Vector-acoustic reverse time migration of Volve ocean-bottomcable data set without up/down decomposed wavefields

Matteo Ravasi1, Ivan Vasconcelos2, Andrew Curtis1, and Alexander Kritski3

ABSTRACT

Methods for wavefield injection are commonly used to extra-polate seismic data in reverse time migration (RTM). Injecting asingle component of the acoustic field, for example, pressure,leads to ambiguity in the direction of propagation. Each recordedwavefront is propagated both upward and downward, and spuri-ous (or ghost) reflectors are created alongside real reflectors in thesubsurface image. Thus, wavefield separation based on the com-bination of pressure and particle velocity data is generally per-formed prior to imaging to extract only the upgoing field frommulticomponent seabed or towed marine seismic recordings.By instead combining vector-acoustic (VA) data with monopole-

and dipole-type propagators in the extrapolation of shot orreceiver gathers, we show that wavefield separation (or deghost-ing) can instead be performed “on-the-fly” at limited additionalcost. This strategy was successfully applied to a line of a NorthSea ocean-bottom cable data set, acquired over the Volve field.We then evaluate additional advantages over standard RTMwithdecomposed fields such as improved handling of the directivityinformation contained in the acquired VA data for clearer shallowsections and better focused space-lag common image gathers, andimaging of the downgoing component without the need for addi-tional finite-difference modeling via mirror migration. Finally, weprove the robustness of our method with respect to sparse andirregular receiver sampling.

INTRODUCTION

Recent advances in seismic streamer technology allow for theacquisition of vector-acoustic (VA) measurements by means of de-vices that record pressure and multicomponent (e.g., vertical andcrossline) particle velocity or acceleration (Robertsson et al.,2008; Cambois et al., 2009). These new systems could representa breakthrough in the way we process and image seismic data. Geo-physical techniques that primarily focus on data-domain processingof such data have already been proposed for improved noise attenu-ation (Cambois et al., 2009), signal reconstruction (Vassallo et al.,2010), 3D deghosting (Özbek et al., 2010), or multiple attenuation(Frijlink et al., 2011). Novel imaging techniques that better honorthe physics of wave propagation through the use of multicomponentdata are emerging. The latter are the focus of this paper.Current standard acoustic migration algorithms, based on either

ray theory or a wave equation formulation (Biondi, 2006), use asingle input data type that is generally the recorded pressure data

or a preprocessed version of it, to create an image of the subsurface.If up/down wavefield separation (i.e., the weighted sum of pressureand velocity data , generally referred to as PZ summation — Barrand Sanders, 1989; Amundsen, 1993) is not included in the prepro-cessing, up- and downgoing (ghost) fields contribute to the gener-ation of real and spurious reflectors in the image, respectively.Ghost fields are in fact erroneously interpreted as upgoing eventsthat travel longer in the subsurface (to compensate for bounces atthe free surface of the water layer) and interact with the directlymodeled source wavefield at incorrect locations below the real re-flectors from which they actually originate. We refer to these, aswell as any other interferences between unrelated events that spa-tially and temporally coincide, as crosstalk.In the context of source-receiver interferometric imaging (Oris-

taglio, 1989; Halliday and Curtis, 2010), Vasconcelos (2013) for-mulates migration in a VA fashion and shows that VA reversetime migration (VARTM) can produce images deprived of ghostreflectors, even when taking as input non-decomposed data. The

Manuscript received by the Editor 24 November 2014; revised manuscript received 25 March 2015; published online 23 June 2015.1The University of Edinburgh, School of GeoSciences, Edinburgh, UK. E-mail: m.ravasi@sms.ed.ac.uk; andrew.curtis@ed.ac.uk.2Schlumberger Gould Research, Cambridge, UK. E-mail: ivasconc@gmail.com.3Statoil, Trondheim, Norway. E-mail: akr@statoil.com.© 2015 Society of Exploration Geophysicists. All rights reserved.

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GEOPHYSICS, VOL. 80, NO. 4 (JULY-AUGUST 2015); P. S137–S150, 12 FIGS.10.1190/GEO2014-0554.1

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decomposition step can be avoided because VA fields, combined bymeans of two-way representation theorems (Wapenaar, 2004; vanManen et al., 2005, 2006; Wapenaar and Fokkema, 2006), enabledirectional wavefield injection at the receiver surface, meaning thatany wavefront is back-propagated only toward the direction fromwhich it arrived (see also Mittet, 1994; Vasconcelos, 2011; Blanch,2012; Vasconcelos et al., 2012; Amundsen and Robertsson, 2014).Zhenhua and van der Baan (2014) use VARTM for microseismicevent localization. El Yadari and Hou (2013) and El Yadari (2015)adapt the VA framework to Kirchhoff and beam migration, respec-tively. Finally, Ravasi and Curtis (2013a) and Ravasi et al. (2014a)extend this methodology to elastic (e.g., land or hard-seabed) datasets, in which wave-mode (P- and S-wave) separation is also em-bedded within wavefield extrapolation (Mittet, 1994).Historically, seafloor seismic data acquisition has advanced ahead

of streamer acquisition in acquiring complementary data compo-nents. This was done partly with the aim of collecting convertedshear-wave (PS) data together with compressional (PP) data (Maver,2011) for improved imaging through gas clouds (Rønholt et al.,2008), density estimation (Leiceaga et al., 2010), and lithology char-acterization (Shahraeeni and Curtis, 2011).Moreover, given the avail-ability of multicomponent measurements at the seafloor, it is standardpractice to combine pressure and particle velocity measurements toattenuate the effect of strong free-surface multiples on the receiverside prior to acoustic (or PP) imaging (Soubaras, 1996; Osen et al.,1999; Schalkwijk et al., 1999; Muijs et al., 2004).Here, we apply VA migration to a receiver line of a North Sea 3D

ocean-bottom cable (OBC) field data set, acquired over the Volvefield (Szydlik et al., 2007) we focus primarily on evaluating its ef-fectiveness in suppressing downgoing waves (ghosts) at the receiverarray by directional injection of VA data as an alternative to wave-field decomposition schemes. Additional advantages of VARTMover standard RTM of decomposed waves such as improved han-dling of directivity information for clearer shallow sections and bet-ter focused space-lag common image gathers (CIGs), and imagingof the downgoing component of the recorded field without the needfor additional finite-difference modeling (via mirror VARTM) arealso demonstrated.The paper proceed as follows: We first briefly review the theory

of VA migration with special focus on the differences that arise inwavefield extrapolation of pressure and velocity data comparedwith standard extrapolation of pressure data only. We then discussthe key characteristics of our field data set and its preprocessing,and we present intermediate results (e.g., receiver wavefield andsingle-shot images) of the application of VARTM as well as multi-shot images and their corresponding interpretation. In so doing, sig-nificant emphasis is placed on identifying and tracking key arrivalsin the data that can be used to assess and illustrate the performanceof VA wavefield extrapolation.

A REVIEW OF VECTOR-ACOUSTIC MIGRATION

Define p and vn to be the time-reversed pressure and normal par-ticle velocity data recorded by multicomponent receivers, withvn ¼ v · n, where v is the particle velocity vector and n is the out-ward pointing normal vector at the recording surface. Similarly, qand fn refer to the propagators used to extrapolate the recorded databy means of monopole- and dipole-type (gradient) sources, respec-tively, where gradients fn are in the n-direction. The ⇒ symbol isused to define the back propagation of a data component (on the

left) with a specific source type (on the right), so p ⇒ q means thatrecorded pressure data are back-propagated with monopole-sourceGreen’s function propagators. When implemented via finite differ-ence methods, ⇒ can alternatively be seen as a boundary conditionthat allows a numerical injection scheme to inject the associateddata components (Robertsson and Chapman, 2000; Robertssonand Amundsen, 2014).Current practice RTM uses only monopole injection sources to

estimate the receiver wavefield (wRTMr ), independently of the choice

of the data to be injected. If the data are the full recorded pressurefield (for example, from single-component acquisition systems),then we write them as

wRTMr ∶p ⇒ q; (1)

and each wavefront injected in the finite-difference scheme propa-gates upward and downward from the injection points, irrespectiveof the direction from which it arrived, thus creating a wavefield ar-tifact for every physical arrival present in the data. Although extrapo-lation can be done with an absorbing boundary (in place of the freesurface) to attenuate the waves propagating upward, each downgoingcomponent of the data that is erroneously back-propagated down-ward instead of upward interacts with the source wavefield and gen-erates crosstalk artifacts in the image.Where multicomponent data are recorded, up/down wavefield

decomposition can be applied prior to imaging to suppress someof the events that are incorrectly handled by the injection procedurein equation 1. Thereafter, it is preferred to use only the upgoing fielddenoted as p− in the extrapolation step as

wu−RTMr ∶p− ⇒ q. (2)

Note that an image of the downgoing field denoted as pþ can alsobe constructed by means of mirror imaging (Grion et al., 2007) by,for example, injection of the ghost data as if it were recorded not onthe seabed but at a sea surface twice as high:

wd−RTMr ∶pþ ⇒ q. (3)

Shot-profile VARTM (see Appendix A for a vector-acoustic der-ivation of receiver-profile migration) solves the injection limitationsof standard RTM without requiring any preprocessing of the data(Vasconcelos, 2013). Simultaneous injection (indicated by the &symbol in the following equation) of the full pressure and (negative)normal particle velocity with two different types of injectionsources, dipole and monopole respectively, enables “on-the-fly”wavefield separation into the up- and downgoing components ofthe recorded field, with upgoing waves that are injected only down-ward and downgoing waves that back-propagate only upward:

wVARTMr ∶ðp ⇒ fnÞ& ð−vn ⇒ qÞ. (4)

Intuitively, the application of dipolar injection sources on pressuredata has the same effect of correcting for the different directions ofarrival of seismic events in the velocity gather (by means of the so-called obliquity factor) when wavefield separation is performed inthe f-k (or τ-p) domain.Independently of the type of wavefield extrapolation used to con-

struct the receiver field, before an image of the subsurface can beconstructed, another wavefield (the so-called source wavefield)

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needs to be computed by propagating forward in time an estimate ofthe source wavelet (s) into the migration model:

ws∶s ⇒ q. (5)

Note that here we use a monopole-type source, but with the recentintroduction of dual-source technology that combines pressure withgradient/dipole marine sources (Robertsson et al., 2012), handlingof ingoing and outgoing waves at the source array will also be pos-sible in the near future by combining the extrapolated receiver fieldstogether with monopole and dipole source side propagators as ex-plained by Vasconcelos (2013).An imaging condition is finally used to map the interaction of

source and receiver wavefields in the subsurface and create a rep-resentation of the earth’s physical property contrasts:

i∶ws � wr; (6)

where * denotes either zero-time crosscorrelation (Claerbout, 1971)or deconvolution (Guitton et al., 2007; Schleicher et al., 2008). Inthis paper, we adopt a crosscorrelation imaging condition.

VOLVE FIELD, OFFSHORE NORWAY

In 2002, a 3D seismic OBC survey was acquired over the Volvefield, offshore Norway, in the gas-/condensate-rich Sleipner area ofthe North Sea (Figure 1a). Volve is a small oil field with a dome-shaped structure formed by the collapse of adjacent salt ridges dur-ing the Jurassic period (Szydlik et al., 2007). Figure 1b shows thereference velocity model, together with the receiver and source linegeometry selected for this study. The receiver line is a 6-km-long 4Ccable placed on the seafloor approximately 92 m below the watersurface containing 235 receivers with an interval of 25 m. The sailline is 12 km long with a shot interval of 50 m.

Preprocessing and single-shot imaging

The acquired data were preprocessed by Statoil. Preprocessingincluded noise suppression, source designature, and vector-fidelitycorrections. Pressure (Figure 2a) and normal (that here we looselycall vertical because the seabed is nearly horizontal) particle veloc-ity (Figure 2b) have been further scaled by the square root of time totransform the data from 3D to 2D geometric spreading (Schalkwijket al., 1999). We also choose a window in the common-receivergather that contains only the direct wave, which is a downgoingfield at the recording array below the seabed (Dash et al., 2009), andwe calibrate the velocity component to the pressure by imposingthat the upgoing field should be zero inside that window (Muijset al., 2004). The VA injection of the calibrated fields is then equiv-alent to acoustic wavefield separation and injection below theseabed. The calibration filter (Figure 2e) has the effect of changingamplitude and phase of the calibrated velocity data (Figure 2c) withrespect to the recorded velocity data (Figure 2b): Note, for example,a slight change in the shape of the first arrival that improves thematching with that in the pressure data (top panel of Figure 2d)and the attenuation of lower frequencies (lower panel of Figure 2d).Figure 3 shows a time snapshot of the wavefields extrapolated

from the data in Figure 2. For illustration purposes, the pressure andvelocity terms of VA extrapolation in equation 4 are injected sep-arately and displayed in panels 3a and 3b. By focusing on the in-

jection of the first-order water-layer multiple (event indicated inFigures 2a and 3c), we see that each single term of VA extrapolationinjects this downgoing field upward and downward. When theseterms are combined together or when VA data are simultaneouslyinjected as shown in Figure 3c, VA extrapolation almost completelysuppresses the wavefront below the receiver surface and enhancesthat above, (i.e., in the direction from which this wavefront arrivedat the receiver array in the physical experiment). If further comparedwith standard extrapolation of the upgoing field (Figure 3d), thereceiver field obtained by means of VA extrapolation is also clearerin the proximity of the receiver array because of the proper handlingof local directionality information from gradients (i.e., from thevelocity data and dipole propagators). This is because standardextrapolation invokes a far-field radiation assumption (e.g., Hallidayand Curtis, 2010; Vasconcelos et al., 2010) that yields amplitude dis-tortions in the near-fields, especially if the curvature of the wavefrontapproaching the receiver array is not negligible (see Wapenaar andFokkema, 2006; Ramírez and Weglein, 2009).In fact, in the absence of particle velocity recordings or dipole

back-propagation sources, the pressure data should be scaled bya frequency-dependent factor that is a function of the angle of in-cidence of each recorded wavefront α, the local medium velocity c,and the angular frequency ω (i.e., −j ω

c cosðαÞ, where j is the imagi-nary unit). This is generally avoided by assuming that the recordedwavefronts are normal to the receiver array so that α ≈ 0 (Wapenaarand Fokkema, 2006; Thorbecke and Wapenaar, 2007). On the otherhand, to be able to perform VA extrapolation, a finite-differencecode that models the set of acoustic first-order partial differential

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equations for pressure and particle velocity (Fokkema and van denBerg, 1993) in a staggered-grid scheme is required. For this reason,the computational cost of such an approach is expected to be higherthan that of standard acoustic extrapolation because not only thepressure component of the extrapolated field (which is visualized

in Figure 3) but also the vertical and horizontal particle velocityfields must be computed and stored in memory.Single-shot images are then computed for the three different

choices of wavefield extrapolation in equations 1, 2, and 4: Cross-talk between the source wavefield and the incorrectly extrapolated

first-order water layer multiple in the receiverwavefield appears in the standard RTM image(indicated in the close-up of Figure 4a), but it issuppressed when VA injection is used (Figure 4b).This artifact is also not present in Figure 4c be-cause the up- and downgoing fields have beenproperly separated in the data domain. A secondstrong artifact appears in the shallow section ofthe standard RTM image; this is caused by theimproper handling of the second-order water-layer multiple.

