Seismic Reconstruction Based on Data Fitting With the l₁-Norm in the Presence of Abnormal Values

In seismic exploration, there are abandoned mines and bad traces, leading to missing data. In addition, unsuitable processing methods will introduce randomized amplitude anomalies. The traditional smoothing term with the $l_{2}$ -norm assumes that the random noise is a Gaussian distribution. For the Gaussian distribution, the representation of the $l_{2}$ -norm has a short tail compared with the $l_{1}$ -norm, abnormal values in the data cannot be suppressed, and the stability of the solution is poor. For the Laplacian distribution, the representation of the $l_{1}$ -norm has a longer tail compared with the $l_{2}$ -norm, and it has a good tolerance for abnormal values. We propose a new method, based on the theory of compressed sensing, which uses the $l_{1}$ -norm as the data fitting term and a sparse reconstruction equation for missing data with abnormal amplitude noise. To solve the equation for the complete data with no abnormal values, we apply the approximate projected subgradient method. Model and field data tests confirm the increased robustness of the proposed method.