Robust f ‐ x projection filtering for simultaneous random and erratic seismic noise attenuation

Linear prediction filters are an effective tool for reducing random noise from seismic records. Unfortunately, the ability of prediction filters to enhance seismic records deteriorates when the data are contaminated by erratic noise. Erratic noise in this article designates non‐Gaussian noise that consists of large isolated events with known or unknown distribution. We propose a robust f ‐ x projection filtering scheme for simultaneous erratic noise and Gaussian random noise attenuation. Instead of adopting the ℓ 2 ‐norm, as commonly used in the conventional design of f ‐ x filters, we utilize the hybridℓ 1 / ℓ 2 ‐norm to penalize the energy of the additive noise. The estimation of the prediction error filter and the additive noise sequence are performed in an alternating fashion. First, the additive noise sequence is fixed, and the prediction error filter is estimated via the least‐squares solution of a system of linear equations. Then, the prediction error filter is fixed, and the additive noise sequence is estimated through a cost function containing a hybridℓ 1 / ℓ 2 ‐norm that prevents erratic noise to influence the final solution. In other words, we proposed and designed a robust M‐estimate of a special autoregressive moving‐average model in the f ‐ x domain. Synthetic and field data examples are used to evaluate the performance of the proposed algorithm.