TY - JOUR
AU - Chen,, Xiaohong
AB - Abstract As the developments of seismic exploration and subsequent seismic exploitation advance, marine acquisition systems with towed streamers become an important seismic data acquisition method. But the existing air–water reflective interface can generate surface related multiples, including ghosts, which can affect the accuracy and performance of the following seismic data processing algorithms. Thus, we derive a deghosting method from a new perspective, i.e. using the transmission matrix (T-matrix) method instead of inverse scattering series. The T-matrix-based deghosting algorithm includes all scattering effects and is convergent absolutely. Initially, the effectiveness of the proposed method is demonstrated using synthetic data obtained from a designed layered model, and its noise-resistant property is also illustrated using noisy synthetic data contaminated by random noise. Numerical examples on complicated data from the open SMAART Pluto model and field marine data further demonstrate the validity and flexibility of the proposed method. After deghosting, low frequency components are recovered reasonably and the fake high frequency components are attenuated, and the recovered low frequency components will be useful for the subsequent full waveform inversion. The proposed deghosting method is currently suitable for two-dimensional towed streamer cases with accurate constant depth information and its extension into variable-depth streamers in three-dimensional cases will be studied in the future. deghosting, transmission matrix, inverse scattering series Introduction As the development of seismic exploration and subsequent seismic exploitation advance, marine acquisition systems with towed streamers are becoming increasingly popular. Because of the existing reflection at the sea surface, observed seismic data is contaminated by source and receiver ghosts. A source ghost can be characterized by an event which starts its propagation upward from the source. A receiver ghost can be expressed by another event ending its propagation by further moving downward at the receiver. Both the source and receiver ghosts can reduce the useful frequency bandwidth, especially decreasing useful low frequency energy and seismic resolution (Amundsen and Zhou 2013). But the low frequency components play an important role in full waveform inversion (FWI), and the inversion results can be far from a true model and lead to unreliable, even wrong interpretations if no low frequency components are available (Shin and Cha 2008, Shin and Cha 2009, Guasch and Warner 2014, Warner and Guasch 2014, Wu et al2014, Luo and Wu 2015). Therefore, deghosting is an important procedure for the subsequent seismic data processing algorithms. Many deghosting methods exist that can be implemented in different domains, such as in the frequency–wavenumber domain (Weglein et al2003, Wang et al2013, Liu and Lu 2016), τ-p domain (Grion et al2015), frequency–space domain (Verschuur et al1992), time-space domain (Amundsen et al2013, Robertsson and Amundsen 2014, Lu et al2017), etc. However, due to the fact that the deghosting methods used in conventional marine acquisition systems lack accuracy and flexibility, non-conventional marine acquisition methods have been developed, such as putting the receivers at two different depths (Posthumus 1993), varying the depth of the streamers along their length (Soubaras and Lafet 2013) and a dual-sensor streamer strategy (Day et al2013). Variable-depth streamer acquisition proposed by Soubaras and Lafet (Soubaras and Lafet 2013) can guarantee notch diversity which can be used for deghosting with higher accuracy and flexibility compared with conventional marine acquisition system with towed streamers at a constant depth. Generally, there exist two ghost parameters (sea surface reflection coefficient and time-shift between primary and its ghost) which should be estimated as a prerequisite when using a ghost-generation model. The estimation accuracy can affect deghosting performance. Under the assumption that the seismic signal has super-Gaussian distribution and the existing ghosts can decrease the super-Gaussian property of the observed seismic data, the sea surface reflection coefficient and time-shift