Solving the eikonal equation for compressional and shear waves in anisotropic media using peridynamic differential operator

Ali Can Bekar, Erdogan Madenci*, Ehsan Haghighat, Umair Bin Waheed, Tariq Alkhalifah

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


Traveltimes of compressional (P) and shear (S) waves have proven essential in many earthquake and exploration seismology applications. An accurate and efficient traveltime computation for P and S waves is crucial for the success of these applications. However, solving the eikonal equation with a complex phase velocity field in anisotropic media is challenging. The eikonal equation is a first-order nonlinear hyperbolic partial differential equation. It represents the high-frequency asymptotic approximation of the wave equation. The fast marching and sweeping methods are commonly used due to their efficiency in numerically solving the 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 for the solution of the eikonal equation by employing the peridynamic differential operator (PDDO). The PDDO provides the non-local form of the eikonal equation by introducing an internal length parameter (horizon) and a weight function with directional non-locality. The operator is immune to discontinuities in the form of sharp changes in field or model variables and invokes the direction of traveltime in a consistent manner. The weight function controls the strength of association among points within the horizon. Solutions are constructed in a consistent manner without upwind assumptions through simple discretization. The robustness of this approach is established 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.

Original languageEnglish (US)
Pages (from-to)1942-1963
Number of pages22
JournalGeophysical Journal International
Issue number3
StatePublished - Jun 1 2022


  • Numerical modelling
  • Numerical solutions

ASJC Scopus subject areas

  • Geophysics
  • Geochemistry and Petrology


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