De-aliased interpolation by low-frequency constrained local slope estimation

It can be demonstrated by tests that the slope estimation is the main factor affecting the sparsity and anti-aliasing ability of the seislet transform. Thus, we can first filter the data using a simple low-pass filter with a very low bound frequency. The bound frequency can be different for different datasets, but is generally below 15 Hz. When interpolating the regularly missing traces using a POCS algorithm, there are three key factors that will affect the final reconstructed results: (1) Slope estimation. (2) Threshold value in the seislet transform. (3) Number of iterations. We re-estimate the dip about every 5 POCS iterations using the low-pass filtered data. In order to set the optimum threshold, we use a percentile strategy, assuming that a certain percentage of coefficients can represent the data. In order to obtain a very good reconstruction, the number of iterations should be relatively large (about 150 iterations). The main computational cost of the proposed approach lays on the forward and inverse seislet transforms Fomel and Liu (2010); Chen et al. (2014b). The seislet transform can be more expensive than the fast Fourier transform and the digital wavelet transform, but still has an $O(N)$ cost and is still efficient in practice.