Imaging of upgoing fields

Images are then computed for the 241 shotsand stacked together. Note that we use shot-pro-file VARTM to generate images throughout thepaper, but images of the same quality canalso be produced by means of receiver-profileVARTM as shown in Appendix A. The incor-rectly extrapolated downgoing events createstructure that is deeper than the real interfacesin the standard RTM image (dashed white arrowsin Figure 5a). Such ghost reflectors are signifi-cantly suppressed in the VARTM image (Fig-ure 5b) and the RTM image of upgoing fieldonly (Figure 5c). A small amount of residual en-ergy is, however, left around those locations:This is not due to inaccuracies in the wavefieldseparation or VA injection, but it is rather the ef-fect of source-side first-order multiples that gen-erally overlap with receiver-side ghosts in OBCdata (see Figure 1 in Xia et al., 2006). Moreover,differences in the quality of the images are visiblein the near surface as a consequence of invoking(RTM) or not (VARTM) a far-field radiationassumption in the extrapolation step.Close-ups of the near surface and two deeper

locations are shown in Figure 6: The maindifferences between the VARTM and upgoingRTM images are indeed in the shallow section,but as a consequence of the different handlingof wavefield curvature, subtle differences be-tween the two images exist also at depths wherethe reservoir of interest is located. However, eventhough most reflectors show better continuity inthe VARTM image, we note that this is not thecase for the reflector indicated by a black arrowin Figure 6b: The additional value provided byVARTM in the appraisal of the deeper structureis the subject of ongoing research.To further study the value of imaging with

VARTM with respect to standard RTM of eitherfull or upgoing pressure, space-lag CIGs (Rickettand Sava, 2002; Sava and Fomel, 2003) are com-

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Figure 3. Fixed-time snapshots of reverse time extrapolated receiver wavefields justafter the injection of the first-order seabed free-surface multiple (see Figure 2a). VAextrapolation of (a) full-pressure component, (b) full-velocity component, and (c) thesum of the two — equation 4. The VA receiver wavefield is compared with (d) standardextrapolation of the upgoing pressure field. The white line denotes the location of thereceiver surface at which the data are injected.

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Figure 4. Single-shot images. (a) RTM of the full-pressure field(equation 1), (b) VARTM (equation 4), and (c) the RTM of upgoingpressure field (equation 2). A line of black arrows pointing upwardindicates that upgoing waves are correctly migrated, and the ghostreflectors are indicated in the RTM close-up.

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Figure 5. Multishot images. (a) RTM of the full-pressure field(equation 1), (b) VARTM (equation 4), and (c) RTM of the upgoingpressure field (equation 2). A line of black arrows pointing upwardindicates that upgoing waves are correctly migrated, solid white ar-rows indicate physical reflectors, whereas the dashed white arrowspoint at ghost reflectors.

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puted as a function of depth at fixed surface coordinates (x ¼ 3, 4,5, 6, 7, 8, and 9 km). Incorrect handling of downgoing waves in RTMis reflected in additional energy in the CIG in Figure 7 that are gen-erally defocused from zero-space lag (white arrows) because they areback-propagated along incorrect wavepaths. Because defocusing fromzero space lag is the criterion used to indicate velocity inaccuracies(Biondi, 2006), including such energy may affect the convergenceof migration velocity analysis algorithms based on space-lag CIGs.As for images in Figure 5, VARTM (Figure 7b) and RTM of upgoingpressure (Figure 7c) construct space-lag CIGs that show better focus-ing around zero space lag from only energy related to upgoing wavesat the recording surface. In the shallow section (from 0 to 0.5 km),however, VARTM takes further advantage of the finite-frequency di-rectivity of the VA injection scheme and greatly attenuates events ofcircular shape that depart from zero space lag in RTM of upgoingwaves. As a consequence of this difference, we conjecture that veloc-ity analysis algorithms based on VARTM CIGs such as those in Fig-ure 7b could provide higher-resolution models of the near surface byincluding also the focusing information of the shallow section thatwould instead be muted out from the RTM CIGs.Extended images (EIs) (Vasconcelos et al., 2010; Sava and Vas-

concelos, 2011) have recently become an alternative tool velocityanalysis (Yang and Sava, 2011; Fleury, 2012) and/or for reservoircharacterization (Thomson, 2012; Vasconcelos and Rickett, 2013;

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Ravasi et al., 2015a). Instead of extending the imaging conditionalong the space or time axis individually, EIs extend the imagingcondition along the space and time axes by constructing an estimateof the data that would have been recorded by carrying a seismicsurvey around or at any location of interest in the subsurface(Vasconcelos et al., 2010; Loer et al., 2014). Figure 8 shows a com-parison between two different EIs at locations x ¼ f6; 2g km andx ¼ f6; 2.7g km with space extension of �1 km, and time exten-sion of 1 s only at positive times. RTM of the full-pressure fieldcreates EIs that are strongly affected by the cross-talk of downgoing waves. Similarly, f-k spectrashow that the ghost effect has propagatedthroughout the model through the receiver propa-gators and acts on the EI as well as on the re-corded data. VARTM cleans up the EIs, and theevents representing the reflection from the top ofthe reservoir (white arrow in the middle plots)can be seen clearly and picked for amplitudeanalysis. Comparison with the EIs from RTM ofthe upgoing field does not show any clear differ-ence from VARTM because the far-field assump-tion made by RTM algorithms holds for thosewaves that contribute to events in the EIs at thedepth level at which they are computed.

Mirror imaging of downgoing fields

Images in Figure 9 correspond to imaging us-ing only the receiver-side ghost fields via mirrorRTM and mirror VARTM, respectively, and com-plement those of Figure 5. To perform mirror im-aging, the free surface is removed and thereceiver extrapolation domain is extended to adepth of −4.5 km with a wavespeed model inthe negative depths that is a mirrored versionof the migration velocity model with respect tothe free surface (Vasconcelos, 2013). This allowsthe arrivals related to the ghost wavefield in Fig-ure 3b to continue propagating upward towardnegative depths; the final ghost receiver wave-field at negative depths is reflected to positivedepths prior to imaging using the same sourcewavefield as that used for the upgoing images.Mirror RTMof the full pressure data (Figure 9a)

creates ghost reflectors that are shallower than thereal interfaces: In fact, upgoing waves, instead ofbeing only back-propagated directly in the subsur-face as in standard RTM, also propagate errone-ously all the way from the receiver array to themirrored domain at negative depths, and this addi-tional path makes them coincide in time with thedirectly modeled source wavefield at shallower lo-cations than those at which they are physically re-flected. These spurious interfaces that are notpresent in the image from mirror RTM of thedowngoing pressure data (Figure 9c) are also suc-cessfully attenuated bymirrorVARTM(Figure 9b)meaning that the on-the-fly wavefield separationinto up- and downgoing waves is successful. Note,however, that, even though mirror VARTM prop-

erly maps first-order receiver-side ghosts in the subsurface, higherorder receiver-side free-surface multiples that are recorded as down-going waves at the receiver array generate spurious structure that isdeeper than the real interfaces in any of the images in Figure 9 asindicated by the dashed black arrows. These artifacts may be correctlyhandled only by novel migration algorithms in which the imagingcondition is modified to overcome the Born (single-scattering)assumption (e.g., Fleury and Vasconcelos, 2012; Ravasi and Curtis,2013b).

