A Structure-Preserving Scheme for Computing the Gradient of the Eikonal Equation

Abstract

We propose a first-order, structure-preserving method for directly computing the normalized traveltime gradient of the eikonal equation in smooth first-arrival regions. The method combines analytical derivatives of spherical linear interpolation (SLERP) in a Godunov-type upwind discretization with spherical forward Euler (SFE) pseudo-time integration. This construction gives a tangent residual and keeps every iterate on $\mathbb{S}^2$ without projection or normalization. The SLERP discretization exactly preserves a family of curved orientation equilibria. Removing the background discretization bias improves absolute orientation accuracy near these equilibria, allowing a prescribed error tolerance to be reached on a coarser grid. We prove the equilibrium-preservation property and, separately, a smaller angular error for SFE than normalized Euler in a one-cell relaxation under the stated step-size condition. Two- and three-dimensional experiments examine convergence, geometric preservation, and computational cost. Comparisons with gradient recovery from scalar traveltimes show more accurate directions on the same grid in the curved media considered. The method provides direction fields for wave propagation and geometric optics, with potential use in adjoint-based inverse problems.