Flux-normalized wavefield decomposition and migration of seismic data

Transcription

1 GEOPHYSICS. VOL. 77, NO. (MAY-JUNE 0); P. S S9, FIGS. 0.90/GEO0-0. Flx-normalized wavefield decomposition and migration of seismic data Bjørge Ursin, Ørjan Pedersen, and Børge Arntsen ABSTRACT Separation of wavefields into directional components can be accomplished by an eigenvale decomposition of the accompanying system matrix. In conventional pressrenormalized wavefield decomposition, the reslting one-way wave eqations contain an interaction term which depends on the reflectivity fnction. Applying directional wavefield decomposition sing flx-normalized eigenvale decomposition, and disregarding interaction between p- and downgoing wavefields, these interaction terms were absent. By also applying a correction term for transmission loss, the reslt was an improved estimate of the p- and downgoing wavefields. In the wave eqation angle transform, a crosscorrelation fnction in local offset coordinates was Forier-transformed to prodce an estimate of reflectivity as a fnction of slowness or angle. We normalized this wave eqation angle transform with an estimate of the planewave reflection coefficient. The flx-normalized oneway wave-propagation scheme was applied to imaging and to the normalized wave eqation angle-transform on synthetic and field data; this proved the effectiveness of the new methods. INTRODUCTION Accrate wavefield traveltimes and amplitdes can be described sing two-way wave eqation techniqes like finite-difference or finite-element methods. However, these methods can often be significantly more comptationally expensive compared to one-way methods. The comptational cost in modeling wavefield extrapolation sing fll wave eqation methods can become a limitation for D applications in particlar. Ray methods based pon asymptotic theory provide effective alternatives to fll wave eqation methods; however, their high-freqency approximations restrict their se in complex sbsrface velocity. One-way wavefield methods based pon a paraxial approximation of the wave eqation provide a robst and comptationally cheap alternative approach for solving the wave eqation. With wavefield propagators based on one-way methods, we can sbstantially increase the speed of comptations compared to fll wavefield methods. Representation of a wavefield sing the one-way wave eqation permits separation of the wavefield into p- and downgoing constitents. This separation is not valid for near-horizontal propagating waves. Schemes for splitting the wave eqation into p- and downgoing parts and seismic mapping of reflectors are discssed by Claerbot (970, 97). Several athors have investigated varios methods for amplitde correction to one-way wave eqations. hang et al. (00, 005, 007) address tre-amplitde implementation of one-way wave eqations in common-shot migration by modifying the one-way wave eqation. This is accomplished by introdcing an axiliary fnction that corrects the leading order transport eqation for the fll wave eqation. Ray theory applied to the modified one-way wave eqations yields p- and downgoing eikonal eqations with amplitdes satisfying the transport eqation. Fll waveform soltions sbstitted with corresponding ray-theoretical approximations provides tre-amplitde in the sense that the imaging formlas redce to a Kirchhoff common-shot inversion expression. Kiyashchenko et al. (005) develop improved estimation of amplitdes sing a mlti-one-way approach. It is developed from an iterative soltion of the factorized two-way wave eqation with a right side incorporating the medim heterogeneities. It allows for vertical and horizontal velocity variations and it is demonstrated that the mlti-one-way scheme redces errors in amplitde estimates compared to conventional one-way propagators. Cao and W (00) reformlate the soltion of the one-way wave eqation in smoothly varying D media based on energy-flx conservations. By introdcing transparent bondary conditions and transparent propagators, their formlation is extended to a general heterogeneos media in the local angle domain tilizing beamlet methods. Manscript received by the Editor 0 Jne 0; revised manscript received October 0; pblished online April 0. Norwegian University of Science and Technology, Department of Petrolem Engineering and Applied Geophysics, Trondheim, Norway. bjorn.rsin@ntn.no; borge.arntsen@ntn.no. Statoil ASA, Arkitekt Ebbels, Trondheim, Norway. orpe@statoil.com. 0 Society of Exploration Geophysicists. All rights reserved. S Downloaded Oct 0 to Redistribtion sbject to SEG license or copyright; see Terms of Use at

