Seismic interferometry of scattered surface waves in attenuative media

Seismic interferometry can be used to estimate interreceiver surface wave signals by cross-correlation of signals recorded at each receiver that are emitted from a surrounding boundary of impulsive or uncorrelated noise sources. We study seismic interferometry for scattered surface waves using a stationary-phase analysis and surface wave Green's functions for isotropic point scatterers embedded in laterally homogeneous media. Our analysis reveals key differences between the interferometric construction of reflected and point-scattered body or surface waves, since point scatterers radiate energy in all directions but a reflection from a finite flat reflector is specular. In the case of surface waves, we find that additional cancelling terms are introduced in the stationary-phase analysis for scattered waves related to the constraint imposed by the optical theorem for surface waves. The additional terms are of second order even for single-scattered waves, and we show that these can be highly significant in multiple-scattering cases. In attenuative media errors are introduced due to amplitude errors in these additional terms. Further, we find that as the distribution of scatterers in a medium becomes more complex the errors in correlation-type interferometry caused by attenuation in the background medium become larger. Convolution-type interferometry has been shown to be effective when considering electromagnetic wavefields in lossy media, and we show that this is also true for scattered surface waves in attenuating elastic media. By adapting our stationary-phase approach to this case, we reveal why convolution-type interferometry performs well in such media: the second-order cancelling terms that appear in the correlation-type approach do not appear in convolution-type interferometry. Finally, we find that when using both correlation- and convolution-type interferometry with realistic source geometries (illustrative of both industrial seismics and 'passive noise' interferometry), we cannot necessarily expect to produce estimates with all dominant scattering events present. This is shown to be especially important if, as proposed previously for electromagnetic applications, the convolution and correlation approaches are compared to help identify errors in the interferometric estimates.