Specularly-reflected multiples, when migrated with the velocity of the primaries, behave as primaries migrated with too slow velocity Alvarez (2005). The tangent-squared approximation can be used to design a Radon transform that focuses the energy of the primaries and the multiples in and ADCIG according to their residual curvature and so can be used to attenuate the multiples in image space Sava and Guitton (2003). This approximation is robust enough that it can even be used to approximate the residual moveout of diffracted multiples, provided that another dimension is added to the Radon transform to account for the shift of the apex of these multiples Alvarez et al. (2004).
Here I show that the approximation of Alvarez 2005 for the residual moveout of the multiples is better than the straight-ray approximation, because it takes into account the non-negligible ray bending of the multiples at the water-bottom interface and by extension any interface in which the velocity of propagation of the primaries and the multiples is substantial, for example at a salt boundary. I show, with both a synthetic and a real ADCIG that the new approximation is more accurate and that it focuses better the multiples in the Radon domain. I then show that this results in an improvement of the estimation of the multiples and therefore in their attenuation.
The first section briefly reviews the theory and shows a comparison of the two residual moveout curves for a given ADCIG. The next section compares both approximations in the Radon domain and the following section compares the results of attenuating the multiples with both approximations on a synthetic and a real ADCIG.