Head-waves and first arrivals

Finite difference solutions of the eikonal equation provide us with continues wavefront surfaces that correspond to the fastest traveling waves. In the presence of inhomogeneity, such wavefront surfaces might include low-curvature regions corresponding, in some cases, to head-waves that often travel faster than the direct waves. Head-waves are low-energy arrivals that are not useful for imaging applications (). The presence of the high velocity salt body in the SEG/EAGE model has intensified the head-waves, and the fast low-energy arrivals problem.

The salt body, because of its very high velocity, acts as a large secondary source that emanates waves from its surface, practically, in all directions. Such waves, typically, have low energy; however, due to their speed through the salt body, they become the fastest arrivals in a big portion of the solution. Most of these waves appear in the solution of the eikonal equation using finite difference methods. Some of these waves correspond to head-waves; others are just low-energy solutions. The energy weakness of such waves is a result of the amount of geometrical spreading they experienced while traveling in the salt body. Typically, the only desired waves emanating through the salt are the ones that travel downward to regions not accessible directly by direct waves.

Figure  shows two types of low-energy arrivals. On the left, we see headwaves emanating from the top of the salt structure. These waves travel partly with the salt wave velocity, and thus beat direct arrivals to areas directly above the salt. On the right of Figure , we see another arrival that penetrated through the salt body and is thus faster than the direct arrivals. The penetration through the salt body, however, has lowered the energy of such arrivals, causing the wavefront to have low curvature, among other things.