“Focusing” eikonal equation and global tomography

The transformation of the eikonal equation from depth coordinates (z, x) into verticaltraveltime coordinates (τ ,ξ ) enables the computation of reflections traveltimes independent of depth-mapping. This separation allows the focusing and mapping steps to be performed sequentially even in the presence of complex velocity functions, that otherwise would “require” depth migration. The traveltimes satisfying the transformed eikonal equation can be numerically evaluated by solving the associated ray tracing equations. The application of Fermat’s principle leads to the expression of linear relationships between perturbations in traveltimes and perturbations in focusing velocity. This linearization, in conjunction with ray tracing, can be used for a tomographic estimation of focusing velocity.