Conventional imaging techniques such as migration cannot provide an accurate picture of poorly illuminated areas Clapp (2005). In such areas, migration artifacts can easily obscure the small amount of signal that exists. One way to solve this problem is to use an inversion formalism introduced by Tarantola (1987) to solve geophysical imaging problems. This procedure computes an image by weighting the migration result 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. That is why the following approximation to the wave-equation inversion must be performed: (1) to compute the Hessian in a target-oriented fashion to reduce the size of the problem; (2) to exploit the sparse structure of the Hessian matrix; and (3) to compute the inverse image following a iterative inversion scheme. The last item renders unnecessary an explicit computation of inverse of the Hessian matrix.
In this paper, I apply the target-oriented wave-equation inversion to the Sigsbee data set. I show postack (zero subsurface-offset) and prestack (subsurface-offset) image space inversion results. I use a customary damping model regularization to make the postack inversion stable. By the other hand, the prestack inversion is stabilized by using a regularization that penalizes energy not focused at zero subsurface-offset Shen et al. (2003).