Seismic ray tracing has many applications and the essential information we try to get from ray tracing is the traveltime. In mining, (time lapsed) traveltime tomography gives a quick and good enough information about velocity structure within the rock. So you can tell where the damage has occurred and where the orebody is located, etc. In mining and in petroleum, automated micro-earthquake locating is a hot topic. You can see (or listen) where the fractures are propagating. However, ray tracing is not easy; it is nonlinear and the first arrival travel time field is not smooth, i.e., even the first order gradient doesn’t exist everywhere.

I propose an alternative method to get the traveltime without tracing the rays: fast sweeping method, a type of finite different method to solve the eikonal equation for traveltime field. It works on tetrahedral elements as well and can handle anisotropic velocity.

Actually there is more information you can get when doing the old school ray tracing, i.e.,the incident angle, the emerging angle, and the traveling distance. This info is not available after solving the eikonal equation but as byproduct after the ray tracing. How to cope with that? Reverse tracing the rays from arbitrary point other than the source point back to the source point is the solution. Here is how:

(1) solve the Eikonal equation
(2) choose the receiver location as the ray initial point
(3) trace the ray backwards using the travel time field till it reaches the element where the source is located.
(4) since the ray path is available, incident angle, emerging angle, and traveling distance can be obtained. Done.

Here is the results:

3D ray tracing based on the first arrival traveltime field. The red dot is the source and red lines shows the rays from the source to the receivers.

Here is the ray skeleton (without the isosurface). It shows direct arrivals, head wave arrivals and refraction arrivals:

3D ray tracing based on the first arrival traveltime field. The red dot is the source and the red lines are the rays. The black dots are the location of receivers.

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