ORALS

Traveltimes due to compressional (P) and shear (S) waves have been proven essential in many applications of earthquake seismology. Therefore, an accurate and efficient traveltime computation approach for P and S waves is essential for successful applications. However, construction of a solution to the Eikonal equation with a complex velocity field in an anisotropic medium is challenging. The Eikonal equation is a first-order, hyperbolic, nonlinear partial differential equation (PDE) that represents a high-frequency approximation of the wave equation. The fast marching method (FMM) and the fast sweeping method (FSM) are the most accepted techniques due to their efficiency for the solution of the Eikonal equation. However, these methods tend to suffer from numerical accuracy in the presence of anisotropic media with sharp heterogeneity, irregular surface topography and complex velocity fields. In order to overcome these difficulties, this study presents a solution method to the Eikonal equation by employing the peridynamic differential operator (PDDO) [1-3]. The PDDO provides the nonlocal form of the Eikonal equation by introducing an internal length parameter (horizon) and a weight function with directional nonlocality. It is immune to discontinuities and invokes the direction of information travel in a consistent manner. It enables numerical differentiation through integration; thus, the field equations are valid everywhere regardless of the presence of discontinuities. The weight function controls the degree of association among points within the horizon. Also, it enables directional nonlocality based on the knowledge of characteristic directions along which information travels. Solutions are constructed in a consistent manner without special treatments through simple discretization. The capability of this approach is demonstrated by considering different types of Eikonal equations with a complex velocity field in anisotropic media. Numerical stability is ensured and solutions compare well with the reference solutions.

References:
[1] E. Madenci, A. Barut, and M. Futch, Peridynamic differential operator and its applications, Comp. Meth. App. Mech. Engng. 304, 408-451, 2016.
[2] E. Madenci, A. Barut, M. Dorduncu and M. Futch, Numerical solution of linear and nonlinear partial differential equations by using the peridynamic differential operator, Num. Meth. Part. Diff. Eqs. 33, 1726–1753, 2017.
[3] E. Madenci, A. Barut, and M. Dorduncu, Peridynamic differential operators for numerical analysis, Springer, Boston MA, 2019.