The most advanced data processing/imaging algorithms aim at optimizing a suitably defined cost function consisting of a data misfit term, posed in the data domain, and a regularization term. For strong phase errors in the data, however, such algorithms fail to achieve an acceptable solution because they force us to match the incorrect phase information. To remedy this deficiency, we replaced the time domain misfit term by a Fourier magnitude fidelity term to recover the desired signal from just the amplitude spectrum of the observations. Such a problem is known as phase retrieval, and it is a subject of interest for those who cannot measure the phase information about the system under study. The proposed phase retrieval problem is solved by a fast iterative shrinkage/thresholding algorithm, and it is used to tackle two common phase problems arising in seismic processing: data recovery in the presence of residual static shifts and deconvolution with a missing wavelet phase. Under sparsity constraints, both problems can be solved using just the amplitude spectrum of the observed data. In both cases, using computer-simulated data and field data, the regularized phase retrieval algorithm was able to obtain better results than the conventional methods in which the misfit term is posed in the time domain.

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