Introduction

Chen et al. (1999) introduces the idea that inversion problems solved with a sparseness constraint on the output model-space can be applied to data sets with an over-complete modeling operator. Over-complete means that the dictionary of transform kernels has (many) more terms than a strict basis transformation such as the standard DFT. For example, this could be realized by using several different modeling operators or even a over-sampled version of a single operator such as a Fourier transform with several times more frequency resolution than the compact support definition of DFT theory. Further, in Donoho and Huo (1999) the mathematics are presented to prove that the combined suite of model space kernels are orthogonal to each other if the inversion satisfies several requirements. Their goal is super-resolution during the analysis of a signal. These investigations are presented within the framework of inversion through linear programming techniques. Futher, the papers imply that these desirable properties are realized with the use of an inversion methodology using a l¹ norm on the model-space. Artman and Sacchi (2003) made preliminary efforts to investigate these properties, but were stalled due to the instability and expense, when applied to the much larger and more complicated data domains of seismic data, of the linear programming technique. Trad et al. (2003) present results of such a combined operator inversion scheme using the Cauchy regularization.

Guitton (2004) explores the novel result that two bound-constrained inversions (bounded by zero from below and above) contain orthogonal null-spaces. Therefore, summing the results of two such inversions can significantly enhance the sparseness of the model space compared to that produced by an inversion which is free to output both positive and negative values for the inverted model. The cost for this improvement is an extra inversion of identical size and cost. This characteristic is shared with the linear programming techniques used for the super-resolution problem, though the inversion algorithm is the L-BFGS.

We will compare the inverted models from several inversion schemes, and their concomitant data-space realizations. Inversion schemes that will be analyzed are:1) bound-constrained (BC), 2) Cauchy norm regularization (Cauchy), 3) Huber norm approximating the l₁ norm (l₁), and 4) the l₂ norm using linear least-squares (l₂). The combined operator of Hyperbolic and Linear Radon Transforms will be used in all the inversion schemes on both synthetic data and three field shot gathers.