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An eikonal based formulation for traveltime perturbation with respect to the source location
Traveltime calculations amount to solving the nonlinear eikonal equation for a given source location. We analyze the relationship between the eikonal solution and its perturbations with respect to the source location and develop a partial differential equation that relates the traveltime field for one source location to that for a nearby source. This linear first-order equation in one form depends on lateral changes in velocity and in another form is independent of the velocity field and relies on second-order derivatives of the original traveltime field. For stable finite-difference calculations, this requires that the velocity field be smooth in a sense similar to ray-tracing requirements. Our formulation for traveltime perturbation formulation has three potential applications. First, it can be used for fast traveltime calculation by source-location perturbation. Second, it can be used for velocity-independent interpolation including datuming as well as velocity estimation. Third, higher-order expansions provide parameters necessary for Gaussian-beam computations.
The traveltime through a linear velocity model that includes a lens at 0.6 km laterally and 0.5 km in depth. The solid curves correspond to the traveltime computed using a source at 0.2 km laterally and in depth (left), and a source at 0.21 km laterally (middle), compared with the traveltime extracted using perturbation theory and Shanks transform (dashed curves). The right plot is a density plot of the difference which is generally small other than small strip on the side of the lens.
Department of Physical Science and Engineering 4700 King Abdullah University of Science and Technology
Thuwal 23955-6900 Saudi Arabia
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