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Marine vibrators (MVibs) are gaining traction as an alternative to conventional airguns for their ability to generate low-frequency, flexible, and repeatable waves with minimal impact on marine wildlife. However, the Doppler shift caused by source motion complicates seismic processing. Phase dispersion simulation or correction can be achieved using a dephasing operator equation based on the instantaneous frequency function, while it requires sufficient spatial sampling to get the proper effect. Besides, surface-related multiple removal is crucial for seismic imaging quality. Despite advances in surface-related multiple elimination (SRME), issues like coarse spatial sampling, missing near-offsets, unequal source and receiver spacing, and time-variant observation positions cause significant discrepancies between predicted and true multiples. We propose a Bayesian primary-multiple separation and Doppler-shift correction iterative algorithm, which assumes approximate shearlet-domain independence of SRME-type moving MVib primaries and multiples, and separate and correct them robustly. In our approach, the shearlet properties determine the energy mismatch between separated and predicted components. Additionally, the F-K domain phase distortion operator based on the instantaneous frequency function incorporates the source motion aspects into the shearlet dictionary, reducing computational costs. Finally, the introduction of linear moveout (LMO) mitigates the Doppler-shift correction sensitivity to spatial aliasing. Synthetic example confirms our approach’s effectiveness.