between the primary and its ghost can be obtained based on the maximization of a non-Gaussian property (Grion et al2015, Liu and Lu 2016, Lu et al2017). With these two estimated parameters, the deghosting algorithm in the frequency-wavenumber domain can be derived using the inverse scattering series (ISS) method (Weglein et al2003, Wang et al2013). The ISS is an infinite series and it is hard to understand its behavior. Therefore, the derivation of the deghosting algorithm is illustrated from a new perspective, i.e. using the transmission matrix (T-matrix) strategy, which has been widely used in quantum physics and introduced into geophysics exploration (Jakobsen et al2003, Jakobsen 2012, Jakobsen and Ursin 2012, Jakobsen and Ursin 2015, Wang et al2016a, 2017, Wu et al2015a, 2015b). In this paper, the T-matrix method is introduced (Wu et al2015b, Wang et al2016a, 2016b) to achieve a deghosting algorithm at a constant depth, and further strategies for varying depth cases is left for future research. First, the validity of the proposed method is demonstrated using a layered synthetic dataset, and the noise-resistant property is also demonstrated. Then, a dataset from the open SMAART Pluto model and a field marine dataset are used to further demonstrate the validity and flexibility of the proposed T-matrix-based deghosting method. Theory The acoustic equation in the frequency domain can be characterized by the following equation (Weglein et al2003): LG=-δ(r-rs),1 where L=ω2K+∇1ρ∇ is the forward modeling operator, G denotes the Green’s operator according to the Dirac delta source function δ. K and ρ represent the true bulk modulus and density, respectively; r and rs are an arbitrary point and the source location, respectively. Given the reference bulk modulus and density K0 and ρ0, the background Green’s function can be obtained by equation (2), L0G0=-δ(r-rs),2 where L0=ω2K0+∇1ρ0∇ and G0 denotes the background Green’s function. Then, the solution of equation (1) can be characterized by the Lippmann–Schwinger equation, which is a fundamental equation of the scattering theory, as shown in equation (3), Ψs=G-G0=G0VG,3 where V=ω21K-1K0+∇1ρ-1ρ0∇ is the scattering potential, and Ψs is the scattered wavefield which is not a Green operator itself. Equation (3) can be characterized by an infinite series using a substitution operation for G, as shown in equation (4): Ψs=G-G0=G0VG=G0VG0+G0VG0VG0+...4 Equation (4) is the well-known Born series, but it fails to guarantee the convergence in the strong perturbation or large perturbation area cases. Introducing the T-matrix, which includes all orders of the scattering effects (Wu et al2015b, Wang et al2016a), can improve the convergence. Denoting TG0=VG, equation (4) can be reformulated as follows: Ψs=G-G0=G0VG=G0TG0.5 In equation (5), all the scattering effects in the Green operator G are transformed into T-matrix T, then T-matrix T contains all the scattering effects. Equation (5) can characterize different types of wavefields, such as pure primary waves, primary waves plus ghosts (source ghost, receiver ghost and source-receiver ghost), primary waves plus internal multiples and primary waves plus free-surface multiples. In this paper, we only focus on the primary waves and the corresponding ghosts and leave the internal multiples and free-surface multiples for future research. When the seismic data is acquired in the marine acquisition system, the ghosts mixed in the seismic data can decrease the resolution of the observed seismic data and generate frequency notch effects which depend on the receiver depth. Both of them can have significant effects on the subsequent seismic data processing algorithms. The marine configuration is shown in figure 1, in which we can notice why the source ghost is generated when firing. According to the reciprocity of the sources and receivers, the receiver ghost can also be easily understood. Actually, there are three types of ghosts, including a source ghost, receiver ghost and source-receiver ghost; the above-mentioned three types of ghosts are illustrated in figure 2. Figures 2(a)–(c) denote the source ghost, receiver ghost and source-receiver ghost, respectively. The above statements relate to the qualitative interpretation of different types of ghosts. Next, we