A new formulation for solving the Eikonal equation is investigated using a time-dependent Hamilton-Jacobi equation. A discontinuous Galerkin (DG) finite element method is proposed for direct reconstruction of the traveltime field as the final stationary solution. Both isotropic and tilted transversely isotropic (TTI) implementations are performed in heterogeneous media with stable and accurate results as long as we honor the high-frequency approximation. We introduce outgoing conditions at edges able to handle complex topography and we deal with singularity at the source through the additive factorization. Expected convergence behavior regarding element interpolation is observed when considering factorization. Comparison between DG and finite difference solutions in the complex BP TTI model with unstructured and structured meshes illustrates the highly accurate traveltime estimation of this DG approach, pointing out perspectives for integrating this accurate local Eikonal solver into efficient methods for getting the stationary solution, such as fast sweeping methods.


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  1. Alkhalifah, T.
    [2000] An acoustic wave equation for anisotropic media. Geophysics, 65, 1239–1250.
    [Google Scholar]
  2. Cerveny, V.
    [2001] Seismic Ray Theory. Cambridge University Press, Cambridge.
    [Google Scholar]
  3. Cheng, Y. and Wang, Z.
    [2014] A new discontinuous Galerkin finite element method for directly solving the Hamilton-Jacobi equations. Journal of Computational Physics, 268, 134–153.
    [Google Scholar]
  4. Crandall, M.G. and Lions, P.L.
    [1983] Viscosity solutions of Hamilton-Jacobi equations. Transactions of the American mathematical society, 277(1), 1–42.
    [Google Scholar]
  5. Fomel, S., Luo, S. and Zhao, H.K.
    [2009] Fast sweeping method for the factored eikonal equation. Journal of Computational Physics, 228, 6440–6455.
    [Google Scholar]
  6. Noble, M., Gesret, A. and Belayouni, N.
    [2014] Accurate 3-D finite difference computation of travel time in strongly hetrogeneous media. Geophysical Journal International, 199, 1572–1585.
    [Google Scholar]
  7. Podvin, P. and Lecomte, I.
    [1991] Finite Difference Computation of Traveltimes in Very Contrasted Velocity Model : a Massively Parallel Approach and its Associated Tools. Geophysical Journal International, 105, 271–284.
    [Google Scholar]
  8. Shah, H.
    [2007] The 2007 BP anisotropic velocity-analysis benchmark. In: Expanded Abstracts. EAGE workshop.
    [Google Scholar]
  9. Slawinski, M.
    [2003] Seismic Waves and Rays in Elastic Media. Elsevier Science.
    [Google Scholar]
  10. TavakoliF. B., Ribodetti, A., Virieux, J. and Operto, S.
    [2015] An iterative factored eikonal solver for TTI media. In: SEG technical program expanded abstracts 2015, 687. 3576–3581.
    [Google Scholar]
  11. Thomsen, L.A.
    [1986] Weak elastic anisotropy. Geophysics, 51, 1954–1966.
    [Google Scholar]
  12. Vidale, D.
    [1990] Finite-Difference Calculation of traveltimes in three dimensions. Geophysics, 55, 521–526.
    [Google Scholar]
  13. Waheed, U.B., Yarman, C.E. and Flagg, G.
    [2015] An iterative, fast-sweeping-based eikonal solver for 3D tilted anisotropic media. Geophysics, 80, C49–C58.
    [Google Scholar]
  14. Zienkewicz, O. and Morgan, K.
    [1983] Finite elements and approximation. J. Wiley and Sons, New York.
    [Google Scholar]

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