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Reverse time migration of internal multiples for subsalt imaging
Yike Liu1, Hao Hu2, Xiao-Bi Xie3, Yingcai Zheng4, and Peng Li5
ABSTRACT
Internal multiple reflections have different propagationpaths than primary reflections and surface-related multiples;thus, they can complement illumination where other wavesare unavailable. Consequently, migration of internal multiplesmay provide vital images in the shadow zone of primary re-flections. We have developed an approach to image structureswith internal multiples. With this method, internal multiplescan be separated and used as data for reverse time migration.Instead of applying the imaging condition directly to thesource and receiver wavefields, we decomposed the extrapo-lated source and receiver waves into up- and downgoingwaves. Then we applied the imaging condition to upgoingsource and receiver waves to form an up-up (UU) image andalso to downgoing source and receiver waves to form a down-down (DD) image. The image of the true reflector and theartifacts behave differently in the DD and UU images. A sim-ilarity analysis is conducted to the UU and DD images to sep-arate true images from artifacts. Numerical examples withsimple velocity models are used to demonstrate how to con-struct UU and DD imaging conditions and eliminate migrationartifacts. Finally, we test the 2D SMAART JV model with saltstructures for migration using internal multiples.
INTRODUCTION
Although primary reflections provide important information forseismic imaging, surface-related and internal multiples are alsovaluable information. Multiples are waves reflected more than onceat interfaces before they reach the receivers. They usually travellonger distances, cover larger areas than primary reflections, and
they can provide additional illumination in shadow zones of pri-mary reflections. In addition, relatively small reflection angles makethem more suitable for imaging rugged reflectors (Berkhout and Ver-schuur, 2003). Even with these advantages, extracting useful infor-mation from multiples is not easy. Weglein (2014) separated currenttreatments of multiples into two categories: (1) those who considerprimaries as signals and multiples as a form of coherent noise thatneeds to be removed and (2) those who consider both primaries andmultiples to be useful and can be used separately or together in seis-mic imaging. To turn multiples into useful data for subsurface imag-ing, several approaches have been proposed (Guitton, 2002; Shan2003; Muijs et al., 2007; Vasconcelos et al., 2008). The idea of ex-tracting valuable events from multiples dates back to the work ofClaerbout (1968) on seismic interferometry, which shows that theGreen’s function on the earth’s surface can be obtained by autocor-relating traces generated by buried sources. In this way, the multiplesare kinematically transformed to primary reflections with virtualsources on the surface, followed by migration using conventionalmethods (Berkhout and Verschuur, 1994; Verschuur and Berkhout,2005). Other approaches to migrating multiples using seismic inter-ferometry can be found in Sheng (2001), Yu and Schuster (2002),Schuster et al. (2004), Jiang et al. (2007), and He et al. (2007).Least-squares migration provides yet another way of using multiplesin imaging (He and Schuster, 2003; Brown and Guitton, 2005) whichutilizes an efficient linear operator to model and image peg-leg multi-ples in a true-amplitude sense. Ocean bottom cable (OBC) data arealso commonly used to image primary reflections and waves re-flected downward from the free surface (Reiter et al., 1991).Reverse time migration (RTM) with the full acoustic wave-equa-
tion can image steeply dipping structures and overhangs (Baysal et al.,1983; Chang and McMechan, 1987). Liu et al. (2011) modify con-ventional RTM to migrate multiples for subsalt imaging, and otherauthors extend this approach (Tu and Herrmann, 2012; Wong et al.,2012; Zuberi and Alkhalifah, 2013; Zhang and Schuster, 2014).
Manuscript received by the Editor 11 September 2014; revised manuscript received 8 April 2015; published online 31 July 2015.1Chinese Academy of Sciences, Institute of Geology and Geophysics, Beijing, China. E-mail: [email protected] Chinese Academy of Sciences, Institute of Geology and Geophysics, Beijing, China, presently University of Houston, Department of Earth and
Atmospheric Sciences, Houston, Texas, USA. E-mail: [email protected] of California at Santa Cruz, Department of Earth and Planetary Sciences, Santa Cruz, California, USA. E-mail: [email protected] of Houston, Department of Earth and Atmospheric Sciences, Houston, Texas, USA. E-mail: [email protected] Research & Development Center, Beijing, China. E-mail: [email protected].© 2015 Society of Exploration Geophysicists. All rights reserved.
