Data-Driven Green's Function Retrieval from Reflection Data - Theory and Example

Recently we introduced a new approach for retrieving the Green's response to a virtual source in the subsurface from reflection data at the surface. Unlike in seismic interferometry, no receiver is needed at the position of the virtual source. Here we present the theory behind this new method. First we introduce the Green's function G and a so-called fundamental solution F of an inhomogeneous medium. Next we derive a relation between G and F, using reciprocity theorems. This relation is used as the basis for deriving a 3D single-sided Marchenko equation. We show that this equation is solved by a 3D autofocusing scheme and that the Green's function is obtained by combining the focusing wave field and its response in a specific way. We illustrate the method with a numerical example.