2 S Ursin et al. By decomposing the wavefield into p- and downgoing waves with an eigenvale decomposition sing symmetry properties of the accompanying system matrix, one can derive simplified eqations for compting the wavefield propagators. This directional decomposition is consistent with a flx-normalization of the wavefield (Ursin, 9). Frther, by neglecting copling terms between the p- and downgoing waves, the reslting system matrix can be sed as a starting point to derive paraxial approximations of the original wave eqation. They also can be sed to derive WKBJ approximations of varios orders (Bremmer, 95; van Stralen et al., 99). In this paper, we derive initial conditions and one-way propagators for flx-normalized wavefield extrapolation in D media and show how this provides accrate amplitde information. We formlate an nbiased estimate of the reflectivity sing the wave eqation angle transform. Frther, we propose an extension to a general heterogeneos media in which the flx-normalization and an approximation to the transmission loss are performed. We accont for the medim pertrbations in the downward propagation sing Forier finite-difference methods (Ristow and Rühl, 99). We apply conventional pressre-normalized and the derived flx-normalized wavefield decomposition and propagation to a field data example from offshore Norway. Using this example, we compare and qantify the estimated reflectivity differences. LATERALLY HOMOGENEOUS MEDIUM We consider acostic waves traveling in a D medim where the principal direction of propagation is taken along the x axis (or depth ), and the transverse axes are ðx ;x Þ. The acostic medim parameters are assmed to be fnctions of depth x only. Let c denote the wave-propagation velocity; v ¼ðv ;v ;v Þ the particle velocity vector; p the pressre; and ρ the density of the medim. With no external volme force acting on the medim, the acostic wavefield satisfies the constittive relation for flids (Pierce, 9) p ¼ ρ t v; () pðt; x ;x ;x Þ¼ ðπþ Pðω;k ;k ;x Þe ið ωtþk x þk x Þ dk dk dω: () The vertical wavenmber k is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω >< c ðk þ k Þ ; if k ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω >: i k þ k c ; if pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k þ k pffiffiffiffiffiffiffiffiffiffiffiffiffiffi k þ k ω c > ω c : (5) We consider a plane wave with wavenmber k ¼ðk ;k ;k Þ T and direction m ¼ðsin θ cos ϕ; sin θ sin ϕ; cos θþ T where θ is the dip angle and ϕ is the azimth. We have k ¼ mω c ¼ ωp; () where p is the slowness vector. In or frther development, it is convenient to introdce the impedance : ¼ ρω ¼ ρ ¼ ρc k p cos θ : (7) Applying the Forier transform to eqations and, the reslting redced linear acostic system of eqations in a horizontally homogeneos flid yield the matrix differential eqation where the system matrix A is given by A ¼ " b ¼ iωab; () ρ 0 ρ # ; (9) 0 k c þk ω and the eqation of motion given by c tp þ ρ v ¼ 0; () and the field vector b by P b ¼ V ; (0) where ¼ð ; ; Þ and t denote the partial derivative with respect to time t. We define the Forier transform with respect to time t and the transverse spatial directions ðx ;x Þ as Pðω;k ;k ;x Þ¼ pðt; x ;x ;x Þe iðωt k x k x Þ dx dx dt; () with the inverse transform with respect to circlar freqency ω and the transverse wavenmbers ðk ;k Þ as where P is the Forier transformation of p and V is the Forier transformation of v with respect to t, x and x. The measred field vector b ¼½P; V Š T can be separated into p- and downgoing waves, denoted U and D, respectively. This separation is accomplished by applying an inverse eigenvector matrix of A, denoted L,onb. We define the transformed field vector containing the directional decomposed wavefield by U w ¼ ¼ L D b: () Moreover, pon sbstittion of w, the matrix differential eqation transforms to Downloaded Oct 0 to Redistribtion sbject to SEG license or copyright; see Terms of Use at