discuss the related ghosts quantitatively. Figure 1. Open in new tabDownload slide The marine configuration and reference Green function. Figure 1. Open in new tabDownload slide The marine configuration and reference Green function. Figure 2. Open in new tabDownload slide Illustration of different ghost events: (a) source ghost, (b) receiver ghost and (c) source–receiver ghost. Figure 2. Open in new tabDownload slide Illustration of different ghost events: (a) source ghost, (b) receiver ghost and (c) source–receiver ghost. When seismic data is acquired using a marine configuration with a reflective free surface, the background Green operator includes two parts: G0fs and G0d, subject to G0=G0d+G0fs,6 where G0d denotes the Green operator from the source direct to the receiver, causal, and G0fs is an additional part of the background Green operator G0 because of the existing free surface (see figure 1). In the absence of a free surface, G0d denotes the reference Green operator, and the forward modeling formula (i.e. equation (5)) shows that the observed data is constructed from the direct propagating Green operator G0d and T-matrix T which includes all scattering effects. When a free surface is present, the forward modeling equation is constructed from G0=G0d+G0fs and the same T-matrix, T. Hence, G0fs is the sole difference between the forward modeling with and without the free surface. Therefore, G0fs should be responsible for generating the ghost events in the free surface cases (see figure 2). Substituting equation (6) into equation (5) can achieve equation (7) which includes ghost effects Ψs=G0TG0=(G0d+G0fs)T(G0d+G0fs)=G0dTG0d+G0fsTG0d+G0dTG0fs+G0fsTG0fs,7 where T denotes the T-matrix, including all the scattering effects, the first term is the primary waves, the second term indicates the receiver ghost, the third term denotes the source ghost and the fourth term represents the source-receiver ghost. The three types of ghosts are shown in figure 2, which provides a simplified interpretation of the ghosts due to the presence of a free surface. In order to attenuate the ghost effects, the deghosted seismic data can be obtained using equation (8): Ψs′=G0dTG0d=G0dG0-1ΨsG0-1G0d.8 From equation (8), we can also obtain the ghosts contaminated seismic data Ψs from the ghost-free seismic data Ψs′, Ψs=G0(G0d)-1Ψs′(G0d)-1G0.9 The deghosting algorithm proposed in this paper can be easily understood compared with the ISS method (Weglein et al2003, Wang et al2013). It is also convergent absolutely because all the scattering effects contained in the T-matrix remain unchanged. The only information that is changed is the Green operator, propagating from the source to the scattering points or propagating from the scattering points to the receiver. The detailed derivation of the deghosting algorithm using the T-matrix strategy is provided as follows. Assuming the air–water interface z=0 is the free surface, the sea water is regarded as the reference medium, (xs,zs), (xg,zg) represent the source and receiver locations, respectively, and x=(x,z) is an arbitrary scattering point in the scattering area. Then, the reference Green operator is subject to (Weglein et al2003) L0G0=∇2ρ0+ω2κ0G0(x,z,xs,zs;ω)=-δ(x-xs)[δ(z-zs)-δ(z+zs)],10 where ρ0 is the reference density and κ0 denotes the reference bulk modulus. Performing a forward Fourier transform towards x in equation (10), we can then obtain equation (11) in the frequency–wavenumber domain 1ρ0∂2∂z2+q2ρ0G0(kx,z,xs,zs;ω)=-12πe-ikxxs[δ(z-zs)-δ(z+zs)].11 The solution to equation (11) can be characterized using equation (12), G0(kx,z,xs,zs;ω)=ρ02πe-ikxxs-2iq(eiq∣z-zs∣-eiq∣z+zs∣),12 where q denotes the vertical wavenumber, subject to q=sgn(ω)(ω/c0)2-kx2, and c0=κ0/ρ0 denotes the acoustic velocity of sea water. Besides, the Green operator from the source direct to each scattering point, G0d, is subject to G0d(kx,z,xs,zs;ω)=ρ02πe-ikxxs-2iqeiq∣z-zs∣.13 In order to give a more detailed derivation, the detailed analysis using integral notation instead of matrix notation for equation (7) is given as Ψs(xr,zr,xs,zs;ω)=∬x,x′G0(xr,zr,x;ω)T(x,x′)×G0(x′,xs,zs;ω)dxdx′.14 Performing the forward Fourier transform toward xr,xs in equation (14) we obtain Ψs(kr,zr,ks,zs;ω)=∬x,x′G0(kr,zr,x;ω)T(x,x′)G0(x′,ks,zs;ω)dxdx′.15 Substituting