Journal of Geophysics and Engineering

PAPER

Deghosting based on the transmission matrixmethodTo cite this article: Benfeng Wang et al 2017 J. Geophys. Eng. 14 1572

 

View the article online for updates and enhancements.

Related contentComparative study of receiver-side ghostwavefield attenuation on different marineacquisition configurationsHao Zhang, Qiancheng Liu, Hongyuan Liet al.

-

Accurate and efficient velocity estimationusing Transmission matrix formalismbased on the domain decompositionmethodBenfeng Wang, Morten Jakobsen, Ru-Shan Wu et al.

-

Extending the inverse scattering seriesfree-surface-multiple-elimination algorithmby accommodating the source property ondata with interfering or proximal seismiceventsLei Zhao, Jinlong Yang and Arthur BWeglein

-

This content was downloaded from IP address 128.114.69.29 on 24/08/2018 at 01:54

Deghosting based on the transmissionmatrix method

Benfeng Wang1,2 , Ru-Shan Wu3 and Xiaohong Chen2,4

1 State Key Laboratory of Intelligent Technology and Systems, Department of Automation, TsinghuaUniversity, Beijing, 100084, People’s Republic of China2 State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Beijing,102249, People’s Republic of China3Department of Earth and Planetary Sciences, University of California, Santa Cruz, CA 95064, UnitedStates of America

E-mail: wbf1232007@126.com and chenxh@cup.edu.cn

Received 6 October 2016, revised 12 July 2017Accepted for publication 28 July 2017Published 29 November 2017

AbstractAs the developments of seismic exploration and subsequent seismic exploitation advance, marineacquisition systems with towed streamers become an important seismic data acquisition method.But the existing air–water reflective interface can generate surface related multiples, includingghosts, which can affect the accuracy and performance of the following seismic data processingalgorithms. Thus, we derive a deghosting method from a new perspective, i.e. using thetransmission matrix (T-matrix) method instead of inverse scattering series. The T-matrix-baseddeghosting algorithm includes all scattering effects and is convergent absolutely. Initially, theeffectiveness of the proposed method is demonstrated using synthetic data obtained from adesigned layered model, and its noise-resistant property is also illustrated using noisy syntheticdata contaminated by random noise. Numerical examples on complicated data from the openSMAART Pluto model and field marine data further demonstrate the validity and flexibility of theproposed method. After deghosting, low frequency components are recovered reasonably and thefake high frequency components are attenuated, and the recovered low frequency components willbe useful for the subsequent full waveform inversion. The proposed deghosting method is currentlysuitable for two-dimensional towed streamer cases with accurate constant depth information and itsextension into variable-depth streamers in three-dimensional cases will be studied in the future.

Keywords: deghosting, transmission matrix, inverse scattering series

(Some figures may appear in colour only in the online journal)

Introduction

As the development of seismic exploration and subsequentseismic exploitation advance, marine acquisition systems withtowed streamers are becoming increasingly popular. Becauseof the existing reflection at the sea surface, observed seismicdata is contaminated by source and receiver ghosts. A sourceghost can be characterized by an event which starts its pro-pagation upward from the source. A receiver ghost can beexpressed by another event ending its propagation by furthermoving downward at the receiver. Both the source andreceiver ghosts can reduce the useful frequency bandwidth,

especially decreasing useful low frequency energy and seis-mic resolution (Amundsen and Zhou 2013). But the lowfrequency components play an important role in full wave-form inversion (FWI), and the inversion results can be farfrom a true model and lead to unreliable, even wrong inter-pretations if no low frequency components are available (Shinand Cha 2008, Shin and Cha 2009, Guasch and Warner 2014,Warner and Guasch 2014, Wu et al 2014, Luo and Wu 2015).Therefore, deghosting is an important procedure for thesubsequent seismic data processing algorithms.

