Dreamlet-based interpolation using POCS method

Journal of Applied Geophysics 109 (2014) 256–265

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Dreamlet-based interpolation using POCS method Benfeng Wang a,b,⁎, Ru-Shan Wu b, Yu Geng b,c, Xiaohong Chen a a b c

State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing 102249, China University of California, Santa Cruz 95060, USA Xi'an Jiaotong University, Xi'an 710049, China

a r t i c l e

i n f o

Article history: Received 11 April 2014 Accepted 18 August 2014 Available online 23 August 2014 Keywords: Dreamlet transform Projection onto convex sets (POCS) Seismic data interpolation Sparsity

a b s t r a c t Due to incomplete and non-uniform coverage of the acquisition system and dead traces, real seismic data always has some missing traces which affect the performance of a multi-channel algorithm, such as Surface-Related Multiple Elimination (SRME), imaging and inversion. Therefore, it is necessary to interpolate seismic data. Dreamlet transform has been successfully used in the modeling of seismic wave propagation and imaging, and this paper explains the application of dreamlet transform to seismic data interpolation. In order to avoid spatial aliasing in transform domain thus getting arbitrary under-sampling rate, improved Jittered under-sampling strategy is proposed to better control the dataset. With L0 constraint and Projection Onto Convex Sets (POCS) method, performances of dreamlet-based and curvelet-based interpolation are compared in terms of recovered signal to noise ratio (SNR) and convergence rate. Tests on synthetic and real cases demonstrate that dreamlet transform has superior performance to curvelet transform. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Due to factors such as presence of obstacles, forbidden areas, feathering and dead traces, real seismic data is always irregularly sampled in spatial coordinates, especially for land acquisition. Since methods, such as surface-related multiple elimination (SRME), wave-equation based migration and inversion, need complete seismic data, irregular sampled seismic data can affect the performance of these algorithms, therefore, seismic data interpolation is an essential stage before multichannel seismic data processing. Methods for interpolation can be divided into three categories (Gao et al., 2013): First, methods based on mathematical transform and signal analysis (Herrmann and Hennenfent, 2008; Naghizadeh and Innanen, 2011; Naghizadeh and Sacchi, 2010; Trad et al., 2002; Wang et al., 2010); second, prediction filter based methods (Naghizadeh and Sacchi, 2007; Spitz, 1991); third, interpolation methods based on wave equation (Ronen, 1987). In the past several years, rank-reduction method (Gao et al., 2013; Kreimer et al., 2013; Ma, 2013) has also played an important role in seismic data interpolation. In this paper, we mainly focus on Projection Onto Convex Sets (POCS) method which belongs in the first interpolation method category. The POCS method was first proposed by Bregman (Bregman, 1965, 1967), and was used in image reconstruction (Stark and Oskoui, 1989; Youla and Webb, 1982). The POCS method was introduced into irregular seismic data interpolation by Abma and Kabir (Abma and Kabir, 2006), however his method is time ⁎ Corresponding author at: State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing, 102249, China. Tel.: +86 15116946918. E-mail address: [email protected] (B. Wang).

http://dx.doi.org/10.1016/j.jappgeo.2014.08.008 0926-9851/© 2014 Elsevier B.V. All rights reserved.

consuming and merely uses linear threshold. With the rapid development of sparse transform, e.g., Fourier transform and Curvelet transform, other researchers have proposed additional strategies based on the POCS method. Gao et al. (Gao et al., 2010) used Fourier transform and POCS method with exponential threshold to interpolate irregular seismic data and they also compared different threshold functions to further improve convergence rate (Gao et al., 2012). Curvelet transform (Candes et al., 2006), which works better for curved seismic events compared with Fourier-based methods, is also used in seismic interpolation (Herrmann and Hennenfent, 2008). Yang et al. (Yang et al., 2012) proves the equivalence of Iterative Shrinkage Threshold Algorithm (IST) and the POCS method with soft threshold when enough iterations are used, and the performance using IST and the POCS method based on Curvelet transform indicates that POCS method with soft threshold is superior over IST in the first few iterations. In order to use the prior information on transform-domain coefficient, Mansour et al. (Mansour et al., 2013) uses the weighted one-norm minimization which seeks the corrections between locations of significant curvelet coefficients, to interpolate the unknown seismic data. There are also some authors who do not use the specific sparse transform, instead they construct the tight frame, which can represent seismic data sparsely, according to the observed seismic data. Liang et al.(Liang et al., 2014) use the tight frame, which is constructed through learning, and the spgl1 method to interpolate unknown data, but this data-driven method is time consuming because a bank of compactly supported filters should be determined first from the observed seismic data. This kind of data-driven method has also been used in seismic data denoising, especially random noise attenuation (Beckouche and Ma, 2014), but it is not widely used because of computational cost. Dreamlet transform (Geng et al., 2009; Wu et al., 2009,

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Fig. 1. Gabor dreamlet atoms.

