Discussion

We have proposed a workflow to decompose full waveform input data into reflections, diffractions and noise by using three cascades of inversions. We also want to highlight regularization through orthogonalization as a novel tool to aid the inversion, in which multiple models are estimated with a single modeling operator. Its ability to track down the leakage of the events from one model space to another appears to be crucial for the second cascade of the inversions. Orthogonalization, combined with shaping regularizations, allows for efficient reflection, diffraction and noise separation.

Shaping regularization allows for a flexible model constraints since models for reflections and diffractions are conditioned independently. For instance, if the regularization terms were incorporated as penalty terms in the objective functions (equations 6, 8 and 9), finding the trade-off between weights for all the terms would be challenging: the two terms would conflict with each other leading to stronger interference between model spaces.

It should be mentioned that the workflow presented here targets diffraction phenomenon in 2D rather than in 3D. In particular, in the case of three dimensions, sparsity constraints can be inadequate for edge diffractions regularization. The latter features are continuous along the edges and act as reflections but at the same time have a diffractive component in the direction perpendicular to the edge (Moser, 2011). To properly regularize edge diffractions sparsity constraints should be accompanied by anisotropic smoothing operator enforcing continuity along the edges (Merzlikin et al., 2018). At the same time, PWD robustly serving the purpose of diffraction extraction in 2D, in 3D should be replaced by AzPWD - workflow properly addressing hybrid signature of edge diffractions (Merzlikin et al., 2016,2017b). These are the modifications, which have to be incorporated into the workflow presented in this paper in order for it to be extended to 3D. At the same time, all the necessary modifications have already been derived and, thus, support the applicability of the reflection-diffraction crosstalk minimization approach presented in this paper in three dimensions.

Finally, the proposed approach (except for analytical time domain path-summation migration describing diffraction probability) is compatible with any migration algorithms operating in pre-stack, post-stack, depth and time domains. In pre-stack domain the PWD filter can be implemented using common-offset sorting of input data. Regularizations are performed in the image domain and, thus, are not dependent on a particular migration algorithm. Analytical expressions exist for path-summation imaging in pre-stack time-migration domain (Merzlikin and Fomel, 2017a), however, path-summation integral migration in the depth domain remains challenging (Landa et al., 2006) and, thus, limits the applicability of the approach presented here. The possible workaround is to incorporate probability weights favouring diffractions (Merzlikin and Fomel, 2017b) into path-summation integral expression and utilize importance sampling strategies for computational load reduction associated with the absence of analytical expressions of path-integral evaluation in the depth domain.