Beyond alias hierarchical scale curvelet interpolation of regularly and irregularly sampled seismic data

We propose a robust interpolation scheme for aliased regularly sampled seismic data that uses the curvelet transform. In a first pass, the curvelet transform is used to compute the curvelet coefficients of the aliased seismic data. The aforementioned coefficients are divided into two groups of scales: alias-free and alias-contaminated scales. The alias-free curvelet coefficients are upscaled to estimate a mask function that is used to constrain the inversion of the alias-contaminated scale coefficients. The mask function is incorporated into the inversion via a minimum norm least-squares algorithm that determines the curvelet coefficients of the desired alias-free data. Once the alias-free coefficients are determined, the curvelet synthesis operator is used to reconstruct seismograms at new spatial positions. The proposed method can be used to reconstruct regularly and irregularly sampled seismic data. We believe that our exposition leads to a clear unifying thread between f-x and f-k beyond-alias interpolation methods and curvelet reconstruction. As in f-x and f-k interpolation, we stress the necessity of examining seismic data at different scales (frequency bands) to come up with viable and robust interpolation schemes. Synthetic and real data examples are used to illustrate the performance of the proposed curvelet interpolation method.