A note on extended full waveform inversion

Full waveform inversion (FWI) aims at estimating subsurface medium properties from measured seismic data. It is usually cast as a non-linear least-squares problem that incorporates uncertainties in the measurements. In exploration seismology, extended formulations of FWI that allow for uncertaties in the physics have been proposed. Even when the physics is modelled accurately, these extensions have been shown to be beneficial because they reduce the non-lineary of the resulting data-fitting problem. In this note, I derive an alternative (but equivalent) formulation of extended full waveform inversion. This re-formulation takes the form of a conventional FWI formulation that includes a medium-dependent weight on the residuals. I discuss the implications of this re-formulation and illustrate its properties with a simple numerical example.