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Imaging of sparsely sampled receivers

Finally, we study the ability of VARTM to migrate data that aresparsely sampled along the receiver coordinate (Figure 10): First,we mimic a crossline acquisition for 3D OBC surveys by taking20 receivers with regular sampling of 250 m; we then further reducethe number of receivers to seven with an irregular space sampling asin ocean-bottom node (OBN) systems. As previously shown byVasconcelos (2013) on synthetic examples, we observe that the abil-ity of VA injection to decompose upgoing from downgoing is notcompromised by either coarse or irregular injection grids: Eventsthat are constructed by the interaction between the downgoing com-ponent of the recorded data and the source wavefield are in factsuccessfully suppressed by VARTM (Figure 10b and 10e) as wellas if the upgoing field is imaged by standard RTM (Figure 10c and10f). Moreover, the use of monopole and dipole back-propagationsources together with pressure and velocity data is again beneficialin producing clearer shallow images.We note that although approaches to up/down wavefield sepa-

ration acting in the frequency-wavenumber domain could be prob-lematic when the spatial sampling is irregular in the source andreceiver coordinates because standard implementations of the fastFourier transform assume data to be regularly sampled (Schone-wille, 2000), VARTM is not as affected by irregular sampling asthe former techniques in that it performs wavefield separation di-rectly in the time-space domain without the need for any Fouriertransform. On the other hand, because VA extrapolation is basedon the theory of representation theorems (Wapenaar, 2004; vanManen et al., 2005, 2006; Wapenaar and Fokkema, 2006), coarseor irregular sampling may lead to spurious energy in the recon-structed wavefields at depth in a similar fashion to seismic inter-ferometry (Mehta et al., 2008). Moreover, given the impossibilityof incorporating recordings of the particle velocity component thatis parallel to the acquisition surfaces in VA extrapolation, we onlyexpect VARTM to slightly reduce migration swinglike artifactsdue to aperture and subsampling by taking advantage of the fi-nite-frequency directivity contained in the normal particle veloc-ity. A more effective handling of the sampling issues may beachieved only if VARTM is applied to data that are previouslyinterpolated by, for example, matching pursuit techniques (Vas-sallo et al., 2010).To gain further insights into the accuracy of VA extrapolation

with respect to the receiver sampling, we now study a single planewave arriving at angle of incidence θ ¼ 11° (Figure 11a) withmaximum frequency fmax ¼ 40 Hz propagating in a constant-velocity medium (c ¼ 1500 m∕s) and extrapolate it in three differ-ent scenarios (Figure 11). Even in the extreme case of a singlereceiver recording (Figure 11b), the use of VA measurementsin wavefield injection discriminates the upgoing event from thedowngoing one, the latter being attenuated in the wavefieldextrapolated below the receiver (and almost completely sup-pressed at zero incidence). Nevertheless, the absence of neighbor-ing receivers prevents VA extrapolation from sending energy onlytoward the correct angle of incidence of the recorded plane wave:Constructive interference across groups of neighboring receiversas in Figure 11c and 11d is crucial to identify the angle of arrivaland send energy in the correct direction. It is important to note thatthis is also the case in the coarsely sampled receiver array(dr ¼ 200 m) in Figure 11c that does not satisfy the samplingcriterion (dr ≤ c∕ð2fmax sin θÞ ¼ 85 m). The uplift provided

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Figure 9. Mirror images. (a) Mirror RTM of the full-pressure field,(b) mirror VARTM, and (c) mirror RTM of the downgoing pressurefield. A line of black arrows pointing downward indicates that in thesecases, downgoing waves are those that are correctly migrated. Thesolid and dashed white arrows indicate physical and ghost reflectors,respectively, and additional dashed black arrows are used to point atspurious reflectors originated from the incorrect mapping in the sub-surface of second- and higher-order free-surface multiples.

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Figure 10. Imaging of sparse receiver acquisition mimicking a crossline in 3D OBC surveys (left) and OBNs systems (right). (a) RTM of thefull-pressure field, (b) VARTM, and (c) RTM of the upgoing pressure field. A line of black arrows pointing upward indicates that upgoingwaves are correctly migrated.

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by a finely sampled array of receivers is visible in Figure 11d inwhich a coherent plane wavefront is visible in the extrapolatedfields.

DISCUSSION

Source-receiver VARTM is an image-domain deghosting tech-nique that exploits the directionality information of combined pres-sure and vertical velocity recordings to extract the upgoing fieldequivalently to the effect of PZ summation (Amundsen, 1993; Sou-baras, 1996) in the data domain. The main difference between theseapproaches is in the handling of seismic events with different anglesof incidence: Data-domain deghosting balances the vertical compo-nent by first combining the information of neighboring receivers viaf-k (or τ-p) transform and subsequently applying a frequency-wavenumber correction factor, whereas VA extrapolation providesthe same balancing directly by injection of the pressure componentas a dipole source at all available receivers, thus without requiringany domain transform (f-k or τ-p) to be applied to the data.This difference can have a significant impact for 3D streamer and

ocean-bottom seismic acquisition, which are usually characterizedby fine sampling along the cable (inline direction) and by very

coarse sampling in the crossline direction, due to the availabilityof only a limited number of cables. Consequently, PZ summation,as well as other standard data-conditioning and processing tech-niques (e.g., the Radon transform or velocity analysis) that musthave access to unaliased representations of the wavefields becausethey are based on plane-wave decomposition, has to rely on addi-tional approximations and assumptions, such as assuming that theseismic events are linear in the crossline direction or even that thecrossline component of the angle of incidence is equal to zero (2Dpropagation approximation). By contrast, VA migration, which isbased on representation theorems that fully account for the 3Dnature of the recorded wavefield (Wapenaar, 2004; van Manenet al., 2005, 2006; Wapenaar and Fokkema, 2006; Curtis and Hal-liday, 2010; Halliday and Curtis, 2010; Vasconcelos, 2013), doesnot require any assumption about the direction of arrival of eventsin the crossline direction (also when poorly sampled or even whenslightly aliased). Although we hypothesize that this, together withcorrect handling of the finite-frequency directionality, could be-come even more evident when VA migration will be applied to thefull 3D OBC survey, a more in-depth study of the benefit of finite-frequency directionality on a 3D data set is the subject of currentresearch.

