Modeling seismic arrivals as inelastically colliding objects

Kristopher A. Innanen

It has been pointed out in several CREWES reports that if we treat the depth of a VSP geophone as an effective time coordinate, and the VSP time coordinate as an effective position coordinate, then the arrivals in the data behave, quantitatively and qualitatively, as the space-time trajectories of objects that move through space, colliding and sticking together. It was also suggested that, if the modeling of these collisions was properly relativistic, i.e., involved ideas objects under proper constant acceleration, then an offset-independent description seemed possible. In this paper we confirm and work through the mathematics for a simple, single interface model. Interestingly, we observe a high degree of accuracy when comparing the two types of model, higher when the velocity contrast across the boundary is small, and with what appears to be essentially an exact result when the offset is equal to the depth of the interface.