Seismic anisotropy in exploration and reservoir ... · Overview of seismic anisotropy 355 Full set...

CWP-642P

Seismic anisotropy in exploration and reservoir

characterization: An overview

I. Tsvankin1, J. Gaiser2, V. Grechka3, M. van der Baan4, & L. Thomsen51Colorado School of Mines, Department of Geophysics, Center for Wave Phenomena, Golden, CO, USA.2Geokinetics, Denver, CO, USA.3Shell Exploration & Production Company, Houston, TX, USA.4University of Alberta, Department of Physics, CEB, 11322-89 Ave., Edmonton, Alberta, T6G 2G7, Canada.5Delta Geophysics, Houston, TX, USA.

ABSTRACTRecent advances in parameter estimation and seismic processing have allowedincorporation of anisotropic models into a wide range of seismic methods. Inparticular, vertical and tilted transverse isotropy are currently treated as an in-tegral part of velocity fields employed in prestack depth migration algorithms,especially those based on the wave equation. Here, we briefly review the stateof the art in modeling, processing, and inversion of seismic data for anisotropicmedia. Topics include optimal parameterization, body-wave modeling methods,P-wave velocity analysis and imaging, processing in the τ–p domain, anisotropyestimation from vertical seismic profiling (VSP) surveys, moveout inversion ofwide-azimuth data, amplitude-variation-with-offset (AVO) analysis, processingand applications of shear and mode-converted waves, and fracture characteri-zation. When outlining future trends in anisotropy studies, we emphasize thatcontinued progress in data-acquisition technology is likely to spur transitionfrom transverse isotropy to lower anisotropic symmetries (e.g., orthorhombic).Further development of inversion and processing methods for such realisticanisotropic models should facilitate effective application of anisotropy parame-ters in lithology discrimination, fracture detection, and time-lapse seismology.

Key words: Seismic anisotropy, velocity analysis, prestack migration, pa-rameter estimation, transverse isotropy, AVO analysis, multicomponent data,mode-converted waves, shear-wave splitting, fracture characterization, verticalseismic profiling

1 INTRODUCTION

The area of applied seismic anisotropy is undergoingrapid transformation and expansion. Whereas the theo-retical foundation for describing anisotropic wave prop-agation had been developed a long time ago, the multi-parameter nature of anisotropic models had precludedtheir widespread application in seismic exploration andreservoir monitoring. The role of anisotropy has dra-matically increased over the past two decades due toadvances in parameter estimation, the transition frompoststack imaging to prestack depth migration, thewider offset and azimuthal coverage of 3D surveys, andacquisition of high-quality multicomponent data. Cur-

rently, many seismic processing and inversion meth-ods operate with anisotropic models, and there is littledoubt that in the near future anisotropy will be treatedas an inherent part of velocity fields.

A detailed historical analysis of developments inseismic anisotropy can be found in Helbig and Thom-sen (2005), so here we mention just several milestones.The work of Crampin (1981, 1985), Lynn and Thomsen(1986), Willis et al. (1986), Martin and Davis (1987),and others convincingly demonstrated that anisotropyhas a first-order influence on shear and mode-convertedPS-waves, which split into the fast and slow modeswith orthogonal polarizations. Shear-wave processing

354 I. Tsvankin, et al.

based on Alford (1986) rotation and its modificationshas helped document ubiquitous azimuthal anisotropyin the upper crust typically caused by near-vertical sys-tems of aligned fractures and microcracks. Acquisitionand processing of high-quality multicomponent offshoresurveys starting in the mid-1990’s clearly showed thatPP- and PS-wave sections could not be tied in depthwithout making the velocity model anisotropic.

In contrast, anisotropy-induced distortions in P-wave imaging (the focus of the majority of explorationsurveys) are less dramatic, especially for poststack pro-cessing of narrow-azimuth, moderate-spread data. Also,incorporating anisotropy into velocity analysis requiresestimation of several independent, spatially variable pa-rameters, which may not be constrained by P-wave re-flection traveltimes. Hence, the progress in P-wave pro-cessing can be largely attributed to breakthroughs inparameterization of transversely isotropic (TI) models,most notably the introduction of Thomsen (1986) no-tation and the discovery of the P-wave time-processingparameter η (Alkhalifah and Tsvankin, 1995). The ex-ploding interest in anisotropy and the importance ofthe parameterization issue have made Thomsen’s clas-sical ’86 article the top-cited paper ever published in thejournal Geophysics.

More recently, the inadequacy of isotropic velocitymodels was exposed by the advent of prestack depthmigration, which is highly sensitive to the accuracy ofthe velocity field. As a result, TI models with a verti-cal (VTI) and tilted (TTI) axis of symmetry have be-come practically standard in prestack imaging projectsall over the world. For instance, anisotropic algorithmsproduce markedly improved images of subsalt explo-ration targets in the Gulf of Mexico, which has long beenconsidered as a region with relatively “mild” anisotropy.

The goal of this paper is to give a brief descriptionof the state of the art in anisotropic modeling, process-ing, and inversion and outline the main future trends.It is impossible to give a complete picture of the fieldin a journal article, and the selection of the materialinevitably reflects the personal research experience andpreferences of the authors. For in-depth discussion oftheoretical and applied aspects of seismic anisotropy, werefer the reader to the books by Helbig (1994), Thomsen(2002), Tsvankin (2005), and Grechka (2009).

2 NOTATION FOR ANISOTROPIC MEDIA

One of the most critical issues in seismic data analysisfor anisotropic media is a proper design of model pa-rameterization. Whereas the stiffness coefficients (cij)are convenient to use in forward-modeling algorithms,they are not well-suited for application in seismic pro-cessing and inversion. An alternative notation for trans-verse isotropy was introduced by Thomsen (1986), whosuggested to describe the medium by the symmetry-direction velocities of P- and S-waves (VP0 and VS0,

respectively) and three dimensionless parameters (ǫ, δ,and γ), which characterize the magnitude of anisotropy.The parameter ǫ is close to the fractional difference be-tween the P-wave velocities in the directions perpendic-ular and parallel to the symmetry axis, so it defines whatis often simplistically called the “P-wave anisotropy.”Likewise, γ represents the same measure for SH-waves.While the definition of δ seems less transparent, thisparameter also has a clear meaning – it governs the P-wave velocity variation away from the symmetry axisand also influences the SV-wave velocity.

Although Thomsen originally used the assump-tion of weak anisotropy (i.e., |ǫ| ≪ 1, |δ| ≪ 1, and|γ| ≪ 1), his notation has since emerged as the bestchoice in seismic processing for TI media with any mag-nitude of velocity variations. Indeed, Thomsen param-eters capture the combinations of the stiffness coeffi-cients constrained by seismic signatures (for details, seeTsvankin, 2005). In particular, P-wave kinematics forTI media with a given symmetry-axis orientation de-pend on just three Thomsen parameters (VP0, ǫ, andδ; the contribution of VS0 is negligible), rather thanfour stiffness coefficients (c11, c33, c13, and c55). Thom-sen notation is especially convenient for reflection dataprocessing because it greatly simplifies expressions fornormal-moveout (NMO) velocity, quartic moveout coef-ficient, amplitude-variation-with-offset (AVO) response,and geometric spreading. Linearization of exact equa-tions in ǫ, δ, and γ provides valuable insight into the in-fluence of transverse isotropy on seismic wavefields andhelps guide inversion and processing algorithms.

Moreover, the contribution of anisotropy to time-domain processing of P-wave reflection data for VTImedia is absorbed by the single “anellipticity” parame-ter η close to the difference between ǫ and δ

[η ≡ (ǫ− δ)/(1 + 2δ)] .

The interval values of η and the NMO velocity forhorizontal reflectors [Vnmo(0)] are sufficient to performnormal-moveout and dip-moveout corrections, prestackand poststack time migration for VTI models witha laterally homogeneous overburden (Alkhalifah andTsvankin, 1995). Most importantly, the time-processingparameters Vnmo(0) and η can be estimated just from P-wave reflection traveltimes using NMO velocity of dip-ping events or nonhyperbolic moveout.

The parameters required for P-wave imaging andAVO analysis in VTI media are listed in Table 1.Whereas ǫ usually quantifies the magnitude of P-wavevelocity variations, the parameters of more importancein seismic processing are δ and η. Laboratory mea-surements of the anisotropy parameters for sedimen-tary rocks from different regions are summarized byWang (2002). Both rock-physics and seismic data indi-cate that vertical and tilted transverse isotropy in sedi-mentary basins are mostly associated with the intrinsicanisotropy of shales caused by aligned plate-shaped clay

Overview of seismic anisotropy 355

Full set Depth imaging Time imaging AVO (intercept, gradient)

VP0 VP0 Vnmo(0) VP0

ǫ or η ǫ or η η –

δ δ – δ

VS0 – – VS0

Table 1. P-wave parameters for imaging and AVO analysis in VTI media. The parameter Vnmo(0) = VP0√

1 + 2δ is the NMOvelocity for horizontal reflectors.

particles. Many sedimentary formations including sandsand carbonates, however, contain vertical or steeply dip-ping fracture sets and should be described by effectivesymmetries lower than TI, such as orthorhombic (seebelow). The effective anisotropy parameters are also in-fluenced by fine layering on a scale small compared toseismic wavelength (Backus, 1962).

The principle of Thomsen notation has been ex-tended to orthorhombic (Tsvankin, 1997; 2005), mono-clinic (Grechka et al., 2000) and even the most general,triclinic (Mensch and Rasolofosaon, 1997) models. Forinstance, Tsvankin’s notation for orthorhombic mediapreserves the attractive features of Thomsen parame-ters in describing the symmetry-plane velocities, travel-times, and plane-wave reflection coefficients of P-, S1-,and S2-waves. It also reduces the number of parametersresponsible for P-wave kinematics and provides a uni-fied framework for treating orthorhombic and TI modelsin parameter-estimation methods operating with wide-azimuth, multicomponent data (Grechka et al., 2005).

Estimation of anisotropy from P-wave vertical seis-mic profiling (VSP) data acquired under a structurallycomplex overburden involves expressing the verticalslowness component in terms of the polarization direc-tion. This problem, discussed in more detail below, leadsto the definition of Thomsen-style anisotropy parame-ters specifically tailored to VSP applications (Grechkaand Mateeva, 2007; Grechka et al., 2007).

Furthermore, Thomsen notation has been general-ized for attenuative TI and orthorhombic media in orderto facilitate analytic description and inversion of body-wave attenuation coefficients (Zhu and Tsvankin, 2006,2007). For a model with VTI symmetry of both the realand imaginary parts of the stiffness matrix, this notation(in addition to Thomsen’s velocity-anisotropy parame-ters) includes the vertical attenuation coefficients of P-and S-waves (AP0 and AS0) and three dimensionless pa-rameters (ǫ

Q, δ

Q, and γ

Q) responsible for attenuation

anisotropy. Linearization of the P-wave phase attenua-tion coefficient in the anisotropy parameters yields anexpression that has exactly the same form as Thom-sen’s (1986) weak-anisotropy approximation for P-wavephase velocity.

Whereas the optimal choice of notation is a pre-

requisite for successful anisotropic parameter estimationand processing, it is also important in forward modeling,which is discussed next.

3 FORWARD MODELING OF BODYWAVES

The ability to compute synthetic seismograms has al-ways been a high priority in geophysics since accu-rate forward modeling can be a valuable aid in seis-mic interpretation and inversion. Unfortunately, a fullyanisotropic (triclinic) Earth is characterized by 21 stiff-ness coefficients (or Thomsen-style parameters) anddensity, all of which may vary in space.

Full-waveform modeling can be implemented bysolving the wave equation for a general 3D (an)elasticmedium using numerical techniques such as finite-difference, finite-element, pseudospectral and spectral-element methods (Kosloff and Baysal, 1982; Virieux,1986; Komatitsch and Tromp, 1999). Although orig-inally many of these approaches were developed forisotropic media, most have been extended to handleanisotropic, anelastic media (Carcione et al., 1988; Ko-matitsch et al., 2000); for a recent review, see Carcioneet al. (2002).

Despite the constantly increasing computationalpower, full anisotropic (i.e., with 21 stiffnesses) model-ing using the above techniques is still rarely attempteddue to the scale of the problem and staggering number ofpossible models. In practice three avenues are commonlyexplored to facilitate interpretation and reduce compu-tation demands: (i) simplifications to theory; (ii) re-duction of information content in the acquired data;and (iii) limitations to considered structures, anisotropicsymmetries, and/or medium types. For instance, seismicwaves are often represented through rays, thereby in-voking a high-frequency approximation (simplificationsto theory). Also, one can choose to analyze traveltimesand/or amplitudes only (reduction in information con-tent), treat only specific types of anisotropy, assumethat wave motion can be described by P-wave propa-gation in acoustic media, and/or impose lateral conti-nuity and consider only vertically heterogeneous struc-


tures (constraints on media and/or structures). Obvi-ously, several approaches can be combined to developan appropriate interpretation strategy.

