3D seismic denoising based on a low-redundancy curvelet transform

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3D seismic denoising based on a low-redundancy curvelet transform

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Journal of Geophysics and Engineering J. Geophys. Eng. 12 (2015) 566–576

doi:10.1088/1742-2132/12/4/566

3D seismic denoising based on a low-redundancy curvelet transform Jingjie Cao1, Jingtao Zhao2 and Zhiying Hu3 1

  Shijiazhuang University of Economics, Shijiazhuang, Hebei 050031, People’s Republic of China   Research Institute of Petroleum Exploration & Development, PetroChina, Beijing 100083, People’s Republic of China 3  Xi’an Fanyi University, Xi’an 710105, People’s Republic of China 2

E-mail: [email protected] Received 11 December 2014, revised 22 April 2015 Accepted for publication 11 May 2015 Published 4 June 2015 Abstract

Contamination of seismic signal with noise is one of the main challenges during seismic data processing. Several methods exist for eliminating different types of noises, but optimal random noise attenuation remains difficult. Based on multi-scale, multi-directional locality of curvelet transform, the curvelet thresholding method is a relatively new method for random noise elimination. However, the high redundancy of a 3D curvelet transform makes its computational time and memory for massive data processing costly. To improve the efficiency of the curvelet thresholding denoising, a low-redundancy curvelet transform was introduced. The redundancy of the low-redundancy curvelet transform is approximately one-quarter of the original transform and the tightness of the original transform is also kept, thus the lowredundancy curvelet transform calls for less memory and computational resource compared with the original one. Numerical results on 3D synthetic and field data demonstrate that the low-redundancy curvelet denoising consumes one-quarter of the CPU time compared with the original curvelet transform using iterative thresholding denoising when comparable results are obtained. Thus, the low-redundancy curvelet transform is a good candidate for massive seismic denoising. Keywords: curvelet transform, denoising, low redundancy, sparse optimization, one-norm (Some figures may appear in colour only in the online journal)

1. Introduction In seismic data processing, field data are often contaminated by environmental noise and interference noise. Successful denoising is an important basis for migration, deconvolution, interpolation, multiple elimination, and enhancing images of the subsurface geology (Hennenfent and Herrmann 2006, Broadhead 2008, Naghizadeh and Sacchi 2010, Yuan and Wang 2013). Generally, noise in seismic data can be classified into coherent noise and incoherent noise. The need to enhance the desired, most informative events motivates the researchers to develop increasingly sophisticated denoising techniques. Some of these methods deal with the coherent events, exhibiting phase consistency from trace to trace; some are more often used to remove random noise. Designing a combination of those techniques adequate to our data and the further 1742-2132/15/040566+11$33.00

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processing scheme is crucial for successful seismic processing and imaging. For random noise attenuation, the sparse transform-based method (Yuan et al 2015) is a commonly used strategy; seismic signals are assumed to be sparse in a transformed domain in which they can be separated from random noise. When this assumption is satisfied, small coefficients in the transformed domain can be removed by using thresholding methods to achieve the denoising results. Because the signals can be denoted by a few large amplitude coefficients in a transformed domain, and because most small coefficients are treated as noise, it is reasonable to eliminate small coefficients by using thresholding methods. Besides the Fourier-based denoising (Naghizadeh 2012), T-X prediction filtering (Abma and Claerbout 1995, Wang 1999, 2002a), Cadzow filtering (Oropeza and Sacchi 2011, Yuan and Wang 2013), and localized slant stack (McMechan © 2015 Sinopec Geophysical Research Institute  Printed in the UK

