Amplitude-preserving reverse time migration: From reflectivityto velocity and impedance inversion

Yu Zhang1, Andrew Ratcliffe2, Graham Roberts2, and Lian Duan2

ABSTRACT

Conventional methods of prestack depth imaging aim at pro-ducing a structural image that delineates the interfaces of thegeologic variations or the reflectivity of the earth. However,it is the underlying impedance and velocity changes that gen-erate this reflectivity that are of more interest for characterizingthe reservoir. Indeed, the need to generate a better product forgeologic interpretation leads to the subsequent application oftraditional seismic-inversion techniques to the reflectivity sec-tions that come from typical depth-imaging processes. Thedrawback here is that these seismic-inversion techniques use ad-ditional information, e.g., from well logs or velocity models, tofill the low frequencies missing in traditional seismic data due tothe free-surface ghost in marine acquisition. We found that with

the help of broadband acquisition and processing techniques,the bandwidth gap between the depth-imaging world and seis-mic inversion world is reducing. We outlined a theory thatshows how angle-domain common-image gathers producedby an amplitude-preserving reverse time migration can estimateimpedance and velocity perturbations. The near-angle stackedimage provides the impedance perturbation estimate whereasthe far-angle image can be used to estimate the velocity pertur-bation. In the context of marine acquisition and exploration, ourmethod can, together with a ghost compensation technique, be auseful tool for seismic inversion, and it is also adaptable to afull-waveform inversion framework. We developed syntheticand real data examples to test that the method is reliable andprovides additional information for interpreting geologic struc-tures and rock properties.

INTRODUCTION

Imaging is conventionally regarded as a process for seeking sub-surface reflectivity. However, when it comes to reservoir characteri-zation, the impedance and velocity embedded in the reflectivity aremore closely related to the physical properties of the hydrocarbon-bearing sediments and are the more desirable result.Assuming that velocity can be separated into a smooth reference

model and a perturbation, Jin et al. (1992) develop a weighted least-squares, linear, iterative elastic inversion in which the forward prob-lem is solved with a combination of ray-theory and the Bornapproximation. The first iteration is essentially a preserved ampli-tude Kirchhoff migration (Thierry and Lambaré, 1995). They con-clude that the success of the inversion depends on the ability of thesmooth reference velocity to provide accurate traveltimes. This workwas further developed by Forgues and Lambaré (1997) who provide

quantitative estimations of the conditioning of a multiparameterinversion. Both of the studies provide numerical examples only.Operto et al. (2000) continue this approach but extend the ray theoryplus Born approximation to multiple arrivals and show with a 2Dsynthetic data case study that impedance perturbations could be im-aged following a ray-theory approach. However, as Forgues andLambaré (1997) note, the method may not work well on real databecause of the lack of computation power and low-frequency infor-mation in the seismic data at the time due to restrictions in recordingbandwidth and the presence of the free-surface ghost in marine data.With recent developments in wave propagation computations,

reverse time migration (RTM) has become the state-of-the-art tech-nique to image and interpret subtle and complex geologic features,not only because it correctly handles complex velocities andpropagates waves without angle limitation but also because ittakes advantage of a complete set of acoustic waves (reflections,

Manuscript received by the Editor 11 December 2013; revised manuscript received 14 June 2014; published online 20 October 2014.1Formerly CGG, Crawley, UK; presently ConocoPhillips, Houston, Texas, USA. E-mail: yu.zhang@conocophillips.com.2CGG, Crawley, UK. E-mail: andrew.ratcliffe@cgg.com; graham.roberts@cgg.com; lian.duan@cgg.com.© 2014 Society of Exploration Geophysicists. All rights reserved.

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GEOPHYSICS, VOL. 79, NO. 6 (NOVEMBER-DECEMBER 2014); P. S271–S283, 8 FIGS.10.1190/GEO2013-0460.1

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transmissions, diffractions, prismatic waves, etc.) to image subsur-face structures (Zhang et al., 2007; Zhang and Sun, 2009; Xu et al.,2011). Furthermore, new broadband acquisition and processingtechniques (Soubaras and Whiting, 2011) have allowed the recov-ery of low-frequency information crucial for the estimation ofacoustic parameters such as impedance. Here, we use a combinationof these two developments, RTM and successful retrieval of lowfrequencies, to build a method for the estimation of impedanceand velocity perturbation from seismic data.Specifically, in this paper, we propose a new method to generate

acoustic velocity and impedance perturbations using an amplitude-preserving, ghost-compensated RTM with a modified imaging con-dition. The context presented here is for marine acquisition, but thebasic principles are just as applicable to land/OBC situations. Webegin with a description of conventional amplitude-preserving RTMfor angle-dependent reflectivity, and then, we discuss how to com-pensate for the ghost effects within an RTM to reliably retrieve thelow-frequency reflectivity information distorted due to free-surfacereflections. Once we have retrieved the low frequencies, we derive amodified imaging condition and show that impedance and velocityperturbations can be estimated from near- and far-angles, respec-tively, of angle domain common image gathers (ADCIGs). Thiswork is a development of the theory developed by Jin et al.(1992) and Forgues and Lambaré (1997). Next, we describe howthe method can be combined with full-waveform inversion(FWI), to provide iterative velocity updates. Finally, we show re-sults from synthetic and real data sets: First, we demonstrate thatghost-compensated RTM with a modified imaging condition reli-ably extracts impedances and velocities. This includes synthetic ex-amples and a real data impedance well match. Second, we showhow the RTM-based velocity perturbation estimation can be incor-porated into an FWI flow using a real data example.

