A novel expression of the spherical-wave reflection coefficient at a plane interface

The spherical-wave reflection coefficient (SRC) describes the reflection strength when seismic waves emanating from a point source impinge on an interface. In this study, the SRC at a plane interface between two infinite half-spaces is investigated. We derive an analytical equation of the SRC when kR -> 0 (k is the wave number and R is the wave propagation distance). It only depends on the density ratio; it is independent of the velocity ratio and incidence angle. On the other hand, we find that the SRCs at different kR lie along an elliptical curve on the complex plane (the complex plane is a geometric representation of the complex numbers established by the real axis and perpendicular imaginary axis). Based on this feature, we construct a new analytical equation for the reflected spherical wave with high accuracy, which is applicable to both small and large kR. Using the elliptical distribution of the SRCs for a series of frequencies recorded at only one spatial location, the density and velocity ratios can be extracted. This study complements the spherical-wave reflection theory and provides a new basis for acoustic parameters inversion, particularly density inversion.