Resolving scaling ambiguities with the ℓ1/ℓ2 norm in a blind deconvolution problem with feedback

Compared to more mundane blind deconvolution problems, blind deconvolution in seismic applications involves a feedback mechanism related to the free surface. The presence of this feedback mechanism gives us an unique opportunity to remove ambiguities that have plagued blind deconvolution for a long time. While beneficial, this feedback by itself is insufficient to remove the ambiguities even with ℓ₁ constraints. However, when paired with an ℓ₁/ℓ₂ constraint the feedback allows us to resolve the scaling ambiguity under relatively mild assumptions. Inspired by lifting approaches, we propose to split the sparse signal into positive and negative components and apply an ℓ₁/ℓ₂ constraint to the difference, thereby obtaining a constraint that is easy to implement. Numerical experiments demonstrate robustness to the initialization as well as to noise in the data.