Topography-dependent eikonal equation and its solver for calculating first-arrival traveltimes with an irregular surface

First-arrival traveltime is commonly used in problems that involve static correction, pre-stack migration, earthquake location, seismic tomography, etc. The classical eikonal equation discretized with regular rectilinear grids is effective for calculating the first-arrival traveltimes in a rectangular domain, but is less efficient for an Earth model that has an irregular surface. Here, we present a topography-dependent eikonal equation in 2-D that makes use of the wave equation in a curvilinear coordinate system, and is equivalent to a direct derivation of the classical eikonal equation together with a transformation from Cartesian to curvilinear coordinates. The topography-dependent eikonal equation is reduced to the classical version when the surface is flat. The topography-dependent equation (in the curvilinear coordinate system) displays the mathematical form of an anisotropic eikonal equation (even though the medium is isotropic in the Cartesian coordinate system). Then, we use a Lax-Friedrichs sweeping scheme, which has been developed as an iterative method for Hamilton-Jacobi equations, to approximate the viscosity solutions (first-arrival traveltimes) of the topography-dependent eikonal equation formulated in the curvilinear coordinate system. Several numerical experiments performed with different models illustrate that the method is stable and accurate in calculating seismic traveltimes with an irregular (non-flat) surface.