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GEOPHYSICS, VOL. 80, NO. 5 (SEPTEMBER-OCTOBER 2015); P. S175–S185, 17 FIGS.10.1190/GEO2014-0429.1
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Internal multiples are usually generated below coal seams, saltdomes, basalt flows, and other geologic formations with high veloc-ity contrasts and behave differently compared to primary and sur-face-related multiples. They can illuminate areas below these high-velocity bodies. Predicting internal multiples is already challenging,and artifacts generated by migrating multiples are difficult to at-tenuate. Thus, relatively few attempts have been made to migrateinternal multiples. Jin et al. (2006) use one return propagator toextrapolate both down- and upgoing duplex waves to image steepstructures. Malcolm et al. (2008) develop an inverse generalizedBremmer coupling series to image triply scattered waves inone-way wave equation based migration. To use the multiply scat-tered data, Fleury (2013) propose nonlinear RTM. Recently, apromising approach based on the Marchenko equation has beenproposed for imaging the total wavefield including internal multi-ples (e.g., Behura et al., 2012; Wapenaar et al., 2012, 2013, 2014;Broggini and Snieder, 2013; Neut et al., 2013; Thorbeckeet al., 2013).In this paper, we present an RTM method that is capable of han-
dling internal multiples. The method first extracts internal multiplesfrom the wavefield, and then it extrapolates those multiples back-ward in time through the subsurface. The extrapolated source andreceiver wavefields are decomposed into up- and downgoing waves.A crosscorrelation imaging condition is applied to these decom-posed waves, and it generates images from waves propagating indifferent directions. By comparing images from upgoing waveswith images from downgoing waves, the artifacts can be recognizedand attenuated. In the rest of this paper, we first review the surface-multiple-based internal multiple prediction method. We then inves-tigate how to migrate internal multiples to form images. Finally, wediscuss the application and efficacy of this method using syntheticexamples.
METHODOLOGY
Predicting and extracting internal multiples
Internal multiples have different kinematic and dynamic charac-teristics compared to primary reflections and surface-relatedmultiples. Verschuur et al. (1992) and Berkhout (1997) providetheoretical insights for wave-equation-based surface-related multi-ple elimination (SRME), which has been widely used in industry
since then (e.g., Moore et al., 2004; Fomel, 2009; Liu et al.,2010). Jakubowicz (1998) extends this technique to attenuate inter-nal multiples. Based on the inverse scattering series (ISS), Wegleinet al. (1997, 2009) develop another method to predict internalmultiples.Both methods are tested in this paper. With the SRME concept, a
first-order internal multiple can be expressed as a combination ofthree primary reflections, i.e., by a convolution between twoprimary reflections, followed by a deconvolution with the thirdprimary reflection (refer to Figure 1). In the frequency domain,it can be estimated by the following integration (Jakubowicz,1998):
IMðrA; rD;ωÞ ¼ZZ
GðrC; rA;ωÞGðrD; rB;ωÞGðrC; rB;ωÞ
drBdrC;
(1)
where r ¼ ðx; zÞ is the location vector, with x and z being the hori-zontal and vertical coordinates; rA and rB are the locations of thesurface sources; rC and rD are the locations of the surface receivers;ω is the frequency; IMðrA; rD;ωÞ is a multiple through the pathðrA; rH; rF; rI; rDÞ in Figure 1; and Gðr2; r1;ωÞ is the frequencydomain Green’s function for the response at r2 generated by asource at r1. The integration in equation 1 is over the surface lo-cations between rA and rD containing at least one stationary point.The major contribution to IMðrA; rD;ωÞ comes from the vicinity ofthe stationary point, i.e., where the multiples and the reflectionspass through rF. Internal multiple