GEOPHYSICS, VOL. 73, NO. 5 共SEPTEMBER-OCTOBER 2008兲; P. VE25–VE34, 17 FIGS. 10.1190/1.2952039

Stereotomography

Gilles Lambaré1

The idea was to use locally coherent events characterized by their slopes in the prestack data volume. Such events can be interpreted as pairs of ray segments and can provide information about the velocity model independently from each other 共Figure 1兲. The idea had been investigated by Riabinkin 共1957兲 and revived by Sword 共1986, 1987兲, who called it the common-directional-reception 共CDR兲 method. Since the time of Sword’s work, variations and improvements of CDR have been proposed. Several velocity-model-building methods, based on the use of locally coherent events, are variations of CDR. For example, in Guillaume et al. 共2001兲, the velocity-model update is based on locally coherent events in common-offset gathers. These events are obtained by picking facets on initial common-offset depth-migrated results. These initial migrated facets are kinematically demigrated into kinematic invariants corresponding to locally coherent events in the common-offset trace gathers. The kinematic invariants are the basis of the tomographic update. They can be kinematically remigrated in any velocity model, allowing an assessment of quality in terms of alignment of migrated facets for different offsets. In a strategy proposed by Fei and McMechan 共2006a, 2006b兲, facets are picked on an initial common-offset depth-migrated image. For each facet, a common-reflection-point trace gather is built by kinematic demigration. Finally, the velocity model is updated to maximize the stack along traveltime curves in common-reflection-point gathers. Fei and McMechan 共2006a兲 mention that maximizing the stack implicitly means fitting the local traveltime shape of the event and consequently its slope, showing strong roots with CDR. Several studies attempt to improve the CDR method. For example, Whiting 共1991, 1998兲 introduces a better constraint of the velocity-model update 共use of entropy constraints and stages of decreasing model-space smoothing兲; whereas, Biondi 共1992兲 combines CDR with a process that maximizes the semblance in beam-stack panels to be less sensitive to the quality of data when picking locally coherent events. Another very important contribution, by Chauris et al. 共2002a, 2002b兲, demonstrates that CDR picking could be done equally in the time-unmigrated or depth-migrated domains, which reconciles time- and depth-domain-picking velocity-model-building approaches.

ABSTRACT Stereotomography was proposed 10 years ago for estimating velocity macromodels from seismic reflection data. Initially, the goal was to retain the advantages of standard traveltime tomography while providing an alternative to difficult interpretive traveltime picking. Stereotomography relies on the concept of locally coherent events characterized by their local slopes in the prestack data cube. Currently, stereotomography has been developed in two and three dimensions, and precious experience has been gained. The expected advantages have been demonstrated fully 共in particular, the efficiency and reliability of the semiautomatic stereotomographic picking strategies兲, and further studies have increased the method’s potential and flexibility. For example, stereotomographic picking can now be done in either the prestack or poststack domain, in either the time 共migrated or unmigrated兲 or depth domain. It appears that the theoretical frame of stereotomography can reconcile, very satisfactorily and efficiently, most methods proposed for velocity-macromodel estimation for depth imaging. Moreover, an extension of the method to full-waveform inversion already exists and opens the way for very interesting developments.

INTRODUCTION We initially proposed stereotomography 共Billette et al., 1997兲 as a robust tomographic method for estimating velocity macromodels from seismic-reflection data. At that time, we recognized the potential efficiency of traveltime tomography 共Bishop et al., 1985; Farra and Madariaga, 1988兲 and difficulties associated with a highly interpretive picking 共which explains why traveltime tomography does not form the base of conventional velocity-model-building today兲. Selected events must be tracked across a large extent of the prestack data cube, which can be challenging for noisy or complex data 共Ehinger and Lailly, 1995兲.

Manuscript received by the Editor 31 December 2007; revised manuscript received 8 February 2008; published online 1 October 2008. 1 CGGVeritas, Massy, France. E-mail: [email protected]. © 2008 Society of Exploration Geophysicists. All rights reserved.

