Simultaneous Primary–Multiple Separation and Doppler-Shift Correction Approach for Marine Vibrator Data Based on Bayesian Theory

Marine vibrators (MVibs) are increasingly used in seismic surveys due to their precise waveform control, flexible signal design, environmental friendliness, and enhanced low-frequency output. However, their long-duration pilot sweeps introduce challenges not present with traditional airguns—most notably, the Doppler effect. The simulation or correction of phase distortion can be achieved using frequency–wavenumber (F–K) domain dephasing operator equation based on the instantaneous frequency function, while they require sufficient spatial sampling to get a reliable effect. Besides, as a typical coherent noise source, the removal of surface-related multiples greatly determines seismic imaging quality. Although the surface-related multiple elimination (SRME) scheme can suppress the interference, moderate phase, timing, and amplitude errors, clutter in predicted signal components is detrimental. We propose a Bayesian primary–multiple separation and Doppler-shift correction iterative framework that assumes the predictions from SRME-type techniques are approximately independent in the shearlet domain and robustly corrects and separates them. In our approach, the energy mismatch between separated and predicted components is effectively controlled. Synthetic and field data examples have shown that its key advancements include: 1) the F–K domain phase distortion operator based on the instantaneous frequency function cleverly matches with the shearlet dictionary to simulate source motion in the sparsity-promoting inversion, reducing computational costs by 35.02%; 2) the introduction of linear moveout (LMO) overcomes the sensitivity of Doppler-shift correction to spatial aliasing and achieves a 18.0742-dB signal-to-noise ratio (SNR) improvement in phase correction; and 3) an improved threshold function is proposed to optimize primary–multiple separation, aiming at the shortcomings of the traditional ones. It outperforms the hard and soft threshold functions regarding average SNR, with improvements of 0.97% and 6.21%, respectively. Our approach only requires about ten iterations to attain the converged solution.