GNGTS 2014 - Atti del 33° Convegno Nazionale

70 GNGTS 2014 S essione 3.1 experiment with sources and receivers placed underneath the formations mainly responsible of the multiply scattered events. This new dataset is constructed using both the linear and non- linear components of the Green’s function, thus accounting for the multiple re ections. We show examples on the synthetic dataset Sigsbee2b. Non-linear datuming. Fig. 1a shows the acquisition setting of a conventional marine seismic experiment. S, R, A and B indicate, respectively, the coordinates of the zero-depth original sources and receivers (( x S , y S , z S = 0), ( x R , y R , z R = 0)) and those of the sources and receivers on a virtual datum surface (( x A , y A , z A ) and ( x B , y B , z B )). Fig. 1 – a) Scheme of the wavefields datuming. Red arrow: source datuming. Blue arrow: receivers datuming. b) Interferometric composition of different types of events. The dotted lines indicate the depth of the virtual receivers. Note that in the example of Fig. 1a, z A = z B , even if all the following derivations also hold in the more general case of z A ≠ z B . Moreover, in the following, with d(y|x) we indicate a wavefield recorded at y and generated by a source placed at x while with capital letters we indicate the same data after a Fourier transformation on the time axis for a selected temporal frequency (e.g. D(y|x)) . We can the re-locate the data recorded by receivers at depth z B (receivers datuming) by means of a backward-in-time and downward-in-space wavefield extrapolation expressed by the following equation: (1) where R indicates the zero-depth original receivers surface and W(B|R) is the Green’s functions between each couple of points R and B . The over-line indicates the complex conjugation operation. The receivers datuming expressed in Eq. (1) is actually the backward-in-time receivers wave eld propagation implemented in the Reverse Time Migration (RTM) imaging algorithms. Once D(B|S) is retrieved, we can proceed and re-locate also the sources (sources datuming). It is possible to perform this step by applying the following equation: (2) where W(A|S) indicates the complex conjugate of the Green’s function between all the points on the surfaces S and A . We can consider the Green’s functions W(A|S) in Eq. (2) to be composed of two different terms: a linear scattered term ( rst-order Born scattering), W l (A|S) , and a non- linear scattered component, W nl (A|S) (higher orders Born terms). Eq. (2) can thus be re-written as follows: