Interpretation of time reversal focal spots based on point-source homogenization

The time reversal method is based on the backpropagation of seismic waveforms recorded at a set of receivers. When this set forms a closed surface and the elastic properties of the medium are correct, the seismic energy focuses at the source location, creating a focal spot. Such a spot is smooth in space, whereas the original wavefield usually shows a displacement discontinuity at the source. The goal of this paper is to discuss the link between the focal spot and the original source using the concept of homogenized point source. We show that the backpropagated wavefield is equivalent to the sum of two low-wavenumber fields resulting from the homogenization of the original point source. In other words, the homogenized point source is the equivalent force for producing the focal spot. In addition to the demonstration in the general 3-D heterogeneous case, we present some numerical examples in 2-D.