Multiple reflection and transmission phases in complex layered media using a multistage fast marching method

Traditional grid-based eikonal schemes for computing traveltimes are usually confined to obtaining first arrivals only. However, later arrivals can be numerous and of greater amplitude, making them a potentially valuable resource for practical applications such as seismic imaging. The aim of this paper is to introduce a grid-based method for tracking multivalued wavefronts composed of any number of reflection and refraction branches in layered media. A finite-difference eikonal solver, known as the fast marching method (FMM) is used to propagate wavefronts from one interface to the next. By treating each layer that the wavefront enters as a separate computational domain, one obtains a refracted branch by reinitializing FMM in the adjacent layer and a reflected branch by reinitializing FMM in the incident layer. To improve accuracy, a local grid refinement scheme is used in the vicinity of the source where wavefront curvature is high. Several examples are presented which demonstrate the viability of the new method in highly complex layered media. Even in the presence of velocity variations as large as 8:1 and interfaces of high curvature, wavefronts composed of many reflection and transmission events are tracked rapidly and accurately. This is because the scheme retains the two desirable properties of a single-stage FMM: computational speed and stability. Local grid refinement about the source also can increase accuracy by an order of magnitude with little increase in computational cost.