Implementing the fast marching eikonal solver: Spherical versus Cartesian coordinates

Spherical coordinates are a natural orthogonal system to describe wavefronts emanating from a point source. While a regular grid distribution in the Cartesian coordinate system tends to undersample the wavefront description near the source (the highest wavefront curvature) and oversample it away from the source, spherical coordinates, in general, provide a more balanced grid distribution to characterize such wavefronts. Our numerical implementation confirms that the recently introduced fast marching algorithm is both a highly efficient and an unconditionally stable eikonal solver. However, its first-order approximation of traveltime derivatives can induce relatively large traveltime errors for waves propagating in a diagonal direction with respect to the coordinate system. Examples, including the infamous Marmousi model, show that a spherical coordinate implementation of the method results in far fewer errors in traveltime calculation than the conventional Cartesian coordinate implementation, and with practically no loss in computational advantages.