Simply denoise: wavefield reconstruction via jittered undersampling

Application to seismic data

Following recent work on Curvelet Reconstruction with Sparsity-promoting Inversion (CRSI - Herrmann and Hennenfent, 2007), seismic wavefields are reconstructed via $\tilde{\ensuremath{\mathbf{f}}} = \tensor{C}^H\tilde{\ensuremath{\mathbf{x}}}$ where

$$\tilde{\ensuremath{\mathbf{x}}} = \arg\min_{\ensuremath{\mathbf{x}}}\vert\vert\ensuremath{\mathbf{x}}\vert\vert _1 \quad\ensuremath{\mathbf{y}}=\tensor{RC}^H\ensuremath{\mathbf{x}}.$$ (9)

In this formulation, $\tensor{C}$ is the discrete wrapping-based curvelet transform (Candès et al., 2005a). Similarly to any other data-independent transforms, curvelets do not provide a sparse representation of seismic data in the strict sense. Instead, the curvelet transform provides a compressible, arguably the most compressible (Hennenfent and Herrmann, 2006), representation. Compressibility means that most of the wavefield energy is captured by a few significant coefficients in the sparsifying domain. Since CS guarantees, for sparse-enough signal representations, the recovery of a fixed number of largest coefficients for a given undersampling factor (Candès et al., 2005b), a more compressible representation yields a better reconstruction, which explains the success of CRSI.

Simply denoise: wavefield reconstruction via jittered undersampling