Preliminary study on Dreamlet based compressive sensing data recovery

Wu, Ru‐Shan; Geng, Yushui; Ye, Lingling

doi:10.1190/segam2013-0800.1

Cited by 15 publications

(13 citation statements)

References 10 publications

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“…; Wu et al . ). The time–space domain seismic data

u (x, z = 0, t)

can be represented in the domain of time–frequency space–wavenumber by dreamlets as

\begin{matrix} left u ((), x, z = 0, t) = \sum_{m, q, k, l} (〈〉, u, {\tilde{d}}_{m, q, k, l}) d_{m, q, k, l} ((), x, t) \\ left = \sum_{m, q, k, l} C_{m, q, k, l} d_{m, q, k, l} ((), x, t), \end{matrix}

where

(〈〉,)

stands for the inner product (Appendix , under equation (B‐1)),

{\overset{d}{true}}_{m, q, k, l} (x, t) = {\overset{g}{true}}_{m, q} (x) {\overset{g}{true}}_{k, l} (t)

is the dual atoms obtained by the tensor product of the dual frames (Appendix , equations (B‐5) and (B‐12)), and

C_{m, q, k, l}

is the coefficient.…”

Section: Seismic Data Representation Using Gaussian Packetmentioning

confidence: 97%

“…; Wu et al . ), which provides us the possibility of achieving an efficient migration method based on Gaussian packet summation.…”

Section: Seismic Data Representation Using Gaussian Packetmentioning

confidence: 99%

“…By discussing the Gaussian packets' initial parameters, we will show how to efficiently decompose seismic data into Gaussian packets through dreamlets that are constructed by a tensor product of Gabor frames, satisfying the causality relation. Dreamlets can represent seismic data accurately and sparsely, and seismic data can be well reconstructed by only a small percentage of coefficients Wu et al 2013), which provides us the possibility of achieving an efficient migration method based on Gaussian packet summation.…”

Section: S E I S M I C D a T A R E P R E S E N T A T I O N U S I N G mentioning

confidence: 99%

“…In addition to space-direction localization, time-frequency localization can also be applied to seismic data decomposition and imaging, such as in the wave packet solution or the pulsed beam solution (Kaiser 2003). Time-frequency spacedirection localization for seismic data decomposition, propagation, and imaging using dreamlets has been introduced and further developed (Wu, Wu, and Geng 2008;Geng, Wu, and Gao 2009;Wu and Wu 2010;Wu, Geng and Wu 2011;Wu, Geng and Ye 2013). Curvelets (Starck, Candes, and Donoho 2002) were reported to be an efficient and sparse decomposition of both seismic wavefield and wave propagation operator (Candes and Demanet 2005) and have been applied to seismic data processing , wavefield extrapolation (Lin and Herrmann 2007), map migration (Douma and de Hoop 2007), and migration and demigration (Chauris and Nguyen 2008).…”

Section: Introductionmentioning

confidence: 99%

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Gabor‐frame‐based Gaussian packet migration

Geng

Gao

2014

Geophysical Prospecting

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We present a Gaussian packet migration method based on Gabor frame decomposition and asymptotic propagation of Gaussian packets. A Gaussian packet has both Gaussian‐shaped time–frequency localization and space–direction localization. Its evolution can be obtained by ray tracing and dynamic ray tracing. In this paper, we first briefly review the concept of Gaussian packets. After discussing how initial parameters affect the shape of a Gaussian packet, we then propose two Gabor‐frame‐based Gaussian packet decomposition methods that can sparsely and accurately represent seismic data. One method is the dreamlet–Gaussian packet method. Dreamlets are physical wavelets defined on an observation plane and can represent seismic data efficiently in the local time–frequency space–wavenumber domain. After decomposition, dreamlet coefficients can be easily converted to the corresponding Gaussian packet coefficients. The other method is the Gabor‐frame Gaussian beam method. In this method, a local slant stack, which is widely used in Gaussian beam migration, is combined with the Gabor frame decomposition to obtain uniform sampled horizontal slowness for each local frequency. Based on these decomposition methods, we derive a poststack depth migration method through the summation of the backpropagated Gaussian packets and the application of the imaging condition. To demonstrate the Gaussian packet evolution and migration/imaging in complex models, we show several numerical examples. We first use the evolution of a single Gaussian packet in media with different complexities to show the accuracy of Gaussian packet propagation. Then we test the point source responses in smoothed varying velocity models to show the accuracy of Gaussian packet summation. Finally, using poststack synthetic data sets of a four‐layer model and the two‐dimensional SEG/EAGE model, we demonstrate the validity and accuracy of the migration method. Compared with the more accurate but more time‐consuming one‐way wave‐equation‐based migration, such as beamlet migration, the Gaussian packet method proposed in this paper can correctly image the major structures of the complex model, especially in subsalt areas, with much higher efficiency. This shows the application potential of Gaussian packet migration in complicated areas.

