Morphological deconvolution

Various effects cause a seismic trace to consist of inhomogeneous (phase-shifted) wavelets, which can be modeled by constant phase rotations of a base wavelet. Analytic deconvolution, an analytic signal formulation of the conventional statistical deconvolution, has been proposed for deconvolution of such traces, but it seems to be ineffective in the presence of wavelet interferences and noise because it has no control over the estimated phase. Morphological deconvolution (MD) allows full control over the estimated phase and reflectivity; hence, it is more effective than analytic deconvolution in deconvolving traces with nonstationary phase variations. MD replaces the convolution model of a seismic trace with the sum of a set of convolutions with phase-rotated wavelets. The solution to the resulting under-determined system of equations provides the reflectivity as a function of the time and phase shifts, and it is found by a sparse plus low-rank optimization. The sparsity constraint forces decomposition of the trace into the least number of wavelets, whereas the low-rank constraint allows for regularizing the phase function. MD is easily applicable in multichannel form (MMD), allowing regularization of the estimated phase in the time and space directions. The efficacy and robustness of the MD and MMD methods are determined on several synthetic and field data sets in the presence of interference and noise.