A hierarchical elastic full-waveform inversion scheme based on wavefield separation and the multistep-length approach

Abstract

Elastic full-waveform inversion (FWI) updates model parameters by minimizing the residuals of the P- and S-wavefields, resulting in more local minima and serious nonlinearity. In addition, the coupling of different parameters degrades the inversion results. To address these problems, we have developed a hierarchical elastic FWI scheme based on wavefield separation and a multistep-length gradient approach. First, we have derived the gradients expressed by different wave modes; analyzed the crosstalk between various parameters; and evaluated the sensitivity of separated P-wave, separated S-wave, and P- and S-wave misfit functions. Then, a practical four-stage inversion workflow was developed. In the first stage, conventional FWI is used to achieve rough estimates of the P- and S-wave velocities. In the second stage, we only invert the P-wave velocity applying the separated P-wavefields when strong S-wave energy is involved, or we merely update the S-wave velocity by matching the separated S-wavefields for the weak S-wave case. The PP and PS gradient formulas are used in these two cases, respectively. Therefore, the nonlinearity of inversion and the crosstalk between parameters are greatly reduced. In the third stage, the multistep-length gradient scheme is adopted. The density structure can be improved owing to the use of individual step lengths for different parameters. In the fourth stage, we make minor adjustments to the recovered P- and S-wave velocities and density by implementing conventional FWI again. Synthetic examples have determined that our hierarchical FWI scheme with the aforementioned steps obtains more plausible models than the conventional method. Inversion results of each stage and any three stages reveal that wavefield decomposition and the multistep-length approach are helpful to improve the accuracy of velocities and density, respectively, and all the stages of our hierarchical FWI method are necessary to give a good recovery of P- and S-wave velocities and density.