AMPLITUDE-PRESERVING REVERSE TIMEMIGRATION

For a zero-phased and designatured shot record at timet; dðxr; yr; xs; ys; tÞ, with the shot at xs ¼ ðxs; ys; zs ¼ 0Þ andreceivers at xr ¼ ðxr; yr; zr ¼ 0Þ, the RTM (Baysal et al., 1983;McMechan, 1983; Whitmore, 1983) based on the theory of ampli-tude-preserving migration, in the simplified case of the acoustic,isotropic, and constant density wave equation, can be summarizedas forward propagation of the source wavefield pF:(�

1v2

∂2∂t2 − ∇2

�pFðx; t; xsÞ ¼ 0;

pFðx; y; z ¼ 0; t; xsÞ ¼ δðx − xsÞRt0 fðt0Þdt 0;

(1)

and backward propagation of the receiver wavefield PB:(�1v2

∂2∂t2 − ∇2

�pBðx; t; xsÞ ¼ 0;

pBðx; y; z ¼ 0; t; xsÞ ¼ dðx; y; t; xsÞ;(2)

where v ¼ vðx; y; zÞ denotes the velocity, fðtÞ denotes the sourcewavelet, ∇2 denotes the Laplacian operator, and x ¼ ðx; y; zÞ de-notes the subsurface imaging location. Zhang et al. (2007) andZhang and Sun (2009) show that the “crosscorrelation” imagingcondition,

Rðx; θÞ ¼ 4

ZZδðθ 0ðx; xsÞ − θÞ

�Z

pBðx; t; xsÞpFðx; t; xsÞdt�dθ 0dxs; (3)

is a proper choice to obtain amplitude-preserving ADCIGs from awave-equation-based migration in 2D. In equation 3, θ is the reflec-tion angle at the imaging location and the term δðθ 0ðx; xsÞ − θÞ rep-resents the conversion from shot gathers to reflection angle gathers.For a 2D application, including details of the conversion of subsur-face offset gathers to angle gathers, please refer to Sava and Fo-mel (2003).In 3D, the following imaging condition provides angle- and azi-

muth-dependent reflectivities when the migration velocity is accu-rate and the common-image gathers are generated in the subsurfaceangle domain (Xu et al., 2011):

Rðx; θ;φÞ ¼ 16π

Z Z Z ZvðxÞsin θ 0 δðθ 0ðx; xsÞ − θÞ

× δðφ 0ðx; xsÞ − φÞðpBpFÞ× ðx; t; xsÞdtdθ 0dφ 0dxs; (4)

where θ and φ are the reflection and azimuth angles, respectively,at the imaging location and the terms δðθ 0ðx; xsÞ − θÞ andδðφ 0ðx; xsÞ − φÞ represent the conversion from shot gathers to re-flection and azimuth angle gathers. Xu et al. (2011) also describehow to generate 3D angle gathers from an RTM.Here, we use the definition of amplitude-preserving migration

(Bleistein, 1987; Hanitzsch, 1997), which intends to remove geo-metric spreading and to recover band-limited angularly dependentreflectivity during the migration process. Our angle-domain, ampli-tude-preserving RTM consists of proper wave-equation formula-tions, the boundary conditions, and the imaging conditions, asshown in equations 1–4, and amplitude-preserving angle gatherconversion (Sava and Fomel, 2003; Xu et al., 2011). If the migrationvelocity is a good approximation to the real earth model, our RTMhandles multipathing automatically and compensates for geometricspreading. However, such an implementation usually assumes theacquired data is well sampled (no data aliasing) and provides goodsubsurface illumination in the interested imaging areas. Also, themethod we have discussed does not compensate for some ofthe other physical effects that effect image amplitude, such as trans-mission losses (Deng and McMechan, 2007), attenuation, anddispersion (Samec and Blangy, 1992).

REVERSE TIME MIGRATION WITH GHOSTCOMPENSATION

In marine acquisition, sources and receivers are placed below thewater surface. The source and receiver ghosts due to free-surfacereflections are thus always recorded in the seismic data and causeangle-dependent frequency and amplitude distortion. In Figure 1a, asynthetic seismic shot record, generated by dynamic ray tracing,models the wave propagation through an isotropic medium withan angle-independent reflectivity containing five horizontal layersat different depths, with uniform amplitude of one. The shot is in thecenter of the section, with depth of 10 m, and the receivers are out toan offset of 7500 m on either side, with depth of 15 m. The water

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velocity is 1500 ms−1. In addition to the effect of geometric spread-ing, the wavelet amplitude and spectrum are distorted depending onthe propagation angles at the shot and the receivers, as described in

equations 5 and 6 below. In particular, the low frequency in the seis-mic data is greatly attenuated due to the ghosts. If we use the con-ventional amplitude-preserving RTM formulation in equations 1–3,

Figure 1. (a) A shot record over five horizontal reflectors in a medium with v ¼ 1500 ms−1. Source and receiver ghosts are recorded. (b) Amigrated ADCIG using conventional amplitude-preserving RTM. (c) The AVA curves picked from the CIG in (b). (d) A migrated ADCIG fromamplitude-preserving RTM with source and receiver ghost compensation. (e) The AVA curves picked from the CIG in panel (d).

