Frequency-domain full-wavefield inversion for ambient seismic surface-wave noise

We pose full wave field inversion of ambient seismic noise as a partial differential equation (PDE) constrained inverse problem resulting in a joint estimation of both the wave field (noise free) and the velocity parameters. Because the ambient seismic field is usually dominated by fundamental mode surface waves, we elect to impose a dispersive Helmholtz equation as a PDE constraint and attempt to retrieve the surface wave dispersion while estimating the wave field that best fits the data. The character of the ambient seismic field is irrelevant, as long as the recordings are non-zero and we can assume the wave field sources to be negligible inside the domain of interest. The boundary conditions of the wave field are explicitly omitted from the PDE constraint and recovered during the inversion. We support the theory with two numerical examples.