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Figure 11. Effect of receiver sampling on wavefield extrapolation of a single plane-wave arrival. (a) A plane wave with incidence angle θ ¼11° is emitted in the subsurface and recorded alongside its ghost reflected off the free surface by a horizontal array of 601 receivers with spacingof 5 m (black line). Panels (b-d) show the extrapolation of the full recorded data using one receiver, a line of 21 receivers with spacing of 200 m,and the entire receiver array, respectively. Each of the left panels represents fixed-time snapshots of standard and VA extrapolation with blackdots used to identify receivers emitting the recorded data, whereas the right panels display the extrapolated wavefields recorded along thedashed gray line in panel (a); the top two panels in each case are for standard monopole wavefield injection, and the bottom two are for VAextrapolation.

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Data-domain decomposition techniques focused on optimizingthe signal-to-noise ratio, such as optimal deghosting (ODG) (Özde-mir et al., 2008; Caprioli et al., 2012), not only cancel receiverghosts but also guarantee minimized residual noise. This is achievedat the cost of adding into the decomposition scheme informationabout the statistics of noise on pressure and velocity data togetherwith their respective ghost models. Even though the ODG solutionis very sensitive to any inaccuracies in the ghost models (e.g., cabledepth, a flat sea assumption, 2D processing), VA migration is a suit-able platform for the implementation of image-domain noise mit-igation strategies that directly weight the pressure and velocityreceiver wavefields or their corresponding prestack images (El Ya-dari et al., 2014) without having to rely on additional acquisitioninformation for the construction of a reliable ghost model.When 4C data are acquired, inline and crossline particle velocity

measurements contain useful additional information for imagingand inversion, as used by map migration methods (Kleyn, 1977)and stereo tomography (Billette and Lambaré, 1998). This informa-tion cannot be readily incorporated in source-receiver VA migrationfor marine acquisition that deploys horizontal streamers and forocean-bottom acquisition with flat, horizontal seabeds: Becausethe normal vector is directed along the vertical direction at the re-cording surface, the vertical particle velocity in fact becomes theonly required component. Although slanted cables (Soubaras andDowle, 2010) could represent a solution in streamer acquisitionto incorporate the horizontal components in source-receiver VA mi-gration, Fleury and Vasconcelos (2013) propose an adjoint-state VAformulation that can incorporate any recording available; this is ul-timately a formulation for reverse time map migration, also repre-senting the starting point for VA full-waveform inversion.Finally, it is important to note that even though VA injection (as

well as its data-domain counterparts — PZ summation, ODG) rep-resents an effective strategy for attenuating water-layer multiples(multiples that never leave the water column) and water-layerpeg legs that are recorded at the receiver array as downgoing waves,this approach does not address all of the effects of the free surface inocean-bottom data. More specifically, source-side ghosts and anyother type and order of free-surface multiples reaching the receiverarray as upgoing waves generate artifacts in the final image. Althoughno attempt to remove these events from the recorded data has beenmade in our field data test, there are a variety of approaches that couldbe applied alongside VARTM to partially or fully mitigate the effectsof the free surface (before VARTM in the prestack domain or afterVARTM in the postimaging domain).First, the source ghost can be removed from the pressure and

velocity recordings as part of the wavelet deconvolution processat an early stage in the processing chain, or directly from the imageafter migration (Caprioli et al., 2014) when data are acquired usingonly monopole sources. If over/under or dual-source data are avail-able, the source ghost attenuation can be performed either in theprestack domain by means of wavefield decomposition techniques(Moldoveanu, 2000; Egan et al., 2007) or directly while imaging byinvoking an imaging condition that combines dual-source data to-gether with dual-source forward propagators or source wavefields(Vasconcelos, 2013).Other types of free-surface multiples (those that do not experi-

ence their final set of multiple bounces only within the water col-umn before being recorded) could be attenuated by adapting theconcept of surface-related multiple elimination (e.g., Verschuur

et al., 1992; Weglein et al. 1997) from streamer geometries toocean-bottom geometries, in which the differences between sourceand receiver depth levels are not negligible and need to be compen-sated for. The key idea behind these methods is to construct a wave-field prediction operator to compensate for the elevation difference.This can be done either analytically (Matson and Weglein, 1996;Pica et al., 2005, 2006), or by making use of the direct wave asthe wavefield continuation operator (Ikelle, 1999). Because the lat-ter procedures are not exclusively valid for pressure data, we foreseethat their application to the pressure and velocity fields could pro-vide a modified input data set for VARTM deprived of free-surfacemultiples. Removal of the residual receiver-side ghosts and carefulhandling of the finite-frequency directionality contained in thevelocity field will be achieved by VARTM.Alternatively, the effects of the free surface can be attenuated

all at once by deconvolving the upgoing field with the downgoingfield (Amundsen, 2001; Wang et al., 2009). If the finite-frequencydirectivity of the retrieved velocity field has not been affectedby the deconvolution process (Ravasi et al., 2015b), VARTMof these fields could still be beneficial in terms of improvingthe imaging of the shallow subsurface. An attractive time-spacedomain extension of this demultiple approach has been recentlyproposed by Vasmel et al. (2014) based on time-domain finite-difference propagators. Their approach relies on the ability to in-ject VA field data to separate up- and downgoing fields as shownin this paper.The suppression of free-surface multiples could also be post-

poned to the imaging domain if the full downgoing wavefield isforward-propagated as the source wavefield (Muijs et al., 2007;Whitmore et al., 2010; Lu et al., 2011; Amundsen and Robertsson,2014) and combined with the backward propagated upgoing wave-field by means of a modified imaging condition, the latter being adeconvolution imaging condition (Guitton et al., 2007; Schleicheret al., 2008). The ability to separate upgoing from downgoing waveswhen VA data are opportunely injected along the receiver array isnot limited to back propagation (or propagation backward in time):The downgoing component can be propagated downward (and theupgoing component upward) by injecting the full recorded VA dataforward in time and interchanging the sign of equation 4 (Robertsonand Chapman, 2000; van Manen et al., 2007; Amundsen and Rob-ertsson, 2014; Vasconcelos et al., 2014; Ravasi et al., 2015c).

CONCLUSIONS

We have developed a workflow for the application of VARTMto a field data set. By ensuring the matching of direct waves in thepressure and vertical velocity components by using a calibrationfilter, and by injecting the full recorded pressure and verticalvelocity data in a VA fashion, up/down separation is performedas part of wavefield extrapolation. The main benefit of VARTMover standard RTM of separated wavefields lies in the overallbetter handling of amplitudes by using a correct combination ofmonopole- and dipole-type propagators, thus resulting in clearershallow sections and better-focused space-lag CIGs. Moreover,up- and downgoing fields can be jointly imaged without the needfor separate finite-difference modeling, and their separation is alsoshown to be successful in the presence of sparse or irregularreceiver sampling.

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ACKNOWLEDGMENTS

The authors thank the Edinburgh Interferometry Project sponsors(ConocoPhillips, Schlumberger Cambridge Research, Statoil, andTotal) for supporting this research. We would like to thank StatoilASA and the Volve license partners ExxonMobil E&P Norway andBayerngas Norge, for the release of the Volve data. We are alsograteful to N. El Yadari, J. Rickett, and N. El Allouche for fruitfuldiscussions. Finally, we greatly appreciate the constructive com-ments of assistant editor J. Shragge, associate editor F. Liu, J. Rob-ertsson, R. van Borselen, and two anonymous reviewers that helpedto improve the manuscript.