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stack all the migrated common image shot gathers to generate subsur-face offset gathers (Sava and Fomel, 2003) and then convert them tosubsurface ADCIGs, we end up with a distortion in the spectrum ofthe migrated image, as shown in Figure 1b and also the incorrectamplitude variation with angle (AVA) trend in Figure 1c.One approach is to compensate for the ghost effect before the

migration: This is a challenging area of research and is often basedon a series of deghosting operators in the τ-p domain (Poole, 2013).On the other hand, because propagation angles for the source andreceiver are well defined and are easily accessible during the wavepropagation part of a migration, it is a fairly simple process to com-pensate for the ghost effects within an RTM (Zhang et al., 2012). Ifwe assume a (perfect) free-surface reflection (i.e., the reflectivity is−1) and denote source depth and water velocity as Δzs and vw, re-spectively, then the source ghost acting on the wavefield with thepropagation angle αs at the source (with respect to the vertical di-rection) takes the following form in the frequency domain:

Gsðω; αsÞ ¼ e−iω cos αsΔzs

vw − eiω cos αsΔzs

vw ¼ −2i sinω cos αsΔzs

vw;

(5)

where GðωÞ represents the temporal Fourier transform of GðtÞand, in the frequency and wavenumber domain, cos αs ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 − v2w

ω2 ðk2x þ k2yÞq

. Similarly, denoting the receiver depth and

receiver propagation angle (with respect to the vertical direction)as Δzr and αr, respectively, the receiver wavelet with ghost canbe expressed as

Grðω; αrÞ ¼ e−iω cos αrΔzr

vw − eiω cos αrΔzr

vw ¼ −2i sinω cos αrΔzr

vw;

(6)

where αr can be computed in the same way as αs. With equations 5and 6, the ghosts can be compensated during the wave propagationof RTM by modifying the boundary conditions in equations 1 and 2as follows:

pFðx; y; z ¼ 0;ω; xsÞ ¼ δðx − xsÞfðωÞ

iωGsðω; αsÞ(7)

and

pBðx; y; z ¼ 0;ω; xsÞ ¼dðx; y;ω; xsÞGrðω; αrÞ

: (8)

Here, the implementation is straightforward. The boundary condi-tion corrections, in equations 7 and 8, can be applied in the fre-quency and wavenumber domain to the source wavelet and inputdata to the RTM. Then, to extrapolate the ghost compensated wave-fields, we solve equations 1 and 2 with the corrected boundary con-ditions. The additional cost to the conventional RTM is minor.After we have compensated for the source and receiver ghosts

using equations 7 and 8, the reflector wavelets on the migrated an-gle gather for the five-horizontal-layer model have wider and morebalanced frequency bandwidth and appear much sharper (Fig-ure 1d). Also, the normalized peak amplitudes along reflectorsin the reflection angle domain converge well, which indicates thatthe reflectivity is well recovered and the AVA relation is more re-liable (Figure 1e).This ghost compensation technique within RTM provides an al-

ternative to processing solutions developed for towed streamermarine acquisition. Zhang et al. (2012) demonstrate that the methodproposed above can retain the low-frequency information in the im-age that is important for seismic inversion.

VELOCITY AND IMPEDANCE PERTURBATIONSUSING REVERSE TIME MIGRATION

Inversions for impedance and velocity perturbations using imag-ing methods have been proposed for Kirchhoff migration (Jin et al.,1992; Forgues and Lambaré, 1997; Operto et al., 2000). Such meth-ods are applicable only when reliable low frequencies are recordedin the seismic data and retained during processing and imaging.Here, we consider the acoustic-wave equation with velocity v and

density ρ variations:

Figure 2. (a) A shot record and (b) a migrated ADCIG using theamplitude-preserving RTM. The dashed blue lines show the se-lected 0° and 66° traces displayed in Figure 3.

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�1

v2∂2

∂t2− ρ∇

1

ρ· ∇

�pðx; t; xsÞ ¼ δðx − xsÞfðtÞ: (9)

For initial velocity v0 and density ρ0 models, the perturbed wave-field δp satisfies

�1

v20

∂2

∂t2− ρ0∇

1

ρ0· ∇

�δpðx; t; xsÞ

≈�2δvv30

∂2

∂t2−�∇δρ

ρ0

�· ∇

�p0ðx; t; xsÞ; (10)

where δv ¼ v − v0 and δρ ¼ ρ − ρ0 denote the velocity and densityperturbations, respectively. Using a similar method developed byJin et al. (1992) and Forgues and Lambaré (1997), with the detailedderivation in Appendix A, we obtain the ray-based relation betweenthe perturbed geologic models and wavefield:

