TY - JOUR
AU - Zhong,, Wei
AB - Abstract Singular value decomposition (SVD) can be used both globally and locally to remove random noise in order to improve the signal-to-noise ratio (SNR) of seismic data. However, it can only be applied to seismic data with simple structure such that there is only one dip component in each processing window. We introduce a novel denoising approach that utilizes a structure-oriented SVD, and this approach can enhance seismic reflections with continuous slopes. We create a third dimension for a 2D seismic profile by using the plane-wave prediction operator to predict each trace from its neighbour traces and apply SVD along this dimension. The added dimension is equivalent to flattening the seismic reflections within a neighbouring window. The third dimension is then averaged to decrease the dimension. We use two synthetic examples with different complexities and one field data example to demonstrate the performance of the proposed structure-oriented SVD. Compared with global and local SVDs, and f–x deconvolution, the structure-oriented SVD can obtain much clearer reflections and preserve more useful energy. Singular value decomposition, random noise attenuation, global SVD, local SVD, structure-oriented SVD 1. Introduction The attenuation of random noise is an important subject in seismic data processing. The enhanced seismic signals with higher signal-to-noise ratio (SNR) can help interpreters to make more accurate decisions. There are generally four different categories of random noise attenuation approaches that exist in the exploration geophysics literature. The first is based on the predictive property of useful signals in small spatial–temporal windows, such as f–x deconvolution (Canales 1984, Chen and Ma 2014), f–x–y prediction filtering (Wang 1999, Wang 2002), t–x prediction error filtering (Abma and Claerbout 1995), and f–x non-stationary polynomial fitting (Liu et al2011). Yang et al (2015) proposed a novel trace-by-trace random noise attenuation approach based on the predictable spectral components property of useful seismic reflections using regularized non-stationary autoregression (Fomel 2013). This approach does not require the spatial coherency assumption and has the potential to be widely used to denoise microseismic signal, the SNR of which is very low. The second is based on the statistical properties of seismic profiles, such as the mean filter (Bonar and Sacchi 2012) or median filter (Liu 2013, Chen 2014a, Chen et al2014b). The third is based on extracting the principal components of seismic data, such as multichannel singular spectrum analysis (MSSA) (Oropeza and Sacchi 2011, Chen et al2015) and EMD based approaches (Chen et al2014c). This type of approach is also related to those rank-reduction based approaches, eigenimage-based approaches. The fourth is based on a transformed domain thresholding strategy (Neelamani et al2008, Fomel and Liu 2010, Chen et al2014a). The transform operator can be a fixed-basis sparsity-promoting transform, and can also be an adaptively learned dictionary: the fixed-basis transform enjoys better efficiency and the learning-based dictionary enjoys better adaptivity. In addition to the aforementioned one-step noise attenuation, Chen and Fomel (2015) proposed a two-step approach for retrieving the lost useful signals from the noise section, which can be seen as residual random noise attenuation. Freire and Ulrych (1988) proposed to carry out rank reduction of seismic images in the t–x domain via singular value decomposition (SVD). They applied SVD to the seismic data matrix and extracted the first singular value in order to remove random noise based on the assumption that the seismic data with only horizontal events have a rank of one. We refer to this method as the global SVD (GSVD), because this SVD method does not require implementation in small windows. The GSVD does not require regularly sampled data because no convolutional operator is used. As long as the seismic profiles are composed of horizontal events, GSVD can obtain a good denoising performance. However, the seismic profiles do not necessarily meet the requirement that there are no dipping events. For these complex profiles, GSVD cannot be applied. Freire and Ulrych (1988) also demonstrated that the GSVD is actually equivalent to the Karhunen–Loève (KL) (or principal component transformation) approach (Jones and Levy 1987). Because of the horizontal-energy selective property of GSVD, Milton et al (2009) proposed to use GSVD filtering to remove dipping interference, such as ground-rolls. The f–x domain rank reduction approaches are independent from dip, and therefore do not require flattening. They require an eigen-decomposition of the spectral matrix of data, which is connected with Cadzow filtering (Cadzow 1988) and multichannel singular spectrum analysis (MSSA) (Vautard et al1992, Oropeza and Sacchi 2011). Although MSSA and Cadzow filtering are equivalent, they originate from different signal analysis subfields. Cadzow filtering was proposed to denoise images, whereas MSSA was proposed to decompose time series arising in the study of dynamical systems (Oropeza and Sacchi 2010). Cadzow filtering and MSSA are also referred to as the f–x SVD (FXSVD). For FXSVD, the eigen-decomposition of the spectral matrix is usually done by applying an SVD to a pre-constructed (block) Hankel matrix. However, this type of method depends on the low-rank property of seismic data; in other words, the seismic profile should contain few linear events (2D) or few plane-wave components (3D). The low-rank and linear-events assumptions of the FXSVD are not always met, which forces the FXSVD to be implemented in local processing windows. The division of local windows and decision of rank value, however, are sometimes an empirical and user-unfriendly process. In this paper, we propose a novel SVD approach for enhancing useful reflections by removing random noise. The novel SVD is implemented obeying the structural information. We first flatten the seismic reflections according to the local slopes by trace prediction. In the flattened domain, we apply a GSVD to horizontal events in order to remove random noise. The flattened traces are then transformed back to the original structural shape by stacking the traces in the flattened domain. We refer to the proposed novel SVD denoising approach as structure-oriented SVD (SOSVD). The SOSVD does not require local processing windows as LSVD and FXSVD do, and it can distinguish signal and noise better than GSVD. Flattened events make the largest singular value components mainly correspond to useful events, which are much easier to separate. However, GSVD may not be applied to flattened events. We organize the paper as follows: we first introduce the well known GSVD, LSVD along with dip steering, and the theoretical aspects of SOSVD. A brief review of the basic theory of the commonly used f–x deconvolution is also provided in the appendix. We then compareapply three different SVDs and commonly used f–x deconvolution to three different synthetic examples (with increasing complexity) and one field data example that comes from the Gulf of Mexico North Sea, and compare their performances. The comparisons show that the SOSVD can obtain much better performance in removing noise and preserving useful signals. 2. Theory 2.1. Noise attenuation using SVD 2D seismic data can be expressed as a data matrix D (M × N), consisting of N traces and M time samples. The SVD of the data matrix D can be expressed as D=UΣ V T.1 Here, U is composed of the eigenvectors of DDT. V is composed of the eigenvectors of DTD. Σ is a diagonal matrix composed of the decreasing singular values. Let us denote U, Σ, and V in the following form: U=[u1,u2,⋯,ur],Σ=diag(σ1,σ,⋯,σr),V=[v1,v2,⋯,vr],2 where r is the rank of D. The vectors uk and vk are also called the propagation vectors and the eigen-wavelets, respectively (Vrabie et al2004). The singular values σk are sorted such that σ1 ⩾ σ2 ⩾ · · · ⩾ σr. They can be obtained by calculating the positive square roots of the eigenvalues of the data covariance matrix DDT. Equation (1) can be expressed as D=∑k=1rλkukvkT,3 where ukvT is the rank-one matrix called the kth eigenimage of D. Thus, from equation (3), the seismic image can be decomposed into r eigenimages, the energy of which corresponds to the value of each element in matrix Σ. We can remove the random noise in seismic