3 Wavefield decomposition S5 w ¼ðiωΛ L LÞw; () where an eigenvale decomposition of A provides the diagonal eigenvale matrix Λ ¼ L p 0 AL ¼ ; () 0 p where p is the vertical slowness. Amplitde-normalized wavefields In the conventional pressre-normalized wavefield separation approach, the eigenvector matrix of A is chosen as (Claerbot, 97; Ursin, 9, 97) L ¼ : () This leads to the inverse eigenvector matrix L ¼ : (5) With the eigenvector matrix defined in eqation, the matrix differential eqation becomes p 0 w ¼ iω w γðx 0 p Þ w; () where γðx Þ¼ log ðx Þ (7) is the reflectivity fnction. Using eqation 7 it can be expressed as γðx Þ¼ ρ þ c ρ x cos : () θ c x The wavefield decomposition described in eqation is referred to as being pressre-normalized in the sense that the pressre field eqals the sm of the p- and downgoing wavefields. We consider a stack of inhomogeneos layers where ρ and c are continos fnctions of x within each layer. At an interface between two layers, the bondary condition reqires that the wave vector b shall be continos. For an interface at x ¼ x k we mst have L þ w þ ¼ L w where L ¼ Lðx k Þ is evalated above the interface, and L þ ¼ Lðx kþ Þ is evalated beneath the interface (the x -axis is pointing vertically downward). We therefore have Eqations and 5 give L þ L ¼ w þ ¼ L þ L w : (9) " þ þ þ # þ þ : (0) þ This can be written as (Ursin, 9, eqation ): T L þ L ¼ R T R T T ; () where T and R are the transmission and reflection coefficients for an pward traveling incident wave at the interface. Flx-normalized wavefields We now derive an alternative directional decomposition by a flx-normalization of the wavefield. The main advantage of flxnormalizing the wavefield is that we obtain a simpler expression of the corresponding directional decomposed matrix differential eqation, as compared to the pressre-normalized approach. Disregarding the interaction between directional components yields a matrix differential eqation independent of the reflectivity fnction. In order obtain a flx-normalized system of eqations, the eigenvector matrix of A is chosen as (Ursin, 9; Wapenaar, 99) " L ¼ pffiffiffi pffiffi p ffiffi and ths the inverse eigenvector matrix becomes pffiffiffi # pffiffi ; () " ffiffi L ¼ p p p ffiffiffi # ffiffiffi pffiffiffi : () pffiffi This provides a flx-normalized representation of the wavefield w ¼ L b; () where w ¼ðŨ; DÞ T, and where Ũ and D denote the flxnormalized directional components of the wavefield. The wavefield is referred to as flx-normalized in the sense that the energy flx in the x -direction is propagation invariant (Ursin, 9; Wapenaar, 99). The pressre-normalized and the flx-normalized decomposition break down for near-horizontally-traveling waves becase the lateral wavenmber k approaches to zero in the horizontal direction. Combining eqations and with eqation yields the transformed matrix differential eqation p 0 w ¼ iω w γðx 0 p Þ 0 0 w: (5) Comparing the flx-normalized system of eqations in eqation 5 with the conventional pressre-normalized system of eqations in eqation, we see that in eqation 5 only the off-diagonal terms (depending on the reflectivity fnction γðx Þ) are present. Frther, by neglecting interaction between the flx-normalized directional decomposed components, the flx-normalized matrix differential eqation becomes independent of the reflectivity fnction γðx Þ. Finally, we note that Downloaded Oct 0 to Redistribtion sbject to SEG license or copyright; see Terms of Use at

4 S rffiffiffi wðω;k ;k ;x Þ¼ wðω;k ;k ;x Þ sffiffiffiffiffiffiffi k ¼ wðω;k ρω ;k ;x Þ: () Ursin et al. x Dðx Þ¼Dðx 0 Þ exp x 0 ðik ðζþþγðζþþdζ sffiffiffiffiffiffiffiffiffiffiffiffiffi ðx ¼ Dðx 0 Þ Þ x exp ðik ðx 0 Þ ðζþþdζ : () x 0 At an interface between two smoothly varying media we have w þ ¼ L L þ w with L þ L ¼ T R T R T T (7) The soltion to eqations 0 and in the zero-order WKB approximation becomes x Dðx Þ¼Dðx 0 ÞTðx Þ exp i k ðζþdζ () x 0 Here, T ¼ sffiffiffiffiffiffi þ T sffiffiffiffiffiffi sffiffiffiffiffiffi ¼ þ þ p ffiffiffiffiffiffiffiffiffiffiffiffi ¼ < 9 þ = þ þ : þ ; ; and x Uðx Þ¼Uðx 0 ÞTðx Þ exp i k ðζþdζ : (5) x 0 and R T () sffiffiffiffiffiffi sffiffiffiffiffiffi ¼ p þ ffiffiffiffiffiffiffiffiffiffiffiffi ¼ < 9 = þ þ : þ ; ; (9) where denotes the impedance at the bottom of the previos thin layer and þ denotes the impedance at the p top of the next layer. hang et al. (005) se the scaling w ¼ ffiffiffiffi k w so they are not sing flx-normalized variables. With their scaling, the transformed matrix differential eqation will only have the same form as eqation 5 for a medim with constant density. ONE-WAY WAVE EQUATIONS We obtain one-way eqations for the p- and downgoing waves by neglecting the interaction terms in eqations and. This gives the zero-order WKBJ approximation (Clayton and Stolt, 9; Ursin, 9) obeying the eqations x with interface conditions U ik þ γ 0 ¼ D 0 ik þ γ U U ¼ T D þ D U D (0) : () In a region with smoothly-varying parameters, the eqation