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In fact, stereotomography is a generalization of CDR, with the added benefits of robustness and simplicity 共the extension to three dimensions, for example, is straightforward 关Chalard et al., 2000兴兲. The introduction of paraxial ray tracing 共Cerveny et al., 1977; Farra and Madariaga, 1987兲 was the other significant improvement because it provided an efficient solution for numerically computing Fréchet derivatives 共the stereotomographic matrix兲 共Billette and Lambaré, 1998兲. Note that Chauris et al. 共2002a, 2002b兲 also introduce paraxial ray tracing in CDR. Ten years after our first paper on stereotomography, we have gained significant experience from practical and theoretical applica-

tions. Stereotomographic algorithms have been developed in two and three dimensions, and they have been applied on various synthetic and field data sets 共Chalard et al., 2002; Alerini et al., 2003; Billette et al., 2003; Lambaré et al., 2004a; Lambaré et al., 2004c; Le Bégat et al. 2004; Lambaré and Alerini, 2005; Shen et al., 2005a; Alerini, 2006; Dümmong et al., 2007兲. Stereotomography also has been extended to the analysis of converted waves 共Alerini et al., 2007, 2008兲, direct arrivals 共Gosselet et al., 2003, 2005兲, and anisotropic propagations 共Barbosa et al., 2006; Nag et al., 2006兲. However, the most interesting extensions certainly revolve around picking concepts. Initially introduced in the prestack time unmigrated domain 共Billette and Lambaré, 1998; Billette et al., 2003, Lambaré et al., 2004b兲, it has been demonstrated that stereotomographic picking could be done in the depth-migrated domain 共Chauris et al., 2002a; Nguyen et al., 2003, 2008兲, in the poststack time domain 共Lavaud et al., 2004; Neckludov et al., 2005, 2006兲, or even in the prestack time migrated domain 共Lambaré et al., 2007兲 共Figure 1兲. In this paper, I first review the basis of stereotomography. Then, I present various developments of the method, leading to a very efficient and general frame for velocity-macromodel estimation. Finally, I discuss advantages and challenges of stereotomography and offer perspectives.

STEREOTOMOGRAPHY A slope tomographic method Figure 1. Stereotomographic picking was initially introduced in the prestack time unmigrated domain 共Billette and Lambaré, 1998兲. It has been shown that it could be done in the depth-migrated domain 共Chauris et al., 2002a兲, in the CRS domain 共Lavaud et al., 2004兲, or even in the prestack time migrated domain 共Lambaré et al., 2007兲. This makes stereotomographic inversion a very powerful and flexible tool for velocity-model building.

Stereotomography belongs to the family of slope tomographic methods 共Riabinkin, 1957; Sword, 1987; Billette and Lambaré, 1998兲. The basis of these methods is to recognize that any locally coherent event in the prestack unmigrated domain, characterized by its traveltime and slopes, provides information on the velocity model. A stereotomographic data set d consists of a set of parameters corresponding to N picked locally coherent events dn: N d ⳱ 关dn兴n⳱1 .

a)

共1兲

Each locally coherent event,

d ⳱ 共s,r,Tsr,ps,pr兲,

共2兲

is described by the source and receiver positions s and r, two-way traveltime Tsr, and slopes of the event in the common-receiver and common-shot directions ps and pr 共Figure 2兲. It is what we call a stereotomographic pick. Any event can be associated in the exact velocity model with a pair of ray segments parameterized by

共X, ␤ s, ␤ r,Ts,Tr兲,

b)

Figure 2. 共a兲 Stereotomographic data. A locally coherent event in the prestack data cube, characterized by its central source and receiver positions s and r along axis sx and rx, slopes ps and pr, and traveltime, Tsr along axis t. 共b兲 Stereotomographic model: The associated pair of ray segments superimposed on the velocity macromodel characterized by the position of the reflector/diffractor point X, ray-shooting angles from X toward s and r, ␤ s, and ␤ r, and the two one-way traveltimes from X toward s and from X toward r, Ts, and Tr.

共3兲

where X denotes the position of the reflector/diffractor; ␤ s, ␤ r are ray-shooting angles from X toward s and r; and Ts, Tr are two oneway traveltimes from X toward s and from X toward r 共Figure 1兲. There are many ways to use information from stereotomographic picks. CDR considers the misfit on a single type of data parameter 共traveltime, slope, or position兲. In stereotomography, the cost function consists of squared misfits on all types of data parameters, i.e., positions, slopes, and traveltimes. This introduces uncertainties on all types of data and theoretically ensures robustness of the local optimization 共Billette and Lambaré, 1998兲. But as a consequence, pairs of ray segments must be optimized jointly with the velocity macromodel, increasing the number of model parameters.

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Stereotomography Then, the stereotomographic model is a combination of the velocity macromodel, described by a set of velocity parameters Vm, 共typically weights associated with a basis of B-spline functions used for describing the velocity model兲 and a set of pairs of ray segments associated with each picked event, M N m ⳱ 关共Vm兲m⳱1 ,关共X, ␤ s, ␤ r,Ts,Tr兲n兴n⳱1 兴.

The stereotomographic algorithm consists of three steps 共Figure 3兲 共Billette et al., 2003兲: 1兲

共4兲 2兲

For any a priori pair of ray segments and velocity model, the set of stereotomographic parameters corresponding to a stereotomographic pick can be calculated:

dcalc ⳱ 共s,r,Tsr,ps,pr兲calc .

共5兲

This computation only requires tracing two ray segments from the common point X toward the surface with initial directions ␤ s and ␤ r and with lengths corresponding to one-way traveltimes Ts and Tr, respectively. The stereotomographic cost function is the squared misfit function between calculated and observed data,

1 ⳮ1 C共m兲 ⳱ 共dcalc共m兲 ⳮ dobs兲TCD 共dcalc共m兲 ⳮ dobs兲, 2 共6兲 where CD denotes the a priori covariance matrix for data parameters 共Tarantola, 1987兲 共typically a diagonal matrix with the square of a priori errors on observed parameters of stereotomographic picks兲 and T denotes the transposition.

Stereotomographic optimization As in standard traveltime tomography, an iterative nonlinear local optimization scheme is used to update the stereotomographic model. Quasi-Newton optimization schemes have been used until now, requiring the computation of Fréchet derivatives of data d 共equation 1兲 with respect to model m 共equation 4兲:

⳵d . ⳵m

共7兲

A derivation of Fréchet derivatives is detailed in Billette and Lambaré 共1998兲. It is based on paraxial ray tracing 共Farra and Madariaga, 1987兲 and requires some smoothness of the velocity macromodel. In our numerical implementations, we use smooth-velocity macromodels defined by cubic-cardinal B-splines 共de Boor, 1978兲. There is no need for interfaces, but they could be introduced if they are sufficiently smooth 共Farra et al., 1989兲. In our local optimization scheme, we first optimize pairs of ray segments fixing the velocity macromodel to its initial value to mitigate the nonlinearity. This step is called the localization step. It can be seen as a generalization of the kinematic migration used, for example, in Guillaume et al. 共2001兲. The same cost function as for the global update is used, but the velocity model is fixed and we independently only optimize the ray segment parameters to fit the source and receiver positions, two-way traveltime, and slopes according to the a priori covariance matrix. In a standard common-offset kinematic migration, only slopes in common-offset gathers are considered, which means introducing very large relative a priori errors on remaining slopes for our localization scheme. Note that in the standard process, the a priori covariance matrix is a diagonal matrix with the square of the error observed for each data parameter.

