Angle-domain parameters computed via weighted slant-stack |

The phase behavior of the offset-to-angle transformation shows that the main contribution for the image in the angle domain comes from the vicinity of the stationary point. Therefore, the use of the weighted stacking strategy (Bleistein, 1987) to map quantities computed in the subsurface-offset domain to the angle domain is straightforward. The mapping of certain attributes can be useful, for instance, to balance amplitudes in the angle domain.

In the following, the aim of the weighted offset-to-angle transformation is to compute weights to be applied on ADCIGs in such a way that amplitude variations due to illumination are attenuated. The weighted offset-to-angle transformation is represented by the computation of ADCIGs from SODCIGs previously multiplied by some parameter -- in the present case, the subsurface-offset Hessian diagonals, -- defined in the subsurface-offset domain. The amplitudes of the resulting ADCIGs, according to the stationary phase results in the previous section, can be represented by

The general formula of the subsurface-offset Hessian in the prestack-inversion problem is

The main diagonal of the Hessian, which is the Laplacian of the cost function related to the model parameters, contains the autocorrelation of the Green's functions and, generally, carries most of the information about illumination. Sometimes, a good and cheap solution is just to approximate the Hessian by its main diagonal and apply its inverse to the migrated image. However, this procedure does not correct for kinematic errors of the migrated image and, depending on the complexity of the illumination pattern, only the least-squares inverse image may be able to provide reasonable results (Clapp, 2005).

Equation 12 shows the structure of the subsurface-offset domain Hessian used in the examples.

Although, in principle, any diagonal of the subsurface-offset Hessian can be transformed to the angle domain by the proposed approach, at present I have conceived a direct application only for the transformed main diagonal, which is to use it as a weight to balance the amplitudes of the ADCIGs. In the next section I show examples of the angle-domain transformed subsurface-offset Hessian diagonals, as well as the comparison of migrated images before and after the amplitude compensation with the transformed main diagonal, for a small portion of the Sigsbee dataset.

Angle-domain parameters computed via weighted slant-stack |

2007-09-15