equations (8), (12) and (13) into equation (15), we can obtain the deghosted data Ψs′(kr,zr,ks,zs;ω)=G0dG0-1Ψs(kr,zr,ks,zs;ω)G0-1G0d=Ψs(kr,zr,ks,zs;ω)(1-e2iqgzg)(1-e2iqszs),16 where 1Q=1(1-e2iqgzg)(1-e2iqszs) can be regarded as a deghosting operator. In a similar way, ghosts contaminated seismic data in the presence of a free surface can be simulated using the ghost-free seismic data and the ghost operator Q=(1-e2iqgzg)(1-e2iqszs), which is the basic foundation for deghosting problems. Ψs(kr,zr,ks,zs;ω)=Ψs′(kr,zr,ks,zs;ω)(1-e2iqgzg)×(1-e2iqszs),17 where qg=sgn(ω)(ω/c0)2-kg2, qs=sgn(ω)(ω/c0)2-ks2 are the receiver and source sides vertical wavenumbers, respectively, corresponding to the horizontal wavenumbers kg, ks. During the deghosting procedure, instability appears when the absolute value of Q is minor enough. Improper treatment of Q values can introduce linear interference with apparent velocity c0. Therefore, the regularization method can be used in equation (16), and then the damped least square solution can be obtained, 1Q=Q*∣Q∣2+εmax(∣Q∣),18 where (•)* is the conjugate operator and ε denotes a regularization factor to stabilize the division. In fact, the deghosting algorithm (equation (16)) can be regarded as a two-step method. Firstly, the receiver side deghosting in the common shot gather; Then, the source side deghosting in the common receiver gather. The above two procedures can help us obtain the deghosted data in the frequency-wavenumber domain. Finally, the deghosted seismic data in the time-space domain can be achieved using the inverse two-dimensional (2D) Fourier transform. Next, a few numerical examples will be used to demonstrate the validity of the proposed method. Numerical examples Firstly, a synthetic dataset simulating from a designed three-layer model is used to demonstrate the validity and effectiveness of the proposed deghosting method, and the deghosted data is consistent with the original ghost-free data which proves the effectiveness of the proposed method, during which the noise-resistant property of the proposed method is also illustrated. Then, a dataset from the open SMARRT Pluto model and a field marine dataset are used to further prove the effectiveness and flexibility of the proposed method. Layered model example Figure 3(a) shows the synthetic dataset simulated from the designed three-layer model, without ghost effects. It includes 101 traces with 10 m as the trace interval and 1001 samples per trace with 2 ms as the time sampling interval. Figure 3(b) shows the seismic data contaminated by the receiver side ghost effects with 6 m as the receiver depth and with 1500 m s-1 as sea water velocity (reference velocity). Figure 3(b) notes that the receiver ghost with reverse polarity is mixed up with the primary waves which decreases the resolution of the observed seismic data. Using the proposed method, we can obtain the deghosted data, as shown in figure 3(c) which is consistent with the reference data (figure 3(a)). In order to see the details, the residual is calculated, as shown in figure 3(d), which is almost zero everywhere. The normalized local frequency spectra are also compared among the three datasets (the ghost-free data, the data contaminated by ghosts and the deghosted data), as shown in figure 4, in which the plus, circle and triangle curves represent the spectra of the data contaminated by ghosts, the deghosted data and the reference data, respectively. The spectrum of the deghosted data is consistent with the reference one, and has enhanced low frequency components compared with the data before deghosting. The above test demonstrates the validity of the proposed T-matrix-based deghosting method. Figure 3. Open in new tabDownload slide Synthetic data test. (a) Reference data; (b) data contaminated by ghosts; (c) deghosted data; (d) residual. Figure 3. Open in new tabDownload slide Synthetic data test. (a) Reference data; (b) data contaminated by ghosts; (c) deghosted data; (d) residual. Figure 4. Open in new tabDownload slide Frequency spectra comparisons before and after deghosting. Triangle: reference data; circle: deghosted data; plus: data before deghosting. Figure 4. Open in new tabDownload slide Frequency spectra comparisons before and after deghosting. Triangle: reference