with Dðx 0 Þ given, has the soltion D x ¼ðik þ γþd () The factor sffiffiffiffiffiffiffiffiffiffiffiffiffi ðx Tðx Þ¼ Þ ðx 0 Þ Y 0<x k <x T sffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx k Þ ðx k Þ ðx kþ Þ () comes from sing eqations and for each layer, starting at the top. Passing from one layer to the next, the factor T ðx k Þ takes the bondary conditions partly into accont (eqation ). The sqareroot factors come from integrating eqation inside each layer sing eqation. This gives sffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx Þ ðx Þ ðx Þ : (7) ðx 0 Þ ðx þ Þ ðx Nþ Þ By combining the sqare roots for each interface from to N we obtain eqation. Using eqation this can be written as: sffiffiffiffiffiffiffiffiffiffiffiffiffi ðx Tðx Þ¼ Þ Y T ðx ðx 0 Þ k Þ () 0<x k <x in terms of the flx-normalized transmission coefficients. For the flx-normalized p- and downgoing waves we obtain from eqations, 5, and sing eqation : Dðx Þ¼ Dðx 0 Þ Y T x ðx k Þ exp i k ðζþdζ (9) 0<x k <x x 0 and Ũðx Þ¼Ũðx 0 Þ Y 0<x k <x T ðx k Þ exp i x x 0 k ðζþdζ : (0) These eqations cold, of corse, also have been obtained directly by neglecting the interaction terms in eqations 5 and 7. Downloaded Oct 0 to Redistribtion sbject to SEG license or copyright; see Terms of Use at

5 HETEROGENEOUS MEDIUM We want to se one-way wave propagators for migration in a heterogeneos medim. Based on the previos discssion, we choose to se flx-normalized variables. The downgoing field from a point sorce is then represented in the wavenmber-freqency domain by D 0 ¼ D 0 ðω;k ;k ;x ¼ 0Þ ¼ πsðωþ ik : () The inverse Forier transform of eqation with respect to k and k is known as the Weyl integral (Aki and Richards, 90). The Forier transform of the effective sorce signatre is SðωÞ. The flx-normalized downgoing field from a point sorce is represented in the wavenmber domain by sffiffiffiffiffiffiffiffiffiffi D 0 ðω;k ;k ;x ¼ 0Þ ¼πi SðωÞ: () ρωk In marine seismic data, we may add the effect of the free srface (the ghost) on the downgoing wavefield (Amndsen and Ursin, 99): sffiffiffiffiffiffiffiffiffiffi D 0 ðω;k ;k ; 0Þ ¼πi ρωk ðexp ½ ik x s Š R 0 exp ½ik x s ŠÞSðωÞ; () where x s is the sorce depth, and the reflection coefficient is theoretically R 0 ¼. If the wavefield is acqired by a conventional streamer configration, only pressre is recorded. The primary pgoing wavefield U 0 can then be estimated by a demltiple procedre (Amndsen, 00; Robertsson and Kragh, 00), where ghost and free-srface mltiples are removed from the data. Hence, sing eqation, the flx-normalized pgoing wavefield can be represented by rffiffiffi Ũ 0 ¼ U 0 : () The pressre and the vertical displacement velocity can be measred in ocean-bottom seismic acqisition. A recent development (Landrø and Amndsen, 007; Tenghamn et al., 00) also allows for both these to be measred on a streamer configration. The flxnormalized pgoing wavefield is then given by Ũ 0 ¼ p ffiffiffiffiffiffi ½P V Š: (5) The downward contination of the wavefields is accomplished by solving the eqation w iĥ 0 ¼ w () x 0 iĥ for x > 0 with wð0þ ¼½Ũ 0 ; D 0 Š T given in eqations 5. Eqation is a generalization of eqation 5 with the copling terms neglected. The operator Ĥ is the sqare-root operator satisfying (Wapenaar, 99) Wavefield decomposition ω Ĥ Ĥ ¼ þ ρ þ ρ : c x ρ x x ρ x (7) Dividing the medim into thin slabs of thickness Δx with negligible variations in the preferred direction x of propagation within each slab, allows s to extend the propagator to a general inhomogeneos medim with small lateral medim variations sing, for example, split-step (Stoffa et al., 990), Forier finite difference (Ristow and Rühl, 99), or a phase-screen (W and Hang, 99) approach depending on the size of the medim heterogeneities in the lateral direction (hang et al., 009). At thin-slab bondaries one may apply a correction term for the transmission loss (see eqation 7): T ¼ s ffiffiffiffiffiffi sffiffiffiffiffiffi þ þ : () þ Cao and W (00) have proposed a similar correction for the downward contination of pressre. The downward-contined wavefields can be sed in a standard way to create an image. One may also apply a crosscorrelation and a local Forier transform to compte common-angle gathers (de Brin et al., 990; Sava and Fomel, 00; de Hoop et al., 00; Sn and hang, 009). This is termed the wave eqation angle transform, and a common-image gather for a single shot is Iðp; x;x Þ¼ Ũ ω; x þ h ;x D ω; x h ;x e iωp h dhdω; (9) where h ¼ðh ;h ; 0Þ is the horizontal-offset