Robust total-variation based geophysical inversion using split Bregman and proximity operators

In this paper, we take advantages of split Bregman and proximity operators to formulate geophysical inverse problems in a convex optimization setting. In this type of formulation, we can efficiently consider a general convex data misfit term in order to incorporate more realistic error probability assumptions than the ordinary Gaussian distribution. Furthermore, we can simply impose convex non-quadratic and non-smooth regularization terms as a prior information. Although the proposed formulation can be extended for incorporating any types of convex regularization and data fidelity terms, here we consider misfit term corresponding to the Huber norm and anisotropic total variation regularization as a prior to an unknown model structure. This type of regularization enables us to construct piecewise-constant solutions. Two-dimensional linear and non-linear seismic travel-time tomography are studied to evaluate the efficiency of the proposed formulation for handling a robust measure of the misfit term in recovering blocky solutions. We consider the first arrival travel-times which are contaminated by outliers and also a mixed combination of Gaussian and outliers. Finally, Seinsfeld cross-hole tomography data set is used to investigate the performance of the robust approach on real data sets. (C) 2016 Elsevier B.V. All rights reserved.