3D interpolation of irregular data with a POCS algorithm

Abstract

Seismic surveys generally have irregular areas where data cannot be acquired. These data should often be interpolated. A projection onto convex sets (POCS) algorithm using Fourier transforms allows interpolation of irregularly populated grids of seismic data with a simple iterative method that produces high-quality results. The original 2D image restoration method, the Gerchberg-Saxton algorithm, is extended easily to higher dimensions, and the 3D version of the process used here produces much better interpolations than typical 2D methods. The only parameter that makes a substantial difference in the results is the number of iterations used, and this number can be overestimated without degrading the quality of the results. This simplicity is a significant advantage because it relieves the user of extensive parameter testing. Although the cost of the algorithm is several times the cost of typical 2D methods, the method is easily parallelized and still completely practical.

Simply denoise: Wavefield reconstruction via jittered undersampling

“…The objective of this paper is to demonstrate that irregular/random undersampling is not a drawback for particular transform-based interpolation methods and for many other advanced processing algorithms as was already observed by other authors (Zhou and Schuster, 1995;Sun et al, 1997;Trad and Ulrych, 1999;Xu et al, 2005;Abma and Kabir, 2006;Zwartjes and Sacchi, 2007). We explain why random undersampling is an advantage and how it can be used to our benefit when designing coarse sampling schemes.…”

“…Sacchi et al, 1998;Xu et al, 2005;Zwartjes and Sacchi, 2007;. Jittered undersampling differentiates itself from random undersampling according to a discrete uniform distribution, which also creates favorable recovery conditions (Xu et al, 2005;Abma and Kabir, 2006;Zwartjes and Sacchi, 2007), by controlling the maximum gap in the acquired data. This control makes jittered undersampling very well suited to methods that rely on transforms with localized elements, e.g., windowed Fourier or curvelet transform (Candès et al, 2005a, and references therein).…”

OC-seislet: Seislet transform construction with differential offset continuation

“…To demonstrate the superior sparseness of the OC-seislet coefficients, we compare the proposed method with the 3-D Fourier transform. Figure 13a displays the interpolated result after a 3-D Fourier interpolation using iterative thresholding (Abma and Kabir, 2006). The Fourier transform cannot provide enough sparseness of coefficients for complex reflections and, therefore, fails in recovering all missing traces.…”

“…Missing traces have been interpolated well even where diffractions are present. For comparison, we applied 3-D Fourier transform with iterative soft-thresholding, as proposed by Abma and Kabir (2006). The same parameters of soft-thresholding for those in the OC-seislet transform were used.…”

“…Xu et al (2005) suggested an iterative Fourier-based matching pursuit (antileakage Fourier transform) for seismic data regularization. Abma and Kabir (2006) used a Fourier-based method with iterative thresholding to solve seismic data interpolation problems. presented a recovery method that exploits the curvelet frame.…”