According to the least-squares approach, the migration operator is constructed as a least-squares inverse of the forward Kirchhoff modeling Tarantola (1987). One can effectively approximate the inverse operator through an application of the conjugate-gradient technique. The conventional migration is then considered as the adjoint of the modeling operator, or, in other words, the first step of a conjugate-gradient iteration Claerbout (1992). A more accurate representation (i.e. additional conjugate-gradient steps) can compensate for irregularities and artifacts of irregular acquisition Nemeth et al. (1999); Nemeth (1996).
A blind least-squares approach cannot, however, compensate for lack of information in the input data. For example, if a particular area in the subsurface is not illuminated by reflection waves, a proper image of that area cannot be resolved by least-squares migration alone. In this case, part of the image will belong to the null space of the least-squares inverse problem. Spotting low-illumination areas is important both for making acquisition decisions and for evaluating the uncertainty of the existing images. Duquet et al. (1998) have proposed to use the inverse diagonal of the Hessian matrix as a measure of illumination in Kirchhoff imaging. Although this measure does provide useful information about the problem's well-posedness, a more rigorous approach to the solution uncertainty would be to estimate the corresponding model resolution operator Jackson (1972).
As shown by Berryman and Fomel (1996), the model resolution matrix can be estimated in an iterative manner. The matrix approximation is constructed from the vectors, already appearing in the conjugate-gradient iteration. Therefore, it requires minimal additional computation with respect to an iterative least-squares inversion. The diagonal of the resolution matrix can serve as a rough direct estimate of the model uncertainty. A similar, although less efficient approach, was proposed by Minkoff (1996) and Yao et al. (1999), who applied it in conjunction with the LSQR method Paige and Saunders (1982).
In this paper, we apply the iterative technique of Berryman and Fomel (1996) for resolution estimation in Kirchhoff imaging. Synthetic and real data tests show that a resolution estimate can indeed provide valuable information about the uncertainty of Kirchhoff images and reveal image areas with illumination problems.