A hybrid fast sweeping method for the isotropic eikonal equation

Eikonal solvers in the Cartesian coordinates often suffer from source singularity due to the plane-wave assumption for the wavefront near the source point. Traveltime errors induced by the source singularity near the source point will spread to the whole domain and reduce the accuracy of subsequent traveltimes. The source singularity can be avoided if the eikonal equation is formulated and solved in the polar/spherical coordinates. However, the grid in the polar/spherical coordinates will cause sparse non-uniform resampling in the region far away from the source point after the traveltime is transformed back to the Cartesian mesh, hence reducing the uniform accuracy in this region. To deal with the source singularity and maintain uniform accuracy, we introduce a hybrid fast sweeping method that computes the traveltime near the source in the polar/spherical coordinates and computes the traveltime far away from the source in the Cartesian coordinates. The source singularity near the source point is resolved in the polar/spherical coordinates, and the uniform accuracy is achieved by switching to the Cartesian coordinates away from the source point. Numerical examples are presented to demonstrate the method.