Fourier reconstruction with sparse inversion

Abstract

A B S T R A C TThe problem of seismic data reconstruction is posed as an inverse problem where the objective is to obtain the Fourier coefficients that synthesize the signal. Once the coefficients have been found, they are used to reconstruct the data on a uniformly spaced grid. A non-quadratic model weight function is included to stabilize the inversion and to provide the additional information required to interpolate through gaps. In the reconstruction of a non-uniformly sampled trace, an image and a marine 3D VSP shot-record, the method shows improved reconstruction in large gaps and is less sensitive to the spatial bandwidth used in the inversion compared to Fourier reconstruction without the non-quadratic model weight function.

Curvelet-based POCS interpolation of nonuniformly sampled seismic records

“…By enforcing a Cauchy-Gaussian a priori sparseness within the Bayesian framework, Sacchi et al (1998) proposed a high-resolution Fourier transform to perform interpolation and extrapolation. Sparseness-constrained Fourier reconstruction was also successfully applied to the irregularly sampled seismic data, using least-squares criterion and minimum (weighted) norm constraint (Liu and Sacchi, 2004;Wang, 2003;Zwartjes and Gisolf, 2007;Zwartjes and Sacchi, 2007). Sparsity-promoting seismic trace restoration was even investigated in Radon domain (Kabir and Verschuur, 1995;Sacchi and Ulrych, 1995;Trad and Ulrych, 2002;Trad et al, 2003).…”

OC-seislet: Seislet transform construction with differential offset continuation

“…If seismic traces in the midpoint direction are missing, the Fourier transform may produce artifacts (spatial leakage) along the midpoint-wavenumber axis (Zwartjes and Gisolf, 2007). Additionally, the missing seismic traces in the offset direction affect the conti-OC-seislet transform nuity of predicted data.…”

“…For input data with nonuniform spatial sampling, one can bin the data to a regular grid first and then use the proposed method for filling empty bins. It is also possible to generalize the method for combining irregular data interpolation with a sparse transform (Zwartjes and Gisolf, 2007). The thresholding iteration helps a sparse transform to recover the missing information.…”

Nonequispaced curvelet transform for seismic data reconstruction: A sparsity-promoting approach

“…Real data example Hindriks and Duijndam (2000), followed by Zwartjes (2005), used a 3-D vertical seismic profile (VSP) dataset to show Fourier reconstruction results of data irregularly sampled along two spatial coordinates. We use the same dataset to illustrate NCRSI.…”

“…They are quite a bit more computationally extensive than transform-based methods though. Transform-based methods (Thorson and Claerbout, 1985;Hampson, 1986;Sacchi et al, 1998;Schonewille, 2000;Trad et al, 2003;Zwartjes, 2005; do not capitalize on any particular geophysical model and may relate to the physics of wave propagation depending on the transform used-e.g., Fourier modes correspond to eigenfunctions of a wave equation with constant velocity. Most of aforementioned methods can also regularize but it is not the case of the curvelet reconstruction with sparsity-promoting inversion (CRSI -Herrmann and Hennenfent, 2008).…”