GNGTS 2013 - Atti del 32° Convegno Nazionale

multiples can be accounted for by using this model. Surface-related multiples generates when primary reflections reach the acquisition surface and undergo to an additional downward bounce that send them again into the subsurface. This mainly happen in the context of marine acquisition, where the interface between the water and the air has an high reflection coefficient. The multiple reflections (at least the surface-related) can thus be viewed as the Earth’s impulse response to the primaries that now act as a source wavefield. These considerations lead to the multiples imaging procedure explained in the next section. WE imaging of multiples. As already said, our goal was to modify pre-existing imaging techniques in order to be able to exploit the information carried by the multiple reflections. Even though Kirchhoff migration is still widely used in the industrial world, our choice fell on the WE (Wavefield Extrapolation) migration algorithms that provide a better accuracy and can handle more complex environments. Both WEM and RTM belong to this class of imaging procedures and can be modified to image multiples as explained here below. The underlying idea of all the WE migration algorithms is to simulate the seismic wavefield that traveled in the subsurface before and after its interaction with the discontinuities, in order to correctly move the recorded reflection events at the location where they were generated. The concept was firstly introduced by Claerbout in his 1971 paper “Towards a unified theory of reflector mapping” where he stated his famous imaging principle : “reflectors exist at points in the ground where the first arrival of downgoing wave is time coincident with an upgoing wave”. The downgoing wavefield is the pressure (or particle velocity) field generated by the source that propagates downward into the subsurface. The upgoing wavefield correspond to reflected energy, traveling upwards towards the recording surface. The whole imaging procedure involved in WE migration algorithms can be summarized by the following three steps: 1) emulation of the downgoing field U D ( t , x ); 2) emulation of the upgoing field U UP ( t , x ); 3) application of the imaging principle ; where x is the vector of the space coordinates x = ( x, y, z ). Depending on the chosen algorithm (WEM or RTM), U D and U UP are extrapolated, respectively, in depth or in time even though the variations needed to correctly image the surface-related multiples are irrespective of the choice of the specific algorithm. If we assume to know the subsurface model (that is the situation when one performs the seismic migration), steps 1 and 2 are simply a modeling exercise. Step 3 implies the analysis of the matching between the two traveling wavefields to determine if they coincide in time and space and to retrieve the information about the reflection coefficient. The imaging principle is implemented by means of an imaging condition . The matching between the two wavefield can be evaluated by using a simple cross-correlation between the two fields, as expressed in the following formula: (1) where the cross-correlation is expressed in the temporal frequency ( ω ) domain. In the conventional migration, the downgoing wavefield is reconstructed by propagating forward-in-time an estimated source function S ( t , x ). The upgoing wavefield is reconstructed by backward-in-time extrapolation of the seismic recording R ( t , x ). Eq. (1), for the conventional migration, can be written as: (2) Following the idea expressed in the previous section, one could migrate the surface-related multiples by simply substituting the data used in step 1 and step 2. For the downgoing field, instead of using S ( t , x ), we inject and forward propagate the seismic recording R ( t , x ). For the 36 GNGTS 2013 S essione 3.1