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Initialization to build the initial model 共pairs of ray segments and velocity model兲. In practice, we use simple initial models, e.g., a homogeneous velocity model and ray segments derived from simple geometric considerations. Localization of events in the initial velocity model. This is done using a quasi-Newton nonlinear optimization. Because all events can be localized independently, we use singular-value decomposition 共SVD兲. The localization step leads to an important reduction of the cost function. Joint iterative inversion of ray segments and velocity-model parameters. This is realized using the LSQR scheme 共Paige and Saunders 1982兲. The LSQR optimization scheme is widely used for seismic tomographic problems because it is well adapted to tomographic problems with large and sparse Fréchet derivative matrices. A Laplacian regularization term is introduced for the velocity model.

For tuning the LSQR algorithm, we use recommendations of Paige and Saunders 共1982兲; for the Laplacian regularization, we use the empirical law given by Wang 共1993兲. Practically, a weighting factor is applied to the Laplacian regularization so that the energy of components of Fréchet derivatives associated with the Laplacian regularization is between one and one-tenth of the energy associated with misfits on the stereotomographic data parameters. In practice, such criteria can be used as a black box. Finally, we also use a multiscale approach for the velocity macromodel to mitigate nonlinearity 共Alerini et al., 2002; Billette et al., 2003; Lambaré et al., 2004a兲.

Stereotomographic picking Stereotomographic picks are locally coherent events in the prestack unmigrated time domain. In this domain, they can be more naturally picked with the advantage of considering the exact acquisition geometry. Because stereotomography aims at deriving velocity macromodels for depth migration, which is a stacking process over traces 共at least in its high-frequency asymptotic assumption兲, it is natural that picking locally coherent events is based on a local stacking process over traces. In fact, in most of our applications of ste-

Figure 3. Illustration of the nonlinear stereotomographic optimization. Starting with an initial stereotomographic model 共velocity Ⳮ rays兲, the first optimization step is a localization in which pairs of ray segments are relocalized iteratively in the initial velocity model. The misfit function generally is reduced significantly and the final joint optimization is done with an iterative gradient-type optimization.

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reotomography, we use stereotomographic picking on local slantstacked panels 共more precisely, the envelope of the local slantstacked traces兲 共Billette et al., 2003兲 共Figure 4兲. Automatic picking can be done, and then various selection criteria can be applied prior to or during the stereotomographic optimization 共Lambaré et al., 2004b兲 共Figure 5兲. Set-up and tuning of the various selection criteria are key points for the success of the stereotomographic update. First, outliers can be eliminated considering the general distribution of picked parameters, but additional parameters can be associated with the picks and also used for the selection. For example, parameters associated with the equivalent medium 共Billette et al., 2003兲, allowing interpretation of any individual stereotomographic pick as a pair of ray segments in

Figure 4. Stereotomographic picking on local slant stack panels in the time domain. The same trace is displayed in its common-shot 共left panel with axis 关sx,t兴 within the left box兲 and common-receiver 共left panel with axis 关rx,t兴 within the right box兲 gathers. Slant stacking allows defining slopes of events. Local slant stack panels are shown at the right of the left and right boxes with axis 共psx,t兲 and 共prx,t兲, respectively. They display the envelope of the local slant stack 共Billette et al., 2003兲.

a homogeneous velocity model, can be used advantageously for selecting the picks 共Lambaré et al., 2004c兲. Moreover, after the localization or joint optimization steps, we have access to local parameters within the depth-migrated cube, which also can be helpful for selecting stereotomographic events during optimization. For example, the local slope in common-image gathers 共CIGs兲 can be used to discriminate primaries from water-bottom multiples 共Figures 6–8兲 or PS from PP events in PP-PS stereotomography 共Lambaré and Alerini, 2005兲. In parallel to developments to improve the stereotomographic picking in the prestack data cube, important studies also have been done to investigate stereotomographic picking in the prestack depthmigrated cube, which is frequently recognized as the most natural domain for the velocity-model-building process. Indeed, the pre-