APPENDIX A

RECEIVER PROFILE VARTM

When data are acquired with ocean-bottom acquisition systems,the number of sources may largely exceed that of receivers. More-over, OBCs are poorly sampled in the crossline direction and OBNsare coarsely and nonuniformly located across the whole survey area.Prestack imaging of ocean-bottom acquisition can alternatively beperformed in the common-receiver domain: In terms of processing,the idea of reciprocity is applied, meaning that the common-receiverdata are injected along the source array, whereas the source waveletis injected at the common-receiver location.To be able to perform receiver-profile VARTM, two receiver

wavefields are separately computed by injecting the common-receiver pressure and velocity data as monopole-type sources:

wp−VARTMr ∶p ⇒ q; wvn−VARTM

r ∶vn ⇒ q. (A-1)

Similarly, two source wavefields are generated, one from a mo-nopole source and the other from a dipole source:

wq−VARTMs ∶s ⇒ q; wfn−VARTM

s ∶s ⇒ fn. (A-2)

The resulting four wavefields are finally combined in the imagingcondition:

iVARTM∶wfn−VARTMs � wp−VARTM

r − wq−VARTMs � wvn−VARTM

r .

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The proper handling of up- and downgoing waves results in twoimages that show ghost reflectors with the same polarity and truereflectors with reverse polarity, such that their difference enhancesthe true reflectors and suppresses the ghost reflectors (Figure A-1).For a comparison with shot-profile VARTM, see Figure 5b.Note that even though shot-profile VARTM has a similar cost as

shot- and receiver-profile RTM (i.e., two forward modeling steps),the cost doubles in receiver-profile VARTM (i.e., four forward-modeling steps), meaning that it is convenient to switch fromshot-profile to receiver-profile VARTM only when the number ofsources is double the number of receivers.

REFERENCES

Amundsen, L., 1993, Wavenumber-based filtering of marine point sourcedata: Geophysics, 58, 1335–1348, doi: 10.1190/1.1443516.

Amundsen, L., 2001, Elimination of free-surface related multiples withoutneed of the source wavelet: Geophysics, 66, 327–341, doi: 10.1190/1.1444912.

Amundsen, L., and J. O. A. Robertsson, 2014, Wave equation processingusing finite-difference propagators, Part 1: Wavefield dissection and im-aging of marine multicomponent seismic data: Geophysics, 79, no. 6,T287–T300, doi: 10.1190/geo2014-0151.1.

Barr, F. J., and J. I. Sanders, 1989, Attenuation of water-column multiplesusing pressure and velocity detectors in a water-bottom cable: 59th An-nual International Meeting, SEG, Expanded Abstracts, 653–656.

Billette, F., and G. Lambaré, 1998, Velocity macro-model estimation fromseismic reflection data by stereotomography: Geophysical JournalInternational, 135, 671–690, doi: 10.1046/j.1365-246X.1998.00632.x.

Biondi, B., 2006, 3D seismic imaging: SEG.Blanch, J., 2012, Is de-ghosting necessary with multi-component data?:

82nd Annual International Meeting, SEG, Expanded Abstracts, doi: 10.1190/segam2012-1094.1.

Cambois, G., D. Carlson, C. Jones, M. Lesnes, W. Söllner, and H. Tabti,2009, Dual-sensor streamer data: Calibration, acquisition QC and attenu-ation of seismic interferences and other noises: 79th Annual InternationalMeeting, SEG, Expanded Abstracts, 142–146.

Caprioli, P. B. A., X. Du, R. P. Fletcher, and I. Vasconcelos, 2014, 3D sourcedeghosting after imaging: 84th Annual International Meeting, SEG, Ex-panded Abstracts, 4092–4096.

Caprioli, P. B. A., A. K. Özdemir, A. Özbek, J. E. Kragh, D. J. van Manen, P.A. F. Christie, and J. O. A. Robertsson, 2012, Combination of multi-com-ponent streamer pressure and vertical particle velocity-theory and appli-cation to data: 74th Annual International Conference and Exhibition,EAGE, Extended Abstracts, A033.

Claerbout, J. F., 1971, Toward a unified theory of reflector mapping: Geo-physics, 36, 467–481, doi: 10.1190/1.1440185.

Curtis, A., and D. Halliday, 2010, Source-receiver wave field interferometry:Physical Review E, 81, 046601, doi: 10.1103/PhysRevE.81.046601.

Dash, R., G. Spence, R. Hyndman, S. Grion, Y. Wang, and S. Ronen, 2009,Wide-area imaging from OBS multiples: Geophysics, 74, no. 6, Q41–Q47, doi: 10.1190/1.3223623.

Egan, M., K. G. El-Kassehand, and N. Moldoveanu, 2007, Full deghostingof OBC data with over/under source acquisition: 77th AnnualInternational Meeting, SEG, Expanded Abstracts, 31–35.

El Yadari, N., 2015, True-amplitude vector-acoustic imaging: Application ofGaussian beams: Geophysics, 80, no. 1, S43–S54, doi: 10.1190/geo2014-0316.1.

El Yadari, N., and S. Hou, 2013, Vector-acoustic Kirchhoff migration and itsmirror extension — Theory and examples: 75th Annual InternationalConference and Exhibition, EAGE, Extended Abstracts, Tu P06 13.

wsfz VARTM wr

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Figure A-1. Receiver-profile VARTM image for migration of up-going waves. A line of black arrows pointing upward indicates thatupgoing waves are correctly migrated.

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El Yadari, N., Y. I. Kamil, and M. Vassallo, 2014, Wavefield extrapolationand imaging using single- or multi-component seismic measurements:U.S. Patent application US20140200820.

Fleury, C., 2012, Migration velocity analysis for nonlinear reverse-time mi-gration: 82nd Annual International Meeting, SEG, Expanded Abstracts,doi: 10.1190/segam2012-0536.1.

Fleury, C., and I. Vasconcelos, 2012, Imaging condition for nonlinear scat-tering-based imaging: Estimate of power loss in scattering: Geophysics,77, no. 1, S1–S18, doi: 10.1190/geo2011-0135.1.

Fleury, C., and I. Vasconcelos, 2013, Adjoint-state reverse-time migration of4C data: Finite frequency map migration for marine seismic imaging:Geophysics, 78, no. 2, WA159–WA172, doi: 10.1190/geo2012-0306.1.

Fokkema, J., and P. van den Berg, 1993, Seismic applications of acousticreciprocity: Elsevier Science.

Frijlink, M., R. van Borselen, and W. Söllner, 2011, The free surfaceassumption for marine data-driven demultiple methods: Geophysical Pro-specting, 59, 269–278, doi: 10.1111/j.1365-2478.2010.00914.x.

Grion, S., R. Exley, M. Manin, X. Miao, A. L. Pica, Y.Wang, P. Granger, andS. Ronen, 2007, Mirror imaging of OBS data: First Break, 25, 37–42.

Guitton, A., A. Valenciano, D. Bevc, and J. Claerbout, 2007, Smoothingimaging condition for shot-profile migration: Geophysics, 72, no. 3,S149–S154, doi: 10.1190/1.2712113.

Halliday, D., and A. Curtis, 2010, An interferometric theory of source-receiver scattering and imaging: Geophysics, 75, no. 6, SA95–SA103,doi: 10.1190/1.3486453.

Ikelle, L. T., 1999, Using even terms of the scattering series for deghostingand multiple attenuation of ocean-bottom cable data: Geophysics, 64,579–592, doi: 10.1190/1.1444565.

