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

Abstract

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 aliasfree 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 bands͒ 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.

Curvelet-based POCS interpolation of nonuniformly sampled seismic records

“…This work was developed further by Naghizadeh and Sacchi (2010b). The authors even employed a curvelet transform and a fast generalized Fourier transform to perform the interpolation task with the hard thresholding (coined a mask function) and ' 2 penalty, where the conjugate gradient (CG) method of Naghizadeh and Sacchi (2010a) and the iterative reweighted leastsquares (IRLS) technique of Naghizadeh and Innanen (2011) are utilized naturally.…”

A fast-convergence POCS seismic denoising and reconstruction method

“…Within the fi rst category, one group of methods relies on transforms, such as the Fourier transform (Duijndamet et al, 1999;Trad, 2009), the Radon transform (Kabir and Verschuur, 1995;Trad et al, 2002), and the curvelet transform (Naghizadeh and Sacchi, 2010;Liu et al, 2011;Kong et al, 2012). Another group of methods relies on prediction-error filtering techniques (Spitz, 1991;Naghizadeh and Sacchi, 2009).…”

On analysis-based two-step interpolation methods for randomly sampled seismic data

“…There exist many effective methods to reconstruct the complete seismic wavefield depending on different acquisition geometries (Trickett et al, 2010;Naghizadeh and Sacchi, 2010). These methods can be roughly classified into two categories:…”

“…(2) Reconstruction with the aid of certain type of transforms such as the Fourier transform (Sacchi et al, 1998;Liu and Sacchi, 2004;Xu et al, 2005;Abma and Kabir, 2006;Naghizadeh, 2007), the Radon transform (Trad and Ulrych, 2002;Trad et al, 2003), the curvelet transform (Hennenfent, 2008;Hennenfent et al, 2010;Naghizadeh and Sacchi, 2010;Yang et al, 2011), etc., The latter is intended to exploit the transform domain sparsity and works well for irregularly sampled seismic records.…”