Wavefield reconstruction and wave equation inversion for seismic surface waves

Surface waves are a particular type of seismic wave that propagate around the surface of the Earth, but which oscillate over depth ranges beneath the surface that depend on their frequency of oscillation. This causes them to travel with a speed that depends on their frequency, a property called dispersion. Estimating surface wave dispersion is of interest for many geophysical applications using both active and passive seismic sources, not least because the speed-frequency relationship can be used to infer the subsurface velocity structure at depth beneath the surface. We present an inversion scheme that exploits spatial and temporal relationships in the scalar Helmholtz (wave) equation to estimate dispersion relations of the elastic surface wave data in both active and passive surveys, while also reconstructing the wavefield continuously in space (i.e. between the receivers at which the wavefield was recorded). We verify the retrieved dispersive phase velocity by comparing the results to dispersion analysis in the frequency-slowness domain, and to the local calculation of dispersion using modal analysis. Synthetic elastic examples demonstrate the method under a variety of recording scenarios. The results show that despite the scalar approximation made to represent these intrinsically elastic waves, the proposed method reconstructs both the wavefield and the phase dispersion structure even in the case of strong aliasing and irregular sampling.