data in order to enhance the seismic reflections by only selecting the first few eigenimages (Freire and Ulrych 1988): D^svd=∑pk=1σkukvkT.4 The random noise attenuation approach by SVD utilizes the property that useful seismic signals are horizontally coherent to separate signal and noise. The decomposition is based on the criterion of horizontal coherency. The energy of coherent signals lies in the first few parts of the decomposition, which is represented by the larger values in Σ. For seismic data, the horizontal coherent part corresponds to useful signal, while random noise or coherent dipping events are not horizontally coherent. Thus, by choosing the first few elements in Σ and making the others equal to zero, we can remove the noise and enhance seismic data quality. SVD acts as a data-driven, low-pass filter by rejecting highly uncorrelated traces (Bekara and van der Baan 2007). Here, we refer to those denoising approaches that simply apply SVD to the original seismic data in order to remove random noise as global SVD (GSVD). 2.2. Dip steering and local SVD Supposing the events in a processing window have one slope, the dip steering algorithm aims to flatten the dipping events by shifting each trace such that the flattened events are suitable for applying a GSVD based denoising approach, as shown in equation (4). This flattening strategy is called dip steering (Bekara and van der Baan 2007). After removing noise, applying an inverse flattening to the data in the processing window can obtain the denoised data. We refer to those denoising approaches that apply a GSVD in the dip steering flattened domain as local SVD (LSVD). The time shifts for each trace in order to flatten the events are obtained by selecting the time delays that can maximize cross-correlation between each trace and a reference trace τn=arg⁡maxτ∑m=min(1−τ,1)max(Mw,Mw−τ)r(m)d(m+τ,n),5 where τn is the optimal time shift for the nth trace in the processing window, Mw denotes the number of time samples in the processing window and r(m) denotes the mth sample of the reference trace and d(m, n) denotes the data value in the mth sample and nth trace. τn > 0 corresponds to shifting the trace downward with respect to the reference trace, and τn < 0 corresponds to shifting the trace upward. As can be seen from equation (5), the selection of the reference trace is crucial for the effectiveness of LSVD. It can be chosen as a random trace in the processing window if the noise level is not high, or can be chosen as a stacked trace after normal moveout (NMO) of common-midpoint gathers. Bekara and van der Baan (2007) proposed an iterative strategy for selecting the optimal reference trace, stating that the resulting shifted traces are stacked after each cross-correlation pass for updating the new reference traces and the process of cross-correlation, shifting, and stacking is repeated until the process converges. Figure 1 shows a demonstration for dip steering and the processes of LSVD. Figure 1(a) is the original noisy dipping event. Figure 1(b) denotes the flattened event after forward dip steering. Figure 1(c) shows the LSVD denoised result by choosing only one singular value. The final denoised result after inverse dip steering in figure 1(c) is shown in figure 1(d). The time shifts after the optimizing equation (5) are shown in figure 2. The reference trace is simply chosen as the first trace of the original data, as shown in figure 1(a). Figure 1. Open in new tabDownload slide Demonstration for dip steering and LSVD. (a) Noisy dip reflector. (b) Flattened event using dip steering. (c) SVD denoised result by selecting only one eigen-image. (d) Denoised data after inverse dip steering. Figure 1. Open in new tabDownload slide Demonstration for dip steering and LSVD. (a) Noisy dip reflector. (b) Flattened event using dip steering. (c) SVD denoised result by selecting only one eigen-image. (d) Denoised data after inverse dip steering. Figure 2. Open in new tabDownload slide Time shifts for each trace for the forward dip steering as shown in figure 1 (reference trace selected as the left trace in figure 1(a)). Figure 2. Open in new tabDownload slide Time shifts for each trace for the forward dip steering as shown in figure 1 (reference trace selected as the left trace in figure 1(a)). 