Many deghosting methods exist that can be implementedin different domains, such as in the frequency–wavenumberdomain (Weglein et al 2003, Wang et al 2013, Liu and

Journal of Geophysics and Engineering

J. Geophys. Eng. 14 (2017) 1572–1581 (10pp) https://doi.org/10.1088/1742-2140/aa82da

4 Author to whom any correspondence should be addressed.

1742-2132/17/0601572+10$33.00 © 2017 Sinopec Geophysical Research Institute Printed in the UK1572

Lu 2016), t - p domain (Grion et al 2015), frequency–spacedomain (Verschuur et al 1992), time-space domain(Amundsen et al 2013, Robertsson and Amundsen 2014, Luet al 2017), etc. However, due to the fact that the deghostingmethods used in conventional marine acquisition systems lackaccuracy and flexibility, non-conventional marine acquisitionmethods have been developed, such as putting the receivers attwo different depths (Posthumus 1993), varying the depth ofthe streamers along their length (Soubaras and Lafet 2013)and a dual-sensor streamer strategy (Day et al 2013). Vari-able-depth streamer acquisition proposed by Soubaras andLafet (Soubaras and Lafet 2013) can guarantee notch diver-sity which can be used for deghosting with higher accuracyand flexibility compared with conventional marine acquisitionsystem with towed streamers at a constant depth.

Generally, there exist two ghost parameters (sea surfacereflection coefficient and time-shift between primary and itsghost) which should be estimated as a prerequisite whenusing a ghost-generation model. The estimation accuracy canaffect deghosting performance. Under the assumption that theseismic signal has super-Gaussian distribution and the exist-ing ghosts can decrease the super-Gaussian property of theobserved seismic data, the sea surface reflection coefficientand time-shift between the primary and its ghost can beobtained based on the maximization of a non-Gaussianproperty (Grion et al 2015, Liu and Lu 2016, Lu et al 2017).With these two estimated parameters, the deghosting algo-rithm in the frequency-wavenumber domain can be derivedusing the inverse scattering series (ISS) method (Wegleinet al 2003, Wang et al 2013). The ISS is an infinite series andit is hard to understand its behavior. Therefore, the derivationof the deghosting algorithm is illustrated from a new per-spective, i.e. using the transmission matrix (T-matrix) strat-egy, which has been widely used in quantum physics andintroduced into geophysics exploration (Jakobsen et al 2003,Jakobsen 2012, Jakobsen and Ursin 2012, Jakobsen andUrsin 2015, Wang et al 2016a, 2017, Wu et al 2015a, 2015b).

In this paper, the T-matrix method is introduced (Wuet al 2015b, Wang et al 2016a, 2016b) to achieve adeghosting algorithm at a constant depth, and further strate-gies for varying depth cases is left for future research. First,the validity of the proposed method is demonstrated using alayered synthetic dataset, and the noise-resistant property isalso demonstrated. Then, a dataset from the open SMAARTPluto model and a field marine dataset are used to furtherdemonstrate the validity and flexibility of the proposedT-matrix-based deghosting method.

Theory

The acoustic equation in the frequency domain can be char-acterized by the following equation (Weglein et al 2003):

d= - -( ) ( )GL r r , 1s

wherew

r= +

⎛⎝⎜

⎞⎠⎟K

L12

is the forward modeling operator,

G denotes the Green’s operator according to the Dirac deltasource function d. K and r represent the true bulk modulusand density, respectively; r and rs are an arbitrary point andthe source location, respectively. Given the reference bulkmodulus and density K0 and r ,0 the background Green’sfunction can be obtained by equation (2),

d= - -( ) ( )GL r r , 2s0 0

wherew

r= +

⎛⎝⎜

⎞⎠⎟K

L1

0

2

0 0

and G0 denotes the background

Green’s function. Then, the solution of equation (1) can becharacterized by the Lippmann–Schwinger equation, which isa fundamental equation of the scattering theory, as shown inequation (3),

