Seismic imaging using non-unitary migration operators Claerbout (1992) often produces 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 apply inversion theory Tarantola (1987), where the image can be obtained by convolving 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.
To reduce the size of the problem Valenciano and Biondi (2004) proposed to compute the Hessian in a target-oriented fashion. By explicitly computing the Hessian, the zero-offset inverse image can be obtained using an iterative inversion algorithm Valenciano et al. (2005a,b). This approach renders unnecessary an explicit computation of inverse of the Hessian matrix.
In this paper we show an example of the computation of the wave-equation angle-domain Hessian. It was defined by Valenciano and Biondi (2005) via an offset-to-angle transformation Fomel (2004); Sava and Fomel (2003). Since the angle-domain Hessian matrix is explicitly computed, a better understanding of the uneven illumination in complex subsurface scenarios can be gained. We illustrate our method by computing the Hessian for a model that contains one reflector under a Gaussian velocity anomaly which creates uneven illumination.