coordinate and p h ¼ p h þ p h. In the Appendix, it is shown that this approach prodces an estimate of the plane-wave reflection coefficient for a horizontal reflector, mltiplied by the energy of the corresponding downgoing plane wave. To obtain an nbiased estimate of the reflection coefficient, it is necessary to divide by this factor (which we will refer to as the sorce correction term). The reslt is R Ũ ω; x þ h ;x D ω; x h ;x e iωp h dhdω Rðp; x;x Þ¼ R j ˆDðω; ωp; x;x Þj ; dω (50) where ˆDðω; ωp; x;x Þ is the local Forier transform as defined in the Appendix. It may be necessary to apply a stabilizing procedre as discssed in Vivas et al. (009). To obtain an estimate of the reflection coefficient for a range of p-vales it is necessary to average the expression in eqation 50 over many shots. NUMERICAL RESULTS S7 Throghot or nmerical examples, we employ a Forier finite-difference approach to accont for lateral medim variations, while correcting for the transmission loss sing the minimm velocity within each thin-slab. Frther, we consider wave-propagation in a D medim. First, the inpt data to migration are modeled over a medim with density contrasts only; hence, the reflection coefficients are independent of angle. Next, we compare conventional pressre normalization to the flx-normalized approach on Downloaded Oct 0 to Redistribtion sbject to SEG license or copyright; see Terms of Use at

6 S Ursin et al. a field data example where we, in a qantitative fashion, compare the estimated reflectivity. For the pressre-normalized approach, we set the transmission correction to nity. Eqation 50 is sed to otpt AVP gathers on selected locations. Imaging in a lateral invariant medim In or first test, we consider a laterally-invariant medim with a constant velocity of 000 m s and with density contrasts in depth at,, and km as illstrated in Figre. We choose this model becase in this particlar case we will have angle-independent reflection coefficients. We create a synthetic split-spread shot gather over the laterally invariant medim sing a finite-difference modeling scheme. In Figre we show the modeled shot, and the migrated shot is shown in Figre. To extract AVP or AVA information at a reflector position, we need information from more than one shot becase each shot gives limited angle information. A schematic representation of angle information available from one shot is shown in Figre. Using information from the wave eqation angle-transform, this can frther be illstrated by plotting Iðp; xþ (in gray-scale) overlaid the sorce correction term (in color-scale) at midpoints x m; ¼ 0.5 km, x m; ¼ 0.0 km, and x m; ¼ 0.5 km shown in Figre 5. By simlating more shots over one midpoint location x m, we can extract angle information for larger angle coverage as shown schematically in Figre. We simlate 00 shots with a shot-distance of 0 m on both sides of x m, in addition to one shot jst above x m. This prodces the angle coverage shown in Figre 7, where we plot Iðp; xþ overlaid the corresponding sorce illmination for the fixed midpoint location x m. We notice that the angle coverage for each reflector in depth is different (as expected). At each reflector depth, we extract the peak amplitde of Rðθ;x m Þ sing eqation 50 with sin θ ¼ pc. The reslt is depicted in Figre. We have plotted the AVA response for the reflector at km p to 50, the reflector at km p to 5, and the reflector at km p to 5. In this example, we expect an angle-independent reflectivity, and from the reslt we see that the reflectivity is recovered relatively accrately for a wide range of angles. De to a limited apertre, edge effects impact the reslts, and the largest angles on each reflector are affected. Distance (km) Offset (km) 0..0 (g/cm) Figre. Migrated shot from the lateral invariant model with flxnormalized wavefields. Figre. Densities sed in the finite-difference modelling over the lateral invariant model example. Offset (km) 0 Receivers Sorce Time (s) Depth 5 Figre. A synthetic shot gather from a finite-difference modeling over the lateral invariant model example. Figre. Schematic representation of angle information contained in one shot, where x m; and x m; are midpoint locations with information arond angles θ and θ. For each midpoint location, each shot only gives limited angle information. Downloaded Oct 0 to Redistribtion sbject to SEG license or copyright; see Terms of Use at