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1983) remove random noise based on linear assumption of seismic events. Additionally, linear/parabolic radon denoising is another method based on the linear/parabolic events assumption of seismic wave fronts (Turner 1990, Kappus et al 1990, Zhou and Greenhalgh 1994). Wang (2002b) suggested the delay-time radon transform for modeling coherent events with any form of move-out curves. Wavelet transforms that can detect local features in the time-frequency domain are another kind of transforms for denoising (Shan et al 2009). Wavelets are non-directional and only capture point-like features; nevertheless, seismic wave fronts are usually curve-like, and thus wavelets are not an ideal choice for seismic denoising. Physical wavelet frame denoising is another wavelet denoising method using the physical wavelets to construct the wavelet frame (Zhang and Ulrych 2003). Filter bank wavelet transform was also introduced in seismic denoising (Cao and Chen 2005). Recently, Parolai (2009) adopted the S-transform for denoising. The dictionary learning method is a newly proposed method that chooses an over-complete dictionary to represent seismic signals (Beckouche and Ma 2014). Because seismic signals are non-stationary, and because noise has a colored spectrum, time frequency analysis methods have been proposed by Xiao and Flandrin (2007). A common feature of the aforementioned noise attenuation methods is that they operate on a single scale; decomposition of seismic signals into multi-resolution elements is not used. In the past decade, multi-scale transforms gained much interest in signal processing field, especially for the curvelet transform. Curvelet transform was proposed by Candes and Donoho (2000) as a new multi-scale, multi-directional, and sparse expression of signals suited for mapping curve-like seismic wave fronts. The curvelets are localized not only in the space-frequency domain but also in the angular orientation, which is an important merit compared with wavelet transform (Mallat 1999). A new important directional parameter provides an angular geometric property with a high degree of orientation that identifies the directional singularities (Candes and Donoho 2000). As a multi-scale, multi-directional, anisotropic tight frame, it is strictly localized in the Fourier domain. Furthermore, it provides an optimal representation of objects that have discontinuities along edges (Candes and Donoho 2000, Starck et al 2002). Hennenfent and Herrmann (2006) combined the thresholding method and curvelet transform to eliminate random noise. Neelamani et al (2008) introduced a curvelet-based noise attenuation method for 3D seismic data corrupted with random and linear noise. Shan et al (2009) compared the results of wavelets, contourlets, and curvelets denoising and proposed a wavelets–curvelets hybrid method for random noise elimination. Wang et al (2010) proposed combining the iterative thresholding method and curvelet transform for denoising; numerical results proved that the curvelet-based thresholding method can achieve a higher signal-to-noise ratio (SNR) and fidelity than the traditional median filter algorithm, the F-X deconvolution algorithm, and the wavelet thresholding algorithm. Gorszczyk et al (2014) applied the curvelet denoising method to 2D and 3D seismic data.

Curvelet denoising was also applied to coherent noise attenuation, such as multiple attenuation (Herrmann et al 2008, Lin and Herrmann 2013) and ground roll attenuation (Kumar et al 2011). More applications of curvelet transform in seismic processing include interpolation (Herrmann et al 2008, Cao et al 2011, Yang et al 2012), deconvolution, and migration (Chauris and Nguyen 2008). Although the curvelet transform has plenty of merits for signal processing, it is a highly redundant transform with redundancy up to a factor of 24–32 for 3D signals, which is a crucial defect of 3D curvelet transform for massive data processing. A low-redundancy curvelet transform was proposed in Woiselle et al (2011), which can reduce the redundancy to a factor of 10 for 3D signals. This article adopts this lowredundancy curvelet transform for 3D data denoising. The low redundancy and the tightness of the low-redundancy transform were utilized to accelerate computation. The structure of this article is as follows: the sparse transform-based denoising model and iterative thresholding method are introduced first, and then the low-redundancy curvelet transform is introduced and the priorities of this transform are illustrated. Numerical examples on synthetic and filed data denoising demonstrate that the low-redundancy curvelet denoising consumes onequarter of the CPU time compared with the original curvelet transform using the iterative thresholding denoising method when comparable results are obtained. Finally, some conclusions and discussion are given.