2.3. Structure-oriented SVD The SOSVD refers to two processes: flattening along the local structure in a local spatial window and applying GSVD in the flattened local spatial window. The procedures can be summarized as D→DjR(j=1,2,⋯,N)→D¯jR→D¯¯jR→dj→D^sosvd.6 Here, DjR denotes the jth spatial window (corresponding to the jth trace) with a radius of R, D¯jR denotes the flattened local spatial window, D¯¯jR denotes the SVD denoised local spatial window, dj denotes the averaged local spatial window, and D^sosvd denotes the output data using SOSVD. As we can see from the workflow, the key step that distinguishes SOSVD from other types of SVD approach is the flattening in the local spatial window. The flattening corresponds to applying a flattening operator to the data (here we use a prediction operator according to local slope) so that the output data have horizontal events: PjDjR=D¯jR.7 where Pj is the jth flattening operator. Here, the flattening operator is chosen as a plane-wave prediction operator related to the local slope. Equation (7) has the following detailed form: [P(1,j)→(1+R,j)⁢ (σ1,j)                                                  ⋱P(1+R,j)→(1+R,j) (σ1+R,j)                                 ⋱   P(1+2R,j)→(1+R,j)  (σ1+2R,j)][d1,j,⋯,d1+R,j,⋯,d1+2R,j]=[d¯1,j,⋯,d¯1+R,j,⋯,d¯1+2R,j].8 Here, P(i, j) → (k, j)(σi, j) denotes the prediction operator from trace i to trace k in the jth spatial window, which is connected to the local slope of the ith trace. Prediction of a trace consists of shifting the original trace along dominant event slopes (Fomel 2010). Prediction of a trace from a distant neighbour can be accomplished by simple recursion (Liu et al2010); i.e. predicting trace k from trace 1 is simply P(1,j)→(k,j)(σ1,j)=P(k−1,j)→(k,j)(σk−1,j)⋯P(2,j)→(3,j)(σ2,j)P(1,j)→(2,j)(σ1,j).9 The prediction operator is a numerical solution of the local plane differential equation ∂D∂x+σ∂D∂t=0,10 for local plane wave propagation in the x direction. The dominant slopes are estimated by solving the following least-square minimization problem using regularized least-square optimization: σ^=arg⁡minσ∥W(σ)D∥22,11 where DW is the destruction operator defined as W=[I00⋯0-P1→2I0⋯00-P2→3I⋯0⋯⋯⋯⋯⋯00⋯-PN-1→NI],12 where I stands for the identity operator, and Pi→k describes prediction of trace k from trace i (the same as the previous version P(i, j) → (k, j)(σi, j) except not in a specific spatial window). The optimization approach as shown in equation (11) for obtaining local slope estimation is called plane wave destruction (PWD) (Fomel 2002). 3. Examples In this section, we first use two synthetic examples with different complexities. Then a field data example is presented for better demonstration of the proposed approach. For all the three examples, we compare the denoising performances for GSVD, LSVD, SOSVD, and the commonly used f–x deconvolution. A brief review of the theory of f–x deconvolution is provided in the appendix. The first synthetic example is a simple seismic profile that has hyperbolic events. It is generated from the SeismicLab library. The clean data and noisy data with simulated Gaussian white noise are shown in figure 3. There are three hyperbolic events in the synthetic data. Two events have small slopes and the other one has high slope. The denoised results using GSVD, LSVD, and SOSVD are shown in figures 4(a), (b), and (d), respectively. As a reference, we also show the denoised result using f–x deconvolution in figure 4(c). From the comparison of denoised results shown in figure 4, we can initially conclude that the GSVD removes the least noise, there is some damage to the events for LSVD, f–x deconvolution does a little damage to the dipping hyperbolic event, and the denoised result using the proposed SOSVD is nearly perfect. The removed noise sections are shown in figure 5. From the noise sections, we can confirm the observation made from the denoised results shown in figure 4. The GSVD cannot remove too much random noise because we cannot remove too many eigenimages, otherwise the damage to useful events is very large. The GSVD causes some damage to the steep event, LSVD causes some damage to areas