2D/3D Paraxial maximum field Green’s funtion generator

Abstract

The time seismographic interpretation can contain errors associated to the incorrect conformation of the seismic events. This fact can even carry economical losses due to exploratory or/and exploitatory well mis-positioning, or still hide a favorable prospecting situation. This is particularly evident in regions of great geological complexity and/or strong lateral velocity variations. The recent advances in the intensive computational technology with production of high computational capacity machines, particularly the parallel architectures, have woken up crescent attention to the depth imaging. The 2D/3D Paraxial Ray-Tracing (Popov, 1977) , (Popov and P˘sen˘cik, 1978b), (Popov and P˘sen˘cik, 1978a) will exert, among other functions, the 2D/3D Green’s function simulator, cting as the heart of a series of applications, such as velocity analysis by reflexion tomographic inversion, AVA, Kirchhoff modeling, amplitude correction, etc. The 3D pre-stack depth migration (PSDM) will use the travel time table obtained by this procedure to implement the image condition to the Kirchhoff’s integral imaging operator and, eventually, to the reverse time migration (RTM). This work presents the results of building the image condition by the solution of a system of 21 non linear first order differential PARAXIAL equations for dynamical ray tracing. This corresponds to a generalization of a kinematic version already developed (Cunha, 1999a), with adaptive step time control between the ray points. These travel times will be subsequently employed in the Kirchhoff’s integral perator for PSDM. The evaluation of the image condition by solving a non linear first order differential equation (EIKONAL’s method) (Faria and Stoffa, 1994) is in general fast, but presents some serious inconveniences, such as: a) High degree of instability in complex media (Popov, 2002) where “Caustic” phenomena will fatally occur, increasing by several orders of magnitude the rate of the error, even in points far from it. b) Determine only the first arrivals. c) Doesn’t allow the determination of the maximum field’s image condition since this requires evaluation of travel time for multiple arrivals. The main motivation for using RTM is to employ the full acoustical - elastic wave equation to propagate the stress field, where it is implicit the presence of a punctual Green’s function centered in the source - receptor stations, which makes the implementation of the maximum field image condition MFIC natural and simple. The experience with real data indicates that the RTM results for complex media are often superior than that obtained by other methods. However, its main limitation is the great computational cost which is proportional to the number of source - receptor points. The parallel processing, particularly LINUX’s Clusters, (Soares, Filho and Bulc˜ao, André and De Bragança, R. S. N., 2002) will gradually favor the conventional RTM, although the computational costs still have some impact. The use of coarse transverse grid (Mufti et al., 1996) (Cunha, 1997) can reduce the computational costs, but at the expense of problems like numerical anisotropy (Alford et al., 1974) (Cunha, 1999b). However, with the development of the “Multi Source” version of the RTM (Cunha, 2002) this seems no longer be true. We believe that part of this qualitative superiority could be credited<br>just to the MFIC, as supported by this work.