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 (e.g., acquisition geometry, obstacles) by weighting the migration result with the inverse of the Hessian. However, the main difficulty with this approach is the explicit calculation of the Hessian and its inverse.
Since accurate imaging of reflectors is more important at the reservoir level, we propose to compute the Hessian in a target-oriented fashion Valenciano and Biondi (2004). This allows us to reduce the Hessian matrix dimensions. We also exploit the sparsity and structure of the Hessian matrix to dramatically reduce the amount of computation while constructing it. After, we compute the least-squares inverse image as the solution of a non-stationary least-squares filtering problem, by means of a conjugate gradient algorithm.
In this paper, we first discuss the structure and sparsity of the target-oriented Hessian. After that, we show how to compute the least-squares inverse image by solving a non-stationary least-squares filtering problem. We illustrate the methodology with two numerical examples, the first in a constant velocity model, and the second in a velocity model with a low velocity Gaussian anomaly.