Curvelet-based POCS interpolation of nonuniformly sampled seismic records

doi:10.1016/j.jappgeo.2011.12.004

Journal of Applied Geophysics

Volume 79, April 2012, Pages 90-99

https://doi.org/10.1016/j.jappgeo.2011.12.004 Get rights and content

Abstract

An exceedingly important inverse problem in the geophysical community is the interpolation of the seismic data, which are usually nonuniformly recorded from the wave field by the receivers. Researchers have proposed many useful methods to regularize the seismic data. Recently, sparseness-constrained seismic data interpolation has attracted much interest of geophysicists due to the surprisingly convincing results obtained. In this article, a new derivation of the projection onto convex sets (POCS) interpolation algorithm is presented from the well known iterative shrinkage–thresholding (IST) algorithm, following the line of sparsity. The curvelet transform is introduced into the POCS method to characterize the local features of seismic data. In contrast to soft thresholding in IST, hard thresholding is advocated in this curvelet-based POCS interpolation to enhance the sparse representation of seismic data. The effectiveness and the validity of our method are demonstrated by the example studies on the synthetic and real marine seismic data.

Highlights

► The connection between POCS interpolation and IST interpolation is established. ► Curvelet transform is introduced into POCS interpolation. ► The effectiveness of curvelet-based POCS interpolation is demonstrated using synthetic and marine seismic data. ► The superior performance of POCS over IST interpolation using curvelet is given.

Introduction

In the process of data acquisition, large volumes of seismic data in the continuous wavefield are recorded by receivers, including many missing traces which are nonuniformly distributed. To utilize these seismic records, excellent interpolation techniques are seriously needed.

Many insightful researchers made the utmost of the linear prediction techniques and the Fourier transform to attack the problem of seismic trace interpolation. Spitz (1991) proposed his classic f–x method, based on the fact that the missing traces spaced equally can be exactly interpolated by a set of linear equations. Following Spitz's way, Porsani (1999) developed half-step f–x method, making trace interpolation significantly more efficient and easier for implementation. Wang (2002) extended f–x to a f–x–y domain, and developed two interpolation algorithms using the full-step and the fractional-step predictions. Gulunay (2003) presented his data adaptive f–k method for spatially aliased data. Recently, Naghizadeh and Sacchi (2007) gave a multistep autoregressive (MSAR) method, trying to predict all the frequencies from the low frequency part.

Parallel to the above approaches, sparseness-constrained seismic data interpolation has attracted much interest of geophysicists. As a matter of fact, sparsity is an old concept that has been first used for inversion of seismic data by Thorson and Claerbout (1985); afterwards, sparse deconvolution became popular in geophysical community to obtain high resolution seismic signals, see Sacchi (1997) and the references therein. 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).

Sparseness promoting seismic data processing methods using curvelets have become popular, and widely investigated with convincing results (Herrmann, 2003, Herrmann, 2004, Hennenfent and Herrmann, 2005, just to name a few). The recent progress of the theory of compressive sensing (CS) (Candès, 2006, Donoho, 2006) provides a nice way to understand sparseness-based interpolation. The data are allowed to be sampled randomly and compressed greatly (Lin and Herrmann, 2007, Neelamani et al., 2010). This leads to a highly reduced cost in the process of data acquisition, barely losing the information of geophysical data. Curvelet-based methods can be well compatible with the context of CS (Hennenfent and Herrmann, 2007, Herrmann, 2010).

This paper will follow the line of sparsity-promoting seismic data reconstruction. We propose a curvelet-based projection onto convex sets (POCS) method for seismic data interpolation, different from the work of Abma and Kabir (2006) who used the Fourier transform. We derive the POCS method from the iterative shrinkage–thresholding (IST) algorithm. In this way, the close relationship between POCS and IST can be revealed clearly, under the sparsity constraint. Hard thresholding is preferred in our approach, instead of the soft one adopted by Hennenfent and Herrmann (2005). We demonstrate the effectiveness and validity of our method with successful interpolated results. Compared with IST, the interpolation of POCS is far better than that of IST according to the SNR measure at the early iterations.