that have conflicting slopes, and f–x deconvolution causes damage to both the steep event and more horizontal events. There are almost no coherent signals lost in the noise section for the proposed SOSVD. Figure 3. Open in new tabDownload slide Hyperbolic-events synthetic seismic profile. (a) Clean data. (b) Noisy data. Figure 3. Open in new tabDownload slide Hyperbolic-events synthetic seismic profile. (a) Clean data. (b) Noisy data. Figure 4. Open in new tabDownload slide Comparison of denoised data for the hyperbolic-events synthetic example. (a) Using GSVD. (b) Using LSVD. (c) Using f–x deconvolution. (d) Using SOSVD. Figure 4. Open in new tabDownload slide Comparison of denoised data for the hyperbolic-events synthetic example. (a) Using GSVD. (b) Using LSVD. (c) Using f–x deconvolution. (d) Using SOSVD. Figure 5. Open in new tabDownload slide Comparison of removed noise for the hyperbolic-events synthetic example. (a) Using GSVD. (b) Using LSVD. (c) Using f–x deconvolution. (d) Using SOSVD. Figure 5. Open in new tabDownload slide Comparison of removed noise for the hyperbolic-events synthetic example. (a) Using GSVD. (b) Using LSVD. (c) Using f–x deconvolution. (d) Using SOSVD. The second synthetic example is a linear-event section. Figure 6 shows the clean and noisy data. There are one horizontal and three dipping events in this section. Two of the dipping events cross each other. After using the GSVD, LSVD, f–x deconvolution, and the proposed SOSVD, we obtain four denoised sections, as shown in figure 7. The GSVD still does a bad job because of the dipping events. The LSVD seems to cause more damage to the useful events than the previous example because in almost every processing window there are more than one slope, which makes the dip steering strategy fail. f–x deconvolution damages both horizontal and dipping events. The proposed SOSVD removes the most noise and preserves the useful energy best. From the noise sections as shown in figure 8, we conclude that the SOSVD achieves an excellent performance except for small damage to the crossing point of the original noisy data. The damage for the crossing point comes from the fact that the PWD algorithm cannot obtain a precise slope estimation in the region and thus makes the flattening along the structure more difficult. In fact, the only disadvantage of the proposed approach is the inability to handle crossing seismic events. The crossing points will appear in the noise section. The problem can be handled by slope-separated processing using the same approach. However, as a byproduct, the proposed approach may have potential to be used as a diffraction detector, which will be helpful for other important tasks in seismic data processing and imaging. This topic regarding the crossing point may be the subject of future investigation. Figure 6. Open in new tabDownload slide Complex-events synthetic seismic profile. (a) Clean data. (b) Noisy data. Figure 6. Open in new tabDownload slide Complex-events synthetic seismic profile. (a) Clean data. (b) Noisy data. Figure 7. Open in new tabDownload slide Comparison of denoised data for the complex-events synthetic example. (a) Using GSVD. (b) Using LSVD. (c) Using f–x deconvolution. (d) Using SOSVD. Figure 7. Open in new tabDownload slide Comparison of denoised data for the complex-events synthetic example. (a) Using GSVD. (b) Using LSVD. (c) Using f–x deconvolution. (d) Using SOSVD. Figure 8. Open in new tabDownload slide Comparison of removed noise for the complex-events synthetic example. (a) Using GSVD. (b) Using LSVD. (c) Using f–x deconvolution. (d) Using SOSVD. Figure 8. Open in new tabDownload slide Comparison of removed noise for the complex-events synthetic example. (a) Using GSVD. (b) Using LSVD. (c) Using f–x deconvolution. (d) Using SOSVD. The third example is a field data example. This example is a part extracted from 3D North Sea data (Lomask et al2006, Fomel 2010). The original image is shown in figure 9. As we can see, the seismic data contain a lot of in-continuousdiscontinuous events caused by a low SNR, which makes the interpretation and evaluation of the useful signals impossible. After using GSVD, LSVD, f–x deconvolution and SOSVD, we obtain four denoised