How to quantify discrepancies between arbitrary polarization states ? Unifying geometrical insight and full-waveform inversion of polarization observables

Summary Polarization observables have long been used to infer properties of the Earth, but the geophysical literature has not explicitly addressed the problem of a general metric for comparing arbitrary polarization states (independently of specific application contexts). To address this question, we first introduce a geometrical representation of polarization based on the Poincaré sphere. This framework allows us to derive a novel polarization discrepancy criterion, defined as the geodesic length between two polarization states represented on the Poincaré sphere. We establish, both analytically and geometrically, the relationship between this intrinsic metric and the main classes of polarization metrics (based on polarization angles, ratios and cross-products). Then, we examine the advantages and limitations of various polarization-based misfit functions from both fundamental and geophysical perspectives. On the fundamental side, we theoretically compare the different misfit functions in light of a few qualitative criteria. From a geophysical perspective, we extend the conventional multicomponent FWI method for inverting the discussed polarization-based criteria. With a simple numerical experiment, we show that misfit functions based on intrinsic and extrinsic distances (Poincaré and normalised cross-product misfit functions, respectively) produce superior inversion results compared to all other polarization-based misfit functions. More generally, inverting polarization observables with a suitable metric can be beneficial for a wide range of geophysical problems, from source estimation to the quantitative imaging of Earth properties based on different types of physical fields (such as seismic or electromagnetic waves).