2011, 2013) is an efficient and alternative sparse transform with respect to curvelet transform. This is because dreamlet is a physical wavelet which is defined on an observation plane and satisfies wave equation. Wu et al. (Wu et al., 2013) did a preliminary study on post-stack seismic data recovery in compressive sensing based on dreamlet transform and L1 constraint, and the performance is superior to curvelet transform. Alias, generated by regular sampling or random sampling with a big gap, will make seismic data interpolation difficult to handle. Antialiasing seismic data interpolation is achieved by Naghizadeh and Sacchi (Naghizadeh and Sacchi, 2007) using a multistep autoregressive algorithm in Fourier transform domain. Using mask function, anti-aliasing seismic data interpolation is obtained in Fourier domain (Gao et al., 2012; Naghizadeh, 2012) and curvelet domain (Naghizadeh and Sacchi, 2010), while performance is easily affected by the parameter setting. A Jittered under-sampling algorithm is proposed (Hennenfent and Herrmann, 2008), which can keep random property as well as control the biggest gap between traces, avoiding the shortcomings of regular sampling and random sampling. Shahidi et al. (Shahidi et al., 2013) extended the Jittered under-sampling strategy from 1D to 2D, and developed Poison Disk sampling and Farthest point sampling strategy which can be used to sample seismic data. In this paper, we propose a novel method which is based on dreamlet (drumbeat-beamlet) transform and POCS method to interpolate irregular sampled seismic data. In order to avoid spatial aliasing, an improved Jittered under-sampling method is proposed to get arbitrary sampling rate. Tests on synthetic and real data approve the validity of this proposed method.

(a)

2. Theory 2.1. Dreamlet (drumbeat-beamlet) theory In dreamlet transform, each atom is a physical wavelet which satisfies wave equation automatically (Wu et al., 2011, 2013). In this paper, we use Gabor frame as the local decomposition atom. The basic dreamlet atom is as follows, dt ω x ξ ðx; t Þ¼g t ω ðt Þbx ξ ðxÞ

ð1Þ


 where g t ω ðt Þ¼W t‐t e‐iωt is time-frequency atom: “drumbeat” with iξx W(t) as a Gaussian window function, and bx ξ ðxÞ ¼ Bðx−xÞe is spacewavenumber atom: “beamlet” with B(x) as a Gaussian window function. Then a dreamlet atom is a localized wavepacket in time-space plane:
 −i dt ω x ξ ðx; t Þ ¼ W t−t Bðx−xÞe






ωt−ξx

ð2Þ

where t, ω, x and ξ are local time, frequency, space and wavenumber, respectively. A time-space atom constructed in this way satisfies the causality automatically, which can represent seismic data efficiently, resulting in sparse coefficient in dreamlet domain. Fig. 1 shows some of the dreamlet basis, and the atom is the local basis which can be used to represent the seismic data locally.

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Table 1 Different threshold functions.

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Threshold function

Expression

Linear threshold Exponential threshold

τkl = τmax ‐ (τmax ‐ τmin)(k ‐ 1)/(N ‐ 1), k = 1, 2, …, N. τke = τmaxec(k ‐ 1)/(N ‐ 1), c = ln(τmin/τmax), k = 1, 2, …, N.

(b) Seismic data decomposition based on dreamlet transform can be expressed as,

(c) uðx; t Þ ¼

XD tωxξ

E X uðx; t Þ; dt ω x ξ ðx; t Þ dt ω x ξ ðx; t Þ ¼ ct ω x dt ω x ξ ðx; t Þ ξ

t ωxξ

ð3Þ

where u(x, t) is seismic data, dt ω x ξ ðx; t Þ is basic dreamlet atom, dt ω x ξ

Fig. 4. Under-sampling strategey demo. (a) Regular sampling; (b) the Jittered undersampling; (c) random sampling.