2. Theory Generally, seismic data with noise can be expressed as the following mathematical model: d + ε = d obs, (1)

where d is the unknown seismic signal, ε is the additive noise, and d obs is the noisy data. Here, the noise ε is assumed to be the incoherent noise. Because the noise and seismic data are combined together, it is difficult to separate them directly in the time-space domain. According to the regularization theory, the ideal solution can be found if some prior information is known at first. Sparsity of signals in certain domain is commonly used; if s = Ψd is sparse, where Ψ is a operator that represents a tight sparse transform, then equation (1) can be transformed to Ψ*s + ε = d obs, (2)

where Ψ* is the conjugate transpose of Ψ. Because of the sparsity constraint of s, it can be obtained by solving the following optimization problem (Chen et al 1998) min s 1 s.t. Ψ*s − d obs 22 ≤ σ , (3)

where σ is an estimation of the noise energy. Here, the denoising problem was changed to an optimization problem. If s is solved, then x can be obtained through x = Ψ*s. For more sparse regularization models of denoising, refer to (Blumensath and Davies 2008, Cao et al 2011, Tang et al 2012, Yuan et al 2012). 567

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For solving problem (3), there are two strategies: the iterative method and the direct method (Daubechies et al 2004). The direct method will generate aliasing, although with fast speed. The iterative method is more preferable to improve denoising quality. The iterative soft thresholding (IST) method (Daubechies et al 2004) is an important method to solve problem (3) and the sequence of iteration is: s = Tλ(s + Ψ(d obs − Ψ*s )), (4)

where Tλ is the soft thresholding function Tλ(s ) = sgn(s ) ⋅ max(0, s − λ ), (5)

and sgn{}means the sign function. The threshold λ is the key to deciding the denoising results. In practice, the curvelet coefficients were sorted into descending curvelet coefficients at first, and then according to a pre-set curvelet coefficients percentage, which was preserved in the first loop to determine the threshold of the first loop. A lower percentage of the coefficients was chosen in the first loop because of the preferable sparsity of seismic data in the curvelet domain. The threshold gradually steps down as the iterations proceed. Therefore, λ is varied with the number of iterations. The curvelet transform is an effective spectral transform that allows sparse expressions of complex data. This spectral technique is based on directional basis functions that represent objects with discontinuity along smooth curves. As a multiscale, multi-directional, and high-dimensional tight frame (Candes et al 2006), the curvelet transform has excellent compression for wave fronts (Herrmann et al 2008). However, its high redundancy with a factor 24–32 for 3D data costs much time in computation. How to improve the efficiency of curvelet transform-based denoising and keep the denoising quality is the starting point of this article. Woiselle et al (2011) proposed a new implementation of the curvelet transform, which can reduce the redundancy to 10 for 3D data and keep the tightness of the curvelet transform. These merits will benefit computation greatly when dealing with massive data. In this article, the low-redundancy transform was utilized for seismic denoising to highlight the computational efficiency of curvelet denoising. Woiselle’s curvelet transform is introduced in the next section. 3.  Low-redundancy curvelet transform implementation In this section, the theory of wrapping-based curvelet transform is introduced. Then, the new implementation of the lowredundancy curvelet transform is illustrated, The reason why the low-redundancy curvelet transform can reduce the extra redundancy is also explained. The 3D curvelet transform (Candes et al 2006) consists of a low-pass approximation sub-band decomposition and a family of curvelet sub-bands carrying the curvelet coefficients indexed by their scale, position, and orientation. There are mainly two steps in the transform: multi-scale separation and angular separation. At first, the input 3D data with size N = (Nx, Ny, Nz ) is separated into a dyadic corona based 568

on the 3D Meyer wavelet transform in the Fourier domain with compactly supported Fourier transform, providing cubes of sizes N , N /2, ..., N /2J , where J is the number of scales. Second, each corona is separated into anisotropic wedges of trapezoidal shape obeying the so-called parabolic scaling law. The curvelet coefficients are obtained by a 3D inverse Fourier transform applied to each wedge appropriately wrapped to fit 3D rectangular parallelepipeds. In the original implementation, high redundancy mainly comes from the angular separation that allows the curvelet to have directional selectivity, but the application of the Meyer wavelet transform in the original 3D curvelet transform added extra redundancy. Woiselle et al (2011) proposed a low-redundancy implementation that can reduce the extra redundancy. There are many differences between the low-redundancy transform and the original one, but the main reason why it can reduce the extra redundancy lies in the way that the Meyer wavelet transform is applied to seismic data. More details of the Meyer wavelet transform implementation are discussed. The extra redundancy of the curvelet transform as implemented in Curvelab (Candes and Donoho 2004) originates mainly from the way the radial window is implemented, especially the finest scale. Taking the 1D Meyer wavelet transform as an example, we call ψj the Meyer wavelet at scale j ∈ {0, ⋯, J − 1} and φJ − 1 the scaling function at the coarsest scale, denoting Mj = ψˆj = 2−3j /2ψˆ (2−j) and MJ = φˆJ − 1 = 2−3(J − 1) /2φˆ (2−2(J − 1)) as their Fourier transforms. The Meyer wavelets ψ (ξ ) are defined in the Fourier domain as follows:

ˆ

⎧ −i2πξ ⎛ π ⎞ sin⎜ ν(6 ξ − 1)⎟, 1/6 < ξ ≤ 1/3 ⎪e ⎝ ⎠ 2 ⎪ ψˆ (ξ ) = ⎨ −i2πξ ⎛ π , (6) ⎞ sin⎜ ν(3 ξ − 1)⎟, 1/3 < ξ ≤ 2/3 ⎪e ⎝2 ⎠ ⎪ ⎩ 0 elsewhere

where ν is a smooth function that goes from 0 to 1 on [0,1], which satisfies ν(ξ ) + ν(1 − ξ ) = 1. The Meyer scaling functions are defined by ⎧ 1, ξ ≤ 1/6 ⎪ ⎪ ⎛π ⎞ ϕ (ξ ) = ⎨ cos⎜ ν(6 ξ − 1)⎟, 1/6 < ξ ≤ 1/3. (7) ⎝ ⎠ 2 ⎪ ⎪ 0 ξ > 1/3 ⎩

ˆ

Figure 1 displays in solid lines the graphs of the Fourier transform of the Meyer scaling and wavelet functions at three scales. The wavelet at the finest scale in the Fourier domain is supported on [−2/3, −1/6] ∪ [1/6, 2/3], therefore exceeding the Shannon band. The original curvelet transform implicitly assumes periodic boundary conditions. Moreover, it is known that computing the wavelet transform of a periodized signal is equivalent to decomposing the signal in a periodic wavelet basis. Thus, the exceeding end of the finest scale is replaced with its mirrored version around the vertical axis at ξ = 1/2, as shown in dashed line at the top of figure 1. Consequently, the support of the data is three-quarters larger than the original one, therefore boosting the redundancy by a factor (3/4)3 in

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Figure 1.  Meyer scaling and wavelet functions in the 1D Fourier domain. Top: Meyer wavelet transform in the Candes curvelet transform. Bottom: Meyer transform in the Woiselle implementation.

3D. In the low-redundancy implementation, the supports of the scaling and wavelet functions were first shrunk by a factor of three-quarters. Furthermore, to maintain the uniformity of the partition of unity, the finest scale wavelet is modified by suppressing its decreasing tail so that the wavelet becomes a constant over [−1/2, −1/4] ∪ [1/4, 1/2]. This added no extra redundancy to the 3D Meyer wavelet transform in the Fourier domain. The redundancy of the original 3D curvelet transform is 24–32, whereas the redundancy of the low-redundancy implementation is only 10 for 3D data (Woiselle et al 2011). Besides the low redundancy, while maintaining the directional selectivity property at the finest scale, the low-redundancy curvelet transform is isometric and has fast exact reconstruction; furthermore, it corresponds to a Parseval tight frame, ie, C *C = I , where C is the curvelet analysis operator and C * is its adjoint operator. Thus, C * turns out to be the inverse operator associated with a fast reconstruction algorithm. For more details of the low-redundancy curvelet transform, refer to Woiselle et al (2011). 4.  Numerical examples To test the redundancy of the low-redundancy curvelet transform numerically, a 3D seismic cube with time sample 64, inline number 64, and crossline number 64 was chosen for numerical experiments. The time sampling ratio is 4 ms and the trace interval is 12.5 m. A slice of the data is shown in figure 2. These data were transformed by these two transforms with different scales and angles. The curvelet coefficient numbers and maximum absolute values are listed in table 1. In the first row of the table, ‘3ʼ indicates the scale number and ‘8ʼ indicates the angle number at the second coarsest scale. In the second row of the table, ‘Candesʼ denotes the Candes curvelet 569

