Seismic imaging using non-unitary migration operators Claerbout (1992) often produce images with reflectors correctly positioned but biased amplitudes Chavent and Plessix (1999); Duquet and Marfurt (1999); Nemeth et al. (1999); Ronen and Liner (2000). One way to solve this problem is to use the inversion formalism introduced by Tarantola (1987) to solve geophysical imaging problems, where the image can be obtained by weighting the migrated image with the inverse of the Hessian matrix. However, when the dimensions of the problem get large, the explicit calculation of the Hessian matrix and its inverse becomes unfeasible.
Valenciano and Biondi (2004) proposed computing the Hessian in a target-oriented fashion to reduce the size of the problem. The zero-offset inverse image can be estimated as the solution of a non-stationary least-squares filtering problem, by means of a conjugate gradient algorithm Valenciano et al. (2005a,b). This approach, renders unnecessary an explicit computation of inverse of the Hessian matrix.
In this paper, we define the wave-equation angle-domain Hessian from the subsurface offset wave-equation Hessian via an angle-to-offset transformation following the Sava and Fomel (2003) approach. To perform the inversion, the angle-domain Hessian matrix can be used explicitly or, implicitly as a chain of the offset-to-angle operator and the subsurface offset Hessian matrix.
The definition of the wave-equation angle-domain Hessian allows the angle-domain regularization required to stabilize the wave equation inversion problem Kuehl and Sacchi (2001); Prucha et al. (2000). It also allows to obtain a prestack inverse image, adding the possibility of doing amplitude vs. angle (AVA) analysis for reservoir characterization.