ðx; t Þ is the unitary dual frame of dt ω x ξ ðx; t Þ and ct ω x ξ is dreamlet coefficient,

2.2. POCS-based interpolation method

D E    ct ω x ξ ¼ uðx; t Þ;dt ω x ξ ðx; t Þ ¼∬uðx; t Þdt ω x ξ ðx; t Þdxdt¼∬uðx; t Þe g ω ðt Þ e bx ðxÞdxdt t

ξ

ð4Þ

Irregular sampled seismic data dobs, can be related to the complete seismic data d0 using sampling matrix R, shown as, dobs ¼Rd0

where * stands for complex conjugate and e g ω ðt Þ, e bx ðxÞ are the unitary t

ξ

dual vectors of g ω ðt Þ and bx ξ ðxÞ, respectively, subjecting to the condition t

(Chen et al., 2006), D E e g ω ðt Þ; g ω ðt Þ ¼ 1 and t

t

D E e b ω ðt Þ; b ω ðt Þ ¼ 1 : t

t

ð5Þ

dreamlet transform used here, is a sparse transform with Gabor frame of redundancy 2 and shift steps 8 in both time and spatial directions, which means that dreamlet transform used in this paper owns redundancy 4 that is lower compared with curvelet transform whose redundancy equals 8 approximately if curvelets are adopted at the finest scale. Lower redundancy can make dreamlet transform more efficient, which is also superior in seismogram decomposition compared with other decomposition methods. Fig. 2(a) shows a shot gather and Fig. 2(b) shows dreamlet-based and curvelet-based recovered signalto-noise ratio (SNR) defined in formula (6) using part of coefficients. We design τ = p * Dmax as the threshold to truncate coefficients, where Dmax is the maximum coefficient value and p is a factor between 0 and 1, then the truncated coefficients are used for reconstruction. In Fig. 2(b), vertical-axis indicates reconstruction signal to noise ratio (SNR), while horizontal-axis shows information of p, that is xp = ‐ log10 (p). From Fig. 2(b), we can see that the recovered SNR of dreamlet transform is a little higher than that of curvelet transform which demonstrates the superiority of dreamlet transform in seismic data recovery to some extent. SNR¼20log10

kd0 k2

. kdrec ‐d0 k2

ð6Þ

Eq. (5) is always ill-posed and due to the sparse property of seismic data in dreamlet domain, objective functional can be constructed as follows,   T 2  ΦðxÞ¼dobs ‐RD x þλP ðxÞ 2

ð7Þ

where x is dreamlet coefficient vector, DT is inverse dreamlet transform and P(x) is sparse constraint which can be chosen as L0 or L1 norm constraint. L0 norm constraint is the sparsest constraint which can be solved through iterative hard threshold algorithm (IHT) (Blumensath and Davies, 2008, 2009; Loris et al., 2010). Equivalence of IST or IHT and POCS with soft or hard threshold has been proved when using enough iterations (Yang et al., 2012) and POCS method is superior in the first few iterations compared with IST or IHT algorithm. Therefore, the POCS method is used to solve Eq. (7), and an interpolation iterative algorithm is as follows, T

Fig. 3. Flow chart of dreamlet-based interpolation using POCS method.

dk ¼dobs þðI‐RÞD T λk ðDdk‐1 Þ

ð8Þ

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Fig. 5. Complete seismic data and incomplete seismic data with different sampling rates. (a) Complete data; (b) 40% traces missing; (c) 50% traces missing; (d) 60% traces missing.

where Tλ(x) is the threshold function. When P(x) = ‖x‖0 (L0 norm), it subjects to,  x; T λ ðxÞ¼ 0;

jxj≥τ jxj<τ

ð9Þ

pffiffiffi where τ¼ λ is the threshold. When P(x) = ‖x‖1 (L1 norm), it subjects to, 8 < x‐τ; x >τ T λ ðxÞ¼ 0; jxj≤τ : xþτ; x<‐τ

function is most frequently used and is superior to the linear one. The corresponding threshold functions are listed in Table 1. Where τmax ¼p1 Dmax ; τmin ¼p2 Dmax and Dmax is the maximum coefficient. A flow chart of dreamlet-based interpolation using POCS is shown in Fig. 3.

