SEPARATION WITH SPARSENESS CONSTRAINTS

A non-linear primary-multiple separation method using curvelets frames is presented. The advantage of this method is that curvelets arguably provide an optimal sparse representation for both primaries and multiples. As such curvelets frames are ideal candidates to separate primaries from multiples given inaccurate predictions for these two data components. The method derives its robustness regarding the presence of noise; errors in the prediction and missing data from the curvelet frame’s ability (i) to represent both signal components with a limited number of multi-scale and directional basis functions; (ii) to separate the components on the basis of differences in location, orientation and scales and (iii) to minimize correlations between the coefficients of the two components. A brief sketch of the theory is provided as well as a number of examples on synthetic and real data. Separation by matched filtering In many complex areas, multiple suppression techniques based on move-out filtering fail due to the fact that the underlying assump-tions are not met, i.e. hyperbolic move-out in the CMP-offset domain, separation of primaries and multiples based on move-out differences. Several attempts have been made to extend the move-out discrimination methods towards handling 3D complexities, e.g. by introducing apex-shifted hyperbolic transforms [5]. In many cases were the move-out filtering based methods fail, wave-equation based predictive methods may provide a solution. However, prediction methods based on 2D input data [see e.g. 18, 1] are often not capable of correctly predicting the surface-related and internal multiples in a complex 3D world