δvv0

þ cos2 θδρ

ρ0

¼ −Z Z Z Z

Wcos αsv0ðxsÞ

cos αrv0ðxrÞ

AsAre−iωðτsþτrÞ

× δdðxr;ω; xsÞδðθ 0ðx; xr; xsÞ − θÞdωdθ 0dxrdxs; (11)

where As and Ar are the amplitudes of the Green’s function from thesource or receiver to the image point, respectively; the amplitudeweight W ¼ 32πv0ðxÞcos2 θ 0∕ sin θ 0, δd is the difference betweenthe seismic data and the predicted data from the forward modeling(using the initial velocity and density); and τs and τr are the trav-eltimes from the source or receiver to the image point, respectively.In the context of RTM defined by Zhang et al. (2007) (equations 6

and 7 therein for 2D one-way, wave-equation migration alsovalid for RTM and easily extended to 3D propagation), theasymptotic forms of pFðωÞ and pBðωÞ for the full-acoustic-waveequation are given in terms of source and receiver traveltimesand amplitudes:

pFðωÞ ¼cos αsv0ðxsÞ

Ase−iωτðx;xsÞ (12)

and

pBðωÞ ¼Z

cos αrv0ðxrÞ

iωAre−iωτðx;xrÞδdðxr;ω; xsÞdxr. (13)

Substituting equations 12 and 13 into equation 11 and replacing thedensity perturbation with impedance where

δðvρÞv0ρ0

¼ δvv0

þ δρ

ρ0. (14)

Equation 11 can be rearranged into the same form as the imagingcondition given by equation 4:

sin2θδvv0

þcos2θδðvρÞv0ρ0

¼−32πZ Z Z

v0ðxÞsin θ 0

cos2θ 0δðθ 0−θÞpBðωÞpFðωÞiω

dωdθ 0dxs:

(15)

Equation 15 shows that for a common shot RTM, if we output sub-surface angle gathers with the above imaging condition, the near-angle images predict the impedance perturbation δðvρÞ∕ðv0ρ0Þ,whereas the far-angle images can be used to estimate the velocityperturbation δv∕v0. Therefore, we can separate the effects of veloc-ity and density by outputting ADCIGs.One may also note that the described process of separating the

velocity and density effects based on a modified imaging conditionprovides a result similar to the well-known traditional amplitudevariation with offset (AVO) or AVA methods. In Appendix B,we show that our approach is consistent with the traditionalAVO/AVA methods upon an angle-domain rescaling.

Figure 3. (a) The impedance and (b) velocity perturbations for thesynthetic example in Figure 2, which present the true perturbations(in blue) and estimated perturbations from the amplitude-preservingRTM (in red).

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If we assume that the density function ρ is a constant, equation 15reduces to

δvv0

¼ −32πZZZ

v0ðxÞsin θ 0 cos

2 θ 0δðθ 0 − θÞ

×pBðωÞpFðωÞ

iωdωdθ 0dxs: (16)

Because equation 16 is independent of θ, integrating equation 16over θ from zero to π∕2, we have

δvv0

¼ −32πZZ

v0ðxÞsin θðx; xsÞ

cos2 θðx; xsÞ

×ðpBpFÞðx;ω; xsÞ

iωdωdxs: (17)

In this case, the stacked image from equation 15 collapses to thevelocity perturbation only, as shown in equation 17. This is consis-tent with the earlier results using Kirchhoff migration (Operto et al.,2000) and RTM (Zhang et al., 2012).

In practice, a band-pass filter is applied to seismic data beforeRTM that matches the effective bandwidth in acquisition, especiallythe low-frequency content. Our RTM is carefully implementedto ensure amplitude and phase correctness during migration, to-gether with a Laplacian filter (Zhang and Sun, 2009) on the outputimages to remove backscattering noise. However, the impedanceand velocity perturbations found using equation 15 are still bandlimited due to the bandwidth restriction of seismic data. Whenthe low-wavenumber information of the impedance and velocity isachievable in the initial velocity v and density ρ, these band-limitedperturbations can be used to invert for more accurate velocity anddensity models, such as the example in the numerical experiment sec-tion using FWI.

RELATIONSHIP OF AMPLITUDE-PRESERVINGREVERSE TIME MIGRATION TO FULL-

WAVEFORM INVERSION

A simple link between RTM and FWI can be seen by consideringa true velocity model that is smoothly varying and in which densitycontrasts generate the reflections. In this case, the forward modeling

Figure 4. (a) TheSigsbee2avelocitymodelwith a black box indicating the zoomarea in (b, c, andd). (b) Themigrationvelocity in the zoomed area.(c) The velocity perturbation computed from RTM. (d) The true velocity perturbation. The dashed lines show the location extracted in Figure 5.

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produces no reflections and the subsequent FWI residuals containthe reflections only. Construction of the FWI gradient then becomesan RTM exercise with an appropriate imaging condition. This givesrise to the classical statement that “migration is the first iteration offull waveform inversion” (Mora, 1989).If we consider marine data with ghosts and ignore the density

variations, then equation 11 with ghost compensation gives thevelocity update as

δvv0

¼ −16π2ZZZ

v0ðxÞsin θðx; xr; xsÞ

cos αsv0ðxsÞ

cos αrv0ðxrÞ

× AsArδde−iωðτsþτrÞ

GsGr

dxrdxsdω: (18)

For a conventional FWI using the steepest descent method, thevelocity is updated by the gradient direction of the least-squares costfunction E ¼ min kδdk2 (Tarantola, 1984). In the high-frequencyapproximation, the gradient calculated in FWI can be expressedas (see, for example, Tarantola, 1984; Mora, 1987)

∂E∂v

¼−Z Z Z

ω2Gðxs;x;ωÞGðx;xr;ωÞδdðxs;xr;ωÞdxrdxsdω

¼−Z Z Z

AsArω2δde−iωðτsþτrÞdxrdxsdω: (19)

Comparing equations 18 and 19, we see that the two formu-lations have similar asymptotic expressions, with the same phase(ωðτs þ τrÞ) and amplitude polarity, but they use different ampli-tude weighting functions. The scaling in the FWI formulationis frequency dependent (ω2) and also without the surface ampli-tude and ghost compensation cos αs cos αr∕ðv0ðxsÞv0ðxrÞGsGrÞ.