7 Wavefield decomposition S9 Marine seismic data example Depth Slowness (s/km) Slowness (s/km) Slowness (s/km) Figre 5. Slowness coverage Iðp; xþ from one shot (gray-scale) overlaid the corresponding sorce correction (color-scale) for (left) x m; ¼ 0.5 km; (middle) x m; ¼ 0.0 km; and (right) x m; ¼ 0.5 km. Sorces We apply conventional pressre-normalized and the derived flxnormalized methods to a field data set from the Nordkapp Basin. The basin is located offshore Finnmark, in the Norwegian sector of the Barents Sea. It is an exploration area which exhibits complex geology and is challenging for seismic imaging. We have extracted a sbset of a D srvey which partially covers two salt dome strctres. Figre 9 shows the velocity model sed in the migration. The data set is composed by collecting and combining streamer data in two directions, providing a split-spread configration. Each shot is separated by 5.0 m, and in or example we have inclded a total of 775 shots. Each streamer has 9 receivers with a hydrophone distance of.5 m and a total offset of abot 00 m on both sides of the sorce location. Notice that no demltiple is applied in the preprocessing step. In Figre 0, we show one extracted shot which is inpt to migration. In the imaging, we have sed a sorce signatre comparable to a Ricker wavelet with a peak freqency of 7 Hz. We sed 5 Hz of the freqency content of the data, and imaged the data down to 0 km. The total apertre of each shot was km. For the pressre-normalized and the flx-normalized wavefield decomposition, we migrate the data set with the same downward contination scheme and the same imaging condition. That is, we se a thirdorder Forier finite-difference migration operator and an imaging condition which estimates the reflectivity by acconting for the sorce illmination. The flx-normalized migration has an approximation to the transmission loss correction applied at thin-slab bondaries sing the minimm velocity at each slab given by the apertre of each migrated shot. In Figres and, the pressre-normalized and the flx-normalized migrated sections are shown, respectively. By inspecting and comparing both sections, we see that we have an apparent similar amplitde response. Figre. Schematic representation of angle coverage θ at one midpoint location x m from a range of shots. To extract a larger range of angle coverage, each midpoint location reqires several shots Angle ( ) Figre. Peak amplitdes at each reflector at km (red), km (green), and km (ble) for one spatial location x m Angle ( ) Figre 7. Angle coverage Iðp; xþ (gray-scale) from one spatial location x m, overlaid the corresponding sorce correction (colorscale), where the contribtion from mltiple shots are inclded. 0 Distance (km) (m/s) Figre 9. The velocity model sed in the Nordkapp field data example. Downloaded Oct 0 to Redistribtion sbject to SEG license or copyright; see Terms of Use at

8 Ursin et al. S90 To qantify the difference between the migrated sections, we compte the difference between the absolte vales of each section. The difference plot is shown in Figre. The colors red or black indicate that the flx-normalized image provides higher or lower amplitdes than the pressre-normalized image, respectively. In the shallower part of the difference image, from the srface to abot Offset (km) 0 Time (s) Figre 0. One extracted shot gather sed in the Nordkapp basin field data example. km, the pressre-normalized image appears to be dominating; however, these parts of the sections are also contaminated by low-freqency-migration noise. In the sediment basin between the two salt-domes, that is, below and arond a distance of km, no coherent energy appears below km. Arond a distance of approximately to km, at abot km depth, the flx-normalized images gives a higher amplitde response on some parts of a few sbsrface reflectors. The peak amplitde difference is arond one-tenth of the reflectivity image amplitdes. Frther, we extract a slowness gather from each of the migration approaches corresponding to a lateral position of. km and these are shown in Figre. Figre a and b shows the otpt from the pressre-normalized and flx-normalized approach, respectively. The gathers look similar. Next, we extract one event at 7. km of depth on these gathers, as shown in Figre 5. For this event, we extract the peak amplitdes for each of the migrated reflectors, and plot these in Figre (top), where the red crve is the flx-normalized peak amplitde and the ble crve is the pressrenormalized peak amplitde. Finally, we take the difference between the normalized peak amplitdes (bottom), where the positive and negative vales correspond to higher and lower peak amplitdes when sing flx-normalized variables. The plot shows differences between the reslts from the different approaches, and explains the difference plot in Figre. Distance (km) Figre. Migrated image of the Nordkapp field example with the pressre-normalized wavefield decomposition and migration approach Distance (km) Figre. Migrated image of the Nordkapp field example with the flx-normalized wavefield decomposition and migration approach. 0 0 Downloaded Oct 0 to Redistribtion sbject to SEG license or copyright; see Terms of Use at