Kleyn, A., 1977, On the migration of reflection time contour maps: Geo-physical Prospecting, 25, 125–140, doi: 10.1111/j.1365-2478.1977.tb01157.x.

Leiceaga, G., J. Silva, F. Artola, E. Marquez, and J. Vanzeler, 2010, En-hanced density estimation from prestack inversion of multicomponent seis-mic data: The Leading Edge, 29, 1220–1226, doi: 10.1190/1.3496912.

Loer, K., G. A. Meles, A. Curtis, and I. Vasconcelos, 2014, Diffracted andpseudo-physical waves from spatially-limited arrays using source-receiver interferometry (SRI): Geophysical Journal International, 196,1043–1059, doi: 10.1093/gji/ggt435.

Lu, S., N. D. Whitmore, A. A. Valenciano, and N. Chemingui, 2011, Im-aging of primaries and multiples with 3D SEAM synthetic: 81st AnnualInternational Meeting, SEG, Expanded Abstracts, 3217–3221.

Matson, K., and A. B. Weglein, 1996, Removal of elastic interface multiplesfrom land and ocean-bottom data using inverse scattering: 66th AnnualInternational Meeting, SEG, Expanded Abstracts, 1526–1530.

Maver, K. G., 2011, Ocean bottom seismic: Strategic technology for the oilindustry: First Break, 29, 75–80.

Mehta, K., R. Snieder, R. Calvert, and J. Sheiman, 2008, Acquisition geom-etry requirements for generating virtual-source data: The Leading Edge,27, 620–629, doi: 10.1190/1.2919580.

Mittet, R., 1994, Implementation of the Kirchhoff integral for elastic wavesin staggered-grid modelling schemes: Geophysics, 59, 1894–1901, doi:10.1190/1.1443576.

Moldoveanu, N., 2000, Vertical source array in marine seismic exploration:70th Annual International Meeting, SEG, Expanded Abstracts, 53–56.

Muijs, R., J. O. A. Robertsson, and K. Holliger, 2004, Data-driven adaptivedecomposition of multicomponent seabed recordings: Geophysics, 69,1329–1337, doi: 10.1190/1.1801949.

Muijs, R., J. O. A. Robertsson, and K. Holliger, 2007, Prestack depth mi-gration of primary and surface-related multiple reflections: Part I — Im-aging: Geophysics, 72, no. 2, S59–S69, doi: 10.1190/1.2422796.

Oristaglio, M. L., 1989, An inverse scattering formula that uses all the data:Inverse Problems, 5, 1097–1105, doi: 10.1088/0266-5611/5/6/015.

Osen, A., L. Amundsen, and A. Reitan, 1999, Removal of water-layer multi-ples from multicomponent sea-bottom data: Geophysics, 64, 838–851,doi: 10.1190/1.1444594.

Özbek, A., M. Vassallo, D.-J. van Manen, and K. Eggenberger, 2010, Cross-line wavefield reconstruction from multicomponent streamer data: Part 2— Joint interpolation and 3D up/down separation by generalized match-ing pursuit: Geophysics, 75, no. 6, WB69–WB85, doi: 10.1190/1.3497316.

Özdemir, K., P. Caprioli, A. Özbek, E. Kragh, and J. Robertsson, 2008, Op-timized deghosting of over/under towed-streamer data in the presence ofnoise: The Leading Edge, 27, 190–199, doi: 10.1190/1.2840366.

Pica, A. L., M. Manin, P. Y. Granger, D. Marin, E. Suaudeau, B. David, G.Poulain, and P. H. Herrmann, 2006, 3D SRME on OBS data using wave-form multiple modelling: 76th Annual International Meeting, SEG, Ex-panded Abstracts, 2659–2663.

Pica, A. L., G. Poulain, B. David, M. Magesan, S. Baldock, T. Weisser, P.Hugonnet, and P. H. Herrmann, 2005, 3D surface-related multiple mod-eling, principles and results: 75th Annual International Meeting, SEG,Expanded Abstracts, 2080–2083.

Ramírez, A. C., and A. B. Weglein, 2009, Greens theorem as a comprehen-sive framework for data reconstruction, regularization, wavefield separa-tion, seismic interferometry, and wavelet estimation: A tutorial:Geophysics, 74, no. 6, W35–W62, doi: 10.1190/1.3237118.

Ravasi, M., and A. Curtis, 2013a, Elastic imaging with exact wavefieldextrapolation for application to ocean bottom 4C seismic data: Geophys-ics, 78, no. 6, S265–S284, doi: 10.1190/geo2013-0152.1.

Ravasi, M., and A. Curtis, 2013b, Nonlinear scattering based imaging inelastic media: Theory, theorems and imaging conditions: Geophysics,78, no. 3, S137–S155, doi: 10.1190/geo2012-0286.1.

Ravasi, M., G. Meles, A. Curtis, Z. Rawlinson, and L. Yikuo, 2015b, Seis-mic interferometry by multi-dimensional deconvolution without wave-field separation: Geophysical Journal International, 202, 1–16, doi: 10.1093/gji/ggv062.

Ravasi, M., I. Vasconcelos, and A. Curtis, 2014a, Beyond conventional mi-gration: Nonlinear elastic subsalt imaging with transmissions and two-sided illumination: Geophysical Journal International, 198, 1187–1199,doi: 10.1093/gji/ggu192.

Ravasi, M., I. Vasconcelos, A. Curtis, and A. Kritski, 2015c, A practicalapproach to vector-acoustic imaging of primaries and free-surface multi-ples: 77th Annual International Conference and Exhibition, EAGE, Ex-panded Abstracts, doi: 10.3997/2214-4609.201413514.

Ravasi, M., I. Vasconcelos, A. Curtis, and G. Meles, 2015a, Elastic extendedimages and velocity-sensitive objective function using multiple reflectionsand transmissions: Geophysical Journal International, in Press.

Rickett, J., and P. Sava, 2002, Offset and angle-domain common image-point gathers for shot-profile migration: Geophysics, 67, 883–889, doi:10.1190/1.1484531.

Robertsson, J. O. A., and L. Amundsen, 2014, Wave equation processingusing finite-difference propagators, Part 2: Deghosting of marine hydro-phone seismic data: Geophysics, 79, no. 6, T301–T312, doi: 10.1190/geo2014-0152.1.

Robertsson, J., and C. Chapman, 2000, An efficient method for calculatingfinite-difference seismograms after model alterations: Geophysics, 65,907–918, doi: 10.1190/1.1444787.

Robertsson, J. O. A., D. F. Halliday, D. J. van Manen, I. Vasconcelos, R. M.Laws, K. Ozdemir, and H. Gronaas, 2012, Full-wavefield, towed-marineseismic acquisitions and applications: 74th Annual InternationalConference and Exhibition, EAGE, Extended Abstracts, doi: 10.3997/2214-4609.20148849.

Robertsson, J. O. A., I. Moore, M. Vassallo, K. Özdemir, D.-J. van Manen,and A. Özbek, 2008, On the use of multicomponent streamer recordingsfor reconstruction of pressure wavefields in the crossline direction: Geo-physics, 73, no. 5, A45–A49, doi: 10.1190/1.2953338.