Y = - = ( )G G G VG, 3s 0 0

where wr r

= - + - ⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜⎛⎝⎜

⎞⎠⎟

⎞⎠⎟⎟K K

V1 1 1 12

0 0

is the scat-

tering potential, andYs is the scattered wavefield which is nota Green operator itself. Equation (3) can be characterized byan infinite series using a substitution operation for G, asshown in equation (4):

Y = - = = + +( )

G G G VG G VG G VG VG ...4

s 0 0 0 0 0 0 0

Equation (4) is the well-known Born series, but it fails toguarantee the convergence in the strong perturbation or largeperturbation area cases. Introducing the T-matrix, whichincludes all orders of the scattering effects (Wu et al 2015b,Wang et al 2016a), can improve the convergence. Denoting

=TG VG,0 equation (4) can be reformulated as follows:

Y = - = = ( )G G G VG G TG . 5s 0 0 0 0

In equation (5), all the scattering effects in the Green operatorG are transformed into T-matrix T, then T-matrix T containsall the scattering effects.

Equation (5) can characterize different types of wavefields,such as pure primary waves, primary waves plus ghosts (sourceghost, receiver ghost and source-receiver ghost), primary wavesplus internal multiples and primary waves plus free-surfacemultiples. In this paper, we only focus on the primary waves andthe corresponding ghosts and leave the internal multiples andfree-surface multiples for future research. When the seismic datais acquired in the marine acquisition system, the ghosts mixed inthe seismic data can decrease the resolution of the observedseismic data and generate frequency notch effects which dependon the receiver depth. Both of them can have significant effectson the subsequent seismic data processing algorithms. Themarine configuration is shown in figure 1, in which we cannotice why the source ghost is generated when firing. Accordingto the reciprocity of the sources and receivers, the receiver ghostcan also be easily understood. Actually, there are three types ofghosts, including a source ghost, receiver ghost and source-

1573

J. Geophys. Eng. 14 (2017) 1572 B Wang et al

receiver ghost; the above-mentioned three types of ghosts areillustrated in figure 2. Figures 2(a)–(c) denote the source ghost,receiver ghost and source-receiver ghost, respectively. Theabove statements relate to the qualitative interpretation of dif-ferent types of ghosts. Next, we discuss the related ghostsquantitatively.

When seismic data is acquired using a marine config-uration with a reflective free surface, the background Greenoperator includes two parts: G0

fs and G ,0d subject to

= + ( )G G G , 60 0d

0fs

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 freesurface (see figure 1). In the absence of a free surface, G0

d

denotes the reference Green operator, and the forward mod-eling formula (i.e. equation (5)) shows that the observed datais constructed from the direct propagating Green operator G0

d

and T-matrix T which includes all scattering effects. When afree surface is present, the forward modeling equation isconstructed from = +G G G0 0

d0fs and the same T-matrix, T.

Hence, G0fs is the sole difference between the forward mod-

eling with and without the free surface. Therefore, G0fs should

be responsible for generating the ghost events in the freesurface cases (see figure 2). Substituting equation (6) intoequation (5) can achieve equation (7) which includes ghosteffects

Y = = + +

= + + +

( ) ( )( )

G TG G G T G G

G TG G TG G TG G TG , 7

s 0 0 0d

0fs

0d

0fs

0d

0d

0fs

0d

0d

0fs

0fs

0fs

where T denotes the T-matrix, including all the scatteringeffects, the first term is the primary waves, the second termindicates the receiver ghost, the third term denotes the sourceghost and the fourth term represents the source-receiver ghost.The three types of ghosts are shown in figure 2, which pro-vides a simplified interpretation of the ghosts due to thepresence of a free surface. In order to attenuate the ghosteffects, the deghosted seismic data can be obtained usingequation (8):

Y Y¢ = = - - ( )G TG G G G G . 8s s0d

0d

0d

01

01

0d

From equation (8), we can also obtain the ghosts contaminatedseismic data Ys from the ghost-free seismic data Y¢ ,s