Figure 6. Application of 2D stereotomography with a semiautomatic picking to the 2004 BP velocity benchmark data set 共Billette and Brandsberg-Dahl, 2005兲. Final stereotomographic model with dip bars 共migrated facets兲 superimposed. Water-bottom multiples were not removed from the data set prior to the stereotomographic picking. A selection of events according to their slopes in CIGs allowed identification and elimination of most multiples from the stereotomographic data set 共Lambaré et al., 2004b兲. Distance (km) 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 0.0

0.5

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1.0

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Figure 5. Selection of stereotomographic picks within the stereotomographic sequence. The selection during optimization loops can be based on information obtained from pairs of ray segments. Figure 4 displays the QC window showing the automatic picking.

Figure 7. Application of 2D stereotomography with a semiautomatic picking to the 2004 BP velocity benchmark data set 共Billette and Brandsberg-Dahl, 2005兲. Final depth-migrated image.

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Stereotomography stack depth-migrated cube, consisting of CIGs, is analogous to the prestack unmigrated data cube 共in three dimensions, hypercubes must be considered兲. In the migrated cube, locally coherent events also can be observed and picked. Schematically, when compared with stereotomographic picking in time, the traveltime is replaced by the depth, positions of source and receiver are replaced by the position and offset of the event, and the two slopes are replaced by the dip in the common-offset image 共COI兲 and the slope in the CIG 共Figure 9兲. The direct connection between the data picked in both spaces is an important contribution by Chauris et al. 共2002a, 2002b兲. It offers a beautiful reconciliation between traveltime-based and image-based approaches of velocity-macromodel estimation for depth imaging. Practically, it also means that the same stereotomographic optimization code could use stereotomographic data picked either in the prestack time 共Lambaré et al., 2007兲 or prestack depth-migrated domains. The benefit of picking in the migrated domain is to ensure a more regular coverage of the model and also to allow an easier interpretation 共i.e., selection兲 of events 共Nguyen et al., 2003, 2008兲. Finally, stereotomography appears to be a very effective kernel for handling many types of kinematic information picked on seismic-reflection arrivals. Indeed, from many parametric representations of traveltimes or depths in various domains, it is possible to build a stereotomographic data set using local information. For example, Lavaud et al. 共2004兲 propose poststack stereotomography in which the stereotomographic data set is built from a CRS profile and CRS attributes. The approach is improved by taking into account residual kinematic variations around CRS curves 共Neckludov et al., 2006兲.

3D stereotomography Stereotomography was first implemented in two dimensions 共Billette and Lambaré, 1998; Billette et al., 2003兲, along with automatic

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slope-picking. As mentioned in the introduction, an advantage of stereotomography, compared with other slope tomographic methods, is the a priori straightforward extension to three dimensions. Although it is straightforward theoretically, it is less evident practically because most acquisition geometries do not provide access to all slopes that would be a priori required for 3D stereotomography. For example, in conventional narrow-azimuth marine acquisitions, the crossline slope at the shotpoint cannot be estimated precisely because the shot-line distance is too large. Therefore, it was important to demonstrate that stereotomography could be extended to 3D, even if only a single lateral slope is available from the acquisition geometry 共Figure 10兲. This is the important contribution of Chalard et al. 共2000兲, who also show the first application of 3D stereotomography to a field data set 共Chalard et al., 2002兲. In fact, 3D stereotomography exhibits very important specificities. Let us consider a set of prestack traces corresponding to an acquisition geometry. We assume this set of traces can be ordered according to parameters n1, n2, and n3. For a single shot-line gather of traces in a narrow-azimuth multistreamer marine acquisition, n1 could be the shot number, n2 the receiver number along a streamer, and n3 the streamer number 共Figure 11兲. For a single-streamer oceanbottom cable 共OBC兲 acquisition, n1 could be the receiver number, n2 the inline shot number, and n3 the crossline shot number 共Figure 12兲. Around a central trace defined by 共n1,n2,n3兲, gathering traces in terms of common 共n2,n3兲, common 共n1,n3兲, or common 共n1,n2兲 allows to pick a stereotomographic pick as

n1,n2,n3,Tobs,

⳵ Tobs ⳵ Tobs ⳵ Tobs , , . ⳵ n1 ⳵ n2 ⳵ n3

共8兲

Such a locally coherent event can be observed on the traces. But for use in a stereotomographic optimization, we need to have access to the geometric information, i.e., shot and receiver positions

a)