data; circle: deghosted data; plus: data before deghosting. In order to demonstrate the noise-resistant property of the proposed deghosting method, the noisy data contaminated by some random noise is used, as shown in figure 5(b), in which the maximum amplitude of the random noise to that of the signal is 0.2. Using the proposed method, we can obtain the deghosted data, as shown in figure 5(c) which is reasonably correct, and proves that the proposed method has the noise-resistant property. The residual shown in figure 5(d) does not contain the coherent events except for some random noise. The noisy data test demonstrates its noise-resistant property. It should be noted that deghosting performance decreases as the noise level increases because the regularization factor ε should be a bigger value in these cases. Figure 5. Open in new tabDownload slide Noisy synthetic data test. (a) Reference data; (b) noisy data contaminated by ghosts; (c) deghosted data; (d) residual between (c) and (a). Figure 5. Open in new tabDownload slide Noisy synthetic data test. (a) Reference data; (b) noisy data contaminated by ghosts; (c) deghosted data; (d) residual between (c) and (a). The above examples are based on the layered model, and in order to further demonstrate the effectiveness of the proposed deghosting method, a dataset from the open SMAART Pluto model is used which contains interference and weak reflections. SMAART Pluto model example This complicated synthetic data is from the open SMAART Pluto model, obtained using finite difference method. The distance between adjacent shots and adjacent receivers are both 22.86 m, and the acquisition system is single-side. There are 361 traces for each shot and 1126 sampling points per trace with 8 ms as the time sampling interval. The depth of the receivers and sources are both 7.62 m, which makes the obtained seismic data contaminated by different kinds of ghosts. In this paper, we mainly concentrate on receiver side ghost elimination, but it can handle other kinds of ghosts. Figure 6(a) shows the scattering data after the elimination of the direct wave. Because of the existing reflecting surface, the ghosts are mixed up with the primary waves and the increasing number of the sidelobes decreases the resolution of the observed data. Besides, the ghosts can change the frequency components distribution of the primary waves, which will be illustrated later. Based on the proposed deghosting method, we can obtain the deghosted data, as shown in figure 6(b). In order to see the details, two local areas marked by rectangles in figure 6 are zoomed in, as shown in figures 7–8. Figures 7(a) and (b) denote the zoomed-in version of the big rectangle in figures 6(a) and (b), respectively. Figure 7 notes that after deghosting, the sidelobes generated by the ghosts are effectively attenuated and the ghost events are also effectively attenuated, especially the locations marked by the arrows, which proves the validity of the proposed method. Figures 8(a) and (b) represent the zoomed-in version of the small rectangle in figures 6(a) and (b), respectively. Figure 8 shows that the ghost events are effectively attenuated, especially where indicated by the arrows. In order to compare the variation of frequency components, the normalized local frequency spectra of seismic data before and after deghosting are shown in figure 9, in which the solid and dashed curves represent the spectra of seismic data before and after deghosting, respectively. Similar to the layered model, after deghosting, the low frequency components are recovered reasonably and will be useful for the subsequent seismic data processing. The dataset from the open SMAART Pluto model demonstrates the validity of the proposed method. Finally, a field marine dataset is used to demonstrate the validity and flexibility of the proposed T-matrix-based deghosting method. Figure 6. Open in new tabDownload slide SMAART Pluto data. (a) Before deghosting; and (b) deghosted data. Figure 6. Open in new tabDownload slide SMAART Pluto data. (a) Before deghosting; and (b) deghosted data. Figure 7. Open in new tabDownload slide Detailed comparisons of the larger rectangles in figure 6. (a) Before deghosting; (b) deghosted data. The arrows are placed in the same position in (a) and (b) to better illustrate the differences. Figure 7. Open in new tabDownload slide Detailed comparisons of the larger rectangles in figure 6. (a) Before deghosting; (b) deghosted data. The arrows are placed in the same position in (a) and (b) to better illustrate the differences. Figure 8. Open in new tabDownload slide Detailed comparisons of the smaller rectangles in figure 6. (a) Before deghosting; (b) deghosted data. The arrows are placed in the same position in (a) and (b) to better illustrate differences. Figure 8. Open in new tabDownload slide Detailed comparisons of the smaller rectangles in figure 6. (a) Before deghosting; (b) deghosted data. The arrows are placed in the same position in (a) and (b) to better illustrate differences. Figure 9. Open in new tabDownload slide Frequency spectra comparisons before and after deghosting. Dashed curve: deghosted data; solid curve: seismic data before deghosting. Figure 9. Open in new tabDownload slide Frequency spectra comparisons before and after deghosting. Dashed curve: deghosted data; solid curve: seismic data before deghosting. Field marine data example The field marine data processing procedure is similar as the SMAART Pluto model. The field marine data is obtained using a conventional acquisition system with towed streamers. There are 384 traces with 12.5 m as the trace interval in each towed streamer and 2501 samples per trace with 2 ms as the time sampling interval. Because of the feathering and fluid flowing effects, the depth of receivers is varying. In order to estimate the accurate depth of the receivers for each shot, Liu and Lu (2016) studied the receiver depth and sea surface reflectivity estimation issue based on non-Gaussian maximization. The optimally estimated receiver depth is 9.8 m and the estimated sea surface reflectivity is -0.88 for the selected shot, as shown in figure 10(a). Figure 10(a) indicates the scattering data after direct wave elimination. Using the proposed T-matrix-based deghosting method, the deghosted data can be obtained, as shown in figure 10(b). For detailed comparisons, two local areas marked by the rectangles in figure 10 are zoomed in, as shown in figures 11–12. Figures 11(a) and (b) indicate the zoomed in versions of the big rectangle in figures 10(a) and (b), respectively. Figure 11 shows that after deghosting, the ghost events are effectively attenuated, especially the locations marked by the arrows. This demonstrates the validity of the proposed method. Figures 12(a) and (b) represent the zoomed-in version of the small rectangles in figures 10(a) and (b), respectively. Similar to figure 11, figure 12 indicates that the ghost events are also effectively attenuated, especially at the places marked by arrows. For the comparisons of local frequency components variation, the normalized frequency spectra before and after deghosting are shown in figure 13, in which the solid and dashed curves represent the spectra of the seismic data before and after deghosting, respectively. After deghosting, the low frequency components are recovered reasonably and can improve the accuracy of the subsequent full waveform inversion. The field marine dataset example demonstrates the validity and flexibility of the proposed T-matrix-based deghosting method. Figure 10. Open in new tabDownload slide Field marine data. (a) Before deghosting; and (b) deghosted data. Figure 10. Open in new tabDownload slide Field marine data. (a) Before deghosting; and (b) deghosted data. Figure 11. Open in new tabDownload slide Detailed comparisons of the larger rectangles in figure 10. (a) Before deghosting; (b) deghosted data. The arrows are placed in the same position in (a) and (b) to better illustrate the differences. Figure 11. Open in new tabDownload slide Detailed comparisons of the larger rectangles in figure 10. (a) Before deghosting; (b) deghosted data. The arrows are placed in the same position in (a) and (b) to better illustrate the differences. Figure 12. Open in new tabDownload slide Detailed comparisons of the smaller rectangles in figure 10. (a) Before deghosting; (b) deghosted data. The arrows are placed in the same position in (a) and (b) to better illustrate the differences. Figure 12. Open in new tabDownload slide Detailed comparisons of the smaller rectangles