Y Y= ¢- -( ) ( ) ( )G G G G . 9s s0 0d 1

0d 1

0

The deghosting algorithm proposed in this paper can beeasily understood compared with the ISS method (Wegleinet al 2003, Wang et al 2013). It is also convergent absolutelybecause all the scattering effects contained in the T-matrixremain unchanged. The only information that is changed is theGreen operator, propagating from the source to the scatteringpoints or propagating from the scattering points to the receiver.The detailed derivation of the deghosting algorithm using theT-matrix strategy is provided as follows. Assuming the air–water interface =z 0 is the free surface, the sea water isregarded as the reference medium, ( )x z, ,s s ( )x z,g g representthe source and receiver locations, respectively, and = ( )x zx ,is an arbitrary scattering point in the scattering area. Then, thereference Green operator is subject to (Weglein et al 2003)

rwk

w

d d d

=

+

=- - - - +

⎛⎝⎜

⎞⎠⎟ ( )

( )[ ( ) ( )] ( )

G G x z x z

x x z z z z

L , , , ;

, 10

s s

s s s

0 0

2

0

2

00

where r0 is the reference density and k0 denotes the referencebulk modulus. Performing a forward Fourier transform towardsx in equation (10), we can then obtain equation (11) in thefrequency–wavenumber domain

r rw

pd d

¶¶

+

= - - - +-

⎛⎝⎜

⎞⎠⎟ ( )

[ ( ) ( )] ( )

z

qG k z x z

z z z z

1, , , ;

1

2e . 11

x s s

k xs s

0

2

2

2

00

i x s

The solution to equation (11) can be characterized usingequation (12),

wrp

=-

--

- +( ) ( )

( )

∣ ∣ ∣ ∣G k z x zq

, , , ;2

e

2ie e ,

12

x s s

k xq z z q z z

00

ii i

x ss s

where q denotes the vertical wavenumber, subject to

w w= -( ) ( )/q c ksgn ,x02 2 and k r= /c0 0 0 denotes the

acoustic velocity of sea water. Besides, the Green operatorfrom the source direct to each scattering point, G ,d

0 issubject to

wrp

=-

--( ) ( )∣ ∣G k z x z

q, , , ;

2

e

2ie . 13d

x s s

k xq z z

00

ii

x ss

In order to give a more detailed derivation, the detailedanalysis using integral notation instead of matrix notationfor equation (7) is given as

w w

w

Y = ¢

´ ¢ ¢¢

∬( ) ( ) ( )

( ) ( )

x z x z G x z

G x z

x T x x

x x x

, , , ; , , ; ,

, , ; d d . 14

s r r s s r r

s s

x x,0

0

Figure 1. The marine configuration and reference Green function.

1574

J. Geophys. Eng. 14 (2017) 1572 B Wang et al

Performing the forward Fourier transform toward x x,r s

in equation (14) we obtain

w w

w

Y = ¢

¢ ¢¢

∬( ) ( ) ( )

( ) ( )

k z k z G k z

G k z

x T x x

x x x

, , , ; , , ; ,

, , ; d d . 15

s r r s s r r

s s

x x,0

0

Substituting equations (8), (12) and (13) intoequation (15), we can obtain the deghosted data

w ww

Y YY

¢ =

=- -

- -( ) ( )( )

( )( )( )

k z k z k z k zk z k z

G G G G, , , ; , , , ;, , , ;

1 e 1 e,

16

s r r s sd

s r r s sd

s r r s sq z q z

0 01

01

0

2i 2ig g s s

where =- -( )( )Q

1 1

1 e 1 eq z q z2i 2ig g s scan be regarded as a

deghosting operator. In a similar way, ghosts contaminatedseismic data in the presence of a free surface can be simu-lated using the ghost-free seismic data and the ghost operator

= - -( )( )Q 1 e 1 e ,q z q z2i 2ig g s s which is the basic foundationfor deghosting problems.

w wY Y= ¢ -

´ -

( ) ( )( )