9 Wavefield decomposition S9 Distance (km) 0 0 Figre. Difference of the absolte vale of the migrated flx-normalized wavefield decomposition and the pressre-normalized wavefield decomposition sections of the Nordkapp field example. Red indicates that the flx-normalized image is dominant, and black indicates that the pressre-normalized image is dominant. 0 a) b) Difference Peak amplitdes Slowness (s/km) Slowness (s/km) Slowness (s/km) Figre. Slowness-gather at a distance of. km from the Nordkapp basin field data example with (a) pressre-normalized, and (b) flx-normalized variables Slowness (s/km) Figre 5. Extracted event at 7. km depth from the slownessgathers in Figre at a distance of. km for (top) flxnormalized variables and (bottom) pressre-normalized variables. Figre. Normalized peak amplitdes along event in 5 (top) with flx-normalized variables (red line) and pressre-normalized variables (ble line). Difference between normalized peak amplitdes (bottom), where positive and negative vales corresponds to higher and lower peak amplitdes in sing flx-normalized variables. CONCLUSIONS By directionally decomposing a wavefield sing a flx-normalized eigenvale decomposition, we have derived initial conditions for prestack depth migration of common-shot data. This decomposition simplifies the system of differential eqations. Frther, by neglecting interaction between directional components, we derive propagators for flx-normalized wavefields where we formlate a transmissionloss compensation approach for flx-normalized wavefield propagation. By sing the wave eqation angle transform, we formlate an estimate of the plane-wave reflection coefficient for a horizontal reflector. From or D nmerical example, we show that a flx-normalized directional decomposition provides accrate amplitde information in a medim where the parameters are fnction of depth only. Finally, we extend or approach to a laterally varying media. From a field data example, we observe some differences in the strength of the estimated reflectivity compared to a pressre-normalized approach. ACKNOWLEDGMENTS The athors wold like to thank Hans-Kristian Helgesen and Robert Fergson for sefl discssions, and Statoil for permission to Downloaded Oct 0 to Redistribtion sbject to SEG license or copyright; see Terms of Use at

10 S9 Ursin et al. pblish data from the Nordkapp basin. This research has received financial spport from the Norwegian Research Concil via the ROSE project. Bjørn Ursin also has received financial spport from VISTA. We also thank the reviewers and the editor for their valable comments and sggestions which helped improve this paper. APPENDIX A THE WAVE EQUATION ANGLE TRANSFORM In terms of local Forier transforms, Ûðω; k r ; x;x Þ¼ Ũ ω; x þ h ;x e ikr h dh and ˆDðω; k s ; x;x Þ¼ D ω; x h ;x e iks h dh; (A-) (A-) the wave eqation angle-transform in eqation can be written Iðp; x;x Þ¼ Ûðω; k r ; x;x π Þ ˆD ðω; k s ; x;x Þ e ikr h e iks h e iωp h dk r dk s dhdω (A-) where k r ¼ðk r ;kr Þ and ks ¼ðk s ;ks Þ. Snell s lawis Ûðω; k r ; x;x Þ¼Rðp; x;x Þ ˆDðω; k s ; x;x Þδðk r k s Þ: (A-) Inserted into eqation A- this gives Iðp; x;x Þ¼ Rðp; x;x π Þj ˆDðω; k; x;x Þj e iðk ωpþ h dkdhdω where k r ¼ k s ¼ k ¼ ωp. Frther simplifications give Iðp; x;x Þ¼ Rðp; x;x π Þj ˆDðω; k; x;x Þj δðk ωpþdkdω and finally Iðp; x;x Þ¼Rðp; x;x Þ j ˆDðω; ωp; x;x Þj dω (A-5) (A-) (A-7) The derivations above are only approximate, becase finite-apertre effects have not been taken into consideration. The reslt is valid only for a horizontally reflecting plane. REFERENCES Aki, K., and P. G. Richards, 90, Qantitative seismology, theory and methods: Volme, W. H. Freeman and Company. Amndsen, L., 00, Elimination of free-srface related mltiples withot need of the sorce wavelet: Geophysics,, 7, doi: 0.90/.9. Amndsen, L., and B. Ursin, 99, Freqency-wavenmber inversion of acostic data: Geophysics, 5, 07 09, doi: 0.90/.. Bremmer, H., 95, The W. K. B. approximation as the first term of a geometric-optical series: Commnications on Pre and Applied Mathematics,, 05 5, doi: 0.00/(ISSN) Cao, J., and R.