in figure 10. (a) Before deghosting; (b) deghosted data. The arrows are placed in the same position in (a) and (b) to better illustrate the differences. Figure 13. Open in new tabDownload slide Frequency spectra comparisons before and after deghosting. Dashed curve: deghosted data; solid curve: seismic data before deghosting. Figure 13. Open in new tabDownload slide Frequency spectra comparisons before and after deghosting. Dashed curve: deghosted data; solid curve: seismic data before deghosting. Conclusions Ghosts from the marine acquisition system have significant effects on the subsequent seismic data processing algorithms. In this paper, we derived the deghosting algorithm using the transmission matrix (T-matrix) strategy which is easily understandable and absolutely convergent. The deghosting performance on the layered model demonstrates the validity of the proposed deghosting method, and its noise-resistant property is also demonstrated using the noisy data contaminated by random noise. The numerical examples of the complicated dataset from the open SMAART Pluto model and the field marine dataset further prove the effectiveness and flexibility of the proposed deghosting method. After deghosting, the low frequency components are recovered reasonably and the fake high frequency is attenuated which is helpful in the subsequent seismic data processing methods, like full waveform inversion. However, the proposed method requires exact depth information on receivers or sources, and the performance will be affected if the depth information is inaccurate. Therefore, further research should be done to study the quantitative effects of inaccurate depth information on the deghosting performance. Besides, the proposed method was used in 2D constant depth streamer cases and the method can be extended into varying depth streamer cases or three-dimensional cases, which will be studied in the future. Acknowledgments This research is financially supported by the National Natural Science Foundation of China (Grant Nos. 41604096, 41674116), the China Postdoctoral Science Foundation (Grant No. 2016M590102), the Key State Science and Technology Project (Grant Nos. 2016ZX05024001-004, 2016ZX05024001-005) and the WTOPI Research Consortium at the University of California, Santa Cruz. References Amundsen L , Zhou H . , 2013 Low-frequency seismic deghosting , Geophysics , vol. 78 (pg. WA15 - 20 ) 10.1190/geo2012-0276.1 Google Scholar Crossref Search ADS WorldCat Crossref Amundsen L , Zhou H , Reitan A , Weglein A B . , 2013 On seismic deghosting by spatial deconvolution , Geophysics , vol. 78 (pg. V267 - V271 ) 10.1190/geo2013-0198.1 Google Scholar Crossref Search ADS WorldCat Crossref Day A , Klüver T , Söllner W , Tabti H , Carlson D . , 2013 Wavefield-separation methods for dual-sensor towed-streamer data , Geophysics , vol. 78 (pg. WA55 - 70 ) 10.1190/geo2012-0302.1 Google Scholar Crossref Search ADS WorldCat Crossref Grion S , Telling R , Barnes J . , 2015 De-ghosting by kurtosis maximisation in practice , SEG Technical Program Expanded Abstracts Tulsa, OK Society of Exploration Geophysicists (pg. 4605 - 4609 ) Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Guasch L , Warner M . , 2014 Adaptive waveform inversion-FWI without cycle skipping-applications 76th EAGE Meeting, Expanded Abstracts Amsterdam, Netherlands E106 14 Jakobsen M . , 2012 T-matrix approach to seismic forward modelling in the acoustic approximation , Stud. Geophys. Geod. , vol. 56 (pg. 1 - 20 ) 10.1007/s11200-010-9081-2 Google Scholar Crossref Search ADS WorldCat Crossref Jakobsen M , Hudson J A , Johansen T A . , 2003 T-matrix approach to shale acoustics , Geophys. J. Int. , vol. 154 (pg. 533 - 558 ) 10.1046/j.1365-246X.2003.01977.x Google Scholar Crossref Search ADS WorldCat Crossref Jakobsen M , Ursin B . , 2012 Nonlinear seismic waveform inversion using a Born iterative T-matrix method , SEG Technical Program Expanded Abstracts Tulsa, OK Society of Exploration Geophysicists (pg. 1 - 5 ) Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Jakobsen M , Ursin B . , 2015 Full waveform inversion in the frequency domain using direct iterative T-matrix methods , J. Geophys. Eng. , vol. 12 (pg. 400 - 418 ) 10.1088/1742-2132/12/3/400 Google Scholar Crossref Search ADS WorldCat