The role of the direct wave and Green's theorem in seismic interferometry and spurious multiples

Abstract

Techniques to estimate the Green's function between two measured points using wavefield correlations and/or crosscorrelations are classified as seismic interferometry. In this paper we provide a unifying framework for understanding a broad class of interferometric techniques using Green's theorem. This framework and foundation allows spurious multiples that occur in certain interferometric approaches to be anticipated and fully explain as a consequence of approximations and compromises made within Green's theorem. We also develop a set of more effective seismic interferometry methods, where fewer compromises effect in a better result. Standard seismic interferometry is based on far‐field and one‐way approximations of the Green's theorem relating two wavefield measurements in the same volume. This method, when applied to surface seismic data, reconstructs the wavefield with a squared source signature and adds spurious multiples, whose amplitudes are comparable to the reconstructed primaries. The artefacts introduced, reduce the method's value. The spurious multiples come from the approximations made to avoid the need of the wavefield's normal derivative. We propose and examine various alternative approaches to seismic interferometry which overcome the appearance of spurious multiples and provides an improvement over traditional methods. One method uses Green's theorem relating a reference Green's function with the measured wavefield. The data are reconstructed without spurious multiples and with a wavelet due to a single source. A synthesized wavefield ought to have a single factor of the wavelet. Using an analytic reference Green's function, data can be extrapolated to positions where no receivers or sources were located. In addition, we provide another form of Green's theorem by imposing a two‐surface Dirichlet boundary condition to the reference Green's function; this method only requires the total wavefield. The normal derivative of the field is not needed.