In seismic data processing, multiples, especially internal ones, are traditionally considered as troublesome noise. Their estimation and elimination are not always straightforward since in most cases they overlap with primaries. Actually, application of internal multiple attenuation approaches may produce erroneous results and distort true primary reflections. In this paper, we propose a new method based on 1D sparse constrained full waveform inversion aiming not in the removal of internal multiples, but instead use them to recover the reflectivity series from the seismic data. Unlike conventional inversion which usually seeks for velocities, our method inverts seismic data for reflectivity. We recover the primaries by using a new iterative shrinkage-thresholding algorithm based on an l1-norm regularized optimization. The shrinkage-thresholding operator provides a good approximation for the sparsest solution. Moreover, we derive an analytical Jacobian, which can be computed by only one forward modeling based on Kennett’s reflectivity method. Our method does not require adaptive subtraction of predicted multiples from the input data and any knowledge of the subsurface, and therefore is fully automatic. This algorithm has been tested with very promising results on 1D synthetic data generated from a field sonic well log.


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