2.3. Under-sampling method ð10Þ

where τ = 0.5λ is the threshold and λ is regularization factor. The threshold can be obtained through linear threshold function or exponential threshold, though the exponential threshold

Random sampling can turn the aliasing effect into lower valued noise in the transform domain when using sparse transforms. Though random sampling has random property compared with regular sampling, it lacks the ability to control maximum gaps between adjacent traces which can change the original spectrum greatly. Random property and the ability to control maximum gaps between adjacent traces can

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Fig. 6. The interpolated results (left) and corresponding residuals (right) of 40% traces missing. Top is for curvelet case and bottom is for dreamlet case (p1 = 0.99 and p2 = 0.000001).

be guaranteed by the Jittered sampling strategy (Hennenfent and Herrmann, 2008). The Jittered under-sampling demo is shown in Fig. 4 together with regular sampling and random sampling strategy. The exact formula for Jittered under-sampling is, yðkÞ¼f ð jk Þ;k¼1;2;…;n

ð11Þ

where jk ¼ 1‐γ 2 þkγþε k ; ε k is an integer subjecting to εk ∈ [‐ ⌊(ε ‐ 1)/ 2⌋, ⌊(ε ‐ 1)/2⌋). Number of sampling traces n, under-sampling factor γ (an odd integer) and total number of traces N subject to n = N/γ; ε subject to 0 ≤ ε ≤ γ. The optimal Jittered under-sampling can be obtained when ε = γ and γ can also be chosen as even integer. The γ, chosen as integer, limits range of under-sampling rate and in order to obtain arbitrary under-sampling rate, formula (11) can be modified into, yðkÞ¼f ð jk Þ;k¼1;2;…;n

ð12Þ

  where, jk ¼ðk‐0:5Þγþεk þ1; εk ∈ ‐ 2ε ; 2ε ; γ¼N=n;N is total number of traces, n is number of observed seismic data and ε subjects to

0 ≤ ε ≤ γ. Optimal Jittered under-sampling can be obtained when ε = γ. 3. Numerical examples Firstly, we design a four-layer model to approve the validity of dreamlet-based interpolation using POCS method and L0 norm constraint. In this synthetic test, traces with 40%, 50% and 60% missing are tested to show the superiority of dreamlet-based method. Then, real seismic data application is also given, in which 50% traces are missing. 3.1. Four-layer model test Fig. 5(a) shows a synthetic seismic data based on a four-layer model including 201 traces (trace interval is 12.5 m) with 1001 samples in each trace (time sampling interval is 2 ms). 40%, 50% and 60% traces are missing based on improved Jitter under-sampling (formula 12), as shown in Fig. 5(b), (c) and (d), respectively. This indicates that random property is satisfied as well as the maximum gap between adjacent

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Fig. 7. The interpolated results (left) and corresponding residuals (right) of 50% traces missing. Top is for curvelet case and bottom is for dreamlet case (p1 = 0.99 and p2 = 0.00005).

traces is controlled using the improved Jittered under-sampling strategy. The interpolated results and residuals of the 40% traces missing, based on dreamlet transform and curvelet transform with hard threshold, are plotted in Fig. 6, and Figs. 7–8 demonstrate the results and residuals of 50% and 60% traces missing, respectively. Figs. 6–8 show that interpolated results are almost the same with the complete seismic data shown in Fig. 5(a). As the missing traces increase, the residual increases. In order to further compare these performances, the SNR curves with respect to iterations are plotted in Fig. 9. Fig. 9 indicates that the dreamlet-based method is superior to the curvelet-based method and the recovered SNR is lower as missing traces increase. The final SNRs are 29.2, 22.3 and 18.6 dB using dreamlet transform with 40%, 50% and 60% traces missing, respectively, while the SNRs corresponding to curvelet transform are 25.4, 19.9 and 17.6 dB, respectively. Performances in this simple synthetic seismic data prove the validity of the proposed method to some extent. To further confirm the

validity of the proposed method, application of a real seismic data from a marine acquisition system is given. 3.2. Real data application Fig. 10(a) shows the complete field data with 180 traces (trace interval is 12.5 m) and 1500 samples in each trace (time sampling interval is 4 ms). Incomplete seismic data with 50% traces missing using improved Jitter under-sampling strategy, shown in Fig. 10(b). Fig. 10(b) indicates that the maximum gap between adjacent traces is controlled and random property is preserved. For convenience of comparisons, dreamlet-based and curvelet-based methods are tested with the same parameters. Interpolated results and residuals are shown in Fig. 11, respectively. The interpolated results shown in Fig. 11(left column) agree well with the complete marine seismic data shown in Fig. 10(a), which