prediction is a data-driven algo-rithm. It does not require any information about the reflectorsthat generate the internal multiples or the medium through whichthe multiples propagate. The algorithm predicts the correct travel-times and approximate amplitudes of all the internal multiples in thedata. Similarly, a higher order internal multiple can be obtained byconvolution between the internal multiples of lower orders. Giventhat the subsurface structure is unknown, we sort the data set intocommon shot gathers and common receiver gathers to form anaperture that decreases with increasing depth. The stationary pathsare then determined by stacking common source and commonreceiver gathers within the aperture. The source and receiverpositions need to be coincident, otherwise interpolation willbe used.The advantage of ISS-based methods is that they only require
background velocity information and are fully data driven. Internalmultiples from all origins, including their arrival times and approxi-mate amplitudes can be predicted without knowing detailed subsur-face structure. The ISS-based 2D equation for predicting internalmultiples can be expressed as (Weglein et al., 1997)
b3ðkg; ks; qg þ qsÞ ¼1
ð2πÞ2Z
∞
−∞
Z∞
−∞dk1e−iq1ðzg−zsÞdk2e−iq2ðzg−zsÞ
×Z
∞
−∞eiðqgþq1Þz1b1ðkg; k1; z1Þdz1
×Z
z1
−∞e−iðq2þq1Þz2b1ðk1; k2; z2Þdz2
×Z
∞
z2
eiðq2þqsÞz3b1ðk2; ks; z3Þdz3; (2)
Figure 1. The values rA and rB are the locations of the surfacesources, and rC and rD are the locations of the surface receivers.Path ðrA; rH; rCÞ indicates a reflected wave from rA to rC, pathðrB; rI ; rDÞ is a reflected wave from rB to rD, and pathðrB; rF; rCÞ is the reflected wave indicated by the dashed line.The three reflected waves compose the internal multiple.
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where b3ðkg; ks; qg þ qsÞ is the predicted internal multiples re-corded at the surface, b1ðkg; ks; zÞ corresponds to f-k migrationin the background velocity (Weglein et al., 1997), kg, ks are hori-zontal wavenumbers, qg and qs are vertical wavenumbers, sub-scripts g and s are for the receiver and source, respectively. zsand zg are the source and receiver depths, zi with i ¼ 1; 2; 3 repre-sents different pseudodepths where reflections happen, and q1 andq2 correspond to vertical wavenumbers for pseudodepths. We usethis method to predict internal multiples in the SMAART Pluto dataset example later.
Image conditions for migrated internal multiples
Conventional prestack RTM uses the zero-lag crosscorrelationimaging condition:
IðrÞ ¼Z
T
0
Sðr; tÞRðr; T − tÞdt; (3)
where Sðr; tÞ and Rðr; T − tÞ are the source- and time-reversedreceiver waves, t is the time, and T is the maximum extrapolationtime. To image internal multiples, we define propagation directions.Given a source and a receiver located at depth z0 ¼ 0 and a series ofinterfaces located at zi−1, zi, ziþ1, with ziþ1 > zi > zi−1 (refer toFigure 2), we name a section of the source-side wave as downgoingwave as SD if it propagates from zi to ziþ1 or the upgoing wave canbe named as SU if it propagates from ziþ1 to zi. Similarly, we canlabel the receiver waves as RD and RU .We modify the conventional imaging condition by separating it
into two groups, those generated by a pair of downgoing waves,which we call down-down (DD) images, and those generated bya pair of upgoing waves, which we call up-up (UU) images. Asshown in Figure 2, by applying the DD and UU imaging conditionsto up- and downgoing waves, we can generate four groups of im-ages:
IðrÞ ¼XTt¼0
Sðr; tÞRðr; T − tÞ
¼XTt¼0
½SUðr; tÞ þ SDðr; tÞ�½RUðr; T − tÞ þ RDðr; T − tÞ�
¼ IUUðrÞ þ IDDðrÞ þ IUDðrÞ þ IDUðrÞ; (4)
where S and R denote the source and receiver waves, subscripts Uand D denote up- and downgoing propagation directions, and
IUUðrÞ ¼XTt¼0
SUðr; tÞRUðr; T − tÞ; (5)
IDDðrÞ ¼XTt¼0
SDðr; tÞRDðr; T − tÞ; (6)
are our UU and DD images. The terms IUDðrÞ and IDUðrÞ mostlycontain crosstalk and will be discarded.
