Random sampling: new insights into the reconstruction of coarsely-sampled wavefields

INTRODUCTION

Dense sampling of seismic data is traditionally understood as evenly-distributed time and space measurements of the reflected wavefield. Moreover, the sampling rate along each axis must be equal to or above twice the highest frequency/wavenumber of the continuous signal being discretized (Shannon/Nyquist sampling theorem). In practice, however, seismic data is often randomly and/or sparsely sampled along spatial coordinates, which is generally considered as a challenge since it breaks one or both previously-stated conditions of dense sampling. It turns out that these acquisition geometries are not necessarily a source of adversity to accurately reconstruct densely-sampled data when using nonlinear optimization promoting sparsity. This new insight, developed in the information theory community, is referred to in the literature by the terms ``compressed sensing'' or ``compressive sampling'' (see e.g. Donoho, 2006; Candes et al., 2005, and references therein).