( )( )

k z k z k z k z, , , ; , , , ; 1 e

1 e ,

17

s r r s s s r r s sq z

q z

2i

2i

g g

s s

where w w= -( ) ( )/q c ksgn ,g g02 2 w w= -( ) ( )/q c ksgns s0

2 2

are the receiver and source sides vertical wavenumbers,respectively, corresponding to the horizontal wavenumbersk ,g k .s

During the deghosting procedure, instability appearswhen the absolute value of Q is minor enough. Impropertreatment of Q values can introduce linear interference withapparent velocity c .0 Therefore, the regularization method canbe used in equation (16), and then the damped least squaresolution can be obtained,

*e

=+∣ ∣ (∣ ∣)

( )Q

Q

Q Q

1

max, 18

2

where *( )• is the conjugate operator and e denotes a reg-ularization factor to stabilize the division. In fact, thedeghosting algorithm (equation (16)) can be regarded as atwo-step method. Firstly, the receiver side deghosting in thecommon shot gather; Then, the source side deghosting in the

common receiver gather. The above two procedures can helpus obtain the deghosted data in the frequency-wavenumberdomain. Finally, the deghosted seismic data in the time-spacedomain can be achieved using the inverse two-dimensional(2D) Fourier transform. Next, a few numerical examples willbe 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 effec-tiveness of the proposed deghosting method, and thedeghosted data is consistent with the original ghost-free datawhich proves the effectiveness of the proposed method,during which the noise-resistant property of the proposedmethod is also illustrated. Then, a dataset from the openSMARRT Pluto model and a field marine dataset are used tofurther prove the effectiveness and flexibility of the proposedmethod.

Layered model example

Figure 3(a) shows the synthetic dataset simulated from thedesigned three-layer model, without ghost effects. It includes101 traces with 10 m as the trace interval and 1001 samplesper trace with 2 ms as the time sampling interval. Figure 3(b)shows the seismic data contaminated by the receiver sideghost effects with 6 m as the receiver depth and with1500 m s−1 as sea water velocity (reference velocity).Figure 3(b) notes that the receiver ghost with reverse polarityis mixed up with the primary waves which decreases theresolution of the observed seismic data. Using the proposedmethod, we can obtain the deghosted data, as shown infigure 3(c) which is consistent with the reference data(figure 3(a)). In order to see the details, the residual is cal-culated, as shown in figure 3(d), which is almost zeroeverywhere. The normalized local frequency spectra are alsocompared among the three datasets (the ghost-free data, thedata contaminated by ghosts and the deghosted data), asshown in figure 4, in which the plus, circle and triangle curves

Figure 2. Illustration of different ghost events: (a) source ghost, (b) receiver ghost and (c) source–receiver ghost.

1575

J. Geophys. Eng. 14 (2017) 1572 B Wang et al

represent the spectra of the data contaminated by ghosts,the deghosted data and the reference data, respectively. Thespectrum of the deghosted data is consistent with the refer-ence one, and has enhanced low frequency componentscompared with the data before deghosting. The above testdemonstrates the validity of the proposed T-matrix-baseddeghosting method.

In order to demonstrate the noise-resistant property of theproposed deghosting method, the noisy data contaminated bysome random noise is used, as shown in figure 5(b), in which

the maximum amplitude of the random noise to that of thesignal is 0.2. Using the proposed method, we can obtain thedeghosted data, as shown in figure 5(c) which is reasonablycorrect, and proves that the proposed method has the noise-resistant property. The residual shown in figure 5(d) does notcontain 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 asthe noise level increases because the regularization factor eshould be a bigger value in these cases.

The above examples are based on the layered model, andin order to further demonstrate the effectiveness of the pro-posed deghosting method, a dataset from the open SMAARTPluto model is used which contains interference and weakreflections.

