Having interpolated the shots to a 25 m grid makes the multiple prediction more accurate. Figure shows a comparison for one offset of the input data and the predicted multiples with and without sparse interpolation. The predicted multiples in Figures b and c look almost identical to the true multiples in Figure a. Some artifacts, shown as A in Figure c, are nevertheless present in the multiple model obtained from the interpolated data without regularization. These artifacts come from the convolution of the noise visible in Figure b before the water-bottom reflection with coherent energy. Some of them could be attenuated by applying a mute on the interpolated shot gathers.
The multiple model looks relatively accurate for the whole section. However, some important discrepancies exist between the true and modeled multiples in some places. For instance, Figure displays the multiple prediction results in an area where the model does not match the observed multiples very well. In Figure b, the water-bottom multiple (shown as 1) is clearly modeled better with the interpolated data with sparseness constraint than in Figure c without regularization. The two circles in Figure b and c highlight aliasing artifacts that are also visible in Figures b and c. These artifacts would disappear by having a smaller sampling of both shot and offset axes. Therefore, these aliasing artifacts are stronger when no Cauchy regularization is applied for the interpolation.
Looking now at shot gathers in Figure , it appears that the modeling of multiples worked well for the main reflection events. Energetic diffracted multiples are almost totally absent in the predictions of Figures b and c, however. Diffracted multiples are generally more difficult to model because they require a very dense surface coverage of sources and receivers in order to recover all the dips (Figure ). This effect is amplified in 3-D. Approximating the sail line as a continuous 2-D survey and ignoring 3-D effects lead to large errors in the model of diffracted multiples.
The estimation of a multiple model with a 2-D prediction scheme for 3-D field data, although being accurate in most places, suffers from kinematic and amplitude errors. For some events, e.g., diffracted multiples, the modeling fails completely. In addition, the shot interpolation technique needs to be chosen carefully to minimize its impact on the final multiple model. Here, the radon-based approach with a sparseness constraint yields the best multiple model. Given this imperfect model, it is now the goal of the subtraction to come up with the best estimated primaries. In the next section, multiple attenuation results are presented with a pattern-based and adaptive subtraction technique.