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“…; Wu et al . ). The time–space domain seismic data

u (x, z = 0, t)

can be represented in the domain of time–frequency space–wavenumber by dreamlets as

\begin{matrix} left u ((), x, z = 0, t) = \sum_{m, q, k, l} (〈〉, u, {\tilde{d}}_{m, q, k, l}) d_{m, q, k, l} ((), x, t) \\ left = \sum_{m, q, k, l} C_{m, q, k, l} d_{m, q, k, l} ((), x, t), \end{matrix}

where

(〈〉,)

stands for the inner product (Appendix , under equation (B‐1)),

{\overset{d}{true}}_{m, q, k, l} (x, t) = {\overset{g}{true}}_{m, q} (x) {\overset{g}{true}}_{k, l} (t)

is the dual atoms obtained by the tensor product of the dual frames (Appendix , equations (B‐5) and (B‐12)), and

C_{m, q, k, l}

is the coefficient.…”

Section: Seismic Data Representation Using Gaussian Packetmentioning

confidence: 97%

“…; Wu et al . ), which provides us the possibility of achieving an efficient migration method based on Gaussian packet summation.…”

Section: Seismic Data Representation Using Gaussian Packetmentioning

confidence: 99%

Section: S E I S M I C D a T A R E P R E S E N T A T I O N U S I N G mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Gabor‐frame‐based Gaussian packet migration

Geng

Gao

2014

Geophysical Prospecting

Self Cite

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“…Seismic signals are sparse or compressible, while there are large numbers of zero, or small, transform coefficients and a few large ones in some transform domain. To date, the sparsity of 2D seismic signals has been investigated by Fourier , wavelet (Herrmann, 2010;Wang et al, 2011), wave packet (Herrmann, 2010;Wang et al, 2010), wave atoms (Herrmann, 2010), curvelet Herrmann et al, 2008;Herrmann and Hennenfent, 2008;Herrmann, 2010), and dreamlet transforms (Wu et al, 2013). Generally, complicated transforms (e.g., curvelet) result in good performance at high computational cost.…”

Section: Introductionmentioning

confidence: 99%

Bernoulli-based random undersampling schemes for 2D seismic data regularization

et al. 2014

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Seismic data regularization is an important preprocessing step in seismic signal processing. Traditional seismic acquisition methods follow the Shannon-Nyquist sampling theorem, whereas compressive sensing (CS) provides a fundamentally new paradigm to overcome limitations in data acquisition. Besides the sparse representation of seismic signal in some transform domain and the 1-norm reconstruction algorithm, the seismic data regularization quality of CS-based techniques strongly depends on random undersampling schemes. For 2D seismic data, discrete uniform-based methods have been investigated, where some seismic traces are randomly sampled with an equal probability. However, in theory and practice, some seismic traces with different probability are required to be sampled for satisfying the assumptions in CS. Therefore, designing new undersampling schemes is imperative. We propose a Bernoulli-based random undersampling scheme and its jittered version to determine the regular traces that are randomly sampled with different probability, while both schemes comply with the Bernoulli process distribution. We performed experiments using the Fourier and curvelet transforms and the spectral projected gradient reconstruction algorithm for 1-norm (SPGL1), and ten different random seeds. According to the signal-to-noise ratio (SNR) between the original and reconstructed seismic data, the detailed experimental results from 2D numerical and physical simulation data show that the proposed novel schemes perform overall better than the discrete uniform schemes.

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Simultaneous seismic data interpolation and denoising with a new adaptive method based on dreamlet transform

Wang

Chen³

et al. 2015

Geophysical Journal International

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128

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Interpolation and random noise removal is a prerequisite for multichannel techniques because the irregularity and random noise in observed data can affect their performances. Projection Onto Convex Sets (POCS) method can better handle seismic data interpolation if the data's signal-to-noise ratio (SNR) is high, while it has difficulty in noisy situations because it inserts the noisy observed seismic data in each iteration. Weighted POCS method can weaken the noise effects, while the performance is affected by the choice of weight factors and is still unsatisfactory. Thus, a new weighted POCS method is derived through the Iterative Hard Threshold (IHT) view, and in order to eliminate random noise, a new adaptive method is proposed to achieve simultaneous seismic data interpolation and denoising based on dreamlet transform. Performances of the POCS method, the weighted POCS method and the proposed method are compared in simultaneous seismic data interpolation and denoising which demonstrate the validity of the proposed method. The recovered SNRs confirm that the proposed adaptive method is the most effective among the three methods. Numerical examples on synthetic and real data demonstrate the validity of the proposed adaptive method.

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Preliminary study on Dreamlet based compressive sensing data recovery

Cited by 15 publications

References 10 publications

Gabor‐frame‐based Gaussian packet migration

Gabor‐frame‐based Gaussian packet migration

Bernoulli-based random undersampling schemes for 2D seismic data regularization

Simultaneous seismic data interpolation and denoising with a new adaptive method based on dreamlet transform

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