Spherical coordinates are the most natural coordinate system in which to solve the eikonal equation in the case of a point source. The radiation of energy from a point source has an overall spherical shape, even in complex media. The fast marching method of solving the eikonal equation, though fast and unconditionally stable, is based on a first-order approximation of the traveltime derivatives. These approximations yield poor results at 45-degree angler wave propagation and at highly curved wavefronts. In Cartesian coordinates, such errors accumulate near the source where the curvature of the wavefront is at its highest. For a point source, both the curvature and the 45-degree angler propagation is reduced in polar coordinates. Even in complex media, wavefronts originating from a point source spend less time traveling diagonally with respect to the polar coordinate system than the Cartesian coordinate system. The Marmousi model is a prime example of the benefits of the polar coordinate system for solving the eikonal equation.

10/9/1997