Valenciano et al. (2005a) solve equation 3 in a poststack image domain (zero subsurface-offset). The authors conclude that a prestack regularization was necessary to reduce the noise in the inversion result. After Valenciano and Biondi (2005), theoretical definition of the prestack wave-equation Hessian (subsurface-offset, and reflection angle) a generalization to the prestack image domain of equation 3 is possible.
Three different regularization schemes for wave-equation inversion have been discussed in the literature. First, an identity operator which is customary in many scientific applications (damping). Second, a geophysical regularization which penalizes the roughness of the image in the offset ray parameter dimension (which is equivalent the reflection angle dimension) Kuehl and Sacchi (2001); Prucha et al. (2000). Third, a differential semblance operator to penalize the energy in the image not focused at zero subsurface-offset Shen et al. (2003). In this paper I compare the first and the third regularization schemes, because a regularization in the subsurface-offset is more attractive in terms of computational cost. The regularization in the reflection angle domain will be a topic of future research.
A generalization to the prestack image domain of equation 3 needs regularization to obtain a stable solution. The first option for regularization is a customary damping that can be stated as follows:
where and are the Green functions from shot position and receiver position to a model space point .
The third regularization option for the prestack generalization of equation 3, is penalizing the energy in the image not focused at zero subsurface-offset. This is obtained using the fitting goals,
In the next section I compare the numerical solution of the inversion problems stated in equations 4 and 5 to the imaging of Sigsbee model.