Frequency-dependent inversion based on spherical-wave reflection coefficient in elastic medium: Theory and methodology

Plane-wave theory based on the Zoeppritz's equations and its various approximations have been extensively utilized in seismic processing and interpretation. However, localized sources used in field acquisition usually generate spherical wave. As a zero-order approximation of spherical reflected wave, plane reflected wave varies with seismic offset, which can only use amplitude information of multiple pre-critical offsets to estimate elastic parameters. While, the change of spherical wavefront curvature across reflected interfaces causes the amplitude and phase of spherical reflected wave to vary with seismic frequency and offset. Therefore, in addition to conducting conventional amplitude variation versus with offset inversion using a single frequency of multiple offsets, spherical reflected wave makes frequency-dependent inversion using multiple low frequencies of a single offset possible. Herein, we derive the integral expression of spherical wave reflection coefficient from the general solution of wave equation and verify the consistency of displacements of reflected wave simulated by numerical integral and finite difference method in elastic medium. Subsequently, we analyze and investigate the feasibility of frequency-dependent inversion using spherical wave reflection coefficient of a single offset in synthetic data with a planar reflection interface. Combining a global optimal algorithm, low frequency-dependent inversion based on spherical wave reflection coefficient can offer accurate velocity and density parameters for further seismic interpretation.