Decomposing internal multiples into up- anddowngoing waves
To decompose the source wavefield in to up- and downgoingwaves, we use the 2D Fourier transform to convert the wavefieldfrom the space-time domain ðx; z; tÞ to the frequency-wavenumberdomain ðx; kz;ωÞ, followed by separation in the f-k domain (Liuet al., 2011):
SDðω; x; kzÞ ¼�Sðω; x; kzÞ; if kz ≥ 00; if kz < 0
; (7)
SUðω; x; kzÞ ¼�0; if kz ≥ 0Sðω; x; kzÞ; if kz < 0 ; (8)
where SU;Dðω; x; kzÞ is the transform of SU;Dðr; tÞ, kz is the verticalwavenumber, and ω is the angular frequency. Finally, we inverseFourier transform SU;Dðω; x; kzÞ and RU;Dðω; x; kzÞ back to thespace-time domain to obtain the required up- and downgoingwaves in equations 5 and 6. Similarly, we can obtain space-time domain up- and downgoing wavefields for the receiverwavefield.
Images and artifacts generated by internal multiples
Unlike one-way wave-equation migration or Kirchhoff migra-tion, RTM uses the full acoustic wave equation to extrapolate sourceand receiver waves. In a realistic velocity model with interfaces, thefull wave equation generates additional reflections, which form ar-tifacts. We use Figure 3 to illustrate how to image multiples andeliminate artifacts. Shown in this Figure are the ðz; tÞ domain wave-fields calculated in a simple three-layer model (refer to the model inFigure 4). The short horizontal bars indicate the depths of the twointerfaces, and the velocities from top to bottom are 1500, 3000, and1000 m∕s, respectively. The top row shows mixed source andreceiver wavefields at x ¼ 1000 m in the (z, t) domain, and themiddle and bottom rows are separated source and time-reversedreceiver waves. Shown in the left column are the total wavefields,in the middle column are the decomposed downgoing waves, and inthe right column are the decomposed upgoing waves. From the
Figure 2. Cartoon showing the up- and downgoing sections ofsource and receiver waves. The black path indicates the forward-propagated wave from a source to a receiver, and the red path in-dicates the back-propagated wave from a receiver to a source. Thearrows indicate the related up- and downgoing source and receiverwaves.
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mixed source and receiver wavefields in Figure 3a, we see that thesource and receiver waves intersect at selected depths at which thetwo waves meet the zero-lag image condition and form images.However, in addition to the image from two real interfaces (indi-cated by white arrows), there are also artifacts (indicated by blackarrows). To eliminate these artifacts, we use equations 7 and 8 todecompose the source and receiver wavefields into up- and down-going waves, and we use equations 5 and 6 to calculate the DD andUU images. Figure 3b is the DD image, in which events related toreal reflectors are labeled DD and the events related to the down-going artifacts are labeled DDA. Figure 3c is the UU image, inwhich events related to real reflectors are labeled UU and eventsrelated to the upgoing artifacts are labeled UUA. Shown in themiddle and bottom rows are the decomposed source and receiverwaves, which explain how these images are formed. ComparingFigure 3b and 3c, the images related to real interfaces appear in
the DD and UU images, whereas the artifacts only tend to appearin one of these images. Hence, by comparing the DD and UU im-ages, we can eliminate artifacts.
Artifacts attenuation based on the similarity betweenUU and DD images
We attempt to attenuate artifacts by comparing the similarity ofthe DD and UU images. To specify, we use a spatial window todecompose the DD and UU images into small areas. Within eachwindow, we have two sequences of discretized images:
IiDD ¼ IDDðriÞ; (9)
IiUU ¼ IUUðriÞ; (10)
Figure 3. Migrated source and receiver wavefields in the ðz; tÞ domain. (a-c) overlapped source and receiver waves, (d-f) source waves, and(g-i) time-reversed receiver waves. (a, d, and g) total wavefield, (b, e, and h) decomposed downgoing waves, and (c, f, and i) decomposedupgoing waves. The short horizontal bars indicate the depths of the interfaces. The arrows indicate the depths at which the source andreceiver waves meet the zero phase image condition. The white arrows are for real reflectors, and the black arrows are for artifacts.The symbols DD and UU are images from the DD and UU image conditions, and DDA and UUA are artifacts from these imageconditions.