The earliest efforts to compute body-wave synthet-ics in anisotropic media focused on either simulation offull waveforms for simple models with at most one in-terface (Buchwald, 1959; Lighthill, 1960) or traveltimecalculations by means of geometric ray theory (Vlaar,1968; Červený, 1972). The former efforts were motivatedby the development of ultrasonic techniques for the mea-surement of dynamic elastic constants of pure crystalsand metals (Musgrave, 1970; Auld, 1973). The latterapproach was largely directed at explaining and under-standing anomalous body-wave properties observed inrefraction experiments and seismic arrays (Hess, 1964).

The ray method is a far-field, high-frequency,asymptotic approximation, which can handle later-ally and vertically heterogeneous anisotropic media un-der the assumption that the medium parameters varysmoothly on the scale of wavelength. In addition tobeing much less computationally intensive than finite-difference schemes and similar numerical methods, raytheory makes it possible to model individual wave typesrather than the whole wavefield. Ray tracing can be usedto generate both traveltimes and amplitudes; yet seriousdifficulties arise (especially for amplitude computations)near singular areas, such as caustics, cusps and conicalpoints on the wavefronts, shadow zones, and propaga-tion directions for which the velocities of the split S-waves are close (Gajewski and Pšenč́ık, 1987). Some ofthese problems are related to wavefront folding whenmany rays pass through a common focal point or focalline – a phenomenon that complicates the evaluationof geometric spreading. Geometric ray theory also ex-cludes head waves. Despite its limitations, ray theoryis still at the heart of many migration algorithms thatemploy ray tracing for efficient generation of traveltimetables. A more detailed discussion of ray theory can befound in the paper by Carcione et al. (2002) and mono-graphs by Červený (2001) and Chapman (2004).

The reflectivity method takes an alternative avenueto compute full-waveform synthetics in laterally homo-geneous media (Kennett, 1983). The technique is basedon plane-wave decomposition of point-source radiationcombined with the solution of the plane-wave reflec-tion/transmission problem for layered media obtainedusing so-called “propagator matrices” (Haskell, 1953;Gilbert and Backus, 1966). It can model both kinematicand dynamic properties of recorded wavefields includ-ing all primary and multiple reflections, conversions andhead waves, as long as the 1D assumption (i.e., the elas-tic properties vary only with depth) is satisfied (Fuchsand Müller, 1971; Kennett, 1972). The anisotropic re-flectivity method, originally developed for VTI mod-els and symmetry-plane wave propagation (Keith andCrampin, 1977; Booth and Crampin, 1983), has been

extended to azimuthally anisotropic media (Fryer andFrazer, 1984; Tsvankin and Chesnokov, 1990).

Despite its 1D model assumption, the reflectivitymethod has proved to be a valuable tool for under-standing and interpreting wave-propagation phenom-ena in both VSP and surface-seismic acquisition geome-tries. For instance, Mallick and Frazer (1991) employthis technique to study P-wave amplitude variationswith offset and azimuth in a medium containing verti-cal fractures and demonstrate how azimuthal amplitudeanomalies can help reveal fracture orientation.

4 P-WAVE VELOCITY ANALYSIS ANDIMAGING

Most isotropic time- and depth-migration algo-rithms [Kirchhoff, Stolt, phase-shift, phase-shift-plus-interpolation (PSPI), Gaussian beam, finite-difference,etc.] have been generalized for VTI and, in many cases,TTI media (e.g., Sena and Toksöz, 1993; Anderson etal., 1996; Alkhalifah, 1997; Ren et al., 2005; Zhu et al.,2007a). The key issue in anisotropic processing, how-ever, is reliable estimation of the velocity model fromreflection data combined with borehole and other infor-mation. The parameter η responsible for time process-ing in VTI media can be obtained by inverting eitherdip-dependent NMO velocity or nonhyperbolic (long-spread) reflection moveout (Alkhalifah and Tsvankin,1995; Alkhalifah, 1997; Toldi et al., 1999; Fomel, 2004;Tsvankin, 2005; Ursin and Stovas, 2006). Then the η-field can be refined in the migrated domain using mi-gration velocity analysis (Sarkar and Tsvankin, 2004) orreflection tomography (Woodward et al., 2008).

Building VTI velocity models in the depth domaintypically requires a priori constraints because the verti-cal velocity VP0 and the parameters ǫ and δ can seldombe determined from P-wave reflection moveout alone.In many cases, VP0 is found from check shots or welllogs at borehole locations and used in combination withthe stacking (NMO) velocity to compute the parame-ter δ. Note that ignoring the contribution of δ to NMOvelocity in isotropic processing leads to misties in time-to-depth conversion. Then the velocity field can be con-structed by interpolating the parameters VP0 and δ be-tween the boreholes and estimating η (and, therefore,ǫ) from reflection data. Integration of seismic and bore-hole data can be facilitated by applying geologic con-straints in the process of interpretive model updating(Bear et al., 2005) or rec asting the generation of a denseanisotropic velocity field as an optimization problem.An efficient tool for building heterogeneous VTI mod-els is postmigration grid tomography based on iterativeminimization of residual moveout after prestack depthmigration (Woodward et al., 2008).

Anisotropic migration with the estimated Thom-sen parameters typically produces sections with betterfocusing and positioning of reflectors for a wide range


of dips including steep interfaces, such as flanks of saltdomes. The 2D line in Figure 1 is used by Alkhalifahet al. (1996) to illustrate the improvements achievedby anisotropic time processing in offshore West Africawhere thick TI shale formations cause serious imagingproblems (Ball, 1995). For example, VTI dip-moveoutand poststack migration algorithms succeeded in imag-ing the fault plane at midpoint 7.5 km and depth 3 km(the right arrow in Figure 1b), which is absent on theisotropic section (Figure 1a). Also, the major fault planebetween the midpoints at 2 km and 8 km (it stretchesup and down from the middle arrow in Figures 1a,b)and gently dipping reflectors throughout the section ap-pear more crisp and continuous. Accurate fault imagingbeneath the shales plays a major role in prospect iden-tification in the area.

