Efficient iterative solutions for the anisotropic eikonal equation in seismic wave propagation analysis

Abstract The computation of traveltimes for compressional (P) waves is significantly important for earthquake and seismic exploration applications. The eikonal equation, functioning as a fundamental tool in seismic wave propagation analysis, plays a crucial role in the computation of traveltimes and the construction of seismic images. However, the inherent nonlinearity of the eikonal equation introduces complexity, particularly when addressing anisotropic media. The eikonal equation derived directly from Christoffel equation is applicable for general anisotropic traveltime calculations. However, its equation is a complex quartic equation, presenting significant challenges in the solving process. The selection of a fast and efficient iterative method is important in addressing these challenges. Additionally, determining the initial point for iterative method is also crucial for their stability and effectiveness. Finally, three numerical examples are presented to test the bisection, Newton, and secant methods for solving anisotropic equations, based on the characteristics of locally constructed solutions.