S U M M A R YThe fast marching method (FMM) is a grid based numerical scheme for tracking the evolution of monotonically advancing interfaces via finite-difference solution of the eikonal equation. Like many other grid based techniques, FMM is only capable of finding the first-arriving phase in continuous media; however, it distinguishes itself by combining both unconditional stability and rapid computation, making it a truly practical scheme for velocity fields of arbitrary complexity. The aim of this paper is to investigate the potential of FMM for finding later arriving phases in layered media. In particular, we focus on reflections from smooth subhorizontal interfaces that separate regions of continuous velocity variation. The method we adopt for calculating reflected phases involves two stages: the first stage initializes FMM at the source and tracks the incident wave front to all points on the reflector surface; the second stage tracks the reflected wave front by reinitializing FMM from the interface point with minimum traveltime. Layer velocities are described by a regular grid of velocity nodes and layer boundaries are described by a set of interface nodes that may be irregularly distributed. A triangulation routine is used to locally suture interface nodes to neighbouring velocity nodes in order to facilitate the tracking of wave fronts to and from the reflector. A number of synthetic tests are carried out to assess the accuracy, speed and robustness of the new scheme. These include comparisons with analytic solutions and with solutions obtained from a shooting method of ray tracing. The convergence of traveltimes as grid spacing is reduced is also examined. Results from these tests indicate that wave fronts can be accurately tracked with minimal computational effort, even in the presence of complex velocity fields and layer boundaries with high curvature. Incident wave fronts containing gradient discontinuities or shocks also pose no difficulty. Further development of the wave front reinitialization scheme should allow other later arrivals such as multiples to be successfully located.

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.

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