Therefore, conventional FWI tends to favor higher frequencies dur-ing a given update. This will slow down the convergence in a time-domain iterative implementation of FWI, when compared with theRTM formulation.Equation 18 gives a direct map from the seismic data to the veloc-

ity perturbation and can also be used to give an iterative update ofthe velocity model in a similar manner to standard implementationsof FWI, but by replacing the FWI-derived, unscaled velocity modelperturbation with one computed by our new RTM formulation. Thisis achieved using the following workflow:

1) Starting from a smooth velocity model, we compute the costfunction of the perturbed wavefield.

2) We use this same starting velocity model in our new RTM for-mulation to generate the necessary ADCIGs.

3) We extract the current velocity model perturbation δv∕v by per-forming an AVA-like fit of equation 15 to the ADCIGs at eachdepth in the velocity model.

4) We compute an optimal scalar to minimize the cost function ofthe perturbed wavefield by a step-length calculation on the ex-tracted current velocity model perturbation (this is a standardprocess; see, e.g., Virieux and Operto, 2009).

5) We update the velocity model with this scaled velocity modelperturbation.

6) We iterate until convergence.

NUMERICAL AND REAL DATA EXAMPLES

The first example is designed to demonstrate the reliability ofusing RTM after ghost compensation to estimate impedance andvelocity perturbations. Here, we construct a horizontally invariantgeologic model to a 4-km depth. The velocity and density are ran-domly generated within their dynamic ranges to make the two seriesuncorrelated. Figure 2a shows the central shot record from the

Figure 5. (a) The velocity perturbation computedfrom amplitude-preserving RTM and (b) the truevelocity perturbation. The trace location is indi-cated by the dashed line shown in Figure 4cand 4d.

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model, with receivers spread on both sides to a maximum offset of12 km. The source and receiver ghosts are recorded at a shot andreceiver depth of 6 and 8 m, respectively. Setting the backgroundvelocity and density to 2000 ms−1 and 2000 kgm−3, respectively,we use the ghost-compensated RTM formulation to propagatethe wavefields, apply the appropriate imaging condition to generatesubsurface offset gathers (Sava and Fomel, 2003), and then convertthem to ADCIGs, as shown in Figure 2b. Based on our theory, the 0°trace directly estimates the impedance perturbation and the velocityperturbation can be calculated from any nonzero-degree trace. How-ever, far-angle data will have fewer numerical errors when separat-ing the perturbations. Thus, we select the 0° and 66° traces forimpedance and velocity perturbation, respectively. After simplerescaling, one can see from Figure 3a and 3b that the estimated per-turbations (in red) give a good match to the true perturbations (inblue), derived by subtracting the constant background impedance orvelocity from the exact impedance or velocity, respectively. Themismatch of the velocity perturbation at depths greater than3 km is due to the lack of large reflection angles over the fixed sur-face offset range and the event stretch on the far-angle ADCIG.Next, we demonstrate the performance of our new RTM formu-

lation to produce velocity perturbations, as described by the modi-fied boundary condition in equations 7 and 8 and the imagingcondition in equation 17. Here, we used the Sigsbee2a model (Paf-fenholz et al., 2002) in Figure 4a to generate the synthetic data set,with the source and receiver depth at 25 ft. In this example, we focuson the area to the left of the salt body, as denoted by the box inFigure 4a. Zooming into this area in Figure 4b, we show the con-stant gradient background velocity used for migrating the data. TheRTM velocity perturbation image is shown in Figure 4c. Figure 4dis the true velocity perturbation, derived by subtracting the migra-tion velocity from the true velocity model. With the ghost compen-sation, the RTM image does a good job in reconstructing thevelocity perturbation. In Figure 5, we choose one imaging location,indicated by the dashed line in Figure 4 and compare the velocityperturbation from RTM (Figure 5a) with the true one (Figure 5b).The overall match of the two is good, except for a scaling differ-ence, which can be calibrated by a linear search.In the third example, we apply the new RTM formulation to the

BP2004 2D model (Billette and Brandsberg-Dahl, 2005). The syn-thetic data are generated by high-order, finite-difference acousticmodeling using variable velocity and density models, with shotspacing of 50 m, receiver spacing of 25 m, and 8-km maximumoffset. Source and receiver ghosts are recorded at a 12.5-m depth.In Figure 6, we compare the impedance perturbation output fromour RTM (Figure 6a) using a smooth migration velocity with thetrue impedance perturbation (Figure 6b), in which the true imped-ance perturbation is derived by subtracting the product of thesmooth migration velocity and an assumed constant density fromthe true impedance. The estimated and true impedance perturba-tions generally agree quite well. For detailed investigation, wepresent the estimated and true impedance perturbations at the loca-tion indicated by the dashed line in Figure 6a and 6b. The estimatedimpedance perturbation does a good job of reproducing the trueperturbation, except at the sharply changing impedance boundaries,e.g., at the water bottom or at the top of salt boundary. This is due tothe large model perturbation that does not fully satisfy the lineari-zation assumption in deriving equation 11 and the lack of ultralowfrequencies in the seismic data. This result demonstrates the

Figure 6. BP2004 model example: (a) The estimated impedanceperturbation computed from RTM. (b) The true impedance pertur-bation. (c) The detailed plot of the estimated (in blue) and exact (inred) impedance perturbations at the location indicated by the dashedline in panels (a and b).