Section snippets

Problem statement

In this section we will give the mathematical framework of seismic data interpolation. Let P_Λ be a diagonal matrix with diagonal entries 1 for the indices in Λ and 0 otherwise. Then the observed seismic data with missing traces can be specified as $d_{obs} = P_{Λ} d,$ where d_obs represents the incomplete seismic image, and d the complete data we intend to recover. As is customary, a discrete seismic image $d = {[d_{t_{1}, t_{2}}]}_{1 \leq t_{1} \leq n_{1}, 1 \leq t_{2} \leq n_{2}} \in R^{n_{1} \times n_{2}}$ will be reordered as a vector $d \in R^{n}$ , n = n₁ × n₂ through lexicographic

POCS solver

As has been said earlier, the sparsity constraint has attracted much interest in recent years. It implies that the $ℓ^{0}$ penalty is employed, say, R(x) = ‖x‖₀. The main difficult we encounter is the nonconvex property of the following problem $m i n J (x) = \frac{1}{2} {‖d_{obs} - P_{Λ} Φ x‖}_{2}^{2} + τ {‖x‖}_{0} .$

Since the innovative theory of compressive sensing (CS) (Candès, 2006, Donoho, 2006), researchers relaxed $ℓ^{0}$ to $ℓ^{1}$ when realizing the presence of the $ℓ^{1}$ term encourages small components of x to become exactly zero, which may result in

Curvelet-based seismic data reconstruction using POCS

It should be noted that 1) the POCS method can be derived from the IST method under the assumption of a tight frame dictionary. In this way the close relationship between POCS and IST can be revealed. 2) The POCS method iterates with our target unknown d, different from the approach in Hennenfent and Herrmann (2005), which resorts to the representation vector x and recovers the data via d = Φx indirectly. 3) Abma and Kabir (2006) have employed the Fourier transform in the POCS method to achieve

Results and discussion

To evaluate the quality of our recovery, we define signal-to-noise-ratio (SNR) as noted in Hennenfent and Herrmann (2006): $S N R = 10 l o g_{10} (\frac{{‖d‖}^{2}}{{‖d - \hat{d}‖}^{2}}) (dB) .$

In what follows we will present our example studies on both synthetic geophysical data and real marine data.

Concluding remarks

Obviously, the mathematical framework in this article is general, not only restricted to the Fourier and curvelet transforms. In fact, many related approaches have been reported in the literature. Sacchi et al. (1998) presented high-resolution discrete Fourier transform to perform the task of interpolation and extrapolation, utilizing a Cauchy–Gaussian a priori regularizer. Liu and Sacchi (2004) designed a minimum weighted norm interpolation (MWNI) algorithm, using weighted norm on the Fourier

Acknowledgments

This work is supported by National Natural Science Foundation of China (40730424, 40674064) and National Science and Technology Major Project (2008ZX05023-005-005, 2008ZX05025-001-009). We wish to thank the authors of SeismicLab (http://www-geo.phys.ualberta.ca/saig/SeismicLab) and CurveLab (http://www.curvelet.org/) for their free available codes. We are grateful for the large amount of valuable comments from Dirk J. Eric Verschuur and another anonymous reviewer, which lead to substantial

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View full text

Curvelet-based POCS interpolation of nonuniformly sampled seismic records

Abstract

Highlights

Introduction

Section snippets

Problem statement

POCS solver

Curvelet-based seismic data reconstruction using POCS

Results and discussion

Concluding remarks

Acknowledgments

Applied and Computational Harmonic Analysis

Journal of Computational Physics

3D interpolation of irregular data with a POCS algorithm

Geophysics

Iterative thresholding for sparse approximations

Journal of Fourier Analysis and Applications

A framelet-based image inpainting algorithm

Applied and Computational Harmonic Analysis

Compressive sampling

The curvelet representation of wave propagators is optimally sparse

Communications on Pure and Applied Mathematics

Curvelets: a surprisingly effective nonadaptive representation for objects with edges

An iterative thresholding algorithm for linear inverse problems with a sparsity constraint

Communications on Pure and Applied Mathematics

Compressed sensing

IEEE Transactions on Information Theory

Shaping regularization in geophysical-estimation problems

Geophysics

Time-lapse image registration using the local similarity attribute

Geophysics

Irregular seismic data reconstruction based on exponential threshold model of POCS method

Applied Geophysics

Convergence improvement and noise attenuation considerations for POCS reconstruction

Nonlinear approximation based image recovery using adaptive sparse reconstructions and iterated denoising—part I: theory

IEEE Transactions on Image Processing

Nonlinear approximation based image recovery using adaptive sparse reconstructions and iterated denoising—part II: adaptive algorithms

IEEE Transactions on Image Processing

Seismic trace interpolation in the Fourier transform domain

Geophysics

Sparseness-constrained data continuation with frames: applications to missing traces and aliased signals in 2/3d

Seismic denoising with nonuniformly sampled curvelets

Computing in Science & Engineering

Irregular sampling: from aliasing to noise

Optimal seismic imaging with curvelets

Randomized sampling and sparsity: getting more information from fewer samples

Geophysics

Curvelet imaging and processing: an overview

Non-linear primary-multiple separation with directional curvelet frames

Geophysical Journal International