# A lifted $\ell_1/\ell_2$ constraint for sparse blind deconvolution

 Title A lifted $\ell_1/\ell_2$ constraint for sparse blind deconvolution Publication Type Conference Year of Publication 2015 Authors Ernie Esser, Tim T.Y. Lin, Rongrong Wang, Felix J. Herrmann Conference Name EAGE Annual Conference Proceedings Month 06 Organization UBC Keywords blind deconvolution, EAGE, EPSI Abstract We propose a modification to a sparsity constraint based on the ratio of $\ell_1$ and $\ell_2$ norms for solving blind seismic deconvolution problems in which the data consist of linear convolutions of different sparse reflectivities with the same source wavelet. We also extend the approach to the Estimation of Primaries by Sparse Inversion (EPSI) model, which includes surface related multiples. Minimizing the ratio of $\ell_1$ and $\ell_2$ norms has been previously shown to promote sparsity in a variety of applications including blind deconvolution. Most existing implementations are heuristic or require smoothing the $\ell_1/\ell_2$ penalty. Lifted versions of $\ell_1/\ell_2$ constraints have also been proposed but are challenging to implement. Inspired by the lifting approach, we propose to split the sparse signals into positive and negative components and apply an $\ell_1/\ell_2$ constraint to the difference, thereby obtaining a constraint that is easy to implement without smoothing the $\ell_1$ or $\ell_2$ norms. We show that a method of multipliers implementation of the resulting model can recover source wavelets that are not necessarily minimum phase and approximately reconstruct the sparse reflectivities. Numerical experiments demonstrate robustness to the initialization as well as to noise in the data. Notes (EAGE, Madrid) URL https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2015/esser2015EAGElcs/esser2015EAGElcs.html DOI 10.3997/2214-4609.201413420 Presentation https://www.slim.eos.ubc.ca/Publications/Public/Conferences/EAGE/2015/es... Citation Key esser2015EAGElcs