SMAART Pluto model example

This complicated synthetic data is from the open SMAARTPluto model, obtained using finite difference method. Thedistance between adjacent shots and adjacent receivers areboth 22.86 m, and the acquisition system is single-side. Thereare 361 traces for each shot and 1126 sampling points pertrace with 8 ms as the time sampling interval. The depth of thereceivers and sources are both 7.62 m, which makes theobtained seismic data contaminated by different kinds of

Figure 4. Frequency spectra comparisons before and after deghost-ing. Triangle: reference data; circle: deghosted data; plus: data beforedeghosting.

Figure 3. Synthetic data test. (a) Reference data; (b) data contaminated by ghosts; (c) deghosted data; (d) residual.

1576

J. Geophys. Eng. 14 (2017) 1572 B Wang et al

Figure 5. Noisy synthetic data test. (a) Reference data; (b) noisy data contaminated by ghosts; (c) deghosted data; (d) residual between (c)and (a).

Figure 6. SMAART Pluto data. (a) Before deghosting; and (b) deghosted data.

1577

J. Geophys. Eng. 14 (2017) 1572 B Wang et al

ghosts. In this paper, we mainly concentrate on receiver sideghost elimination, but it can handle other kinds of ghosts.Figure 6(a) shows the scattering data after the elimination ofthe direct wave. Because of the existing reflecting surface, theghosts are mixed up with the primary waves and theincreasing number of the sidelobes decreases the resolution ofthe observed data. Besides, the ghosts can change the fre-quency components distribution of the primary waves, whichwill be illustrated later. Based on the proposed deghostingmethod, we can obtain the deghosted data, as shown infigure 6(b). In order to see the details, two local areas marked

by rectangles in figure 6 are zoomed in, as shown infigures 7–8. Figures 7(a) and (b) denote the zoomed-in ver-sion of the big rectangle in figures 6(a) and (b), respectively.Figure 7 notes that after deghosting, the sidelobes generatedby the ghosts are effectively attenuated and the ghost eventsare also effectively attenuated, especially the locationsmarked by the arrows, which proves the validity of the pro-posed method. Figures 8(a) and (b) represent the zoomed-inversion of the small rectangle in figures 6(a) and (b),respectively. Figure 8 shows that the ghost events are effec-tively attenuated, especially where indicated by the arrows. Inorder to compare the variation of frequency components, thenormalized local frequency spectra of seismic data before andafter deghosting are shown in figure 9, in which the solidand dashed curves represent the spectra of seismic data beforeand after deghosting, respectively. Similar to the layeredmodel, after deghosting, the low frequency components arerecovered reasonably and will be useful for the subsequentseismic data processing. The dataset from the open SMAARTPluto model demonstrates the validity of the proposedmethod. Finally, a field marine dataset is used to demonstratethe validity and flexibility of the proposed T-matrix-baseddeghosting method.

Figure 7. Detailed comparisons of the larger rectangles in figure 6.(a) Before deghosting; (b) deghosted data. The arrows are placed inthe same position in (a) and (b) to better illustrate the differences.

Figure 8. Detailed comparisons of the smaller rectangles in figure 6. (a) Before deghosting; (b) deghosted data. The arrows are placed in thesame position in (a) and (b) to better illustrate differences.

Figure 9. Frequency spectra comparisons before and after deghost-ing. Dashed curve: deghosted data; solid curve: seismic data beforedeghosting.

1578

J. Geophys. Eng. 14 (2017) 1572 B Wang et al

Field marine data example

The field marine data processing procedure is similar as theSMAART Pluto model. The field marine data is obtained usinga conventional acquisition system with towed streamers. Thereare 384 traces with 12.5m as the trace interval in each towed

