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Discussion

In this paper, I presented a method that intends to correct migrated images by approximating the Hessian of the imaging operator with non-stationary matching filters. These filters are estimated from two migration results. One migrated image, ${\bf
 m_1}$, corresponds to the first migration result. The second image, ${\bf m_2}$, is computed by re-modeling the data from ${\bf
 m_1}$ and then by re-migrating it. It turns out that the relationship between ${\bf
 m_1}$ and ${\bf m_2}$ is similar to the relationship that exists between the least-squares inverse ${\bf \hat{m}}$ and ${\bf
 m_1}$. In the proposed approach, this relationship is simply ``captured'' by matching filters.

I demonstrate with the Marmousi dataset that this approach gives a better image than does least-squares without regularization and at a lower cost. In addition this approach can be used on images migrated at zero-offset or in the angle domain. As opposed to Hu et al. (2001), the correction in the image is completely data driven, does not depend on the velocity, and can be applied with any migration operator. It also works in the poststack or prestack domain without any extra effort. Providing the data and the ability to run at least two migrations to estimate ${\bf m_2}$, this method would be easy to apply with 3-D migrated images.

Compared to Rickett (2001), this proposed approach does not need reference images. In addition, the multi-dimensional filters offer more degrees of freedom for the correction than does a simple zero-lag weight: in that we might also correct for kinematic changes and move energy locally in the image. In the limit case where we choose one filter per point and only one coefficient (zero-lag) per filter, the matching filter approach would theoretically perform better than Rickett's method because the weights would be optimal in a least-squares sense without ad-hoc smoothing. In the future, it would be valuable to go beyond 2-D filters by extending them to 3-D and to test this 3-D approach with more field data. In addition, these filters could be used as preconditioning operators providing faster convergence for iterative inversion. The stability of the inverse filters still needs to be addressed, however.


next up previous print clean
Next: acknowledgments Up: Guitton: Approximating the Hessian Previous: Angle domain results
Stanford Exploration Project
7/8/2003