A conventional amplitude variation with offset (AVO) inversion is based on geometrical seismics which exploit plane-wave reflection coefficients to describe the reflection phenomenon. Widely exploited linearizations of plane-wave coefficients are mostly valid at pre-critical offsets for media with almost flat and weak-contrast interfaces. Existing linearizations do not account for the seismic frequency range by ignoring the frequency content of the wavelet, which is a strong assumption. Plane-wave reflection coefficients do not fully describe the reflection of seismic waves at near-critical and post-critical offsets, because reflected seismic waves are typically generated by point sources. We propose an improved approach to AVO inversion, which is based on effective reflection coefficients (ERCs). ERCs generalize plane-wave coefficients for seismic waves generated by point sources and therefore more accurately describe near-critical and post-critical reflections where head waves are generated. Moreover, they are frequency-dependent and incorporate the local curvatures of the wavefront and the reflecting interface. In our study, we neglect the effect of interface curvature and demonstrate the advantages of our approach on synthetic data for a simple model with a plane interface separating two isotropic half-spaces. A comparison of the inversion results obtained with our approach and the results from an AVO inversion method based on the exact plane-wave reflection coefficient suggests that our method is superior, in particular for long-offset ranges which extend to and beyond the critical angle. We thus propose that long offsets can be successfully exploited in an AVO inversion under the correct assumption about the reflection coefficient. Such long-offset AVO inversion shows the potential of outperforming a conventional moderate-offset AVO inversion in the accuracy of estimated model parameters.

Widely exploited in the industry, amplitude-variation-withoffset (AVO) inversion techniques are based on weak-contrast approximations of the plane-wave reflection coefficients. These approximations are valid for plane waves reflected at almost flat interfaces with weak contrasts in seismic parameters and for reflection angles below the critical angle. Regardless of the underlying assumptions, linearized coefficients provide a simple and physically adequate tool to accurately invert AVO data for seismic parameters at precritical angles. However, the accuracy of linearized coefficients drastically decreases with increasing incidence angle. Limitations occur around and beyond the critical ray, where the effect of wavefront curvature becomes prominent and thus can no more be neglected. The effective reflection coefficients generalize the plane-wave reflection coefficients for waves generated by point sources and reflected at curved interfaces. They account for the wavefront curvature and are adequate at any incidence angle. Our previous studies have shown that including the reflections around and beyond the critical angle in the AVO inversion significantly improves the accuracy of estimated parameters. However, the interface curvature also must have its contribution to the longoffset AVO inversion. We find that the interface curvature affects the energy propagation along the ray tube and the energy diffusion across the ray tube. The energy propagation along the tube is characterized by the geometrical spreading, which is strongly affected by interface curvature. The transverse diffusion is captured by the effective reflection coefficients which are less influenced by interface curvature. The long-offset AVO inversion is thus sensitive to interface curvature through a combination of several wave propagation factors.

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