Geophysical data are notoriously contaminated by noise, some of which may behave as outlier to the rest of the data. Least absolute deviations can successfully handle such outliers, however it requires the L1-norm of the data to be minimized, which is much harder to minimize than the L2- norm commonly used in least-squares. The problem lies in the non-differentiability of the L1-norm, which prohibits the efficient use of gradient based methods. In recent years, algorithms based on the proximal operator have gained great popularity in the mathematical community, due to their ability to efficiently minimize objective function consisting of both a smooth and a non-smooth part. The Alternating Direction Method of Multipliers (ADMM) is one such algorithms that is able to solve a large class of problems. In this work, we present a new ADMM-type solver that can handle objective functions that are the sum of an L1 data fidelity term and an L1 regularization term. We apply our algorithm to the problem of seismic deblending and compare it against the IRLS method of Ibrahim and Sacchi (2014).