Abstract: When using the grid methods, such as the finite difference method for solving the eikonal equation satisfied by seismic traveltimes, the traveltimes obtained at the adjacent grid points may not have the smoothness needed for computing ray trajectories. This is because of the discretization of the velocity model by grids. To solve this problem, some researchers proposed a backward gradient scheme based on a B-spline interpolation. However, the ray trajectories calculated by the B-spline interpolation scheme will have large errors when the velocity changes abruptly in some model region. Aiming at this problem, instead of the B-splines, we first utilized the Chebyshev polynomials, which are optimal for solving the zero deviation and the least square problem, to obtain the formulas that can optimally approximate the grid point traveltimes resulted from the finite different solution of the eikonal equation. Then, we used a computation procedure which is similar to that used in the B-spline scheme for obtaining smooth ray trajectories. The numerical test showed that the use of Chebyshev polynomials for approximating traveltimes led to smooth ray trajectories of multiply reflected waves with high accuracy. It should be investigated further in the future work.

Key words: multiple reflections, ray traveltimes, ray trajectories, traveltimes approximation, Chebyshev polynomials