It is even more critical to properly account foranisotropy in prestack depth migration because the re-sults of prestack imaging are highly sensitive to thequality of the velocity model. The section in Figure 2bwas produced by applying VTI migration velocity anal-ysis (MVA) and Kirchhoff prestack depth migration tothe line from Figure 1 (Sarkar and Tsvankin, 2006).MVA was carried out by dividing the section into fac-torized VTI blocks, in which the parameters ǫ and δ areconstant, while the velocity VP0 is a linear function ofthe spatial coordinates. Factorized VTI is the simplestmodel that allows for both anisotropy and heterogeneityand requires minimal a priori information to constrainthe relevant parameters (Sarkar and Tsvankin, 2004). Inthe absence of pronounced velocity jumps across layerboundaries, knowledge of the vertical velocity at thetop of a piecewise-factorized VTI medium is sufficientto estimate the parameters VP0, ǫ, and δ along with thevelocity gradients throughout the section using only P-wave data (Figures 3a,c,d).

The velocity analysis revealed significant lateral ve-locity gradients in some of the layers (Figure 3a), whichcould not be handled by time-domain techniques. Asa result, the depth-domain parameter estimation pro-duced a more reliable, laterally varying η-field (Fig-ure 3b). The depth imaging facilitated structural inter-pretation of the deeper part of the section by remov-ing the false dips seen in Figure 2a. Also, most anti-thetic faults that look fuzzy on the time section arewell focused, and subhorizontal reflectors within theanisotropic layers are better positioned and stacked.This and many other published case studies demon-strate that a major advantage of anisotropic depthimaging is in providing accurate well ties without sacri-ficing image quality.

In tectonically active areas or in the presence of dip-ping fracture sets the symmetry axis of TI formationscan be tilted, and the VTI model becomes inadequate.Tilted transverse isotropy is common in the CanadianFoothills, where shale layers are often bent and mayhave steep, variable dips (e.g., Vestrum et al., 1999).

Figure 1. Comparison between isotropic and VTI timeimaging (after Alkhalifah et al., 1996, and Sarkar andTsvankin, 2006). A 2D line from West Africa after (a)isotropic and (b) anisotropic time imaging. The processingsequence included NMO and DMO corrections and post-stack phase-shift time migration. Both time sections arestretched to depth. The arrows point to the main improve-ments achieved by taking anisotropy into account.

Also, uptilted shale layers near salt domes may causeserious difficulties in imaging steeply dipping segmentsof the salt flanks (Tsvankin, 2005). A detailed descrip-tion of distortions caused by applying VTI algorithmsto data from typical TTI media can be found in Beheraand Tsvankin (2009), who extend MVA to TI modelswith the symmetry axis orthogonal to reflectors. In theircase studies from the Gulf of Mexico, Huang et al. (2009)and Neal et al. (2009) demonstrate that accounting forthe tilt of the symmetry axis produces significant im-provements in imaging of steep dips, fault resolution,and spatial positioning of reflectors. The inadequacy ofVTI models for many subsalt plays in the Gulf of Mex-ico has become especially apparent with acquisition ofwide-azimuth surveys. In complicated structural envi-ronments, the full benefits of TI imaging can be realizedwith reverse time migration (RTM) based on solving thetwo-way wave equation (Huang et al., 2009). RTM withTTI or VTI velocity models is already widely used inGoM subsalt imaging projects.

Despite the recent successes, parameter estimationfor heterogeneous TTI media remains a highly challeng-ing problem, even with the common assumption that thesymmetry axis is orthogonal to reflectors. Methods cur-rently under development combine TI ray-based reflec-tion tomography with check shots and walkaway VSP


0.0 4.0 8.0Midpoint (km)

2.0

4.0Dep

th (

km)

2.0

Dep

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km)

4.0

0.0 4.0 8.0Midpoint (km)

Figure 2. Comparison between VTI time and depth imag-ing (after Sarkar and Tsvankin, 2006). The line from Figure 1after (a) anisotropic time processing (same as Figure 1b) and(b) anisotropic MVA and prestack depth migration. The ar-rows point to the main differences between the two sections.

surveys (e.g., Bakulin et al., 2009). It is likely that pro-cessing of high-quality wide-azimuth surveys in some areas will require employing more complicated (but morerealistic), orthorhombic velocity models. A promisingdirection for high-resolution anisotropic velocity anal-ysis is full-waveform inversion, which so far has beendeveloped mostly for acoustic models.

5 SLOWNESS-BASED PROCESSING ANDINVERSION

Processing and inversion of surface seismic data ismostly done in the time-offset domain; yet other do-mains may offer advantages in terms of noise sup-pression and/or inversion for anisotropy parameters.It is especially beneficial to transform seismic datainto the slowness domain, with applications as diverseas common-conversion-point sorting, anisotropy estima-tion, geometric-spreading correction, and amplitude andfull-waveform inversion.

Snell’s law states that the horizontal slowness pdoes not change along a ray in 1D models. This facthelps identify correlated pure-mode (PP) and converted(PS) reflections from the same interface that have com-mon downgoing P-wave ray segments (and, therefore,the same horizontal slownesses). Identification of com-mon ray segments leads to a straightforward common-conversion-point sorting scheme in both the time-offsetand intercept time - horizontal slowness (τ -p) domains

80 4Midpoint (km)

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80 4Midpoint (km)

2

0

4

Dep

th (

km)

δ

0.05

0.00

(d)

Figure 3. Estimated parameters (a) VP0; (b) η; (c) ǫ; and(d) δ used to generate the depth-migrated section in Fig-ure 2b (after Sarkar and Tsvankin, 2006).


(Van der Baan, 2005). In the presence of lateral and ver-tical velocity variations one can still identify PP and PSreflections with the same takeoff angles (i.e., the samehorizontal slownesses) at the source by matching theirtime slopes in common-receiver gathers. This more gen-eral scheme called “PP+PS=SS” (discussed in more de-tail below) is designed to construct the traveltimes ofpure-mode SS reflections solely from acquired PP- andPS-waves (Grechka and Tsvankin, 2002).

Slowness-based traveltime inversion can also in-crease the accuracy of interval parameter estimation.Anisotropic traveltime inversion is often based on move-out approximations that become unnecessary if seismicdata are processed by means of a plane-wave decom-position, such as the τ -p transform. Indeed, plane-wavepropagation is directly governed by phase rather thangroup velocities because the interval τ (p) curves repre-sent rescaled versions of the slowness functions (Hake,1986). Note that phase velocities generally are less com-plex mathematically compared to group velocities.

