Efficient seismic ray tracing based on the shortest path method

Limiting the number of valid nodes around a reference ray can greatly reduce the calculation time of the shortest-path method (SPM). The calculation is executed by increasing the number of cells and/or nodes in the target area, step by step, until the rays converge. The ray obtained in the previous step is used as a reference ray, and the initial reference rays are given by the ordinary SPM. The Dijkstra algorithm and binary heap sorting method are used, as in the ordinary SPM. As the cell and node numbers increase, the calculation time for the modified SPM (mSPM) is reduced compared to that for the ordinary SPM. In the 3-D (100 x 100 x 100 km(3)) checkerboard velocity pattern model, the relative calculation time becomes two to four orders of magnitude smaller. The calculation time for the mSPM itself is approximately proportional to E log(2)(V), where E is the edge (ray path segment) number and V is the vertex (node) number, as seen in the heap sorting algorithm. The mean traveltime and ray path differences between the mSPM and pseudo-bending method (PBM) are small, less than 0.005 s and around 0.6 km, respectively, and slightly larger than those with respect to the ordinary SPM. The total differences from the exact solution are estimated to be less than 0.01 s and 1.0 km, which are sufficiently small for traveltime tomography. The traveltime and ray path can be improved by utilizing iterative calculations, shifting of the starting point and more neighbour nodes. The ray path obtained by the mSPM can be a local minimum, according to velocity models. The relative mean traveltime and ray path differences between adjacent cells and nodes generally show trends similar to those of the traveltime and ray path differences from the PBM. Hence, these relative differences can help reveal the behaviour of the differences from the PBM.