Extraction of the 1D Green's function in a layered medium by three interferometry techniques: Theory and a case study

This paper presents a comparison of three interferometry techniques; cross-correlation, deconvolution and cross-coherency, when applied on a theoretical model of a layered medium. Explicit equations for the interferometry wavefields are obtained from the equations that govern displacements in a single-layered medium under the incidence of a plane SH-wave. A propagation matrix is developed in order to calculate the deconvolved wavefield in a medium with N layers without the need to know the displacement equations. It is herein demonstrated that cross-coherency is equivalent to the sign function of deconvolution; applied on deconvolved wavefields the cross-coherency is obtained by using the propagation matrix. The interferometry response from seismic data in the Roma C borehole localized in the valley of Mexico is compared with the theoretical response generated with the geological profile of the borehole, by deconovolution and cross-coherency.