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applicability of our theory to geologic models with complex struc-tures and density variations.Next, we present ghost-compensated RTM images for a shallow

water data set from the central North Sea. The source and receiverdepths are 6 and 7 m, respectively. The geology of this area shows afairly flat velocity profile down to the Top Balder horizon at approx-imately 3-km depth. The velocities then kick strongly here andagain in a high-velocity chalk section. These are typical geologicfeatures in this area. Figure 7a shows a stacked reflectivity image,and Figure 7b shows an impedance perturbation section — thelow-frequency components (lost due to ghost effects) are compen-sated in both of the images, which give the sections a more con-tinuous and textured appearance characteristic of broadband data.The inset in Figure 7b compares the estimated impedance pertur-bation from RTM against impedance the well-log data filtered tothe seismic bandwidth at the location given by the red vertical line.It can be seen that the estimated impedance provides a good

approximation to the measured data. Figure 7c shows a colored in-version result (Lancaster and Whitcombe, 2000), using the RTMreflectivity of Figure 7a and the well log. Figure 7d shows theRTM impedance perturbation only, obviously generated withoutthe use of the well log, but replotted from Figure 7b with the samecolor map as the colored inversion in Figure 7c. The filtered well-logimpedance is included at the well location to aid comparison here. Itis clear that there is good agreement between the colored inversionand the RTM impedance. This shows that the new RTM formulationis not only a technique for achieving images with broad bandwidth,but also a useful tool to estimate band-limited impedance of the real-world subsurface geology, achieved without using well data.Finally, using the same shallow water data set from the central

North Sea, we demonstrate how the velocity perturbation can beused in an iterative manner to update the velocity field. Becauseof the limited dynamic range available in a color map, we chooseto zoom into the shallow and deep sections separately. Figure 8a

Figure 7. Central North Sea data example: (a) ghost-compensated stacked RTM reflectivity image, (b) impedance perturbation section with acomparison of filtered well-log impedance (in black) and estimated RTM impedance (in red) at the location of the vertical line, (c) coloredinversion generated using the RTM reflectivity from (a) and the well log, and (d) RTM impedance perturbation from (b), but replotted with thesame color map as in (c). The filtered well-log impedance is included at the well location to aid comparison here.

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Figure 8. Central North Sea data example: (a) shallow section display of the initial velocity overlaid with an RTM image, (b) shallow sectiondisplay of the inverted velocity overlaid using the proposed iterative method with RTM. Some interesting geologic features at this depth arenow present in the RTM-updated velocity model. (c) Deeper section display of the initial velocity overlaid with an RTM image, (d) deepersection display of the inverted velocity overlaid using the proposed iterative method with RTM. Velocity structure within the chalk package isclearly improved by the update; in particular, note the lower velocity Hod formation. (e) Deeper section display of ghost-compensated RTMimage migrated with initial (left) and RTM-updated (right) velocity model. The red circles highlight some areas of difference, albeit a subtleimprovement.

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shows the shallow section for the initial velocity model overlaidwith an RTM image. This overlay is done to aid the visual identi-fication between the features in the velocity model and those in theseismic stack section. Figure 8b shows the same shallow sectionafter an iterative application of the velocity perturbations derivedfrom RTM as described earlier. The extra detail in the updatedvelocity model is clear and correlates with the main geologic fea-tures in this shallow section, namely, the contourites and dewateringfaults. Figure 8c shows the deeper section for the initial velocitymodel overlaid with an RTM image — the chalk package is clearlyidentifiable on this section. Figure 8d shows the same deeper sec-tion after an iterative application of the velocity perturbations. Wenow see internal structure in the chalk section, in particular, thelower velocity Hod formation (a typical feature in the NorthSea) within the chalk package. Also, the vertical extent of the chalksection is better defined with sharper boundaries. Figures 8e com-pares ghost-compensated RTM images for the deeper section mi-grated with the initial (left) and updated velocity (right) model,respectively. Although it is a subtle effect, there is an improvementin the image obtained with the velocity model as updated by ourRTM velocity perturbations, and this improvement is observedin the areas in which we have the strongest velocity change. Overall,many high-frequency detailed features are formed in our RTM-updated velocity model. These features seem geologically plausibleand agree with our knowledge of the area. In summary, the real dataimpedance and velocity example shown here confirm the appli-cability and potential uses of our theory to geologic models withcomplex structures.

CONCLUSIONS

We have developed a theory of amplitude-preserving RTM todelineate the impedance and velocity perturbations of subsurfacestructures. The near-angle stacked image provides the impedanceperturbation estimate, whereas far-angle images can be used to es-timate the velocity perturbation. Together with the ghost compen-sation technique, the method described here can be a useful tool ofseismic inversion for marine exploration and used to link and bridgethe gap between RTM and FWI. Synthetic and real data exampleshave shown that the method is reliable and provides additional geo-logic information for interpretation and inversion.