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where ri is the location of an image pixel, i ¼ 1;2; 3: · · · :n, and nis the number of pixels in the window. The similarity between thetwo sequences can be measured by the correlation coefficient
R ¼P
ni¼1½ðI 0DD − ĪDDÞ − ðI 0UU − ĪUUÞ��P
ni¼1½ðI 0DD − ĪDDÞ2
�1∕2
�Pni¼1½ðI 0UU − ĪUUÞ2
�1∕2 ;
(11)
where
ĪDD ¼1
n
Xni¼1
ðIiDDÞ (12)
and
ĪUU ¼1
n
Xni¼1
ðIiUUÞ (13)
are the mean values of the images. The correlation coefficient Rvaries between −1 and þ1, where R ¼ þ1 means that the two se-quences are perfectly positively correlated, whereas R ¼ −1 meansthat they are perfectly negatively correlated, and R ¼ 0 means theyare totally unrelated. We set a positive threshold; if the correlationcoefficient of the UU and DD images in the spatial window is abovethe threshold, this part of the image will be retained. Otherwise, itwill be judged as an artifact and rejected from the final image. Thisprocess can be expressed as
IðrÞ ¼X
window
MðrÞWðrÞ½IDDðrÞ þ IUU�; (14)
where IðrÞ is the final image andWðrÞ is a spatial window function,which is nonzero near location r, and the summation is over all thespatial windows. The valueMðrÞ is a mask function for the windowand can be expressed as
MðrÞ ¼�1 if R ≥ RT0 if R
; (15)
where RT is a threshold, with its value chosen between 0 and 1. Toobtain a smoothed image, the window function can be tapered andpartially overlapped with other windows.
NUMERICAL EXPERIMENTS
Migration of internal multiples in a three-layer model
We use the simple three-layer model, shown in Figure 4, to dem-onstrate our approach. The model is 2-km wide and 1.5-km deep,and it is partitioned with a grid interval of 5 m in the x- and z-di-rections. The layers are separated by a flat upper interface and acurved lower interface with velocities (from top to bottom) of1500, 3000, and 1000 m∕s, respectively. We use a total of 100 shotsto illuminate the model, with the first one starting at a horizontaldistance of 505 m. From the left, each shot has 101 split-spreadreceivers. The source and receiver intervals are 10 m. The totalrecord time T ¼ 2.5 s, and the sampling rate is 1 ms. A 25-HzRicker wavelet is used as the source time function. Illustrated inFigure 5 is a common shot record, in which primary reflectionsand surface-related multiples have been removed, leaving only in-ternal multiples. There are four orders of internal multiples that canbe identified.To attenuate the artifacts generated by the internal multiples, fol-
lowing the process mentioned previously, we decompose theextrapolated wavefields into up- and downgoing waves. Figure 6illustrates the processes of image and artifact elimination, in which,in the central column, are RTM images, and the left and right col-umns are the ðz; tÞ-domain wavefields for selected vertical profiles
Figure 4. A three-layer velocity model with two interfaces. Theupper interface is flat, and the lower interface is curved.
Figure 5. A common shot record with primary reflections elimi-nated. More than four orders of internal multiples can be identified.
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A and B indicated in the image as thick vertical lines. The imagesrelated to the true interfaces and artifacts are similarly labeled asthose in Figure 3. Shown in the top, middle, and bottom rowsare the images and related wavefields for total wavefields Sþ R,downgoing waves SD þ RD and upgoing waves SU þ RU . In Fig-ure 6e and 6h, the true interfaces appear in the DD and UU images,whereas the artifacts appear only in the DD or UU image, with theartifacts in the DD image appearing below the two interfaces andthe artifacts in the UU image appearing above the two interfaces.The related wavefields in Figure 6d, 6f, 6g, and 6i explain the mech-anisms forming these images and artifacts. To attenuate artifactsgenerated by internal multiples, we use equation 11 to measure thesimilarity of the DD and UU images (i.e., Figure 6e and 6h). Then,we use equations 14 and 15 to eliminate the unwanted part. Afterthis process, the result is shown in Figure 7. Where artifacts inter-sect an interface event, the overlapping part will be preserved,whereas the other part will be removed.
