Seismic AVO and the inverse Hessian in precritical reflection full waveform inversion

We analyse the role played by amplitude-variation-with-offset (AVO) information in the construction of full waveform inversion (FWI) updates from pre-critical seismic reflection data. A mixed continuous-discrete formulation of frequency-domain FWI is found to conveniently expose issues such as parameter cross-talk to quantitative interpretation. Two types of approximate inverse Hessian operator, differing in which off-diagonal elements are retained and which are neglected, are derivable within this formulation. The first, which emphasizes correlations between different parameters collocated in space, is referred to as an approximation of parameter-type. The second, which emphasizes correlations between spatially separated elements, is referred to as an approximation of space-type. The two approximations are such that, if they are both made simultaneously, the result is an update involving only the diagonal elements of the Hessian. The parameter-type inverse Hessian approximation appears to be capable of performing the kind of data manipulations through which linearized AVO-inversion and inverse scattering methods avoid cross-talk. This is confirmed with an analytic calculation of the first iteration in the reconstruction of a single acoustic boundary. The parameter-type FWI update, in this special case, reduces to linear admixtures of reflection coefficient versus angle which are consistent with those produced by direct AVO inversion. The parameter-type components in the full inverse Hessian represent, in this sense, the generalization of standard linearized AVO inversion within FWI.