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 inversion (FWI) technologies to deeper targets and more complex geologies remains challenging making FWI an expensive and challenging proposition.
So far, efforts to overcome these challenges have mainly been directed towards extended formulations. These include Symes’s and Biondi’s subsurface offset extensions; Mike Warner/LLuis Guasch’s Wiener filters in their adaptive full-waveform inversion (AWI); van Leeuwen/Herrmann’s penalty formulation wavefield reconstruction inversion (WRI) where the wave-equation appears as an extension; and Symes recent source extension where the source wavefield entails the extension. While these efforts — including approaches such as different data-misfit objective functions — have shown some success in mitigating FWI’s cycle-skip issues, none of these approaches makes a concerted effort to include prior information on the unknown control parameters–i.e., the Earth model.
Following recent work by the late Ernie Esser, we present an optimization framework to regularize FWI with constraints. Contrary to Tikhonov-like quadratic regularization with penalties or gradient filtering, imposing prior information in the form of constraints has several key advantages, namely *(i)* (convex) constraint sets can be expressed in simple human understandable terms; *(ii)* as long as their intersection in non-empty, multiple (convex) constraint sets can readily and uniquely be combined; and *(iii)* these constraints can be imposed in a separate inner loop, which does not affect gradients and Hessians. The latter property is important because certain types of regularization (including TV-norm regularization) in the form of penalties may be ill-conditioned adversely affecting the conditioning of the Hessian.
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