A B S T R A C TA strategy for multiple removal consists of estimating a model of the multiples and then adaptively subtracting this model from the data by estimating shaping filters. A possible and efficient way of computing these filters is by minimizing the difference or misfit between the input data and the filtered multiples in a least-squares sense. Therefore, the signal is assumed to have minimum energy and to be orthogonal to the noise. Some problems arise when these conditions are not met. For instance, for strong primaries with weak multiples, we might fit the multiple model to the signal (primaries) and not to the noise (multiples). Consequently, when the signal does not exhibit minimum energy, we propose using the L 1 -norm, as opposed to the L 2 -norm, for the filter estimation step. This choice comes from the well-known fact that the L 1 -norm is robust to 'large' amplitude differences when measuring data misfit. The L 1 -norm is approximated by a hybrid L 1 /L 2 -norm minimized with an iteratively reweighted leastsquares (IRLS) method. The hybrid norm is obtained by applying a simple weight to the data residual. This technique is an excellent approximation to the L 1 -norm. We illustrate our method with synthetic and field data where internal multiples are attenuated. We show that the L 1 -norm leads to much improved attenuation of the multiples when the minimum energy assumption is violated. In particular, the multiple model is fitted to the multiples in the data only, while preserving the primaries.

Obtaining migrated images with meaningful amplitudes is a challenging problem when the migration operator is not unitary. One possible solution to this problem is iterative inversion. However, inversion is an expensive process that can be rather difficult to apply, especially with 3D data. In this paper, I propose estimating migrated images similar to the least-squares inverse images by approximating the inverse Hessian, thus avoiding the need for iterative inversion. The inverse Hessian is approximated with a bank of nonstationary matching filters. These filters are not exact impulse responses and are limited in their ability to mimic the full effects of least-squares inversion. Tests on two data sets show that this filtering approach gives results similar to iterative least-squares inversion at a lower cost. This technique is flexible enough to be applied to images migrated from zero-offset or angle-domain common-image-point gathers.

The "Huber function" (or "Huber norm") is one of several robust error measures which interpolates between smooth (l 2 ) treatment of small residuals and robust (l 1 ) treatment of large residuals. Since the Huber function is differentiable, it may be minimized reliably with a standard gradient-based optimizer. We propose to minimize the Huber function with a quasi-Newton method that has the potential of being faster and more robust than conjugate-gradient methods when solving nonlinear problems. Tests with a linear inverse problem for velocity analysis with both synthetic and field data suggest that the Huber function gives far more robust model estimates than does a least-squares fit with or without damping.

We introduce an analytical method for integrating dip information to flatten uninterpreted seismic data. First, dips are calculated over the entire seismic volume. The dip is then integrated in the Fourier domain, returning for each sample a time shift to a flat datum. Then each sample is shifted in the seismic data to remove all structural folding deformation in a single non-interpretive step. Using the Fourier domain makes it a quick process but requires that the boundaries are periodic. This method does not yet properly handle faults because of their discontinuous nature, but is presently very effective at removing warping and folding.