Constraints versus penalties for edge-preserving full-waveform inversion

Full-waveform inversion is arguably one of the most challenging inverse problems out there. For this reason, it is remarkable that so much progress has been made over the years. While we can look back with confidence at an increasing number of success stories, the application of full-waveform technologies to deeper targets and complex geologies remains challenging making it an expensive proposition. So far, efforts to overcome some of these challenges have mainly been directed towards extended formulations such as Symes's and Biondi's subsurface offset/rau-parameters, Mike Warner/Lluis Guasch's Wiener filters in their adaptive full-waveform inversion, and van Leeuwen/Herrmann's Wavefield Reconstruction Inversion, where the wave-equation appears as a least-squares term in the objective. While this recent work –- including other approaches such as different data-misfit objective functions –- has removed some of FWI cycle-skip issues it does, with the exception perhaps of box constraints, not incorporate rudimentary information on the geology on the unknown model. Following recent work by Ernie Esser, we present a general framework how to incorporate this type of information in the form of constraints. Contrary to Tikhonov-like regularization or gradient filtering, imposing constraints has several key advantages, namely (i) they can be expressed in simple human understandable terms such as lower and upper limits for the velocity without the necessity to introduce additional parameters and assumptions on the probability distribution of the model; (ii) they can consist of several constraints, e.g. box and smoothness constraints, as long as their intersection in non-empty; and (iii) they can be imposed in a separate inner loop, which does not affect gradients and Hessians of full-waveform inversion itself. By means of examples, we will demonstrate the advocacy of constrained formulations as they apply to full-waveform inversion. This is joint work with Bas Peters.

https://slim.gatech.edu/Publications/Public/Conferences/SEG/2016/herrman...