Figure 6. RTM images of internal multiples. Shown in the central column are different images, and in the left and right columns are the ðz; tÞ-domain wavefields sampled at vertical lines A and B. The top row is for the mixed wavefield, the middle row is for the downgoing source andreceiver waves, and the bottom row is for the upgoing source and receiver waves. The arrows indicate the depths at which the source andreceiver waves meet the zero-phase image condition. The white arrows are for the real reflectors, and black arrows are for the artifacts. Thesymbols DD and UU are the images from the DD and UU image conditions, and DDA and UUA are the artifacts from these image conditions.
Figure 7. Final image for the three-layer model after the artifactsare removed.
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Illumination analysis of primary reflections and inter-nal multiples
As an example, we use a model shown in Figure 8b to illustratehow internal multiples can improve the image in areas poorly illu-minated by primary reflections. The velocity model is 4.0-km wideby 2.0-km deep, and it is partitioned with a grid interval of 10 m inthe x- and z-directions. A high-velocity triangle of 4500 m∕s is em-bedded in a two-layer background with velocities of 1500 and3000 m∕s. A synthetic shot gather (Figure 8a) is generated witha surface source located at distance 1000 m from the left edgeand a right-side receiver array ranging from offset 0 to 3000 m withan interval of 10 m. The source time function is a 25-Hz Rickerwavelet. The primary reflections and internal multiples are denotedin the figure.To investigate the illumination of primary reflections and internal
multiples, we calculate the energy of the source and receiver wave-fields. Shown in Figure 9a is the total energy from the source wave-field. We see that the primary waves can illuminate the left flank ofthe high-velocity prism and the horizontal interface. Below theprism, there is a shadow zone for the primary waves. However,the internal multiples between the bottom of the prism and the hori-zontal interface can provide effective illumination in the shadowzone. The total energy from the back-propagated receiver wavefieldis shown in Figure 9b. Again, we can see contributions from theprimary reflections and internal multiples. Figure 9c and 9d showsthe energy from the back-propagated wavefields but with the pri-mary reflections and internal multiples separated. We see clearly theprimary reflections illuminate the left flank of the prism and part ofthe horizontal interface, but it cannot illuminate the shadow zone.On the contrary, internal multiples provide effective illumination inthe shadow zone.The corresponding images are shown in Figure 10. The image
from the primary reflections is shown in 10a, where the left flankof the prism and part of the horizontal interface are properly imaged,but the structures within the shadow zone are totally missing. On thecontrary, the internal multiples properly image the structures in theshadow zone (Figure 10b). Figure 10c and 10d shows images frominternal multiples using the UU and DD imaging conditions.
RTM image using internal multiples in the SMAARTPluto salt data
The 2D SMAART JV model (Stoughton et al., 2001) represents asequence of sedimentary layers, several normal and thrust faults,and several salt bodies (Figure 11). The free-surface and velocitycontrasts at the sea bottom and the top and bottom salt boundariesgenerate strong free-surface and internal multiples. The salt bodiesgenerate significant multipathing and nonhyperbolic moveout,which result in imaging problems for a conventional migration us-ing only primary reflections. A subset of the SMAART JV model(horizontally from 11 km to 42 km and vertically from the surface toa 9-km depth) is used to demonstrate our method. A6960 × 1201 grid, with an interval of 7.62 m in the horizontaland vertical directions, is used to partition the velocity model. Atotal of 1387 shots, with an interval of 22.86 m, are used to generatea synthetic data set. Each source has 540 receivers. The recordinglength is 9 s, and the sampling rate is 8 ms. The source and receiverdepths are 7.62 m. The source time function is a 15-Hz Rickerwavelet. Surface-related multiples can be attenuated by SRME.
Sea-bottom and salt boundaries are responsible for most internalmultiples, which can be predicted with equation 2. Compared inFigure 12 is a typical shot gather, in which in the left panel arethe original traces composed of primary reflections, surface-relatedmultiples, and internal multiples, and in the right panel are the pre-dicted internal multiples. By stacking shot gathers composed of in-ternal multiples only, a zero-offset profile is obtained and is shownin Figure 13. Applying prestack RTM to shot gathers of internalmultiples and using the imaging condition of equation 6, the DDimage is shown in Figure 14, which shows more balanced and con-tinuous interfaces in the sedimentary layers from horizontal coor-dinates 5 to 22 km. The images below the salt bodies are correct, butthey are contaminated by strong specular and nonspecular artifacts.Below the salt bodies, there are crossing events caused by internalmultiples generated inside the salt bodies. To attenuate these arti-facts, the UU image is calculated using equation 5, and the result isshown in Figure 15. Three salt bodies are correctly imaged. How-ever, due to the lack of strong deep interfaces to provide upwardillumination, the image in the subsalt region is weak comparedto the salt boundaries and can barely be seen. As discussed inthe “Methodology” section, the DD and UU images have similarpatterns for the real interfaces but different patterns for the artifacts.
