Introduction

Chen et al. (1999) introduces the idea of ``basis pursuit'' (BP) as a principle to aid in precision analysis of complex signals. The central tenet of the presentation lies in the assumption that a minimal number of constituent members of a model space dictionary are responsible for the signal being analyzed. Therefore, the distribution of energy across the coefficients associated with dictionary atoms (be they sinusoids, wavelets, chirps, velocities, etc.) during the analysis of a signal should be uneven and sparse. This idea of a sparse model space is counter to a smoothly distributed l² norm inversion, and thus a different algorithm needs development to satisfy these requirements.

The development of this inversion principle into an algorithm can take any number of forms. Guitton and Symes (2003) choose the Huber norm to effect an l¹-like measure of the inversion error. We will cast the problem through the primal-dual Linear Programming (LP) structure resulting in a methodology wherein the concept of convergence is central to the algorithm. This fact has two important consequences. Firstly, the precision of the output model space is one of (the very few) input parameters. Secondly, the parameter space is insensitive to manipulation as compared to $\epsilon$ in regularized least squares problems or the cutoff value needed for Huber norm approaches.

Conventionally, LP methods deal almost exclusively in a small world of conveniently short time signals such as bursts of speech. Application of these methods to geophysical problems of much larger size may prove prohibitive. While at its best the complexity of this method can be comparable to IRLS, in practice the method is usually several times slower to produce optimal solutions.

As an example of the method, the hyperbolic radon transforms are used as analysis operators of seismic and synthetic seismic data. An exploration of the method comparable to Guitton and Symes (1999) will be used to highlight the strengths and weaknesses of the method compared to conventional least squares and Huber norm inversion for velocity from seismic data.