Which data residual norm for robust elastic frequency-domain full waveform inversion?

Elastic full-waveform inversion is an ill-posed data-fitting procedure that is sensitive to noise, inaccuracies of the starting model, definition of multiparameter classes, and inaccurate modeling of wavefield amplitudes. We have investigated the performance of different minimization functionals as the least-squares norm ℓ2, the least-absolute-values norm ℓ1, and combinations of both (the Huber and so-called hybrid criteria) with reference to two noisy offshore (Valhall model) and onshore (overthrust model) synthetic data sets. The four minimization functionals were implemented in 2D elastic frequency-domain full-waveform inversion (FWI), where efficient multiscale strategies were designed by successive inversions of a few increasing frequencies. For the offshore and onshore case studies, the ℓ1-norm provided the most reliable models for P- and S-wave velocities (VP andVS), even when strongly decimated data sets that correspond to few frequencies were used in the inversion and when outliers polluted the data. The ℓ2-norm can provide reliable results in the presence of uniform white noise for VP and VS if the data redundancy is increased by refining the frequency sampling interval in the inversion at the expense of computational efficiency. The ℓ1-norm and the Huber and hybrid criteria, unlike the ℓ2-norm, allow for successful imaging of the VS model from noisy data in a soft-seabed environment, where the P-to-S-waves have a small footprint in the data. However, the Huber and hybrid criteria are sensitive to a threshold criterion that controls the transition between the criteria and that requires tedious trial-and-error investigations for reliable estimation. The ℓ1-norm provides a robust alternative to the ℓ2-norm for inverting decimated data sets in the framework of efficient frequency-domain FWI.