A τ -p domain approach has been applied by Gaiser(1990) to analysis of VSP data, by Hake (1986) andVan der Baan and Kendall (2002, 2003) to inversionof reflections traveltimes, and by Mah and Schmitt(2003) to processing of ultrasonic measurements. Sim-ilar gains in accuracy can be obtained by formulat-ing slowness-based inversion algorithms directly in thetime-offset domain (Douma and Van der Baan, 2008;Fowler et al., 2008; Dewangan and Tsvankin, 2006a;Wang and Tsvankin, 2009). It should be mentioned thatthe velocity-independent layer-stripping method of De-wangan and Tsvankin (2006a) is valid for an arbitrarilyheterogeneous target horizon. An important feature ofslowness-based algorithms is that they replace Dix-typedifferentiation of moveout parameters with traveltimestripping (Figure 4), which increases the accuracy andstability of interval parameter estimates.

Plane-wave decomposition also represents a conve-nient tool for geometric-spreading correction in horizon-tally layered media. Indeed, plane waves in 1D mod-els are not subject to geometric spreading (in contrastto spherical waves); this is implicitly used in the re-flectivity method discussed above (Fuchs and Müller,1971; Kennett, 1983; Fryer and Frazer, 1984). Wangand McCowan (1989), Dunne and Beresford (1998) andVan der Baan (2004) explicitly employ plane-wave de-composition to remove the geometric-spreading factorfrom the amplitudes of all primary and multiple re-flections (including mode conversions) simultaneously.Subsequent moveout correction and stacking in the τ -p domain (Stoffa et al., 1981, 1982) generates higher-quality stacked sections compared to those produced bythe standard time-offset stacking process (Figure 5). In-terestingly, stacking in the slowness domain preserveshead waves suppressed by conventional processing (Vander Baan, 2004).

Finally, plane-wave decomposition may facilitate

amplitude analysis at far offsets near and beyond thecritical angle, where the wavefield in the time-offset do-main cannot be described by plane-wave reflection co-efficients (Van der Baan, 2004; Van der Baan and Smit,2006; Tsvankin, 1995a). Thus, slowness-based process-ing and inversion has many advantages for data fromanisotropic media. Indeed, it has been suggested in theliterature that seismic inversion techniques should beapplied to slant-stacked data obtained after a plane-wave decomposition (Müller, 1971; Fryer, 1980; Trei-tel et al., 1982). This approach is also suitable for full-waveform inversion in stratified media (Kormendi andDietrich, 1991; Martinez and McMechan, 1991; Ji andSingh, 2005) and separation of interfering PP and PSwaves (Van der Baan, 2006).

6 ANISOTROPY ESTIMATION FROM VSPDATA

The concept of operating with slowness measurementsis also essential in processing of vertical seismic profilingsurveys. Although anisotropic velocity models have tobe built based on surface reflection data for most appli-cations in seismic exploration, VSP can often provideuseful complementary anisotropy estimates. Becausethose estimates are made at seismic frequencies, theyhave an important advantage over well-log anisotropymeasurements, which pertain to the frequency range of103 − 104 Hz and require upscaling for use in seismicprocessing. The anisotropy parameters constrained byVSP data strongly depend on both the acquisition de-sign and the magnitude of lateral heterogeneity of theoverburden.

The simplest VSP experiment involves a single geo-phone placed in a well. First-break P-wave times pickedfrom such VSP data reflect the influence of effectiveanisotropy between the earth’s surface, where the seis-mic sources are located, and the geophone’s depth. Be-cause this depth is known, P-wave walkaway VSP data(or the combination of surface reflection data and checkshots) for laterally homogeneous VTI media yield the ef-fective Thomsen parameter δ. Whether or not it is pos-sible to estimate another anisotropy parameter (η or ǫ)governing P-wave kinematics depends on the presenceof sufficiently large offsets in the data. While the useful-ness of such low-resolution anisotropy estimates mightbe questioned, P-wave traveltimes recorded in any VSPgeometry help build an exact depth-migration operatorsuitable for constructing a subsurface image near thegeophone.

The opposite “end member” is a wide-azimuth,multicomponent VSP survey recorded by a string of geo-phones placed beneath a laterally homogeneous over-burden. Such VSP data make it possible to obtain acomplete (triclinic) local stiffness tensor near the bore-hole. This is demonstrated by Dewangan and Grechka(2003) who apply the so-called slowness-polarization


a) b) c)

Figure 4. Layer stripping in the τ -p domain (after Van der Baan and Kendall, 2002). A hyperbolic moveout curve in thetime-offset domain maps onto an ellipse in the τ -p domain, and nonhyperbolic moveout manifests itself by a deviation fromthe ellipse. (a) Moveout curves in the τ -p domain are created by summing the contributions of the individual layers. Removingthe influence of (b) the top layer or (c) the two top layers yields the moveout in the corresponding interval. The first and thirdlayers are isotropic, whereas the second layer is anisotropic, as evidenced by the strong deviation of its interval moveout froman ellipse on plot (b).

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00.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0

a) b)

Figure 5. Comparison of stacking techniques on field data (after Van der Baan, 2004). (a) A conventional t-x stacked sectionafter the geometric-spreading correction. (b) The same section obtained by stacking in the τ -p domain after applying plane-wavedecomposition to remove geometric spreading. The two sections are structurally similar but τ -p processing produced a highersignal-to-noise ratio at small traveltimes and somewhat different reflector amplitudes between 1.3 s and 1.6 s.

method (White et al., 1983; de Parscau, 1991; Hsu et al.,1991; Horne and Leaney, 2000) to estimate anisotropyfrom the traveltimes and polarization directions of P-,S1-, and S2-waves recorded at Vacuum Field (New Mex-ico, USA). They conclude that the VSP measurementscan be well-described by an orthorhombic model witha near-horizontal symmetry plane. Unfortunately, theslowness-polarization method can be successfully imple-mented only when lateral heterogeneity of the overbur-den is negligible. Then the horizontal slowness compo-nents, which are measured on common-receiver gathers

and pertain to seismic sources at the earth’s surface, canbe used to reconstruct the slowness surface at geophonelocations (Gaiser, 1990; Miller and Spencer, 1994; J́ıleket al., 2003).

Strong lateral heterogeneity (for instance, due tothe presence of salt in the overburden) renders recon-struction of the slowness surfaces inaccurate and of-ten makes shear-wave arrivals too noisy for anisotropicinversion. Consequently, anisotropy has to be inferredfrom P-waves only, which leads to the introduction ofThomsen-style parameters for P-wave VSP inversion.


