Workflow

The workflow we propose to decompose the input data into reflections, diffractions and noise is outlined in the flowchart (Figure 1) and is as follows:

1. run the first inversion (equation 6);
2. run the second inversion (equation 8) and use the output from the first inversion cascade as the starting model;
3. restore high wavenumbers in reflections (equation 10); use the output of the second inversion as the starting model;
4. restore high wavenumbers in diffractions (equation 11); use the output of the second inversion as the starting model for diffractivity; use reflectivity from the third step to subtract it from the data.

For shaping regularization of reflections ( $\mathbf{S}_{\sigma})$ dominant local slopes should be estimated in the image domain (conventional Kirchhoff migration image should suffice). For PWD filter ( $\mathbf {D}_{data}$) dominant local slopes should be estimated in the data domain (zero-offset or stacked section). Both slopes should be precomputed before the inversion.

scheme-4-lsq
Figure 1. Flowchart illustrating the workflow for reflection, diffraction and noise separation. Internal iterations correspond to “Conjugate Gradients” minimizing the misfit term. External iterations correspond to “Shaping Regularization” performing reflection/diffraction separation. First, the objective function given by the equation 6 is minimized (“First Inversion” in the chart). The data to be matched in the “First Inversion” is filtered with PWD followed by path-summation integral filter. The output from the “First Inversion” is input to the “Second Inversion” cascade (equation 8). Orthogonalization of updates to reflections and to diffractions is introduced. The data to be input to the “Second Inversion” is filtered with path-summation integral filter. PWD is disabled. Estimated reflectivity is then input to high-wavenumber restoration inversion denoted by “Restore Reflections' Wavenumbers". Reflections with high-wavenumbers restored and diffractions from the second inversion are then input to the “final” inversion restoring high-wavenumbers in diffractions. The data to be matched by the last two inversions is zero-offset data $\mathbf {d}$ with no PWD or path-summation integral filter applied.