Rønholt, G., H. A. Aronsen, T. Hellmann, and S. Johansen, 2008, Improvedimaging of the Snøhvit field through integration of 4C OBC and dual-azi-muth streamer seismic data: First Break, 26, 61–66.

Sava, P., and S. Fomel, 2003, Angle-domain common-image gathers bywavefield continuation methods: Geophysics, 68, 1065–1074, doi: 10.1190/1.1581078.

Sava, P., and I. Vasconcelos, 2011, Extended imaging condition for wave-equation migration: Geophysical Prospecting, 59, 35–55, doi: 10.1111/j.1365-2478.2010.00888.x.

Schalkwijk, K. M., C. P. A. Wapenaar, and D. J. Verschuur, 1999,Application of two-step decomposition to multicomponent ocean-bottom data: Theory and case study: Journal of Seismic Exploration,8, 261–278.

Schleicher, J., J. C. Costa, and A. Novais, 2008, A comparison of imagingconditions for wave-equation shot profile migration: Geophysics, 73,no. 6, S219–S227, doi: 10.1190/1.2976776.

Schonewille, M., 2000, Fourier reconstruction of irregularly sampled seis-mic data: Ph.D. thesis, Delft University of Technology.

Shahraeeni, M., and A. Curtis, 2011, Fast probabilistic nonlinear petrophys-ical inversion: Geophysics, 76, no. 2, E45–E58, doi: 10.1190/1.3540628.

Soubaras, R., 1996, Ocean-bottom hydrophone and geophone processing:66th Annual International Meeting, SEG, Expanded Abstracts, 24–27.

Soubaras, R., and R. Dowle, 2010, Variable depth streamer— A broadbandmarine solution: First Break, 28, 89–96.

Szydlik, T., P. Smith, S. Way, L. Aamodt, and C. Friedrich, 2007, 3D PP/PSprestack depth migration on the Volve field: First Break, 25, 43–47.

Thomson, C., 2012, The form of the reflection operator and extendedimage gathers for a dipping plane reflector in two space dimensions:Geophysical Prospecting, 60, 64–92, doi: 10.1111/j.1365-2478.2011.00971.x.

Thorbecke, J., and K. Wapenaar, 2007, On the relation between seismicinterferometry and the migration resolution function: Geophysics, 72,no. 6, T61–T66, doi: 10.1190/1.2785850.

Van Manen, D.-J., A. Curtis, and J. O. A. Robertsson, 2006. Interferometricmodeling of wave propagation in inhomogeneous elastic media usingtime reversal and reciprocity: Geophysics, 71, no. 4, SI47–SI60, doi:10.1190/1.2213218.

Vector-acoustic RTM of Volve S149

Dow

nloa

ded

07/0

5/15

to 2

17.1

55.2

05.1

31. R

edis

trib

utio

n su

bjec

t to

SEG

lice

nse

or c

opyr

ight

; see

Ter

ms

of U

se a

t http

://lib

rary

.seg

.org

/

Van Manen, D.-J., J. Robertsson, and A. Curtis, 2007. Exact wavefield sim-ulation for finite-volume scattering problems: Journal of the AcousticalSociety of America, 122, EL115–EL121, doi: 10.1121/1.2771371.

Van Manen, D.-J., J. O. A. Robertsson, and A. Curtis, 2005, Modeling ofwave propagation in inhomogeneous media: Physical Review Letters, 94,164301, doi: 10.1103/PhysRevLett.94.164301.

Vasconcelos, I., 2011, Source-receiver reverse-time imaging of vector-acoustic seismic data: 81st Annual International Meeting, SEG, ExpandedAbstracts, 3184–3189.

Vasconcelos, I., 2013, Source-receiver reverse-time imaging of dual-source,vector-acoustic seismic data: Geophysics, 78, no. 2, WA123–WA145, doi:10.1190/geo2012-0300.1.

Vasconcelos, I., M. Ravasi, and J. van der Neut, 2014, An interferometry-based, subsurface-domain objective function for waveform inversion:76th Annual International Conference and Exhibition, EAGE, ExtendedAbstracts, We 106 11.

Vasconcelos, I., and J. Rickett, 2013, Broadband extended images by jointinversion of multiple blended wavefields: Geophysics, 78, no. 2, WA147–WA158, doi: 10.1190/geo2012-0475.1.

Vasconcelos, I., J. Robertsson, M. Vassallo, and D. J. van Manen, 2012,Reverse-time imaging of dual-source 4C marine seismic data using pri-maries, ghosts, and multiples: 74th Annual International Conference andExhibition, EAGE, Extended Abstracts, X045.

Vasconcelos, I., P. Sava, and H. Douma, 2010, Nonlinear extended imagesvia image-domain interferometry: Geophysics, 75, no. 6, SA105–SA115,doi: 10.1190/1.3494083.

Vasmel, M., J. O. A. Robertsson, and L. Amundsen, 2014, A new solution toeliminate free surface related multiples in multicomponent streamer re-cordings: 76th Annual International Conference and Exhibition, EAGE,Extended Abstracts, Tu E102 01.

Vassallo, M., A. Özbek, K. Özdemir, and K. Eggenberger, 2010, Crosslinewavefield reconstruction from multicomponent streamer data: Part 1 —

Multichannel interpolation by matching pursuit (MIMAP) using pressureand its crossline gradient: Geophysics, 75, no. 6, WB53–WB67, doi: 10.1190/1.3496958.

Verschuur, D., A. Berkhout, and C. Wapenaar, 1992, Adaptive surface-re-lated multiple elimination: Geophysics, 57, 1166–1177, doi: 10.1190/1.1443330.

Wang, Y., S. Grion, and R. Bale, 2009, What comes up must have gonedown: The principle and application of up-down deconvolution for multi-ple attenuation of ocean bottom data: CSEG Recorder, 34, 16–20.

Wapenaar, K., 2004, Retrieving the elastodynamic Green’s function of anarbitrary inhomogeneous medium by cross correlation: Physical ReviewLetters, 93, 254301, doi: 10.1103/PhysRevLett.93.254301.

Wapenaar, K., and J. Fokkema, 2006, Green’s function representations forseismic interferometry: Geophysics, 71, no. 4, SI33–SI46, doi: 10.1190/1.2213955.

Weglein, A., F. Gasparotto, P. Carvalho, and R. Stolt, 1997, An inverse-scat-tering series method for attenuating multiples in seismic reflection data:Geophysics, 62, 1975–1989, doi: 10.1190/1.1444298.

Whitmore, N. D., A. A. Valenciano, W. Sollner, and S. Lu, 2010, Imaging ofprimaries and multiples using a dual-sensor towed streamer, 80th AnnualInternational Meeting, SEG, Expanded Abstracts, 3187–3192.

Xia, G., R. Clarke, J. Etgen, N. Kabir, K. Matson, and S. Michell, 2006,OBSmultiple attenuation with application to the deepwater GOMAtlantisOBS nodes data: 76th Annual International Meeting, SEG, Expanded Ab-stracts, 2654–2658.

Yang, T., and P. Sava, 2011, Waveform inversion in the image domain: 73rdAnnual International Conference and Exhibition, EAGE, Extended Ab-stracts, F028.

Zhenhua, L., and M. van der Baan, 2014, Acoustic representation theorembased time-reversal-extrapolation for microseismic event localization:84th Annual International Meeting, SEG, Expanded Abstracts, 2240–2244.

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ight

; see

Ter

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://lib

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