transform and ‘Woiselleʼ denotes the low-redundancy implementation. In the table, the coefficient numbers of the lowredundancy curvelet transform are approximately two-fifths of the Candes curvelet coefficients for different scales and angles, which will benefit computational efficiency greatly for massive data processing. When the divided scale was 3 and the angle number at the second coarsest scale was 8, curvelet coefficient amplitudes were sorted in descending order at first, and different percentages of large coefficients were chosen to restore the data; results are given in table  2. It is clear that, with the same percentage of coefficients, restorations of the low-redundancy curvelet transform outperform the Candes’ curvelet transform. Three examples are provided to verify the denoising ability of the low-redundancy curvelet transform; the iterative soft thresholding (IST) method in section 2 was adopted to realize denoising. The iteration number of the IST method is set as 60 and the thresholding reduction strategy is the exponential method. The least thresholds are chosen from data to data. The SNR with definition SNR = 10 log10

x orig

2 2

x orig − x rest

2 2

is used to

measure the denoising ability, where xorig is the true signal and x rest is the denoised signal. These examples were run on a laptop with an i3 processor and 2G inner memories. Synthetic data (Data 1) with time sample 100, inline number 100, and crossline 100 are shown in figure  3(a); the time sampling ratio is 4 ms and the trace distance is 12.5 m. Different scales of Gaussian noise are added to this data. Figure  3(b) shows a noisy version of the original data with SNR =  −1.3903 db. Figure 4 presents two noisy data with SNR of 2.6853 and 4.6263, respectively. The denoised result using these two transforms are shown in figures  5–7. The detailed SNRs of the denoised data are given in table 3. It can be seen that results of the low-redundancy curvelet transform denoising

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Figure 2.  Slice of a synthetic data cube with size 64 × 64 × 64. Table 1.  Comparison of Candes curvelet transform and the low-redundancy curvelet transform.

(3,8)

(4,8)

(5,8)

Scales and angles

Candes

Woiselle

Candes

Woiselle

Candes

Woiselle

No. of coefficients

6,932,397

2,958,281

7,199,027

3,008,529

8,380,925

3,015,701

Table 2.  Restoration results with different percentages of coefficients.

Percentage

10%

20%

30%

40%

50%

60%

70%

80%

Candes Woiselle

7.224 11.903

11.663 18.244

15.416 22.838

18.618 26.826

21.749 30.847

25.217 35.110

29.344 40.133

34.730 46.515

Figure 3.  (a) Slice of synthetic data with size 100 × 100 × 100. (b) A noisy version of figure 3(a) with SNR =  −1.3903. 570

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Figure 4.  (a) Two noisy versions of figure 3(a) with SNR 2.6853 db and 4.6263 db, respectively.

Figure 5.  Denoising results of figure 3(b) with Candes transform and the low-redundancy transform, respectively. (a) Candes curvelet denoising. (b) The low-redundancy curvelet denoising.

Figure 6.  Denoising results of figure 4(a) with Candes transform and the low-redundancy transform, respectively. (a) Candes curvelet denoising. (b) The low-redundancy curvelet denoising.

are comparable to the original curvelet denoising. Furthermore, the low-redundancy curvelet denoising costs approximately one-quarter of the CPU time of the Candes curvelet denoising. To test the denoising ability and computational efficiency of the low-redundancy transform further, other synthetic data with

time sample 150, inline number 61, and crossline number 61 are used for the numerical experiment. The original data cube (Data 2) is shown in figure 8(a); the time sampling ratio is 4 ms and the trace distance is 12.5 m. Figure 8(b) shows a noisy version of figure 8(a) with SNR =  −2.1169 db. Figure 9 gives two 571

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Figure 7.  Denoising results of figure 4(b) with Candes transform and the low-redundancy transform, respectively. (a) Candes curvelet denoising. (b) The low-redundancy curvelet denoising. Table 3.  Denoising results of data 1.

