://doi.org/10.1190/geo2016-0160.1.
Håvie T., 1979. Generalized Neville type extrapolation schemes. Bit, 19, 204–213, DOI: https://doi.org/10.1007/bf01930850.
Levin D., 1973. Development of non-linear transformation for improving convergence of sequences. Int. J. Comput. Math., B3, 371–388, DOI: https://doi.org/10.1080/00207167308803075.
Masmoudi N. and Alkhalifah T., 2016. Traveltime approximations and parameter estimation for orthorhombic medium. Geophysics, 81, C127–C137, DOI: https://doi.org/10.1190/geo2015-0367.1.
Popovici M., 1991. Finite-difference travel time maps. Stanford Exploration Project Report, 70, 245–256.
Roy D., 2009. Global approximation for some functions. J. Comput. Phys. Commun., 180, 1315–1337, DOI: https://doi.org/10.1016/j.cpc.2009.02.010.
Sarkar D., Castagna J.P. and Lamb W., 2001. AVO and velocity analysis. Geophysics, 66, 1284–1293, DOI: https://doi.org/10.1190/1.1487076.
Sarkar D., Baumel R.T. and Larner K.L. 2002. Velocity analysis in the presence of amplitude variation. Geophysics, 67, 1664–1672, DOI: https://doi.org/10.1190/1.1512814.
Schneider W.A., 1995. Robust and efficient upwind finite-difference traveltime calculations in three dimensions. Geophysics, 60, 1108–1117, DOI: https://doi.org/10.1190/1.1443839.
Shanks D., 1955. Non-linear transformation of divergent and slowly convergent sequences. J. Math. Phys., 34, 1–42, DOI: https://doi.org/10.1002/sapm19553411.
Shuey R.T., 1985. A simplification of the Zoeppritz equations. Geophysics, 50, 609–614.
Sidi A., 1980. Analysis of convergence of the t-transformation for power series. Math. Comput., 151, 833–850, DOI: https://doi.org/10.2307/2006198.
Sidi A. and Levin D. 1983. Prediction properties of the t-transformation. SIAM J. Numer. Anal., 20, 589–598, DOI: https://doi.org/10.1137/0720039.
Stovas A. and Alkhalifah T., 2012. A new traveltime approximation for TI medium. Geophysics, 77, C37–C42, DOI: https://doi.org/10.1190/geo2011-0158.1.
Stovas A., Masmoudi N. and Alkhalifah T., 2016. Application of perturbation theory to a P-wave eikonal equation in orthorhombic medium eikonal equation in orthorhombic medium. Geophysics, 81, C309–C317, DOI: https://doi.org/10.1190/geo2016-0097.1.
Thomsen L., 1986. Weak elastic anisotropy. Geophysics, 51, 1954–1966, DOI: https://doi.org/10.1190/1.1442051.
Tsvankin I. and Thomsen L., 1995. Inversion of reflection traveltimes for transverse isotropy. Geophysics, 60, 1095–1107, DOI: https://doi.org/10.1190/1.1443838.
Tsvankin I., 2005. Seismic Signatures and Analysis of Reflection Data in Anisotropic Medium. 2nd Edition. Elsevier Science, Oxford, U.K.
van Trier J. and Symes W.W., 1991. Upwind finite-difference calculation of traveltimes. Geophysics, 56, 812–821, DOI: https://doi.org/10.1190/1.1443099.
Vidale J.E., 1988. Finite-difference traveltime calculation. Bull. Seismol. Soc. Amer., 78, 2062–2076.
Vidale J.E., 1990. Finite-difference calculation of traveltimes in three dimensions. Geophysics, 55, 521–526, DOI: https://doi.org/10.1190/1.1442863.
Waheed U.B., Alkhalifah T. and Stovas A., 2013. Diffraction traveltime approximation for TI medium with an inhomogeneous background. Geophysics, 78, WC103–WC111, DOI: https://doi.org/10.1190/geo2012-0413.1.
Weniger E.J., 1989. Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series. Comput. Phys. Rep., 10, 189–371, DOI: https://doi.org/10.1016/0167-7977(89)90011-7.
Xu T., Gao E., Xu G.M. and Zhu L.B., 2004. Block modeling and shooting ray tracing in complex 3D media. Chinese J. Geophys., 47, 1261–1271.
Xu T., Xu G.M., Gao E.G., Li Y.C., Jiang X.Y. and Luo K. Y., 2006. Block modeling and segmentally iterative ray tracing in complex 3D media. Geophysics, 71, T41–T51.
Xu T., Zhang Z., Zhao A., Zhang A., Zhang X. and Zhang H., 2008. Sub-triangle shooting ray tracing in complex 3D VTI media. J. Seism. Explor., 17, 133–146.
Xu T., Zhang Z.J., Gao E.G., Xu G.M. and Sun L., 2010. Segmentally iterative ray tracing in complex 2D and 3D heterogeneous block models. Bull. Seismol. Soc. Amer., 100, 841–850, DOI: https://doi.org/10.1785/0120090155.
Xu S., Stovas A. and Hao Q., 2017. Perturbation-based moveout approximations in anisotropic medium. Geophys. Prospect., 65, 1218–1230, DOI: https://doi.org/10.1111/1365-2478.12480.
Acknowledgments
We would like to thank the reviewers for their thoughtful comments and insightful suggestions for improving our manuscript. This work was supported by Research and Development Center for Oil and Gas Technology “LEMIGAS” and Bandung Institute of Technology.
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Purba, H., Priyono, A., Triyoso, W. et al. Improving the accuracy of the expanded anisotropic eikonal equation at larger offsets using Levin T-transformation. Stud Geophys Geod 64, 349–372 (2020). https://doi.org/10.1007/s11200-020-0610-3
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DOI: https://doi.org/10.1007/s11200-020-0610-3