ACKNOWLEDGMENTS

We thank CGG’s Multi-Client and New Ventures division forproviding the North Sea data sets, our colleagues in CGG for theirsupport, especially R. Wombell, A. Smith, and A. Khalil for theirhelp with the data examples, and S. Gray and J. Sun for their help torevise the paper. Finally, we are grateful to the associate editor B.Tang and also to other anonymous reviewers for their help in im-proving the final quality of this paper.

APPENDIX A

DERIVATION OF EQUATION 11

We assume that the investigated subsurface area of interest is anisotropic medium with velocity v and density ρ such that the propa-gation of wavefield pðx; t; xsÞ can be modeled by using the acous-tic-wave equation and the recorded seismic data dðxr; t; xsÞ at the

receiver location xr and source at location xs. This is consistent withthe amplitude-preserving RTM in equations 1 and 2 and can bemodeled as follows:

(�1v2

∂2∂t2 − ρ∇ 1

ρ · ∇�pðx; t; xsÞ ¼ δðx − xsÞδðtÞ;

dðxr; t; xsÞ ¼ pðxr; t; xsÞ:(A-1)

Introducing small perturbations to the velocity and density pro-files using vþ δv and ρþ δρ, respectively, the resultant wavefieldpþ δp and observed data perturbation δd must satisfy the sameacoustic wave equation A-1, as follows:

(�1

ðvþδvÞ2∂2∂t2 − ðρþδρÞ∇ 1

ρþδρ ·∇�ðpþδpÞ¼ δðx−xsÞδðtÞ;

δdðxr; t;xsÞ¼ δpðxr; t;xsÞ:ðA�2Þ

By substituting equation A-1 into A-2, applying Taylor’s expan-sion, and ignoring the higher order terms, we have

�1

v2∂2

∂t2− ρ∇

1

ρ· ∇

�δpðx; t; xsÞ

¼�2δvv3

∂2

∂t2−�∇δρ

ρ

�· ∇

�pðx; t; xsÞ: (A-3)

Although the inhomogeneous part of equation A-3 is quite com-plicated, the Green’s function Gðx; x 0; t − t 0Þ for solving it is onlyrequired to satisfy the homogeneous equation:

�1

v2∂2

∂t2− ρ∇

1

ρ· ∇

�Gðx; x 0; t − t 0Þ ¼ δðx − x 0Þδðt − t 0Þ:

(A-4)

Using the boundary condition in equation A-2, the solution ofequation A-3 can then be expressed at the receiver locations, xr,

δdðxr; t;xsÞ ¼ZZ

2δvðxÞvðxÞ3

∂2pðx; t 0;xsÞ∂t 02

Gðxr;x; t− t 0Þ

þ δρðxÞρðxÞ ∇ · ðGðxr;x; t− t 0Þ∇pðx; t 0;xsÞÞdt 0dx;

(A-5)

or equivalently in the frequency domain

δdðxr;ω;xsÞ¼Z

−2ω2δvðxÞvðxÞ3 pðx;ω;xsÞGðxr;x;ωÞ

þδρðxÞρðxÞ ∇ · ðGðxr;x;ωÞ∇pðx;ω;xsÞÞdx: (A-6)

We denote the traveltime and amplitude from a location x to an-other location y as τðx; yÞ and aðx; yÞ, respectively. Asymptotically,the pressure and Green’s functions satisfy the following equations:

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pðx;ω; xsÞ ¼ aðx; xsÞeiwτðx;xsÞ (A-7)

and

Gðxr; x;ωÞ ¼ aðxr; xÞeiwτðxr;xÞ; (A-8)

and the gradient of the asymptotic pressure function is given by

∇pðx;ω; xsÞ ≈ iω∇τðx; xsÞaðx; xsÞeiωτðx;xsÞ: (A-9)

Substituting equations A-7–A-9 into A-6, the high-frequencyapproximation of the second term on the right side can be expressed as

∇ · ðGðxr; x;ωÞ∇pðx;ω; xsÞÞ¼ −ω2∇τðx; xsÞ · ð∇τðx; xrÞþ ∇τðxs; xÞÞaðx; xsÞaðxr; xÞeiωðτðx;xsÞþτðxr;xÞÞ

¼ −ω2cos 2θ þ 1

vðxÞ2 aðx; xsÞaðxr; xÞeiωðτðx;xsÞþτðxr;xÞÞ

¼ −ω22 cos2 θ

vðxÞ2 Aðxs; xr; xÞeiωTðxs;xr;xÞ; (A-10)

where Tðxs; xr; xÞ ¼ τðx; xsÞ þ τðxr; xÞ and Aðxs; xr; xÞ ¼ aðx; xsÞaðxr; xÞ denote the ray traveltime summation and amplitude productof the wavefield from a source location xs to a subsurface location xand reflected back to the receiver location xr, respectively, and θ isthe incident angle at the subsurface location x.

