Seismic singularity characterization with redundant dictionaries

We consider seismic signals as a superposition of waveforms parameterized by their fractional- orders. Each waveform models the reflection of a seismic wave at a particular transition between two lithological layers in the subsurface. The location of the waveforms in the seismic signal corresponds to the depth of the transitions in the subsurface, whereas their fractional-order constitutes a measure of the sharpness of the transitions. By considering fractional-order transitions, we generalize the zero-order transition model of the conventional deconvolution problem, and aim at capturing the different types of transitions. The goal is to delineate and characterize transitions from seismic signals by recovering the locations and fractional-orders of its corresponding waveforms. This problem has received increasing interest, and several methods have been proposed, including multi- and monoscale analysis based on Mallat’s wavelet transform modulus maxima, and seismic atomic decomposition. We propose a new method based on a two-step approach, which divides the initial problem of delineating and characterizing transitions over the whole seismic signal, into two easier sub-problems. The algorithm first partitions the seismic signal into its major components, and then estimates the fractional-orders and locations of each component. Both steps are based on the sparse decomposition of seismic signals in overcomplete dictionaries of waveforms parameter- ized by their fractional-orders, and involve $\ell_1$ minimizations solved by an iterative thresholding algorithm. We present the method and show numerical results on both synthetic and real data.