Noisy data Candes Woiselle

  −  1.3903 dB 10.9150 dB 11.2453 dB

2.6853 dB 14.1326 dB 13.7713 dB

4.6263 dB 15.7639 dB 14.9435 dB

8.7075 dB 18.1486 dB 17.3940 dB

Figure 8.  (a) Slice of synthetic Data 2 with size 150 × 61 × 61. (b) A noisy version of figure 8(a) with SNR of  −2.1169 db.

Figure 9.  Two noisy versions of figure 8(a) with SNR of 0.7981 db and 5.2436 db, respectively. 572

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Figure 10.  Denoising results of figure 8(b) with Candes transform and the low-redundancy transform, respectively. (a) Candes curvelet

denoising. (b) The low-redundancy curvelet denoising.

Table 4.  Denoising results of data 2.

Noisy data Candes Woiselle

  −  2.1169 dB 8.5528 dB 9.9650 dB

0.7981 dB 10.3810 dB 11.9018 dB

5.2436 dB 13.2265 dB 15.1624 dB

8.7553 dB 15.4754 dB 16.6268 dB

Figure 11.  Denoising results of figure 9(a) with Candes transform and the low-redundancy transform, respectively. (a) Candes curvelet

denoising. (b) The low-redundancy curvelet denoising.

Figure 12.  Denoising results of figure 9(b) with Candes transform and the low-redundancy transform, respectively. (a) Candes curvelet

denoising. (b) The low-redundancy curvelet denoising.

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Figure 13.  Field data with size 200 × 200 × 200.

Figure 14.  Denoising result of the field data using the low-redundancy transform. 574

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Figure 15.  Residual of the field data after low-redundancy curvelet denoising.

more noisy versions of figure 8(a) with SNR = 0.7981 db and 4.6263 db, respectively. Figures 10–12 are the denoised results using these two transforms. The detailed SNRs can be seen in table 4. It can be concluded that results of these two kinds of curvelet denoising are comparable. However, the CPU time of the low-redundancy curvelet transform is still approximately one-quarter of the Candes curvelet, so this low-redundancy curvelet transform is more efficient with respect to computational efficiency compared with the original one. To verify the denoising ability of the low-redundancy transform for field data, field data with time sample 200, inline number 200, and crossline number 200 are shown in figure 13. The time sampling ratio is 4 ms and the trace distance is 12.5 m. Some coherent and random noise can be seen in the profile, and the events appear blurred, so we need to perform denoising. The denoised results using the low-redundancy curvelet is shown in figure 14. It can be seen that most of the random noise has been attenuated and the quality of the original field data is obviously improved. The events become distinct and continuous. Figure 15 shows the residual of the low-curvelet denoising. The residual shows a few scattered signals and the result is relatively satisfactory. Thus, the lowredundancy curvelet transform is able to meet the high SNR and fidelity demands in seismic data processing.

the efficiency of curvelet-based denoising. The redundancy can reduce to a factor of 10 for 3D data, and the tightness of the curvelet transform is also kept. These merits render it suitable for massive data processing. Numerical examples proved that the low-redundancy transform can save three-quarters of the CPU time compared with the Candes curvelet transform when iterative thresholding denoising strategy is adopted. For denoising problems, a threshold value related to the energy of the noise is the key to deciding the denoising results. However, it is difficult to estimate this parameter, and we can only decide according to experience. How to estimate this parameter adaptively is still undecided.

5.  Conclusions and discussion

References

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Acknowledgments We are grateful for the valuable suggestions and comments raised by three reviewers and the editor. In addition, we thank the authors of Curvelab for making their codes available. This work is supported by National Natural Science Foundation of China (grant number 41204075), a grant from the China Scholarship Council, Natural Science Foundation of Hebei Province (grant number D2014403007), and Young Talents of Universities in Hebei Province (grant number BJ2014049).

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