Seismic imaging (migration) operators are non-unitary Claerbout (1992) because they depend on: (1) the seismic experiment acquisition geometry Duquet and Marfurt (1999); Nemeth et al. (1999); Ronen and Liner (2000), (2) the complex subsurface geometry Kuehl and Sacchi (2001); Prucha et al. (2000) and (3), the bandlimited characteristics of the seismic data Chavent and Plessix (1999). Often, they produce images with reflectors correctly positioned but with biased amplitudes.
Attempts to solve this problem have used the power of geophysical inverse theory Tarantola (1987), which compensates for the experimental deficiencies (acquisition geometry, obstacles, etc.) by weighting the migration result with the inverse of the Hessian. The main difficulty with this approach is the explicit calculation of the inverse of the Hessian. However, in most of the situations, the direct computation of its inverse for the entire model space is practically unfeasible.
Three different paths have been followed to practically approximate the inverse of the Hessian. The first approach approximates the Hessian as a diagonal matrix Chavent and Plessix (1999); Rickett (2003), which makes its inversion trivial. The second approach makes use of iterative algorithms like conjugate-gradient Duquet and Marfurt (1999); Kuehl and Sacchi (2001); Nemeth et al. (1999); Prucha et al. (2000); Ronen and Liner (2000) to implicitly calculate the inverse of the Hessian. The third approach Guitton (2004) approximates the inverse of the Hessian with a bank of nonstationary matching filters.
Since accurate imaging of reflectors is more important at the reservoir level, we propose calculating the Hessian in a target-oriented fashion. This can be done in practice, since the new dimensions of the Hessian (in the target region alone) are smaller than the dimensions of the whole image. By knowing the characteristics of the exact Hessian, an educated choice can be made regarding how to approximate its inverse.
In this paper, we first discuss how the target-oriented Hessian can be calculated from precomputed Green functions. We also show three numerical examples of target-oriented computed Hessians, the first in a constant velocity model, the second in the Sigsbee model (to study the effects of poor illumination in the Hessian), and the third in the Marmousi model.