Distance (km) 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68

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Depth (km)

1.0

b)

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Figure 8. Application of 2D stereotomography with a semiautomatic picking to the 2004 BP velocity benchmark data set 共Billette and Brandsberg-Dahl, 2005兲. CIGs in angle domain obtained with the final stereotomographic model 共Figure 6兲. We see that water-bottom multiples, which were not removed from the data set, have a significant curvature, whereas primaries are rather flat.

Figure 9. Stereotomographic picking in the depth-migrated domain. There is a link between a locally coherent event in the prestack unmigrated time domain 共a兲 and a locally coherent event in the prestack migrated depth domain 共b兲. The slope in the common-offset time domain is connected to the dip in the common-offset depth-migrated domain 共indicated in orange in both domains兲. Then, given previous slope and dip, the slope in the common-shot 共or common midpoint兲 time domain is connected to the slope of the event in the CIG 共indicated in yellow in both domains兲.

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s ⳱ 共sx,sy,sz兲 and r ⳱ 共rx,ry,rz兲, and differential quantities

⳵ sr ⳵ sr ⳵ sr , , . ⳵ n1 ⳵ n2 ⳵ n3

共9兲

With this information, calculated stereotomographic slopes are obtained from

⳵ Tcalc ⳵s ⳵r ⳱ ps · Ⳮ pr · , ⳵ ni ⳵ ni ⳵ ni

i ⳱ 1,2,3,

共10兲

where ps and pr are slowness vectors at ending points of the pair of ray segments 共Figure 13兲. Equation 10 allows us to understand how stereotomography can be extended to data sets in which slopes of the event are not considered in the horizontal plane 共Lambaré et al., 2005兲. In this case, topographical variations have to be introduced within differential quantities 共equation 10兲. Finally, note that for full azimuth acquisition, when source and receiver positions cover independently the entire acquisition surface,

Figure 10. 3D stereotomography. Slopes 共psx, prx, pry兲 correspond to the value of the projection of source and receiver rays slowness vectors at the surface on directions of the acquisition geometry. A single lateral slope is sufficient to constrain a 3D velocity macromodel 共here pry兲. Three-dimensional stereotomography can then be applied to 3D narrow-azimuth marine acquisitions.

Figure 11. 3D acquisition ordered as n1, n2, and n3 for a single-line multistreamer narrow-azimuth marine acquisition; n1 is the shot number, n2 the receiver number along a streamer, and n3 the streamer number.

an additional numbering parameter n4 can be introduced and 3D stereotomography can then be done in the more natural way, considering four slopes instead of three. Very few 3D stereotomographic results have been published up to now 共Chalard et al., 2000; Chalard et al., 2002兲. I reproduce on Figures 14–17 some results presented in Chalard and Lambaré 共2005兲 that demonstrate the ability of 3D stereotomography to recover a 3D velocity model when just a single lateral slope is available. Figure 14 shows the exact pairs of ray segments computed from a regular grid of reflecting points in a smoothed version of the EAGE/SEG overthrust model 共Aminzadeh et al., 1997兲. Figure 15 shows an inline and a crossline section of the velocity model with the location and dip of the regular grid of reflecting points. Slope components used in the stereotomographic data set are psx, prx, and pry 共psy is not considered兲. Figure 16 shows the same sections for the final stereotomographic model. The general structure of the velocity model has been recovered except for the deeper part and the boundaries of the model that were not sufficiently constrained by the data set. Figure 17 shows the final pairs of ray segments to be compared with the exact ones on Figure 14. We observed some lateral shift of the reflecting point positions when compared with the exact ones 共see Figures 16 and 17兲. Again, this is because of limited constraints on the stereotomographic model from the stereotomographic data, as for any tomographic tool addressing velocity-model building from limited-aperture seismic-reflection data. Appropriate a priori information should be introduced for better matching.