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demonstrates the validity of sparse transform based interpolation method and the performance is satisfactory. In order to further compare the performances, the SNR curves with iterations for this marine datasets are shown in Fig. 12. Fig. 12 indicates that results based on dreamlet are superior to the curvelet-based ones. From the SNR curves shown in Fig. 12, we can conclude that dreamlet transform is sparser and the performance is better compared with the curvelet-based ones in this real data example, which demonstrates the validity of the dreamlet-based POCS method. From Fig. 9 and Fig. 12, we can draw a rough conclusion that the POCS method is efficient when using sparse transform with L0 norm as sparse constraints. When the maximum iteration number is set as 50, the recovered SNR is stable and the curve becomes horizontal after nearly 20–30 iterations which means that the quality of the recovered seismic data improves very little as the iteration increases after 20–30 iterations in the synthetic data test. While for the real seismic date, the recovered seismic date quality improves very little after 30–40 iterations. Therefore, the iteration can be terminated reasonably if we want to improve

the efficiency further. Synthetic and real seismic data applications demonstrate the validity of the proposed method.

4. Conclusion Interpolated seismic data is obtained from the newly developed sparse transform: dreamlet transform, using POCS method. Tests on synthetic and real seismic data show that the performances based on dreamlet transform are better than those based on curvelet transform in terms of sparsity and recovered SNR. The results in this paper are based on improved Jittered under-sampling strategy, which controls the maximum gap between adjacent traces and keeps random property. If the gap between adjacent traces is big or alias occurs in transform domain when using regular sampling strategy, anti-aliasing methods should be used (Gao et al., 2012; Naghizadeh, 2012; Naghizadeh and Sacchi, 2010). Our future research topic is anti-aliasing interpolation using dreamlet transform and sparse constraints.

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Acknowledgment This research work was conducted during Benfeng Wang's visit to University of California, Santa Cruz as a visiting student. It is supported by the National Natural Science Foundation of China (grant no. U1262207), the Key State Science and Technology Project (grant nos. 2011ZX05023-005-005, 2011ZX05019-006) and the WTOPI Research Consortium at the University of California, Santa Cruz. The first author would like to thank the China Scholarship Council for their financial support to study abroad. References Abma, R., Kabir, N., 2006. 3D interpolation of irregular data with a POCS algorithm. Geophysics 71 (6), E91–E97. Beckouche, S., Ma, J., 2014. Simultaneous dictionary learning and denoising for seismic data. Geophysics 79 (3), A27–A31.

Blumensath, T., Davies, M.E., 2008. Iterative thresholding for sparse approximations. J. Fourier Anal. Appl. 14 (5–6), 629–654. Blumensath, T., Davies, M.E., 2009. Iterative hard thresholding for compressed sensing. Appl. Comput. Harmon. Anal. 27 (3), 265–274. Bregman, L., 1965. The method of successive projection for finding a common point of convex sets (theorems for determining common point of convex sets by method of successive projection). Sov. Math. 6, 688–692. Bregman, L., 1967. The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7 (3), 200–217. Candes, E., Demanet, L., Donoho, D., Ying, L., 2006. Fast discrete curvelet transforms. Multiscale Model. Simul. 5 (3), 861–899. Chen, L., Wu, R., Chen, Y., 2006. Target-oriented beamlet migration based on GaborDaubechies frame decomposition. Geophysics 71 (2), S37–S52. Gao, J., Chen, X., Li, J., Liu, G., Ma, J., 2010. Irregular seismic data reconstruction based on exponential threshold model of POCS method. Appl. Geophys. 7 (3), 229–238. Gao, J., Stanton, A., Naghizadeh, M., Sacchi, M.D., Chen, X., 2012. Convergence improvement and noise attenuation considerations for beyond alias projection onto convex sets reconstruction. Geophys. Prospect. 61 (Suppl.1), 138–151. Gao, J., Sacchi, M., Chen, X., 2013. A fast reduced-rank interpolation method for prestack seismic volumes that depend on four spatial dimensions. Geophysics 78 (1), V21–V30.

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Fig. 11. The interpolated results (left) and corresponding residuals (right). Top is for curvelet case and bottom is for dreamlet case.

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