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Data interpolation for 3D Marchenko Green’s function retrieval

Marchenko redatuming enables the retrieval of the Green’s function using singlesided seismic reflection response in media in 1D, 2D or 3D dimensions. The retrieved Green’s function contains both primaries and multiples of the seismic waves. Papers have been published to demonstrate various aspects or applications of the Marchenko methods, however, the majority of them are carried out with a 1D or 2D wave propagation. In this study, we investigate Marchenko redatuming in the three dimensions with ideal datasets and with marine streamer type datasets. We reformulate the Marchenko-type equations in the three dimension Cartesian coordinate system, which can be directly implemented in our 3D Marchenko algorithm. Based on the 3D Marchenko equations and the stationary phase method, we propose an approach to efficiently determine the aperture for surface seismic survey and the spatial sampling of the source/receiver pairs required by 3D Marchenko redatuming. To overcome the sparsity of the cross-line spatial sampling in a 3D marine towed-streamer survey, we revisit a forward interpolation method, based on B-splines, for cross-line data reconstruction. We test our 3D Marchenko redatuming algorithm with a 2.5D Gulf of Mexico salt model and a 3D trapezoidal and dipping model, demonstrating that 3D Marchenko Green’s function retrieval is able to retrieve both primaries and multiples with a dataset acquired from an ideal geometry by the comparison with the finite-difference modelled Green’s function. We then apply the B-spline interpolation method to a synthetic towed-steamer dataset for the reconstruction of the cross-line data and demonstrate that we successfully retrieve the Green’s function with the dense 3D dataset reconstructed by the B-spline interpolation. We discuss our computational optimization for the Marchenko algorithm for saving memory and CPU time and formulate a comparison between the surface-related multiple elimination (SRME) method and the Marchenko method.