Crossref Liu L , Lu W . , 2016 Adaptive F-K deghosting method based on non-Gaussianity , J. Appl. Geophys. , vol. 127 (pg. 14 - 22 ) 10.1016/j.jappgeo.2016.02.004 Google Scholar Crossref Search ADS WorldCat Crossref Lu W , Xu Z , Fang Z , Wang R , Yan C . , 2017 Receiver deghosting in the T-X domain based on super-Gaussianity , J. Appl. Geophys. , vol. 136 (pg. 134 - 141 ) 10.1016/j.jappgeo.2016.11.004 Google Scholar Crossref Search ADS WorldCat Crossref Luo J , Wu R . , 2015 Seismic envelop inversion: reduction of local minima and noise resistance , Geophys. Prospect. , vol. 63 (pg. 597 - 614 ) 10.1111/1365-2478.12208 Google Scholar Crossref Search ADS WorldCat Crossref Posthumus B J . , 1993 Deghosting using a twin streamer configuration , Geophys. Prospect. , vol. 41 (pg. 267 - 286 ) 10.1111/j.1365-2478.1993.tb00570.x Google Scholar Crossref Search ADS WorldCat Crossref Robertsson J O A , Amundsen L . , 2014 Wave equation processing using finite-difference propagators: II. Deghosting of marine hydrophone seismic data , Geophysics , vol. 79 (pg. T301 - T312 ) 10.1190/geo2014-0152.1 Google Scholar Crossref Search ADS WorldCat Crossref Shin C , Cha Y H . , 2008 Waveform inversion in the Laplace domain , Geophys. J. Int. , vol. 173 (pg. 922 - 931 ) 10.1111/j.1365-246X.2008.03768.x Google Scholar Crossref Search ADS WorldCat Crossref Shin C , Cha Y H . , 2009 Waveform inversion in the Laplace-Fourier domain , Geophys. J. Int. , vol. 177 (pg. 1067 - 1079 ) 10.1111/j.1365-246X.2009.04102.x Google Scholar Crossref Search ADS WorldCat Crossref Soubaras R , Lafet Y . , 2013 Variable-depth streamer acquisition: broadband data for imaging and inversion , Geophysics , vol. 78 (pg. WA27 - 39 ) 10.1190/geo2012-0297.1 Google Scholar Crossref Search ADS WorldCat Crossref Verschuur D J , Berkhout A J , Wapenaar C P A . , 1992 Adaptive surface-related multiple elimination , Geophysics , vol. 57 (pg. 1166 - 1177 ) 10.1190/1.1443330 Google Scholar Crossref Search ADS WorldCat Crossref Wang B , Jakobsen M , Wu R-S , Lu W , Chen X . , 2017 Accurate and efficient velocity estimation using Transmission matrix formalism based on the domain decomposition method , Inverse Prob. , vol. 33 035002 10.1088/1361-6420/aa5998 OpenURL Placeholder Text WorldCat Crossref Wang B , Wu R , Chen X . , 2016a Non-linear parameter estimation method based on T-matrix , Chin. J. Geophys. , vol. 59 (pg. 2257 - 2265 ) OpenURL Placeholder Text WorldCat Wang B , Wu R-S , Chen X , Lu W , Liu G . , 2016b Nonlinear deghosting based on the T-matrix method 78th EAGE Conf. & Exhibition We P1 01 Wang F , Li J , Chen X . , 2013 Deghosting method based on inverse scattering series , Chin. J. Geophys. , vol. 56 (pg. 1628 - 1636 ) OpenURL Placeholder Text WorldCat Warner M , Guasch L . , 2014 Adaptive waveform inversion: theory 84th Annual Int. Meeting, SEG Expanded Abstracts (pg. 1089 - 1093 ) Weglein A B , Araújo F V , Carvalho P M , Stolt R H , Matson K H , Coates R T , Corrigan D , Foster D J , Shaw S A , Zhang H . , 2003 Inverse scattering series and seismic exploration , Inverse Prob. , vol. 19 (pg. R27 - 83 ) 10.1088/0266-5611/19/6/R01 Google Scholar Crossref Search ADS WorldCat Crossref Wu R-S , Luo J , Wu B . , 2014 Seismic envelope inversion and modulation signal model , Geophysics , vol. 79 (pg. WA13 - 24 ) 10.1190/geo2013-0294.1 Google Scholar Crossref Search ADS WorldCat Crossref Wu R-S , Wang B , Jakobsen M . , 2015a Green’s function and T-matrix reconstruction using surface data for direct nonlinear inversion , SEG Technical Program Expanded Abstracts Tulsa, OK Society of Exploration Geophysicists (pg. 1286 - 1291 ) Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Wu R-S , Wang B , Hu C . , 2015b Renormalized nonlinear sensitivity kernel and inverse thin-slab propagator in T-matrix formalism for wave-equation tomography , Inverse Prob. , vol. 31 115004 10.1088/0266-5611/31/11/115004 OpenURL Placeholder Text WorldCat Crossref © 2017 Sinopec Geophysical Research Institute
TI - Deghosting based on the transmission matrix method
JO - Journal of Geophysics and Engineering
DO - 10.1088/1742-2140/aa82da
DA - 2017-12-01
UR - https://www.deepdyve.com/lp/oxford-university-press/deghosting-based-on-the-transmission-matrix-method-jJqGi0cCc1
SP - 1572
VL - 14
IS - 6
DP - DeepDyve
ER -