streamer and 2501 samples per trace with 2 ms as the timesampling interval. Because of the feathering and fluid flowingeffects, the depth of receivers is varying. In order to estimatethe accurate depth of the receivers for each shot, Liu and Lu(2016) studied the receiver depth and sea surface reflectivityestimation issue based on non-Gaussian maximization. Theoptimally estimated receiver depth is 9.8 m and the estimatedsea surface reflectivity is −0.88 for the selected shot, as shownin figure 10(a). Figure 10(a) indicates the scattering data afterdirect wave elimination. Using the proposed T-matrix-baseddeghosting method, the deghosted data can be obtained, asshown in figure 10(b). For detailed comparisons, two localareas marked by the rectangles in figure 10 are zoomed in, asshown in figures 11–12. Figures 11(a) and (b) indicate thezoomed in versions of the big rectangle in figures 10(a) and (b),respectively. Figure 11 shows that after deghosting, the ghostevents are effectively attenuated, especially the locationsmarked by the arrows. This demonstrates the validity of theproposed 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 theghost events are also effectively attenuated, especially at theplaces marked by arrows. For the comparisons of local fre-quency components variation, the normalized frequencyspectra before and after deghosting are shown in figure 13, inwhich the solid and dashed curves represent the spectra of theseismic data before and after deghosting, respectively. Afterdeghosting, the low frequency components are recovered rea-sonably and can improve the accuracy of the subsequent fullwaveform inversion. The field marine dataset exampledemonstrates the validity and flexibility of the proposedT-matrix-based deghosting method.

Figure 10. Field marine data. (a) Before deghosting; and (b) deghosted data.

Figure 11.Detailed comparisons of the larger rectangles in figure 10.(a) Before deghosting; (b) deghosted data. The arrows are placed inthe same position in (a) and (b) to better illustrate the differences.

1579

J. Geophys. Eng. 14 (2017) 1572 B Wang et al

Conclusions

Ghosts from the marine acquisition system have significanteffects on the subsequent seismic data processing algorithms.In this paper, we derived the deghosting algorithm using thetransmission matrix (T-matrix) strategy which is easilyunderstandable and absolutely convergent. The deghostingperformance on the layered model demonstrates the validityof the proposed deghosting method, and its noise-resistantproperty is also demonstrated using the noisy data con-taminated by random noise. The numerical examples of thecomplicated dataset from the open SMAART Pluto modeland the field marine dataset further prove the effectivenessand flexibility of the proposed deghosting method. Afterdeghosting, the low frequency components are recoveredreasonably and the fake high frequency is attenuated which ishelpful in the subsequent seismic data processing methods,like full waveform inversion.

However, the proposed method requires exact depthinformation on receivers or sources, and the performance willbe affected if the depth information is inaccurate. Therefore,further research should be done to study the quantitativeeffects of inaccurate depth information on the deghostingperformance. Besides, the proposed method was used in 2D

constant depth streamer cases and the method can be extendedinto varying depth streamer cases or three-dimensional cases,which will be studied in the future.

Acknowledgments

This research is financially supported by the National NaturalScience Foundation of China (Grant Nos. 41604096,41674116), the China Postdoctoral Science Foundation(Grant No. 2016M590102), the Key State Science andTechnology Project (Grant Nos. 2016ZX05024001-004,2016ZX05024001-005) and the WTOPI Research Con-sortium at the University of California, Santa Cruz.

ORCID iDs

Benfeng Wang https://orcid.org/0000-0002-5743-0664

References

Amundsen L and Zhou H 2013 Low-frequency seismic deghostingGeophysics 78 WA15–20

Amundsen L, Zhou H, Reitan A and Weglein A B 2013 On seismicdeghosting by spatial deconvolution Geophysics 78 V267–71

Day A, Klüver T, Söllner W, Tabti H and Carlson D 2013Wavefield-separation methods for dual-sensor towed-streamerdata Geophysics 78 WA55–70

Grion S, Telling R and Barnes J 2015 De-ghosting by kurtosismaximisation in practice SEG Technical Program ExpandedAbstracts (Tulsa, OK: Society of Exploration Geophysicists)pp 4605–9

Guasch L and Warner M 2014 Adaptive waveform inversion-FWIwithout cycle skipping-applications 76th EAGE Meeting,Expanded Abstracts (Amsterdam, Netherlands) E106 14

Jakobsen M 2012 T-matrix approach to seismic forward modellingin the acoustic approximation Stud. Geophys. Geod. 56 1–20

Jakobsen M, Hudson J A and Johansen T A 2003 T-matrix approachto shale acoustics Geophys. J. Int. 154 533–58

Figure 12. Detailed comparisons of the smaller rectangles in figure 10. (a) Before deghosting; (b) deghosted data. The arrows are placed inthe same position in (a) and (b) to better illustrate the differences.

