Seismic interferometry and stationary phase at caustics

Waves that propagate between two receivers can be extracted by cross correlating noise recorded at these receivers if the noise is generated by sources on a closed surface surrounding the receivers. This concept is called seismic interferometry. Of all these noise sources, those for whom the travel time difference for propagation to the two receivers is in the stationary phase zone, give the dominant contribution. In this paper we analyze the stationary phase properties when one receiver is at a caustic for waves leaving the other receiver. A simple model for a waveguide and a general treatment of caustics show that at a caustic the curvature of the travel time difference at the stationary phase zone vanishes. As a result the stationary phase region is considerably wider at a caustic than at other points and it is more likely that noise sources are present in the stationary phase region. This explains why it is easier to extract waves from seismic interferometry when the receivers are at caustics such as antipodal receivers on the earth.