Seismic reflections from space-, time- and mixed-boundaries

Kristopher A. Innanen

It has been pointed out that if a homogeneous medium that is supporting a propagating wave were to suddenly undergo a change in medium properties, the propagating wave would immediately partition into reflected and transmitted components–exactly as if it had impinged on a spatial boundary. Bacot et al. in 2016 in Nature Physics published laboratory examples of this, referring to the change causing the reflection as a “time mirror”. Here we model this phenomenon numerically, speculate on a practical usage of it in monitoring reservoirs undergoing rapid pressure changes, and offer a modest extension of the theoretical description. In the extension we point out that the time-mirror reflections are essentially non-oblique, and then, motivated by the explorationists’ tendency to think about obliquely incident waves, we ask the question of how we could force a wave to impinge on a time-boundary at an angle. The answer requires the introduction of boundaries with both space- and time features. Upon setting up such a problem basic rules for reflection angles and transmission angles are derivable by appeal to Huygens’ principle.