6.40

6.46

6.52

6.58

6.70

6.64

Dep

th (

km)

0 0.05 0.1 0.15

21

21.1

21.2

21.3

21.4

21.5

21.6

21.7

21.8

21.9

22

δVSP

Dep

th (

kft)

a

40 60 80 100 120

21

21.1

21.2

21.3

21.4

21.5

21.6

21.7

21.8

21.9

22

Gamma−ray log

b

shale

shale

sand

sand

Figure 6. Comparison of (a) the anisotropy parameter δ VSP (bold line) estimated from VSP for subsalt sediments in the Gulfof Mexico with (b) a gamma-ray log (after Grechka and Mateeva, 2007). The thin lines on plot (a) mark the standard deviationof δ VSP.

The measured quantities include the P-wave verticalslowness component, q, expressed as a function of thepolar (ψ) and azimuthal (ϕ) angles of the polarizationvector. The values of q, ψ, and ϕ do not depend on thestructural complexity of the overburden and correspondto the vicinity (with the spatial extent approximatelyequal to the wavelength) of downhole geophones. If themedium around the borehole is VTI, the vertical slow-ness q is independent of the polarization azimuth ϕ,and the weak-anisotropy approximation for q(ψ) takesthe form (Grechka and Mateeva, 2007)

q(ψ) =cosψ

VP0

`

1 + δVSP sin2 ψ + ηVSP sin

4 ψ´

, (1)

where

δVSP =δ

(VP0/VS0)2 − 1and

ηVSP = η(VP0/VS0)

2 + 1

(VP0/VS0)2 − 1(2)

are the anisotropy parameters responsible for the P-wave slowness-of-polarization dependence.

The pairs {δVSP, ηVSP} and {δ, η} play compara-ble roles for processing of P-wave VSPs acquired alongvertical boreholes and of P-wave surface reflection data,respectively, in VTI media. Indeed, equation 1 revealsthat δVSP is responsible for the near-vertical variation ofq(ψ), while ηVSP governs the vertical slowness at largerpolarization angles. Importantly, δVSP and ηVSP absorbthe shear-wave velocity VS0, rendering its value unneces-sary for fitting the P-wave slowness-of-polarization func-tions. Grechka and Mateeva (2007) illustrate this pointand present estimates of δVSP and ηVSP in a salt bodyand subsalt sediments in the deepwater Gulf of Mexico.Wheras the salt proved to be nearly isotropic, δVSP inthe subsalt sediments exhibits a clear correlation withlithology (Figure 6). As is usually the case for the pa-rameter δ, the value of δVSP is larger in shales than ina predominantly sandy interval.

This technique of anisotropy estimation from P-


wave VSP surveys has been extended to azimuthalanisotropy. For example, Grechka et al. (2007) invertwide-azimuth VSP data acquired at tight-gas RulisonField (Colorado, USA) for Tsvankin’s (1997) parame-ters of orthorhombic media and show that the estimatedanisotropic model is consistent with the presence of gas-filled vertical fractures in a VTI host rock.

7 AZIMUTHAL MOVEOUT ANALYSIS

Azimuthal variation of traveltimes, amplitudes, and at-tenuation coefficients of reflected waves can providevaluable information about anisotropy associated withnatural fracture systems, nonhydrostatic stresses, ordipping TI layers (e.g., Lynn et al., 1999; Rüger, 2002).Wide-azimuth P-wave data are often acquired on landfor purposes of fracture characterization via azimuthalmoveout and AVO analysis. The rapid advent of wide-azimuth offshore technology, designed primarily for bet-ter imaging of subsalt exploration targets, is expected tofurther stimulate development of processing algorithmsfor azimuthally anisotropic models.

Moveout analysis of wide-azimuth, conventional-spread data is based on the concept of the NMO ellipseand on the generalized Dix-type averaging equations(Grechka and Tsvankin, 1998; Grechka et al., 1999).The normal-moveout velocity of pure (non-converted)reflected waves expressed as a function of the azimuthα of the CMP line is given by the following quadraticform:

V −2nmo(α) = W11 cos2 α + 2W12 sinα cosα

+W22 sin2 α , (3)

where W is a symmetric 2×2 matrix determined bythe medium properties around the zero-offset ray. If thetraveltime increases with offset in all azimuthal direc-tions (i.e., in the absence of reverse move out), Vnmo(α)traces out an ellipse even for arbitrarily anisotropic,heterogeneous media. Furthermore, equation 3 can beapplied to mode-converted waves, if their moveout inCMP geometry is symmetric with respect to zero offset(this is the case for horizontally layered models with ahorizontal symmetry plane).

The equation of the NMO ellipse provides a simpleway to correct for the azimuthal variation in stackingvelocity often ignored in conventional processing. Evenmore importantly, the semiaxes and orientation of theNMO ellipse can be used in anisotropic parameter esti-mation and fracture characterization. A critical issue inmoveout analysis of wide-azimuth data is separation ofthe influence of anisotropy and lateral heterogeneity (inthe form of velocity gradients, dipping interfaces, veloc-ity lenses, etc.) on reflection traveltimes (e.g., Jenner,2009).

A data-driven correction of the NMO ellipse for lat-eral velocity variation in horizontally layered media is

suggested by Grechka and Tsvankin (1999), who presenta complete processing sequence for azimuthal moveoutinversion that also includes 3D “global” semblance anal-ysis and generalized Dix differentiation of effective NMOellipses. They show that the orientation of the P-waveinterval NMO ellipses produced by this methodologyin the Powder River Basin (Wyoming, USA) is well-correlated with the depth-varying fracture trends in thefield.

P-wave azimuthal moveout analysis has proved tobe effective in predicting the dominant fracture orien-tation in many other exploration regions (e.g., Corriganet al., 1996; Lynn et al., 1999; Tod et al., 2007). Jen-ner (2001), who has developed a trace-correlation ap-proach for estimating the NMO ellipse, shows that thefast NMO-velocity direction at Weyburn field in Canadais aligned with the dominant fracture strike and the po-larization vector of the fast S-wave; this implies thatthe medium symmetry is HTI or orthorhombic. Still, insome cases the NMO ellipse is rotated with respect tothe shear-wave polarization directions, which may indi-cate the presence of lower symmetries.

