3-D traveltimes and amplitudes by gridded rays

Seismic imaging in three dimensions requires the calculation of traveltimes and amplitudes of a wave propagating through an elastic medium. They can be computed efficiently and accurately by integrating the eikonal equation on an elemental grid using finite-difference methods. Unfortunately, this approach to solving the eikonal equation is potentially unstable unless the grid sampling steps satisfy stability conditions or wavefront tracking algorithms are used. We propose a new method for computing traveltimes and amplitudes in 3-D media that is simple, fast, unconditionally stable, and robust. Defining the slowness vector as p = (p(x), p(y), p(z)) and assuming an isotropic medium, the ray velocity v is related to the slowness vector by the relation v = p/(p (.) p). Rays emerging from gridpoints on a horizontal plane are propagated downward a single vertical grid step to a new horizontal plane. The components of the slowness vector are then interpolated to gridpoints on this next horizontal plane, This is termed regridding; the process of downward propagation of rays, one vertical grid step at a time, is continued until some prescribed depth is reached. Computation of amplitudes is achieved using a method similar to that for obtaining the zero-order approximation in asymptotic ray theory. We show comparisons with a full-wave method on readily accessible 3-D velocity models.