Seismic Laboratory for Imaging and Modeling

Geophysical inverse problems typically involve a trade off between data misfit and some prior. Pareto curves trace the optimal trade off between these two competing aims. These curves are commonly used in problems with two-norm priors where they are plotted on a log-log scale and are known as L-curves. For other priors, such as the sparsity-promoting one norm, Pareto curves remain relatively unexplored. We show how these curves lead to new insights in one-norm regularization. First, we confirm the theoretical properties of smoothness and convexity of these curves from a stylized and a geophysical example. Second, we exploit these crucial properties to approximate the Pareto curve for a large-scale problem. Third, we show how Pareto curves provide an objective criterion to gauge how different one-norm solvers advance towards the solution. INTRODUCTION Many geophysical inverse problems are ill posed (Parker, 1994)—their solutions are not unique or are acutely sensitive to changes in the data. To solve this kind of problem stably, additional information must be introduced. This technique is called regularization (see, e.g., Phillips, 1962; Tikhonov, 1963). Specifically, when the solution of an ill-posed problem is known to be (almost) sparse, Oldenburg et al. (1983) and others have observed that a good approximation to the solution can be obtained by using one-norm regularization to promote sparsity. More recently, results in information theory have breathed new life into the idea of promoting sparsity to regularize ill-posed inverse problems. These results establish that, under certain conditions, the sparsest solution of a (severely) underdetermined linear system can be exactly recovered by seeking the minimum one-norm solution (Candès et al., 2006; Donoho, 2006; Rauhut, 2007). This has led to tremendous activity in the newly established field of compressed sensing. Several new one-norm solvers have appeared in response (see, e.g., Daubechies et al., 2004; van den Berg and Friedlander, 2008, and references therein). In the context of geophysical applications, it is a challenge to evaluate and compare these solvers against more standard approaches such as iteratively reweighted least-squares (IRLS Gersztenkorn et al., 1986), which uses a quadratic approximation to the one-norm regularization function. In this letter, we propose an approach to understand the behavior of algorithms for solving one-norm regularized problems. The approach consists of tracking on a graph the Seismic Laboratory for Imaging and Modeling, Department of Earth and Ocean Sciences, The University of British Columbia, 6339 Stores Road, Vancouver, V6T 1Z4, BC, Canada Scientific Computing Laboratory, Department of Computer Science, The University of British Columbia, 2366 Main Mall, Vancouver, V6K 1Z4, BC, Canada