Frequency-Dependent Spherical-Wave Reflection in Acoustic Media: Analysis and Inversion

Spherical-wave reflectivity (SWR), which describes the seismic wave reflection in real subsurface media more accurately than plane-wave reflectivity (PWR), recently, again attracts geophysicists' attention. The recent studies mainly focus on the amplitude variation with offset/angle (AVO/AVA) attributes of SWR. For a full understanding of the reflection mechanism of spherical wave, this paper systematically investigates the frequency-dependent characteristics of SWR in a two-layer acoustic medium model with a planar interface. Two methods are used to obtain SWR. The first method is through the calculation of classical Sommerfeld integral. The other is by 3D wave equation numerical modeling. To enhance computation efficiency, we propose to perform wave equation simulation in cylindrical coordinates, wherein we for the first time implement unsplit convolutional perfectly matched layer as the absorbing boundary. Both methods yield the same results, which demonstrate the validity and accuracy of the computation. From both the numerical tests and the theoretical demonstration, we find that the necessary condition when frequency dependence of SWR occurs is that the upper and lower media have different velocities. At the precritical small angle, the SWR exhibits complicated frequency-dependent characteristics for varying medium parameters. Especially when the impedance of upper medium equals that of lower one, the PWR is zero according to geometric seismics. Whereas the SWR is nonzero: the magnitude of SWR decreases with growing frequency, and approaches that of the corresponding PWR at high frequency; the phase of SWR increases with growing frequency, but approaches 90 degrees or -90 degrees at high frequency. At near-and post-critical angles, large difference exists between SWR and PWR, and the difference is particularly great at low frequencies. Finally, we propose a nonlinear inversion method to estimate physical parameters and interface depth of media by utilizing the frequency-dependent SWR at a precritical small angle. Tests show that good inversion results can be obtained when the input data contain both low and moderate frequencies.