GNGTS 2015 - Atti del 34° Convegno Nazionale

124 GNGTS 2015 S essione 3.3 Eq. 1 can be recasted to a fully data-driven approach, by substituting the travel-time with a third OBS data contribution D(t, x 2 ,x 1 ) , as proposed in (Ma et al. , 2010). Nonetheless, the proposed “hybrid” approach (where the knowledge of the bathymetry and the velocity model for the water layer are required) compares favorably to the fully data-driven approach: it proves to be less computationally expensive and more accurate than the convolution of three different traces (as time delay is replaced by full trace convolution, then additional I/O is mandatory to retrieve the seismic trace at x 2 ,x 1 ). Furthermore, estimated higher order multiples are less prone to source wavelet and relative amplitude distorsion. Finally, the fully data-driven approach strictly relies on the quality of direct wavefield recorded in OBS data to correctly estimate both water-layer reverberations and first-order multiples). Practical implementation issues. As well-known, for an efficient 3DSRME implementation (Moore et al. , 2008), an optimized interpolation strategy that allows a simple on-the-fly data regridding to required geometries s-x 1 , x 2 -r is implemented. This strategy must allow to cope with sparse and irregular sampling and avoid operator aliasing (Bienati et al. , 2012). The interpolation scheme involves the retrieval of one (or more) neighbouring traces, and the computation of a differential correction (i.e., differential moveout) or a weighted stack (i.e., continuation operators). The neighbourhood selection rule must minimize the sensitivity of the interpolation kernel to model errors, taking into account both azimuth and offset differences: (2) where m i is the midpoint, h i is the offset and θ i is the azimuth of i-th trace; α, β and γ are weights that must be chosen in some heuristic way. Note that any data regridding strategy (implied in all 3D SRME practical implementations) may introduce a mild model dependency, even for fully data-driven approaches. Furthermore, another subtle dependency on a-priori knowledge of subsurface structure is implied in the choice of the integration area. However, both the selection of neighboring traces, and the computation of a differential correction must be modified according to OBS geometry. First of all, the different depths of sources (at sea surface) and receivers (at sea bottom) must be taken into account in Eq. 2. While for surface data the azimuth terms ( θ i ) is considered without signum within the range [0°,180°] (i.e. it is possible to exchange sources with receivers and viceversa), for OBS data the same terms specify the azimuth with signum within the full range [-180°,180°].Also the values of weights �� � ��� � ��� �� �������� �������� ��� ���������� α � � ��� �� �������� �������� ��� ���������� , β and γ can be slightly modified for minimizing the interpolation error of OBS traces. Then, an adapted differential correction must be defined in order to compensate the different depths of input and output trace receivers, as water bottom is a generally varying surface. A simple differential correction with constant velocity is implemented, where the interpolated traces time t out is approximated by: (3) where V is the constant reference velocity, t in , h in and z in are the time, the offset and the receiver depth of input trace respectively, and t out , h out and z out are the time, the offset and the receiver depth of output trace respectively. In practice, in order make the algorithm less expensive, the terms of summation 1 associated to highly unlikely x 1 - x 2 patterns that give incoherent contributions to the result can be discarded, as shown in Fig. 2b. The correct choice of valid patterns depends on both the depth and smoothness of water bottom and the complexity of the subsurface. It must be noted that direct wavefield contributes to both source and receiver sides ghost thereby it must not be removed from the data before the processing: in fact, the proposed approach still relies on the quality of direct wavefield recorded in data to correctly estimate