Figure 13. Frequency spectra comparisons before and afterdeghosting. Dashed curve: deghosted data; solid curve: seismic databefore deghosting.

1580

J. Geophys. Eng. 14 (2017) 1572 B Wang et al

Jakobsen M and Ursin B 2012 Nonlinear seismic waveforminversion using a Born iterative T-matrix method SEGTechnical Program Expanded Abstracts (Tulsa, OK: Societyof Exploration Geophysicists) pp 1–5

Jakobsen M and Ursin B 2015 Full waveform inversion in thefrequency domain using direct iterative T-matrix methodsJ. Geophys. Eng. 12 400–18

Liu L and Lu W 2016 Adaptive F-K deghosting method based onnon-Gaussianity J. Appl. Geophys. 127 14–22

Lu W, Xu Z, Fang Z, Wang R and Yan C 2017 Receiver deghostingin the T-X domain based on super-Gaussianity J. Appl.Geophys. 136 134–41

Luo J and Wu R 2015 Seismic envelop inversion: reduction of localminima and noise resistance Geophys. Prospect. 63 597–614

Posthumus B J 1993 Deghosting using a twin streamer configurationGeophys. Prospect. 41 267–86

Robertsson J O A and Amundsen L 2014 Wave equation processingusing finite-difference propagators: II. Deghosting of marinehydrophone seismic data Geophysics 79 T301–12

Shin C and Cha Y H 2008 Waveform inversion in the Laplacedomain Geophys. J. Int. 173 922–31

Shin C and Cha Y H 2009 Waveform inversion in the Laplace-Fourier domain Geophys. J. Int. 177 1067–79

Soubaras R and Lafet Y 2013 Variable-depth streamer acquisition:broadband data for imaging and inversion Geophysics 78WA27–39

Verschuur D J, Berkhout A J and Wapenaar C P A 1992 Adaptivesurface‐related multiple elimination Geophysics 57 1166–77

Wang B, Jakobsen M, Wu R-S, Lu W and Chen X 2017 Accurateand efficient velocity estimation using Transmission matrixformalism based on the domain decomposition method InverseProb. 33 035002

Wang B, Wu R and Chen X 2016a Non-linear parameter estimationmethod based on T-matrix Chin. J. Geophys. 59 2257–65

Wang B, Wu R-S, Chen X, Lu W and Liu G 2016b Nonlineardeghosting based on the T-matrix method 78th EAGE Conf. &Exhibition We P1 01

Wang F, Li J and Chen X 2013 Deghosting method based on inversescattering series Chin. J. Geophys. 56 1628–36

Warner M and Guasch L 2014 Adaptive waveform inversion: theory84th Annual Int. Meeting, SEG Expanded Abstractspp 1089–93

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 and Zhang H2003 Inverse scattering series and seismic exploration InverseProb. 19 R27–83

Wu R-S, Luo J and Wu B 2014 Seismic envelope inversion andmodulation signal model Geophysics 79 WA13–24

Wu R-S, Wang B and Jakobsen M 2015a Green’s function andT-matrix reconstruction using surface data for direct nonlinearinversion SEG Technical Program Expanded Abstracts (Tulsa,OK: Society of Exploration Geophysicists) pp 1286–91

Wu R-S, Wang B and Hu C 2015b Renormalized nonlinearsensitivity kernel and inverse thin-slab propagator in T-matrixformalism for wave-equation tomography Inverse Prob. 31115004

1581

J. Geophys. Eng. 14 (2017) 1572 B Wang et al