sections that are shown in figure 10. It is obvious that GSVD and LSVD both cause some damage to the useful signals. Although f–x deconvolution preserves the useful reflections well, there seems to be no improvement for SNR of the data. By using SOSVD, we can obtain a very good denoised image (figure 10(d)). The image becomes clean and seismic reflections becomes continuous, which is beneficial for the following interpretation. From the noise sections shown in figure 11, we can come to the similar conclusion that SOSVD preserves the most useful energy while removing the most noise. Figure 9. Open in new tabDownload slide Field data example. Figure 9. Open in new tabDownload slide Field data example. Figure 10. Open in new tabDownload slide Comparison of denoised data for field data. (a) Using GSVD. (b) Using LSVD. (c) Using f–x deconvolution. (d) Using SOSVD. Figure 10. Open in new tabDownload slide Comparison of denoised data for field data. (a) Using GSVD. (b) Using LSVD. (c) Using f–x deconvolution. (d) Using SOSVD. Figure 11. Open in new tabDownload slide Comparison of removed noise for field data. (a) Using GSVD. (b) Using LSVD. (c) Using f–x deconvolution. (d) Using SOSVD. Figure 11. Open in new tabDownload slide Comparison of removed noise for field data. (a) Using GSVD. (b) Using LSVD. (c) Using f–x deconvolution. (d) Using SOSVD. 4. Conclusions We have proposed a novel structure-oriented singular value decomposition (SOSVD) for enhancing useful reflections by removing random noise of seismic data. The SOSVD locally flattens the useful reflections by predicting each trace from its neighbouring traces. The plane-wave destruction (PWD) operator serves as the slope estimator for the prediction operator. In the flattened local processing window, we can apply a traditional SVD to remove incoherent noise. The SVD denoised windows are then stacked to output the denoised trace. Compared with GSVD, the SOSVD can preserve many more dipping events by slope flattening. Compared with LSVD, the SOSVD skips the use of dip steering, that depends on an uniform shift for each trace and assumes a uniform-slope processing window, thus is more robust and effective in handling complex structures. Compared with f–x deconvolution, the SOSVD can be significantly more effective in both the removal of random noise and the preservation of useful reflections. Acknowledgments We would like to thank three anonymous reviewers for very constructive suggestions, which helped us to improve the manuscript greatly. Appendix. Review of f–x deconvolution f–x deconvolution is one of the most widely used approaches for random noise attenuation. In this appendix, we briefly review the theory of f–x deconvolution. Let st(x) be the seismic signal located at trace x and time t. If the slope of a linear event with constant amplitude in a seismic section is ψ, then st(x+1)=st−xψΔh(1),A.1 where Δh denotes the spatial interval. equation (A.1) can be directly transformed into the frequency domain utilizing the time-shift property of the Fourier transform: Sf(x+1)=Sf(1)e−i2πfxψΔh.A.2 For a specific frequency f0, from equation (A.2) we can obtain a linear recursion, which is given by Sf0(x+1)=af0(1)Sf0(x),A.3 where af0(1)=e−i2πf0ψΔh ⁠. This recursion is also known as an auto-regressive (AR) model of order 1 (Canales 1984). Similarly, superposition of p linear events in the t–x domain can be represented by an AR model of order p (Tufts and Kumaresan 1982, Harris and White 1997) as in the following equation: Sf0(x+1)=af0(1)Sf0(x)+af0(2)Sf0(x-1)+⋯        +af0(p)Sf0(x+1-p).A.4 Equation (A.4) can be formulated as a convolutional form: d=f∗a,A.5 where d denotes the vector composed of Sf0(x+1) (x = 1, 2, · · ·, X), f denotes the vector composed of Sf0(x) (x = 1, 2, · · ·, X), a denotes the vector composed of af0(x) (x = 1, 2, · · ·, X), and X denotes the number of traces. Equation (A.5) can be formulated as a matrix vector form: d=Fm,A.6 where F is the convolution matrix composed by f. Suppose p = 4; the detailed form of equation (A.6) can be expressed as [f2f3⋮fX000]=[f1000f2f100⋮f2f10fX−1⋮f2f10fX−1⋮f200fX−1⋮000fX−1][a1a2a3a4].A.7 However, equation (A.6) is based on the clean signal model. In reality, the seismic data