Full-waveform inversion (FWI) aims at a broadband reconstruction of the subsurface physical properties by fitting the entire recorded wavefield. In realistic exploration seismic surveys, however, conventional FWI often fails to retrieve the deep velocity model due to the limited penetration depth of diving waves. Joint FWI (JFWI) unifies reflection-waveform inversion and early arrival waveform inversion to reconstruct simultaneously the shallow and deep subsurface kinematics. However, several factors limit the appeal of JFWI velocity-model building: (1) conflict between fixed reflectivity and evolving kinematics, creating phase ambiguity at short offsets; (2) susceptibility to cycle skipping at mid-to-long offsets, thus reliance on the quality of the starting model; and (3) cost of building and updating the reflective model. We have developed a fully operational JFWI-based methodology that systematically addresses the aforementioned issues. JFWI is reformulated in the pseudotime domain, to enforce consistency between velocity and reflectivity in a cost-effective fashion, without repeated least-squares migrations. A JFWI graph-space optimal transport (GSOT) objective function is designed to avert cycle skipping, whereas nonuniqueness is mitigated at no extra cost by smoothing the velocity gradient along the structures extracted from the reflective model. A dedicated asymptotic-based preconditioner is developed for impedance waveform inversion, making it possible to obtain sharp and balanced reflective images in a fraction of the time. We determine that pseudotime GSOT-JFWI retrieves complex velocity macromodels from limited-offset data sets with minimal preprocessing, starting from noninformative initial solutions. Compared with depth-domain JFWI, the computing cost is reduced significantly, along with a simpler and less subjective design of data weighting and inversion strategy. Pseudotime GSOT-JFWI provides FWI with the necessary low wavenumbers to converge to the broadband model, reducing the need for accurate starting models, on the road to a fully waveform-based imaging workflow.

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