  The traveltime of compressional (P) and shear (S) waves have proven essential
in many applications of earthquake and exploration seismology. An accurate and
efficient traveltime computation for P and S waves is crucial for the success
of these applications. However, solutions to the Eikonal equation with a
complex phase velocity field in anisotropic media is challenging. The Eikonal
equation is a first-order, hyperbolic, nonlinear partial differential equation
(PDE) that represents the high-frequency asymptotic approximation of the wave
equation. The fast marching and sweeping methods are commonly used due to their
efficiency in numercally solving Eikonal equation. However, these methods
suffer from numerical inaccuracy in anisotropic media with sharp heterogeneity,
irregular surface topography and complex phase velocity fields. This study
presents a new method to solving the Eikonal equation by employing the
peridynamic differential operator (PDDO). The PDDO provides the nonlocal form
of the Eikonal equation by introducing an internal length parameter (horizon)
and a weight function with directional nonlocality. The operator is immune to
discontinuities in the form sharp changes in field or model variables and
invokes the direction of traveltime in a consistent manner. The weight function
controls the degree of association among points within the horizon. Solutions
are constructed in a consistent manner without upwind assumptions through
simple discretization. The capability of this approach is demonstrated by
considering different types of Eikonal equations on complex velocity models in
anisotropic media. The examples demonstrate its unconditional numerical
stability and results compare well with the reference solutions.