Substituting equation A-10 into equation A-6, we have

δdðxr;ω; xsÞ ¼ −Z

2ω2

vðxÞ2�δvðxÞvðxÞ þ cos2θ

δρðxÞρðxÞ

�× Aðxs; xr; xÞeiωTðxs;xr;xÞdx; (A-11)

which describes the relationship of the data perturbation in the fre-quency (time) domain to the perturbations in the model domain.However, for imaging algorithms, we aim for the (inverse) relation-ship of equation A-11 that describes the model perturbations interms of the data perturbation. Following Bleistein (1987), we in-troduce the inverse operator Bðxr;ω; xs; x; θ;φÞ, and the model per-turbations must satisfy

δvvþ cos2 θ

δρ

ρ

¼ZZZ

Bðxr;ω; xs; x; θ;φÞδdðxr;ω; xsÞ

× e−iωTðxs;xr;xÞdxrdxsdω; (A-12)

where φ is subsurface azimuth angle. By substituting equation A-11into equation A-12, we have

δvvþ cos2 θ

δρ

ρ

¼ −ZZZ

Bðxr;ω; xs; x; θ;φÞ

×Z

2ω2

vðx 0Þ2�δvðx 0Þvðx 0Þ þ cos2 θ 0 δρðx 0Þ

ρðx 0Þ�

× Aðxs; xr; x 0ÞeiωðTðxs;xr;x 0Þ−Tðxs;xr;xÞÞdx 0dxrdxsdω:

(A-13)

Changing the integration variable from the surface coordinates,ðxr; xsÞ, to the subsurface angles, ðθ 0;φ 0Þ, equation A-13 can beapproximated as

δvvþ cos2 θ

δρ

ρ

≈ −ZZZ

Bðxr;ω; xs; x; θ 0;φ 0Þδðθ − θ 0Þδðφ − φ 0Þ

×Z

2ω2

vðx 0Þ2�δvðx 0Þvðx 0Þ þ cos2 θ 0 δρðx 0Þ

ρðx 0Þ�

× Aðxs; xr; x 0Þeikðx 0−xÞ���� ∂ðxr; xs;ωÞ∂ðk; θ 0;φ 0Þ

����dx 0dkdθ 0dφ 0:

(A-14)

The approximation in equation A-14 is only valid when the high-frequency asymptotic approximation ωðTðxs;xr;x0Þ−Tðxs;xr;xÞÞ≈kðx0−xÞ holds (Bleistein, 1987). The Jacobian in equation A-14includes the Beylkin determinant (Beylkin, 1985) and providesthe connection between local direction vectors at an image locationand the actual acquisition geometry. In the current case of isotropicmedia, it is derived in Xu et al. (2011) as follows:���� ∂ðk; θ;φÞ

∂ðxr; xs;ωÞ����

¼ 2ð4πÞ4 ω2ðcos θÞ2Aðxs; xr; xÞ2

vðxÞ sin θ

cos αsvðxsÞ

cos αrvðxrÞ

; (A-15)

where αs and αr (again) denote the source and receiver propagationangle with respect to the vertical direction, respectively.To make equation A-14 valid, we require the inverse operator

Bðxr;ω; xs; x; θ;φÞ to satisfy

2ω2

v2Aðxs; xr; xÞBðxr;ω; xs; x; θ;φÞ

���� ∂ðxr; xs;ωÞ∂ðk; θ 0;φ 0Þ���� ¼

�1

2π

�3

:

(A-16)

By substituting equation A-15 into equation A-16, we have

Bðxr;ω; xs; x; θ 0;φ 0Þ ¼ −Wcos αsvðxsÞ

cos αrvðxrÞ

Aðxs; xr; xÞ;(A-17)

whereW ¼ 32πvðxÞcos2 θ 0∕ sin θ 0. Substituting equation A-17 intoequation A-12, we find the desired relationship of equation 11 as

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δvv0

þ cos2 θδρ

ρ0

¼ −ZZZ

Wcos αsvðxsÞ

cos αrvðxrÞ

Aðxs; xr; xÞδdðxr;ω; xsÞ

× δðθ − θ 0Þe−iωTðxs;xr;xÞdxrdxsdωdθ 0. (A-18)

APPENDIX B

RELATION TO AMPLITUDE VARIATIONWITH OFFSET

The AVO studies are often based on approximations to the Zoep-pritz equations. These equations describe the reflection and trans-mission coefficients of a plane-wave incident at a planar interface.The approximations are valid for small changes in elastic propertiesacross the interface and, depending on the approximation used, forincidence angles up to 30° or so. In the acoustic case, the approxi-mation to the P-wave reflectivity derived by Aki and Richards(1980) for an incident plane P-wave is given by

RppðθaÞ ¼1

2

�δðVP ρÞVP ρ

þ 1

2tan2 θa

�δVP

VP

; (B-1)

where VP and θa are the P-wave velocity and the mean of the in-cident and transmission angles, respectively. Equation 15 indicatesthat the image from an amplitude-preserving RTM provides

IðθÞ ¼ cos2 θ

�δðVP ρÞVP ρ

þ sin2 θ

�δVP

VP

: (B-2)

To reconcile these two formulas, we notice that an angle-dependentscaling of cos2 θ is applied to the ADCIGs from an RTM. Dividingequation B-2 by this factor gives the Aki and Richards AVOapproximation, so equation B-1, which is commonly used inAVO analysis (Debski and Tarantola, 1995; Li, 2005), is consistentwith our proposed RTM in equation 15.

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