Kirchhoff scattering series for sources and receivers located in an elastic medium: Another insight into the concept of virtual events

The convolution-type and correlation-type representation theorems are building blocks of wave-scattering theory whose usefulness expands in many seismological applications. For example, the Kirchhoff scattering series currently used for attenuating free-surface multiples has been derived from the convolution-type representation theorem. The recently introduced concept of virtual events, which allows us to put virtual sources and virtual receivers inside the subsurface based on the data collected at the sea surface, has been derived by a combined use of the convolution-type and correlation-type representation theorems. The formulation of inverse Kirchhoff scattering series and virtual events has been limited so far to the cases in which sources or receivers, or both, are located in the water. Unfortunately, this assumption is not valid, especially in the context of virtual events, in which both sources and receivers will often be located in a solid. We here redescribe the Kirchhoff scattering series and reformulate the concept of virtual events for the cases in which sources and receivers are in a solid. Moreover, we describe a new form of Kirchhoff series based on the correlation-type representation theorem and new formulae for computing virtual events which do not include the complex renormalization operation of the previous formulation.