-S. W, 00, Stdy of the inflence of propagator amplitde correction on image amplitde sing beamlet propagator with local WKBJ approximation: 7th Annal International Meeting, SEG, Expanded Abstracts, Cao, J., and R.-S. W, 00, Amplitde compensation for one-way wave propagators in inhomogeneos media and its application to seismic imaging: Commnications in Comptational Physics,, 0. Claerbot, J., 97, Fndamentals of geophysical data processing: McGraw-Hill. Claerbot, J. F., 970, Coarse grid calclations of waves in inhomogeneos media with application to delineation of complicated seismic strctre: Geophysics, 5, 07, doi: 0.90/.00. Claerbot, J. F., 97, Toward a nified theory of reflector mapping: Geophysics,, 7, doi: 0.90/.05. Clayton, R. W., and R. H. Stolt, 9, A Born-WKBJ inversion method for acostic reflection data: Geophysics,, , doi: 0.90/.. de Brin, C. G. M., C. P. A. Wapenaar, and A. J. Berkhot, 990, Angledependent reflectivity by means of prestack migration: Geophysics, 55,, doi: 0.90/.9. de Hoop, M., R. van der Hilst, and P. Shen, 00, Wave eqation reflection tomography: Annihilators and sensitivity kernels: Geophysical Jornal International, 7, 5, doi: 0./gji.00.7.isse-. Kiyashchenko, D., R.-E. Plessix, B. Kashtan, and V. Troyan, 005, Improved amplitde mlti-one-way modeling method: Wave Motion,, 99 5, doi: 0.0/j.wavemoti Landrø, M., and L. Amndsen, 007, Fighting the ghost: Geo ExPro, 7 0. Pierce, A. D., 9, Acostics: An introdction to its physical principles and applications: McGraw-Hill. Ristow, D., and T. Rühl, 99, Forier finite-difference migration: Geophysics, 59,, doi: 0.90/.575. Robertsson, J. O. A., and E. Kragh, 00, Rogh-sea deghosting sing a single streamer and a pressre gradient approximation: Geophysics, 7, 005 0, doi: 0.90/ Sava, P., and S. Fomel, 00, Angle-domain common-image gathers by wavefield contination methods: Geophysics,, 05 07, doi: 0.90/.507. Stoffa, P. L., J. T. Fokkema, R. M. de Lna Freire, and W. P. Kessinger, 990, Split-step Forier migration: Geophysics, 55, 0, doi: 0.90/.50. Sn, J., and Y. hang, 009, Practical isses of reverse time migration: Tre amplitde gathers, noise removal and harmonic-sorce encoding: 9th Annal International Meeting, SEG, Expanded Abstracts, 0 07, doi: 0.90/.079. Tenghamn, R., S. Vaage, and C. Borresen, 00, GeoStreamer, a dal-sensor towed marine streamer: 70th Annal International Conference and Exhibition, EAGE, Extended Abstracts, B0. Ursin, B., 9, Review of elastic and electromagnetic wave propagation in horizontally layered media: Geophysics,, 0 0, doi: 0.90/.59. Ursin, B., 9, Seismic migration sing the WKB approximation: Geophysical Jornal of the Royal Astronomical Society, 79, 9 5. Ursin, B., 97, The plane-wave reflection and transmission response of a vertically inhomogeneos acostic medim, in M. Bernabini, P. Carrion, G. Jacovitti, F. Rocca, S. Treitel, and M. Worthington, eds., Deconvoltion and inversion: Blackwell, van Stralen, M. J. N., M. V. de Hoop, and H. Blok, 99, Generalized Bremmer series with rational approximation for the scattering of waves in inhomogeneos media: Jornal of the Acostical Society of America, 0, 9 9, doi: 0./.5. Vivas, F. A., R. C. Pestana, and B. Ursin, 009, A new stabilized leastsqares imaging condition: Jornal of Geophysics and Engineering,,, doi: 0.0/7-///005. Wapenaar, K., 99, Reciprocity properties of one-way propagators: Short note: Geophysics,, , doi: 0.90/.7. W, R.-S., and L.-J. Hang, 99, Scattered field calclation in heterogeneos media sing a phase-screen propagator: 5nd Annal International Meeting, SEG, Expanded Abstracts, 9 9. hang, J., W. Wang, and. Yao, 009, Comparison between the Forier finite-difference method and the generalized-screen method: Geophysical Prospecting, 57, 55 5, doi: 0./gpr isse-. hang, Y., S. X, N. Bleistein, and G. hang, 007, Tre-amplitde, angledomain, common-image gathers from one-way wave eqation migrations: Geophysics, 7, no., S9 S5, doi: 0.90/.997. hang, Y., G. hang, and N. Bleistein, 00, Tre amplitde wave eqation migration arising from tre amplitde one-way wave eqations: Inverse Problems, 9,, doi: 0.0/0-5/9/5/07. hang, Y., G. hang, and N. Bleistein, 005, Theory of tre-amplitde oneway wave eqations and tre-amplitde common-shot migration: Geophysics, 70, no., E E0, doi: 0.90/.9. Downloaded Oct 0 to Redistribtion sbject to SEG license or copyright; see Terms of Use at