A seismic trace corresponding to an anelastic layered earth model can be sparsely represented by a structured dictionary of properly attenuated wavelets, through a nonstationary sparse deconvolution. The sparseness of the coefficients (reflectivity series), however, considerably decreases when the wavelets are incorrectly modeled due to an incorrect Q model. Mathematically, the wavelets, as the elements of the dictionary, are nonlinearly related to the earth quality factor Q (the inverse of the attenuation coefficient). A parametric dictionary-learning strategy enables interval-Q estimation and compensation by training the dictionary atoms from the input trace adaptively to provide a sparse representation of it. We assumed a piecewise Q model by dividing the dictionary elements into several groups, each containing several wavelets whose temporal supports are close to each other and can be described by a single Q-value. The dictionary is learned iteratively where at each iteration only one group of the wavelets is optimized by searching for the corresponding optimum Q-value, leading to an iterative construction of the Q model. The main advantages of our method for interval-Q estimation are its stability because it performs in a forward-modeling manner and its accuracy because of the resolved interferences by the sparsity constraint. Our method is tested on synthetic and field data sets, and the results that we obtained demonstrate the stability and accuracy of the method for interval-Q estimation and compensation.

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