Robust and efficient upwind finite‐difference traveltime calculations in three dimensions

Abstract

First‐arrival traveltimes in complicated 3-D geologic media may be computed robustly and efficiently using an upwind finite‐difference solution of the 3-D eikonal equation. An important application of this technique is computing traveltimes for imaging seismic data with 3-D prestack Kirchhoff depth migration. The method performs radial extrapolation of the three components of the slowness vector in spherical coordinates. Traveltimes are computed by numerically integrating the radial component of the slowness vector. The original finite‐difference equations are recast into unitless forms that are more stable to numerical errors. A stability condition adaptively determines the radial steps that are used to extrapolate. Computations are done in a rotated spherical coordinate system that places the small arc‐length regions of the spherical grid at the earth’s surface (z = 0 plane). This improves efficiency by placing large grid cells in the central regions of the grid where wavefields are complicated, thereby maximizing the radial steps. Adaptive gridding allows the angular grid spacings to vary with radius. The computation grid is also adaptively truncated so that it does not extend beyond the predefined Cartesian traveltime grid. This grid handling improves efficiency. The method cannot compute traveltimes corresponding to wavefronts that have “turned” so that they propagate in the negative radial direction. Such wavefronts usually represent headwaves and are not needed to image seismic data. An adaptive angular normalization prevents this turning, while allowing lower‐angle wavefront components to accurately propagate. This upwind finite‐difference method is optimal for vector‐parallel supercomputers, such as the CRAY Y-MP. A complicated velocity model that generates turned wavefronts is used to demonstrate the method’s accuracy by comparing with results that were generated by 3-D ray tracing and by an alternate traveltime calculation method. This upwind method has also proven successful in the 3-D prestack Kirchhoff depth migration of field data.