Figure 8. A synthetic shot record is shown in panel (a), in a velocitymodel in which a high-velocity triangle is embedded in a two-layerbackground, shown in panel (b). The different arrivals are labeled inthe shot record.
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Figure 9. Illumination for the model shown in Figure 8, with the (a) total energy from the source wavefield, (b) total energy from the back-propagated receiver wavefield, (c) receiver wavefield energy related to the primary reflections, and (d) receiver wavefield energy related tomultiples.
Figure 10. RTM images from a single-shot record in the prism model, (a) from primary reflections only, (b) from internal multiples, (c) frominternal multiples with the UU image condition, and (d) with the DD image condition
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We use this feature and equation 11 to calculate the similarity be-tween the DD and UU images, and we use equation 15 to removethe unwanted artifacts. The result is shown in Figure 16. ComparingFigure 16 to Figure 14 and 15, certain artifacts in the subsalt regionare attenuated, e.g., those in the dashed ellipses. To further inves-tigate the details, we zoom in to different regions in the image. Theregions within the dashed boxes are shown in Figure 17a–17d. Inareas above the top salt boundaries (17a and 17b), the images arepromising due to strong internal multiples generated between theseafloor and the top salt boundaries. However, in Figure 17c, theimages are poor between the two salt bodies because there is notop salt boundary to generate internal multiples. In the area belowthe right salt body, the image is clear and easier to see, with mostartifacts properly removed. Certain artifacts can still be seen in thesubsalt area, partially due to the limitation of the multiple predictionprocess.
Figure 11. Velocity model for the SMAART Pluto data set.
Figure 12. A typical shot gather for the SMAARTPluto data set: (a) traces with primaries and inter-nal multiples and (b) the predicted internal multi-ples, in which the surface-related multiples havebeen removed.
Figure 13. Zero-offset profile obtained by stacking 1387 internalmultiple shot gathers.
Figure 14. Prestack RTM of the internal multiple data for theSMAART Pluto model. Shown here is the DD image.
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CONCLUSION
In regions in which primary reflections provide effective illumi-nation, satisfactory images can be obtained. However, in complexsubsurface environments such as the shadow zones below the saltbodies, illumination from primary reflections is limited and the im-age from internal multiples may provide vital information. We havedeveloped a new approach for internal multiple migration and sub-
salt imaging. The new method could properly migrate internal mul-tiples to their correct subsurface locations. In this process, weseparated the image into DD and UU parts, in which the true imageand the artifact behave differently. Then, by comparing their simi-larities, we separated the artifacts from the true images. Correctlypredicting various orders of internal multiples, particularly incomplicated velocity models, is vital to develop a robust migrationalgorithm.
Figure 15. Prestack RTM of internal multiples compare to the Fig-ure 14 image obtained using the UU image condition
Figure 16. Prestack RTM image of internal multiple data with ar-tifacts and crosstalk attenuated. Rectangular boxes mark the mag-nified areas detailed in Figure 17.
Figure 17. Magnified images: (a-d) correspond torectangular boxes A to D in Figure 16.
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ACKNOWLEDGMENTS
We thank Gerard T. Schuster and Yi Luo for helpful suggestionsand insightful comments. We would also like to acknowledge theassociate editor John Etgen and the reviewers, Arthur Weglein, JinLee, and one anonymous reviewer for their constructive commentsand suggestions, which greatly helped to improve this manuscript.We are grateful to SMAART JV consortium for offering SMARRTPluto data. The research was funded by the National Nature ScienceFoundation of China (grant nos. 41430321, 41374138), and alsosupported by Statoil through contract (grant no. 4502794579).
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