Frequency-dependent spherical wave reflectivity

Spherical wave reflectivity (SWR), a more precise description of seismic wave reflection in real subsurface media than plane wave reflectivity (PWR), recently again attracts geophysicists’ attention. However, the studies mainly focus on the AVO attributes of SWR and the frequency-dependent characteristics are neglected. In this paper, frequency-dependent SWR is investigated in two semi-infinite media with a planar interface for varying parameters. We employ classical Sommerfeld-integral to construct SWR, and use adaptive Gauss-Kronrod quadrature to compute it. Tests show that although SWR and PWR are almost the same at high frequency as the classic geometric ray theory declares, there exists obvious difference between them at low source frequency. Noticeably, if velocities of both media are equivalent, SWR just equals PWR, meaning that SWR here is frequency-independent. However, when the velocity of upper medium is different from that of lower one, which is the necessary condition for SWR dependence on frequency, frequency-dependent SWR varies with incidence angle, physical properties of the medium, and wave propagation distance. Small velocity contrast, large upper medium density, and short propagation distance lead to great deviation between SWR and PWR. The above properties may be used to improve and extend current AVO analysis.