GNGTS 2015 - Atti del 34° Convegno Nazionale

GNGTS 2015 S essione 3.3 123 the multiple estimation in a single pass, without the need of any intermediate output. Moreover the data-driven nature of this approach allows to inherently handle both P and P-S converted waves. Similarly to 3D SRME, for each input trace s-r, OBS multiple estimation involves the convolution and subsequent stacking of OBS trace couples s-x 1 , x 2 -r, where x 2 is any possible Downward Reflection Point (DRP) belonging to a regularly sampled surface, while x 1 belongs to an analogous surface laid on the sea bottom (Fig. 2a). Thus, for each trace two sets of points must be defined: one set of Downward Reflection Points (DRPs) at the sea level, between the source and the receiver, and one set of Projected DRPs (P-DRPs) with the same horizontal coordinates of DRPs and the same depth of the water bottom. As shown in Fig. 2a, the path of any surface-related multiple can be split in three parts: s-x 1 , x 1 -x 2 , x 2 -r (where x 1 identifies the P-DRPs and x 2 the DRPs). A possible OBS multiple contribution is given by the convolution of interpolated traces s-x 1 and x 2 -r (blue and red solid lines in Fig. 2a respectively) shifted by the travel-time of the direct ray in the water layer from x 1 to x 2 (green dashed line in Fig. 2a), which can be easily computed on-the-fly (assuming a known water-layer velocity model). The estimated multiples M are computed as a summation of the contributions of all possible P-DRPs and DRPs combinations as: (1) where D is the OBS data, * represent the convolution along time axis, δ is the Dirac Delta function, s, x 1 , x 2 and r are the source, the P-DRP, the DRP and the receiver respectively, and τ x1,x2 is the travel time in the water layer from x 1 to x 2 (that is simply computed as ratio between the distance x 1 -x 2 and the water velocity V w when a constant velocity assumption can be done for the water layer). The computational cost of the double summation involved in Eq. 1 is greater than SRME approach for conventional streamer data, as it implies multiple contribution gathers with double dimensionality. The multiple contribution gather for 2D OBS multiple estimation is a 3D sampled volume, while 3D OBS multiple estimation requires a 5D MCG data volume. One can notice similarities between OBS SRME MCGs and the data driven techniques for internal multiples estimation, that share the same increased dimensionality. However, stationary points are minima for OBS SRME MCGs (all terms in Eq. 1 delay the events), while they are saddle points for internal multiple MCGs (as a cross-correlation term, or time advance is implied in the corresponding expression). Fig. 2 – Single-pass OBS SRME algorithm schematics: a) path of multiple decomposed in three contributions, with a set of DRPs x 2 at sea surface and a set of P-DRPs x 1 at sea bottom; b) practical implementation of multiple estimation for a 3D OBS geometry: for each x 1 point, only a sub-area of x 2 is taken into account in the summation.