I use the local ray-tracing method to compute the traveltime field in a two-layer velocity model. The velocities are 1.5 km/s in the top layer and 4.5 km/s in the bottom layer. After the critical angle, each point in the top layer has two arrivals: an arrival propagating directly from the source and an arrival refracted at the boundary of two layers. Therefore, the traveltime field is multi-valued in this region. I compute the traveltime field of multiple arrivals by applying the local ray-tracing method with three different selections: the first one selects the first arrivals, the second one selects the most energetic arrivals, and the third one selects upgoing arrivals. Figure shows the contours of the three single-valued traveltime field. In the top layer, the first arrivals are the combinations of direct arrivals and diffracted arrivals, the most energetic arrivals are purely direct arrivals, and the upgoing arrivals have more diffracted arrivals than the first arrivals. Three traveltime fields are identical in the bottom layer because each point there has only one arrival. The discontinuities of the contours are easy to identify. Figure compares these results calculated by the local ray-tracing method with the wave field simulated by wave-equation modeling. Because a minimum phase wavelet is used in wave-equation modeling, the contour lines of the traveltime fields closely follow the first breaks of the wavelets in the corresponding snapshots of the wave fields.
As a final experiment, I compute the traveltime field in the Marmousi model shown in Figure . This synthetic velocity model has a complex geological structure with large velocity contrasts. Many regions in the model have multiple arrivals. The computation is conducted on a triangular grid with a grid size of 16 meters. Figure shows the traveltime field of first arrivals in the Marmousi model. From the contour plot, we see that the circular wavefronts generated at the source point are severely distorted after propagate through the complex structure. The wavefronts propagate rapidly in the thin layers of high interval velocities. The comparison with the wave field computed by wave-equation modeling confirms accurate result of the local ray-tracing method. This experiment also shows that the local ray-tracing method is robust.
Figure shows that the first arrivals at some regions are considerably weaker than later arrivals. Therefore, I compute another traveltime field by selecting most energetic arrivals. As shown in Figure, the result on this model is not completely satisfactory. Although energetic arrivals at the shallow regions of the model are correctly computed, the method still picks up the weak arrivals diffracted at the boundaries of high-velocity thin layers. An explanation of this result is that the local ray-tracing method does not ensure global selections. The weak arrival propagating in the high-velocity thin layer may be strong at the early stage of the propagation.