A Single-sided Representation for Virtual Sources and Virtual Receivers

Virtual sources can be created in several ways. In seismic interferometry, a virtual source is created by crosscorrelating responses at different receivers, which are illuminated from all directions. Seismic interferometry can be mathematically described by the homogeneous Green's function representation, which is a closed boundary integral. Virtual sources can also be created with the Marchenko method. For the Marchenko method it is sufficient that the position of the virtual source is illuminated from one side. We derive a single-sided homogeneous Green's function representation, which is an open boundary integral along reflection measurements at the surface. Applying this representation, we obtain virtual sources and virtual receivers in the subsurface from real sources and receivers at the surface (note that in our earlier work on the Marchenko method the response to the virtual source was only obtained for receivers at the surface). The retrieved virtual data show the entire evolution of the response to a virtual source in the subsurface, including primary and multiple scattering at unknown interfaces.