Seismic denoising with nonuniformly sampled curvelets

Adaptive curvelet-domain primary-multiple separation

“…with C the 2D discrete curvelet transform ͑see, e.g., Candes et al, 2006;Hennenfent and Herrmann, 2006͒, w the curvelet-domain scaling vector, and M the index set of curvelet coefficients. Because we are using the curvelet transform based on wrapping, which is a tight frame, C T C ‫ס‬ I, and the transpose, denoted by the symbol T, equals the pseudoinverse.…”

“…For field data, these factors preclude iterative SRME, resulting in amplitude errors that vary for different multiple orders ͑see, e.g., Verschuur and Berkhout, 1997;Paffenholz et al, 2002͒. In practice, the second separation stage appears to be particularly challenging because adaptive ᐉ 2 -matched-filtering techniques are known to lead to residual multiple energy, high-frequency clutter, and deterioration of the primaries ͑Chen et al, 2004;Abma et al, 2005;Herrmann et al, 2007a͒. By employing the ability of the curvelet transform ͑Candes et al., 2006;Hennenfent and Herrmann, 2006͒ to detect wavefronts with conflicting dips ͑e.g., caustics͒, Herrmann et al ͑2007a͒ and Herrmann et al ͑2008b͒ derived a nonadaptive separation scheme ͑independent of the total data͒ that uses the original data and SRME-predicted multiples as input and produces an estimate for the primaries. This threshold-based method proved to be robust with respect to moderate errors ͑sign, phase, and timing͒ in the predicted multiples and derived its success from the sparsifying property of curvelets for data with wavefronts.…”

“…This threshold-based method proved to be robust with respect to moderate errors ͑sign, phase, and timing͒ in the predicted multiples and derived its success from the sparsifying property of curvelets for data with wavefronts. Despite recent advances in thresholding by a Bayesian formulation ͑Saab et al, Wang et al, 2007͒ and mitigation of the effects of missing data ͑Hennenfent and Herrmann, 2008;Herrmann et al, 2007b͒, curvelet-domain separation deteriorates when predicted multiples have significant amplitude errors. Thresholding in these cases can give rise to inadvertent removal of primary energy or to remnant multiple energy.…”

3-D Data Denoising and Inpainting with the Low-Redundancy Fast Curvelet Transform

“…However, the major issue that rises with such a mask is that it exhibits a strong coherence with some curvelet atoms 6 . Indeed, curvelets are plate-shaped atoms, and those oriented at 45 degrees are very correlated with the inclined planes formed by the alternating even-odd lines along the time axis.…”

“…In fact, the curvelet transform provides atoms that are well localized in space and frequency and exhibit a strong directional selectivity. This transform has found a wide spectrum of applications including denoising [4,5,6], contrast enhancement [7], inpainting [8,9] or deconvolution [10,11].…”