are composed of random noise. Thus, we have to solve a from the noise corrupted observation d based on some optimization schemes. Based on equation (A.4), we can formulate an optimization problem based on the minimum prediction error energy assumption. The predictive error filter can be solved by minimizing the following objective function: J=||Fm−d||22,A.8 where ||⋅||22 denotes the squares of L2 norm. Taking derivatives of the cost function (A.8) with respect to m, and setting the result to zero, we can obtain the following equation: F Td=F TFm,A.9 where [·]T denotes the transpose. Note that FTF is a Toeplitz form and thus can be efficiently solved using Levinson’s recursion. In order to stabilize the recursion for solving a^ ⁠, we need to add a small perturbation to the diagonal of the Toeplitz matrix: â=(F TF+μI)-1F Td.A.10 Finally, the estimated clean data (denoised data) can be expressed as d^=Fa^.A.11 It is worth mentioning that the f–x deconvolution approach introduced here only applies a forward AR model to estimate the signal. A hybrid forward and backward version of AR model was proposed in Wang (1999). References Abma R , Claerbout J . , 1995 Lateral prediction for noise attenuation by t - x and f - x techniques , Geophysics , vol. 60 (pg. 1887 - 1896 ) 10.1190/1.1443920 Google Scholar Crossref Search ADS WorldCat Crossref Bekara M , van der Baan M . , 2007 Local singular value decomposition for signal enhancement of seismic data , Geophysics , vol. 72 (pg. V59 - V65 ) 10.1190/1.2435967 Google Scholar Crossref Search ADS WorldCat Crossref Bonar D , Sacchi M . , 2012 Denoising seismic data using the nonlocal means algorithm , Geophysics , vol. 77 (pg. A5 - A8 ) 10.1190/geo2011-0235.1 Google Scholar Crossref Search ADS WorldCat Crossref Cadzow J A . , 1988 Signal enhancement: a composite property mapping algorithm , IEEE Trans. Acoust. Speech Signal Process. , vol. 1 (pg. 49 - 62 ) 10.1109/29.1488 Google Scholar Crossref Search ADS WorldCat Crossref Canales L . , 1984 Random noise reduction 54th Annual Int. Meeting, SEG, Expanded Abstracts (pg. 525 - 527 ) Chen Y . , 2014 Deblending using a space-varying median filter , Explor. Geophys. doi: 10.1071/EG14051 OpenURL Placeholder Text WorldCat Chen Y , Fomel S . , 2015 Random noise attenuation using local signal-and-noise orthogonalization , Geophysics at press OpenURL Placeholder Text WorldCat Chen Y , Fomel S , Hu J . , 2014a Iterative deblending of simultaneous-source seismic data using seislet-domain shaping regularization , Geophysics , vol. 79 (pg. V179 - V189 ) 10.1190/geo2013-0449.1 Google Scholar Crossref Search ADS WorldCat Crossref Chen Y , Gan S , Liu T , Yuan J , Zhang Y , Jin Z . , 2015 Random noise attenuation by a selective hybrid approach using f–x empirical mode decomposition , J. Geophys. Eng. , vol. 12 (pg. 12 - 25 ) 10.1088/1742-2132/12/1/12 Google Scholar Crossref Search ADS WorldCat Crossref Chen Y , Ma J . , 2014 Random noise attenuation by f–x empirical mode decomposition predictive filtering , Geophysics , vol. 79 (pg. V81 - V91 ) 10.1190/geo2013-0080.1 Google Scholar Crossref Search ADS WorldCat Crossref Chen Y , Yuan J , Jin Z , Chen K , Zhang L . , 2014b Deblending using normal moveout and median filtering in common-midpoint gathers , J. Geophys. Eng. , vol. 11 045012 10.1088/1742-2132/11/4/045012 OpenURL Placeholder Text WorldCat Crossref Chen Y , Zhou C , Yuan J , Jin Z . , 2014c Application of empirical mode decomposition to random noise attenuation of seismic data , J. Seism. Explor. , vol. 23 (pg. 481 - 495 ) OpenURL Placeholder Text WorldCat Fomel S . , 2002 Application of plane-wave destruction filters , Geophysics , vol. 67 (pg. 1946 - 1960 ) 10.1190/1.1527095 Google Scholar Crossref Search ADS WorldCat Crossref Fomel S . , 2010 Predictive painting of 3d seismic volumes , Geophysics , vol. 75 (pg. A25 - A30 ) 10.1190/1.3453847 Google Scholar Crossref Search ADS WorldCat Crossref Fomel S . , 2013 Seismic data decomposition into spectral components using regularized nonstationary autoregression , Geophysics , vol. 78 (pg. O69 - O76 ) 10.1190/geo2013-0221.1 Google Scholar Crossref Search ADS WorldCat Crossref Fomel S , Liu Y . , 2010 Seislet transform and seislet frame , Geophysics , vol. 75 (pg. V25 - V38 ) 10.1190/1.3380591 Google Scholar Crossref Search ADS WorldCat Crossref Freire S L M , Ulrych T J . , 1988 Application of singular value decomposition to vertical seismic profiling , Geophysics , vol. 53 (pg. 778 - 785 ) 10.1190/1.1442513 Google Scholar Crossref Search ADS WorldCat Crossref Harris P , White R . , 1997 Improving the performance of f–x prediction at low signal-to-noise ratios , Geophys. Prospect. , vol. 45 (pg. 269 - 302 ) 10.1046/j.1365-2478.1997.00347.x Google Scholar Crossref Search ADS WorldCat Crossref Jones I F , Levy S . , 1987 Signal-to-noise ratio enhancement in multichannel seismic data via the Karhunen–Loeve transform , Geophys. Prospect. , vol. 35 (pg. 12 - 32 ) 10.1111/j.1365-2478.1987.tb00800.x Google Scholar Crossref Search ADS WorldCat Crossref Liu Y . , 2013 Noise reduction by vector median filtering , Geophysics , vol. 78 (pg. V79 - V87 ) 10.1190/geo2012-0232.1 Google Scholar Crossref Search ADS WorldCat Crossref Liu G , Chen X , Du J , Song J . , 2011 Seismic noise attenuation using nonstationary polynomial fitting , Appl. Geophys. , vol. 8 (pg. 18 - 26 ) 10.1007/s11770-010-0244-2 Google Scholar Crossref Search ADS WorldCat Crossref Liu Y , Fomel S , Liu G . , 2010 Nonlinear structure-enhancing filtering using plane-wave prediction , Geophys. Prospect. , vol. 58 (pg. 415 - 427 ) 10.1111/j.1365-2478.2009.00840.x Google Scholar Crossref Search ADS WorldCat Crossref Lomask J , Guitton A , Fomel S , Claerbout J , Valenciano A A . , 2006 Flattening without picking , Geophysics , vol. 71 (pg. P13 - P20 ) 10.1190/1.2210848 Google Scholar Crossref Search ADS WorldCat Crossref Milton P , Silva M G , Melo E M , Ursin B . , 2009 Ground-roll attenuation based on SVD filtering 79th Annual Int. Meeting, SEG, Expanded Abstracts (pg. 3381 - 3385 ) Neelamani R , Baumstein A , Gillard D , Hadidi M , Soroka W . , 2008 Coherent and random noise attenuation using the curvelet transform , Leading Edge , vol. 27 (pg. 240 - 248 ) 10.1190/1.2840373 Google Scholar Crossref Search ADS WorldCat Crossref Oropeza V , Sacchi M . , 2010 A randomized SVD for multichannel singular spectrum analysis (MSSA) noise attenuation 80th SEG Expanded Abstracts: 83rd Annual Int. Meeting (pg. 3539 - 3544 ) Oropeza V , Sacchi M . , 2011 Simultaneous seismic data denoising and reconstruction via multichannel singular spectrum analysis , Geophysics , vol. 76 (pg. V25 - V32 ) 10.1190/1.3552706 Google Scholar Crossref Search ADS WorldCat Crossref Tufts D , Kumaresan R . , 1982 Estimation of frequencies of multiple sinusoids: making linear prediction perform like maximum likelihood , Proc. IEEE , vol. 70 (pg. 975 - 989 ) 10.1109/PROC.1982.12428 Google Scholar Crossref Search ADS WorldCat Crossref Vautard R , Yiou P , Ghil M . , 1992 Singular-spectrum analysis: a toolkit for short, noisy chaotic signals , Phys. D: Nonlinear Phenom. , vol. 58 (pg. 95 - 126 ) 10.1016/0167-2789(92)90103-T Google Scholar Crossref Search ADS WorldCat Crossref Vrabie V D , Mars J L , Lacoume T L . , 2004 Modified SVD by means of ICA , Signal Process. , vol. 84 (pg. 645 - 652 ) 10.1016/j.sigpro.2003.12.007 Google Scholar Crossref Search ADS WorldCat Crossref Wang Y . , 1999 Random noise attenuation using forward–backward linear prediction , J. Seism. Explor. , vol. 8 (pg. 133 - 142 ) OpenURL Placeholder Text WorldCat Wang Y . , 2002 Seismic trace interpolation in the f–x–y domain , Geophysics , vol. 67 (pg. 1232 - 1239 ) 10.1190/1.1500385 Google Scholar Crossref Search ADS WorldCat Crossref Yang W , Wang R , Chen Y , Wu J , Qu S , Yuan J , Gan S . , 2015 Application of spectral decomposition using regularized non-stationary autoregression to random noise attenuation , J. Geophys. Eng. , vol. 12 (pg. 175 - 187 ) 10.1088/1742-2132/12/2/175 Google Scholar Crossref Search ADS WorldCat Crossref © 2015 Sinopec Geophysical Research Institute
TI - Structure-oriented singular value decomposition for random noise attenuation of seismic data
JF - Journal of Geophysics and Engineering
DO - 10.1088/1742-2132/12/2/262
DA - 2015-04-01
UR - https://www.deepdyve.com/lp/oxford-university-press/structure-oriented-singular-value-decomposition-for-random-noise-kFqD6Y0D45
SP - 262
VL - 12
IS - 2
DP - DeepDyve
ER -