Seismic imaging can be formulated as an optimization problem (Nemeth et al., 1999; Ronen and Liner, 2000):
Rather than fitting seismic data as in equation 5, we search for models minimizing a misfit weighted by path-summation migration operator. Path-summation operator preceded by plane-wave destruction filter applied in the data domain ( ) for reflection energy elimination can be treated as diffraction probability at a certain location. Therefore, model fitting becomes constrained to those samples, in which diffractions are most probable. The terms of the misfit in equation 6 can be rearranged to stress the misfit weighting by diffraction probability ( ):
Incorporation of PWD ( ) as a misfit weight into the inversion (equation 6) puts some of the reflected energy in the null space. PWD is a crucial part of the inversion. By weighting the residual with path-summation integral and PWD we guide the inversion to extract diffractions from the full wavefield, which is predominated by reflections. To restore reflections from the null space and account for their leakage to the diffraction image domain PWD can be disabled and another inversion cascade can be run:
Path-summation integral acts as a dip filter suppressing events with high wavenumbers (Merzlikin and Fomel, 2017a). High wavenumbers can be restored by running the third cascade of the inversion with path-summation integral filter disabled:
Reflections restored by equation 10 ( ) are subtracted from the full wavefield data to allow for diffraction-only restoration . No interference is expected between reflections and diffractions provided accurate starting models from the previous inversions (equations 6 and 8).
Instead of implementing , , and as penalty terms for reflections and diffractions correspondingly, the problem is reformulated in the shaping framework (see “Regularization” subsection for details).