Two-step full waveform inversion of diving and reflected waves with the diffraction-angle-filtering-based scale-separation technique

Full waveform inversion (FWI) is a highly nonlinear optimization problem that aims to reconstruct high-resolution subsurface structures. The success of FWI in reflection seismology relies on appropriate updates of low-wavenumber background velocity structures, which are generally driven by the diving waves in conventional FWI. On the other hand, the reflected waves mainly contribute to updating high-wavenumber components rather than low-wavenumber components. To extract low-wavenumber information from the reflected waves in addition to the diving waves, we propose a two-step FWI strategy that separates a given model into the reflectivity and background velocity models and then alternately update them using the scale-separation technique based on diffraction-angle filtering (DAF; which was proposed to effectively control wavenumber components of the FWI gradient). Our strategy first inverts the high-wavenumber reflectivity model by suppressing energy at large diffraction angles, which are necessary to compute the reflection wavepaths (i.e., the rabbit-ears-shaped kernels) for low-wavenumber updates in the subsequent stage. Then, we extract low-wavenumber components due to the diving (banana-shaped kernels) and reflected waves (rabbit-ears-shaped kernels) from the gradient by suppressing energy at small diffraction angles. Our strategy is similar to reflection waveform inversion (RWI) in that it separates a given model into high- and low- wavenumber components and uses the rabbit-ears-shaped kernels for low-wavenumber updates. The main difference between our strategy and RWI is that our strategy adopts the DAF-based scale-separation technique in the space domain, which makes our algorithm of using both the banana-shaped and rabbit-ears-shaped kernels computationally attractive. By applying our two-step inversion strategy to the synthetic data from the Marmousi-II model and the real ocean-bottom cable (OBC) data from the North sea, we demonstrate that our method properly reconstructs low-wavenumber structures even if initial models deviate from the true models.