Introduction

Radon transforms are popular operators for velocity analysis Guitton and Symes (2003); Taner and Koehler (1969), noise attenuation Foster and Mosher (1992), and data interpolation Hindriks and Duijndam (1998); Trad et al. (2002). One property that is often sought in radon domains is sparseness, where the energy in the model space is well focused for each corresponding event in the data space. Sparseness is especially useful for multiple attenuation and interpolation. In practice, depending on the radon transform, sparseness can be achieved either in the Fourier Herrmann et al. (2000) or time domain Sacchi and Ulrych (1995). To estimate sparse radon panels in the time domain, a regularization operator that enforces long-tailed probability density functions for the model parameters is often used. This regularization operator can be the $\ell^1$ norm Nichols (1994) or the Cauchy norm Sacchi and Ulrych (1995).

In this paper, a new time-domain method is presented that yields sparse radon panels. This method estimates a sparse model by adding two models estimated independently with only positive or negative values obtained with a bound-constrained optimization technique. Therefore, by forcing the model to fall within a certain range of values, the null space and its effects are decreased.

In the section following this introduction, I introduce the problem of finding a bound-constrained model and its resolution by presenting the limited memory L-BFGS-B technique Zhu et al. (1997). This method aims at finding a solution with simple bounds for linear or non-linear problems. Then, I introduce a method that estimates sparse radon domains. Finally, this technique is tested on a synthetic and field data examples and compared to the Cauchy regularization Sacchi and Ulrych (1995). They demonstrate that the proposed strategy yields sparse radon panels comparable to the Cauchy approach. One advantage of this new strategy is that the choice of parameters is much simpler; for instance, no Lagrange multiplier is needed. One drawback is that two inversions need to be carried out as opposed to one for the Cauchy method.