The seislet transform was proposed in Fomel and Liu (2010). In this paper, we first compare the sparsity of different well-known transforms widely used in the seismic data processing community, including the Fourier transform, the wavelet transform, the curvelet transform, and the seislet transform. The comparison, to the best of our knowledge, is never done in the literature. This sparse comparison offers us a new view in selecting the sparsest transform for related applications in seismic data processing. The seislet transform has found successful applications in noise attenuation (Liu et al., 2009b; Fomel and Liu, 2010). However, the successful application of the seislet transform in iterative interpolation, especially from the industry, is barely reported. One of the drawbacks that impede the wide application of seislet based interpolation is the efficiency. The seislet transform itself does not slow down the efficiency too much. The efficiency of seislet transform is about 2-4 times slower than the fast Fourier transform, and is about 4-8 times slower than the fast wavelet transform Fomel and Liu (2010). However, the slope estimation that is required by the seislet transform is much slower. In order to accelerate the process, the slope estimation is commonly estimated every several iterations. In this paper, the slope estimation is iterated every 5 iterations. Even though, the computational cost is still much heavier than the widely used Fourier transform. In this paper, the FPOCS approach can greatly accelerate the efficiency by reducing a large number of iterations. According to the performance of the two examples in the paper, about two thirds iterations can be saved using the fast iterative approach. The cost saving in the paper is only obtained from 2D data examples. The application of 3D seislet-based POCS approach can even offer more cost savings, which will potentially allow the wide application of the seislet transform in the industry.
The widely used POCS and IST algorithms can be both considered as the simplest and most effective iterative approaches for seismic data interpolation. In a mathematical sense, the IST is a type of POCS, the multiple projections include the weighted projection (model update), the forward sparse transform, the soft thresholding, and the inverse sparse transform. However, in the community of exploration geophysics, the IST algorithm and the POCS algorithm are different. The most apparent difference is whether to use the known data as a part of the model. However, there is no published literature discussing the performance difference using the two approaches. We use a group of two simple but convincing tests with and without strong random noise to show the slight difference between the two approaches. The selection of the two approach simply depends on the noise level in the seismic data. When the noise level is high, we should use the IST based approach (FIST), otherwise, we should use the POCS based approach (FPOCS).
How to measure data recovery performance is another long-standing argument in the seismic data processing community. In the case of simulated test, in which we know the true answer, the traditionally used signal-to-noise ratio (SNR) seems to be the best choice. However, the SNR can only obtain a global measurement of the recovery quality while the local performance is not measured effectively. For example, an extrema (or huge error) in a local area will also result in a small global average. We use the local similarity (Fomel, 2007a) as a way to measure the data reconstruction performance in this paper. This evaluation is based on the assumption that the true signal and the estimated should have high local similarity and the true signal and the estimation error should have low local similarity. From the local similarity maps, we can get more details of different approaches. We can observe clearly that, even in the case of very close SNRs, the local similarity can still show slight but obvious difference, which makes it more sensitive in comparing different state-of-the-art approaches.