Simply denoise: wavefield reconstruction via jittered undersampling

Generalization of the concept of undersampling artifacts

Undersampling artifacts are only one particular case of MAI that specifically occurs in the interpolation problem, i.e., $\tensor{A}\; {\buildrel\rm def\over=}\; \tensor{RS}^H$ . The study we have done on these artifacts as a function of the restriction operator $\tensor{R}$ can be extended to more general cases (see e.g. Lustig et al., 2007, in magnetic resonance imaging). For example, when $\tensor{A}$ is defined as $\tensor{A}\; {\buildrel\rm def\over=}\; \tensor{RMS}^H$ with $\tensor{M}$ a modeling/demigration-like operator (Herrmann et al., 2007; Wang and Sacchi, 2007). In this case, $\ensuremath{\mathbf{x}}_0$ is the sparse representation of the Earth model in the $\tensor{S}$ domain and $\ensuremath{\mathbf{y}}$ incomplete seismic data. The study of the MAI now determines which coarse spatial sampling schemes are more favorable than others in the context of sparsity-promoting migration/inversion. Based on observations in Zhou and Schuster (1995) and Sun et al. (1997), we believe that discrete random, optimally-jittered, and continuous random undersamplings will also play a key role.