A practical grid-based method for tracking multiple refraction and reflection phases in three-dimensional heterogeneous media

Marthijn de Kool, Nicholas Rawlinson, Malcolm Sambridge

Research output: Contribution to journalArticlepeer-review

157 Citations (Scopus)


We present a practical grid-based method in 3-D spherical coordinates for computing multiple phases comprising any number of reflection and transmission branches in heterogeneous layered media. The new scheme is based on a multistage approach which treats each layer that the wave front enters as a separate computational domain. A finite-difference eikonal solver known as the fast-marching method (FMM) is reinitialized at each interface to track the evolving wave front as either a reflection back into the incident layer or a transmission through to the adjacent layer. Unlike the standard FMM, which only finds first arrivals, this multistage approach can track those later arriving phases explicitly caused by the presence of discontinuities. Notably, the method does not require an irregular mesh to be constructed in order to connect interface nodes to neighbouring velocity nodes which lie on a regular grid. To improve accuracy, local grid refinement is used in the neighbourhood of a source point where wave front curvature is high. The method also provides a way to trace reflections from an interface that are not the first arrival (e.g. the global PP phase). These are computed by initializing the multistage FMM from both the source and receiver, propagating the two wave fronts to the reflecting interface, and finding stationary points of the sum of the two traveltime fields on the reflecting interface.

A series of examples are presented to test the efficiency, accuracy and robustness of the new scheme. As well as efficiently computing various global phases to an acceptable accuracy through the ak135 model, we also demonstrate the ability of the scheme to track complex crustal phases that may be encountered in coincident reflection, wide-angle reflection/refraction or local earthquake surveys. In one example, a variety of phases are computed in the presence of a realistic subduction zone, which includes several layer pinch-outs and a subducting slab. Our numerical tests show that the new scheme is a practical and robust alternative to conventional ray tracing for finding various phases in layered media at a variety of scales.
Original languageEnglish
Pages (from-to)253-270
JournalGeophysical Journal International
Issue number1
Publication statusPublished - Oct 2006


  • fast-marching method
  • inite-difference methods
  • ray tracing
  • seismic phases
  • seismic wave propagation
  • traveltime


Dive into the research topics of 'A practical grid-based method for tracking multiple refraction and reflection phases in three-dimensional heterogeneous media'. Together they form a unique fingerprint.

Cite this