Imaging using compressive sensing

Introduction

Reverse Time Migration is quickly becoming the standard high-end imaging technique. The last few years has seen numerous papers on how to speed up the finite-difference kernel on various platforms (Nemeth et al., 2008; Micikevicius, 2009; Clapp et al., 2010; Nguyen et al., 2010). These techniques was sufficient when RTM was simply used to produce a final image.

When RTM is used to construct angle gathers for velocity or rock property analysis, the finite difference kernel becomes a secondary concern. The construction of angle gathers, particularly 3-D angle gathers, through sub-surface offset correlation (Sava and Fomel, 2006) or time-shift gathers (Sava and Fomel, 2003) becomes the dominant cost. Some have proposed reducing the cost of 3-D angle gathers by constructing angle gathers along only a few azimuths. While these techniques are significantly less costly than full 3-D angle gathers, they are still expensive and not ideal. Compressive sensing (Donoho, 2006) offers a potential solution to this computation and storage problem. In compressive sensing, a random sub-set of the desired measurements are made. An inversion problem is then set up to estimate in an $\ell_1$ , or preferably $\ell_0$ , sense, a sparse basis function that fully characterizes the desired signal. For compressive sensing to work, a signal must be highly compressible. For compressive sensing to be worthwhile, the cost of inverting for the basis function must be significantly less than the cost of acquiring the full signal.

In this paper, I show how angle gather construction fits the criteria for compressive sensing. I demonstrate how angle gathers are highly compressible in the multi-dimensional wavelet domain. Further, I demonstrate how the cost of constructing a sub-set of the sub-surface offsets and then performing an $\ell_1$ inversion is significantly less expensive.

Imaging using compressive sensing