Raytracing based on the Sympletic algorithm

Abstract

The raytracing is an essential problem in seismic travel-time tomography. The reliability of its result relied on both spatial trajectory and travel-time of the ray, which is widely used in seismology. Since the raytracing in Cartesian coordinate system is not suit for regional and globe seismic tomography, i.e., we should consider the eccentricity of the earth when we build the initial velocity model and interface structure. Three-dimensional seismic ray tracing in heterogeneous spherical is an effective way to avoid the affected of Earth shape, thus we introduce the Hamiltonian structure to approach the eiknoal equation [1] in spherical polar coordinate (r, θ, ϕ) system. Then we can apply the symplectic algorithm to solving the Hamiltonian system [2] of ray equations. In this paper, we apply the Sympletic algorithm method (SAM) with bi-cubic convolution algorithm [3] to solve the Hamilton System in the raytracing problem. Compared with the Runge-Kutta method, the result shows that SAM can keep the stability of the solution for the eikonal equation. Due to the use of SAM, it can produce a reliable seismic wavefront with an accurate ray trajectory. Meanwhile, the numerical modeling shows that the SAM can not only keep the stability of the Hamilton System with a fast computation but also improve the precisely of the seismic waves propagation.