Figure 12. 3D acquisition ordered as n1, n2, and n3 for a single receiver line 3D OBC acquisition; n1 is the receiver number, n2 the inline shot number, and n3 the crossline shot number.

Figure 13. Computation of calculated slopes in 3D stereotomography. Slowness vectors associated with source and receiver rays at the acquisition surface are ps and pr. Differential quantities ⳵ s/⳵ ni and ⳵ r/⳵ ni required by equation 10, the expression of calculated stereotomographic data, are indicated in orange.

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Stereotomography

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• Easier and denser picking when compared with standard traveltime tomography 共Lambaré et al., 2004a兲.

DISCUSSION AND PERSPECTIVES Several years of developments, investigations, and practice of stereotomography significantly changed our assessment of the method. We successfully demonstrated these expected advantages of the method: • Robustness and easy extension to 3D when compared with original CDR 共Chalard et al., 2000兲;

In addition, a powerful theoretical and practical framework has emerged. Now, stereotomography allows reconciling time- and depth-domain methods, and poststack and prestack methods. The concept allowing these reconciliations is the local aspect of data analysis. Practically, most types of information used in the other ve-

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z (km)

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Figure 14. Exact pairs of ray segments for 3D stereotomography. They were built shooting upward from a regular grid of reflecting points with purely inline ray shooting angles at the reflecting point.

Figure 16. Estimated velocity model and dip bars using 3D stereotomography. Figures can be compared with those in Figure 15. We see the good quality of the estimated velocity model with the exception of areas with poor ray coverage, i.e., the deep part and boundaries of the model. We also observe some shifting of recovered dip bars that cannot be perfectly constrained from the stereotomographic data set.

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Figure 15. Exact velocity model and dip bars for 3D stereotomography. Inline 共a兲 and crossline 共b兲 section of the exact smooth-velocity model. Dip bars corresponding to the exact position and dip of reflecting points are superimposed. No dips are considered and reflecting points are located on a regular grid.

10Y (km) m) y (k

20

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Figure 17. Estimated pairs of ray segments using 3D stereotomography. When compared with Figure 14, we see the good quality of recovered pairs of ray segments, except for some slight shifts in depth where the ray coverage is not sufficient to constrain the stereotomographic model.