Since the P-wave NMO ellipse constrains only threecombinations of the medium parameters, its inversionfor the physical properties of fractures (e.g., fracturecompliances) suffers from ambiguity, which can be re-duced by using the NMO ellipses of the split S-waves,nonhyperbolic moveout, or other (amplitude, borehole)information. For instance, joint inversion of the NMO el-lipses of P- and S-waves with a priori constraints helpsbuild even orthorhombic and monoclinic velocity models(Grechka et al., 2000; Vasconcelos and Grechka, 2007);more details are given in the section on fracture charac-terization.

Among the first to recognize the benefits of em-ploying nonhyperbolic (long-spread) reflection moveoutin anisotropic parameter estimation was Sena (1991),whose analytic traveltime expressions for multilayered,weakly anisotropic media are based upon the “skewed”hyperbolic moveout formulation of Byun et al. (1989).Long-spread, wide-azimuth P-wave traveltime in az-imuthally anisotropic media can be accurately describedby generalizing the nonhyperbolic moveout equationsof Tsvankin and Thomsen (1994) and Alkhalifah andTsvankin (1995) originally designed for VTI media.Vasconcelos and Tsvankin (2006) develop a moveout-inversion algorithm for horizontally layered orthorhom-bic media based on the extended Alkhalifah-Tsvankinequation:

t2(x, α) = t20 +x2

V 2nmo(α)(4)

−2η(α) x4

V 2nmo(α) [ t20 V

2nmo(α) + (1 + 2η(α)) x2]

,

where t0 is the zero-offset time, Vnmo(α) is the NMOellipse (equation 3), and η(α) is the azimuthally vary-ing anellipticity parameter. Equation 5 can be combined


with the velocity-independent layer-stripping method(Dewangan and Tsvankin, 2006a) to compute the in-terval traveltime in the target layer and estimate theinterval NMO ellipse and anellipticity parameters η(1),η(2), and η(3) (Wang and Tsvankin, 2009). Nonhyper-bolic moveout inversion of wide-azimuth data not onlyrepresents a promising fracture-characterization tech-nique (see the case study in Vasconcelos and Tsvankin,2006), but also provides the input parameters for P-wave time imaging and geometric-spreading correctionin layered orthorhombic media.

8 PRESTACK AMPLITUDE ANALYSIS

Angle-dependent reflection and transmission coefficientscontain valuable information about the local mediumproperties on both sides of an interface. Therefore,analysis of amplitude variations with incidence angle(usually called AVO – amplitude variation with offset)and/or azimuth is often used in reservoir characteriza-tion. Because reflection coefficients are determined bythe elastic properties averaged on the scale of seismicwavelength, AVO analysis can achieve a much highervertical resolution than traveltime methods.

Exact equations for plane-wave reflection coeffi-cients are cumbersome even for isotropy and, therefore,rarely used in processing. Whereas exact reflection co-efficients for VTI media and symmetry planes of or-thorhombic media can still be obtained in closed form(Daley and Hron, 1977; Rüger, 2002), for lower sym-metries it is necessary to apply computational schemes(e.g., Fryer and Frazer, 1984; J́ılek, 2002a,b). Impor-tant insight into anisotropic reflectivity is provided bylinearized weak-contrast, weak-anisotropy approxima-tions, which have a much simpler form and often reducethe number of free parameters. The approximate P-wavereflection coefficient for VTI media depends on the con-trasts in the vertical P- and S-wave velocities (VP0 andVS0), density, and the parameters δ and ǫ (Banik, 1987;Thomsen, 1993; Rüger, 1997). Although the contribu-tion of δ distorts the AVO gradient, the P-wave AVOsignatures in isotropic and VTI media are generally sim-ilar, which complicates amplitude inversion for the fiveindependent parameters. Indeed, as shown by de Nico-lao et al. (1993), only two parameters can be resolvedfrom the isotropic reflection coefficient.

Analysis of azimuthal amplitude variations showsconsiderably more promise, in particular for estima-tion of dominant fracture directions in naturally frac-tured (e.g., tight-gas and tight-oil) reservoirs (Mallickand Frazer, 1991; Gray et al., 2002). Linearized P-wave reflection coefficients were derived for HTI mediaand symmetry planes of orthorhombic media by Rüger(1997, 1998) and for arbitrary anisotropy by Vavryčukand Pšenč́ık (1998); for details, see Rüger’s (2002) com-prehensive monograph. Application of these analytic ex-pressions in quantitative AVO inversion, however, is hin-

dered by nonuniqueness in parameter estimation. In-stead, it is more common to reconstruct the azimuthalvariation (which is close to elliptical) of the magni-tude of the AVO gradient (Gray et al., 2002; Hall andKendall, 2003). For HTI and orthorhombic media, theextrema of the AVO gradient lie in the orthogonal ver-tical symmetry planes of the model.

If azimuthal anisotropy is caused by one set of verti-cal fractures, the maximum AVO gradient may be eitherparallel or perpendicular to the fractures, which gener-ally leads to a 90◦-uncertainty in the fracture azimuth.Despite this ambiguity, the azimuthally varying P-waveAVO response has been successfully used for estimatingthe dominant fracture orientation and, in some cases,mapping “sweet spots” of intense fracturing (e.g., Grayet al., 2002; Gray and Todorovic-Marinic, 2004; Xu andTsvankin, 2007). For instance, Hall and Kendall (2003)demonstrate that the direction of the minimum AVOgradient at Valhall field is well-aligned with faults in-ferred from coherency analysis (Figure 7).

For HTI and orthorhombic media with a singlefracture set, the difference between the symmetry-planeAVO gradients is proportional to the fracture density(which is close to the shear-wave splitting parameter)and also depends on the fracture infill (Rüger, 2002).Therefore, even for such simple models the inversionof the P-wave AVO gradient for the fracture proper-ties is generally nonunique. In principle, fracture den-sity and saturation can be constrained by combiningthe P-wave AVO response and NMO ellipse, but thisapproach is applicable only to relatively thick, weaklyheterogeneous reservoirs (e.g., Xu and Tsvankin, 2007).Additional complications may be caused by multiplefracture sets (which lower the symmetry to at least or-thorhombic) and the presence of fractures on both sidesof the target reflector. For such realistic fractured reser-voirs, it is highly beneficial to employ multicomponentdata in azimuthal AVO analysis (Bakulin et al., 2000;J́ılek, 2002a,b; DeVault et al., 2002). In particular, J́ılek(2002b) presents a methodology for joint nonlinear AVOinversion of wide-azimuth PP and PS reflections for TIand orthorhombic media.

