L1 norm inversion method for deconvolution in attenuating media

In order to perform a good pulse compression, the conventional spike deconvolution method requires that the wavelet is stationary. However, this requirement is never reached since the seismic wave always suffers high‐frequency attenuation and dispersion as it propagates in real materials. Due to this issue, the data need to pass through some kind of inverse‐Q filter. Most methods attempt to correct the attenuation effect by applying greater gains for high‐frequency components of the signal. The problem with this procedure is that it generally boosts high‐frequency noise. In order to deal with this problem, we present a new inversion method designed to estimate the reflectivity function in attenuating media. The key feature of the proposed method is the use of the least absolute error (L1 norm) to define both the data and model error in the objective functional. The L1 norm is more immune to noise when compared to the usual L2 one, especially when the data are contaminated by discrepant sample values. It also favours sparse reflectivity when used to define the model error in regularization of the inverse problem and also increases the resolution, since an efficient pulse compression is attained. Tests on synthetic and real data demonstrate the efficacy of the method in raising the resolution of the seismic signal without boosting its noise component.