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locity-macromodel-estimation methods 共residual curvature of CIGs, stacking velocities, prestack times, etc.兲 can be used in stereotomography. The only requirement is that the data set contains sufficient local information, i.e., local derivatives. This great characteristic of the approach results from the very simple and direct connection between the data and velocity model through concepts of locally coherent events and pairs of ray segments. Several variations of slope tomography have been proposed 共Biondi, 1992; Guillaume et al., 2001; Duveneck, 2004; Lavaud et al. 2004; Fei and McMechan, 2006a and 2006b兲, somehow dropping the local aspect of the methods. I am convinced that this characteristic should be preserved, at least in the inner core of the algorithms. I encourage further developers of numerical algorithms for velocity-macromodel estimation to base them on stereotomography 共or at least on CDR兲, and to use conversion schemes to transfer kinematic information into stereotomographic data. As it was introduced, stereotomography is a ray-based method, with the well-known limitations of ray-based methods in complex media. However, the principle of a local analysis of the data is not limited to picked data, and the potential extension of the method to full waveform inversion is recognized by Symes 共1998兲 or Chauris 共2000兲. They emphasize in particular the connection of stereotomography with differential semblance optimization 共Symes and Carrazone, 1991; Symes, 1998; Chauris and Noble, 2001; Mulder and ten Kroode, 2002; Foss et al., 2005; Shen et al., 2005b; Khoury et al., 2006; Li and Symes, 2007兲, which, therefore, could be the basis for a natural extension of the method to full waveform inversion. Certainly, much remains to be done for the practical use of stereotomography. For example, we have never fully succeeded in recovering really complex velocity models such as Marmousi 共Billette et al., 1998, Chauris et al., 2001兲. Is it because of the nonlinearity of the problem, an intrinsic limitation of ray theory, or the inefficiency of the stereotomographic picking? Further investigations are required. Both steps of stereotomography certainly can be improved: stereotomographic picking and stereotomographic optimization. Picking still remains a serious bottleneck 共even if it improved significantly compared with standard traveltime tomography兲, and investigations should continue toward semiautomatic strategies mixing various types of kinematical information. Concerning optimization, improvements can be achieved through the introduction of reliable a priori information 共on model and data兲 and more general velocity models including anisotropy and discontinuities. In addition, the combination of various types of data as wide angle 共Trinks et al., 2003兲, vertical seismic profiling data 共Gosselet et al., 2003兲, or multicomponent data in PP-PS stereotomography 共Alerini et al., 2002 and 2003; Lambaré et al., 2004c; Lambaré and Alerini, 2005; Lambaré et al., 2005兲 already have demonstrated its potential and should be investigated further. Finally, it is particularly interesting to see the present interest in the introduction of locally coherent events at all other steps of the seismic-processing sequence. In a very interesting paper, Fomel 共2007兲 describes, for example, the potential of using the local slope of seismic events to improve time imaging. In the depth-migration arena, the success of Gaussian-beam migration proposed by Hill 共1990 and 2001兲 offers great motivation for the development of various directional 共Takahashi, 1995; Sun and Schuster, 2000兲 and beam migrations 共Sun et al., 2000; Albertin et al., 2004; Notfors et al., 2006; Zhu et al., 2006兲. Gao et al. 共2006兲 presents a fast-beam-migration method based on kinematic migration of stereotomographic

events and is, without a doubt, a very serious step toward interactive imaging. From my point of view, this trend exhibits the potential of considering locally coherent events in seismic imaging, which is likely to continue to increase in the future, with benefits even for methods not based on high-frequency asymptotic approximations 共Jilek et al., 2007兲.

CONCLUSION Stereotomography now has been established as a robust method for velocity-model building from seismic-reflection data. Its ability to handle numerous types of kinematic information picked in unmigrated or migrated, prestack or poststack domains makes it particularly attractive as the kernel for a velocity-model-building tool. Some work certainly remains to be done, e.g., the definition of criteria for a semiautomatic selection of stereotomographic events, introduction of appropriate a priori information, combination of various types of seismic data, or introduction of anisotropy. Fundamentally, the main advantage of stereotomography 共and other slope-tomography methods兲 comes from using locally coherent events. We already see a significantly increasing interest in all steps of the seismic-imaging flow from the exploration geophysics community. In this context, stereotomography should definitely be combined with beam-migration methods, but it also should be advantageous when used ahead of high-frequency asymptotic assumptions in which benefits of using locally coherent events will certainly be demonstrated in the coming years.

ACKNOWLEDGMENTS Stereotomography has been developed in the Center of Geophysical Research of the School of Mines of Paris. I am very grateful to my colleagues and to students of our laboratory for their enthusiastic adherence to the stereotomographic method. Stereotomography was initially developed in the frame of the Joule European projects RODT and 3D FOCUS. Development of stereotomography then was supported by FSH 共French Ministry of Industry兲, CGG, and from 2001–2005 by sponsors of the DIG consortium. I thank them for their scientific and financial support, and for various data sets provided by Norsk Hydro, CGG, IFP, Elf, Total, and BP. Finall,y I thank CGGVeritas for allowing me to publish this paper, and the associate editor and two reviewers for their suggestions and comments that helped me improve the paper.

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