Computation of travel-times and its spatial derivatives is at the heart of many seismic imaging techniques. We shall discuss shortly the Lagrangian, semi-lagrangian and Eulerian approaches which have been investigated in the past. Eulerian approaches enjoy recent advances when considering anisotropic media with complex topographies while having the attractive regular sampling needed for slope tomography and migration workflow. While this Eulerian approach is quite efficient for firstarrival travel-time/slope tomography and has replaced ray tracing approach, it has not yet replaced standard ray tracing tools (in spite of its irregular sampling of the medium) in migration where multiple arrivals are important. An efficient and accurate discontinuous Galerkin method for solving the non-linear Eikonal partial differential equation providing travel-times might be an attractive proposition both for tomography and migration. Thanks to this Eulerian approach, data- and domainspace definitions involve only subsurface parameters such as velocity parameters and imaging point positions for slope tomography while the forward modeling has a computational complexity depending only on the acquisition design and not on the picking density. Still bottle-necks exist when considering multiple arrivals and their identification will help to find solutions in the future.


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