Another interesting possibility is to combine az-imuthal AVO and attenuation analysis, which helps re-move the uncertainty in estimating the fracture ori-entation for HTI media (Clark et al., 2009). Fur-thermore, body-wave attenuation coefficents are highlysensitive to anisotropy and fracturing and may po-tentially provide powerful fracture-characterization at-tributes (Chapman, 2003; Zhu et al., 2007b; Chichin-ina et al., 2009; Maultzsch et al., 2009). On the otherhand, in some cases azimuthally varying attenuation (ifunaccounted for) may distort the AVO signature. Effi-cient velocity-independent techniques for estimating in-terval offset- and azimuth-dependent attenuation fromfrequency-domain reflection amplitudes are suggestedby Behura and Tsvankin (2009) and Reine et al. (2009).


N

Figure 7. Application of P-wave azimuthal AVO analysisto fracture detection at Valhall field (after Hall and Kendall,2003). The fracture azimuths (ticks) estimated from the az-imuthally varying AVO gradient for the top-chalk horizonare compared with interpreted fault traces. Note the gen-eral alignment of fractures with large-scale faulting, espe-cially near the faults trending from northwest to southeast.In the southeast corner, fractures also appear to be perpen-dicular to the surface curvature defined by the time contours(the contours are plotted at 20-ms intervals, with the redcolor indicating deeper areas).

AVO analysis is designed to operate with the plane-wave reflection coefficient at the target interface. Therecorded amplitude of reflected waves, however, also de-pends on the source/receiver directivity and such propa-gation factors as geometric spreading, transmission co-efficients, and attenuation (Martinez, 1993; Maultzschet al., 2003). Anisotropic layers in the overburden focusor defocus seismic energy like an optical lens, thus dis-torting the amplitude distribution along the wavefrontand causing pronounced angle variations of geometricspreading (Tsvankin, 1995b, 2005; Stovas and Ursin,2009). In that case, robust reconstruction of the angle-dependent reflection coefficient requires an anisotropicgeometric-spreading correction.

Geometric spreading in the time-offset domain isrelated to the convergence or divergence of ray beams(Gajewski and Pšenč́ık, 1987) and, therefore, can becomputed directly from the spatial derivatives of trav-eltime (Vanelle and Gajewski, 2003). This ray-theoryresult is exploited in the moveout-based geometric-spreading correction devised for horizontally layeredVTI models by Ursin and Hokstad (2003) and extendedto wide-azimuth, long-spread PP and PS data from az-imuthally anisotropic media by Xu and Tsvankin (2006,2008). In particular, this correction has proved to be es-sential in azimuthal AVO analysis of reflections from thebottom of relatively thick fractured reservoirs (Xu andTsvankin, 2007).

On the whole, recent developments have laid thegroundwork for transforming anisotropic AVO analysisinto a valuable reservoir-characterization tool.

9 PROCESSING AND APPLICATIONS OFMULTICOMPONENT DATA

Early applications of shear-wave seismology had to copewith erratic and unpredictable data quality and mistiesbetween SS-wave reflections at the intersection of 2D ac-quisition lines (Lynn and Thomsen, 1986; Willis et al.,1986). This caused serious difficulties in generating in-terpretable shear-wave sections and using multicompo-nent data in lithology discrimination and fracture char-acterization. Alford (1986) suggested that these prob-lems are related to shear-wave splitting due to azimuthalanisotropy and proposed simple rotation operators totransform SS data into two principal sections contain-ing the fast and slow modes. Likewise, Martin and Davis(1987) discuss the need to rotate converted PS-wavesacquired for fracture-characterization purposes at SiloField (Colorado, USA).

Shear waves in anisotropic media exhibit birefrin-gence (shear-wave splitting) and travel as two separatemodes with different velocities and orthogonal (for thesame phase direction) polarizations. If the medium isHTI or orthorhombic with a horizontal symmetry plane,the vertically traveling split S-waves are polarized in thesymmetry planes of the model. The magnitude of shear-wave splitting at vertical incidence is described by theparameter γ(S), which is close to the fractional differ-ence between the velocities of the fast (S1) and slow(S2) modes and can be estimated as γ

(S) ≈ (ts − tf )/tf ,where ts and tf are the traveltimes of the waves S2 andS1, respectively. After separating the split shear waveson prestack data, it may be possible to evaluate theirNMO ellipses and AVO signatures.

9.1 Pure-mode SS-waves

Processing surface shear-wave data for azimuthalanisotropy analysis has primarily involved 1D compen-sation for splitting at near-vertical propagation direc-tions. Alford’s (1986) rotation algorithm operates onfour-component, stacked (supposed to be equivalent tozero-offset) data excited by two orthogonal sources andrecorded by two orthogonal receivers. Data can be ac-quired on a 2D line, with sources and receivers orientedparallel (inline) and perpendicular (crossline) to the ac-quisition azimuth. The four recorded S-wave displace-ment components can be represented in the form of thefollowing 2×2 matrix:

D =

„

DXX DXYDY X DY Y

«

, (5)

where X denotes inline and Y crossline; the first letterin the subscript refers to the source orientation, and thesecond letter to the receiver orientation. Prestack shear-wave data depend on the anisotropic velocity field andhave polarization properties controlled by the azimuthof the line with respect to the symmetry planes. How-ever, stacking or performing AVO inversion of 4C data


Figure 8. (a) Polarization azimuth of the PS1-wave and (b) the shear-wave splitting coefficient (in percent) above the GessosoSolfifera formation at Emilio Field (after Gaiser et al., 2002). The north direction is rotated about 15◦ clockwise.

can provide an estimate of the difference between thenormal-incidence reflection coefficients (i.e., AVO inter-cepts) of the split S-waves, which is governed by theparameter γ(S) regardless of the original propagationazimuth.

Thus, even for 2D acquisition geometry, pureshear modes can yield information about azimuthalanisotropy. When the acquisition line is parallel to avertical symmetry plane and the medium is laterallyhomogeneous, no reflection energy should be presenton the off-diagonal components in equation 5. Out-of-plane (obliquely oriented) lines, however, may containsignificant coherent energy on DXY and DY X . Alford’s(1986) 4C operator simultaneously rotates the sourcesand receivers in order to estimate the symmetry-planeazimuths and traveltime difference between the fast andslow S-waves:

D′ = RS D RTR , (6)

where R is a 2×2 matrix of rotation around the verticalaxis for sources (RS) and receivers (RR), and T denotestranspose. Rotation is applied to each CDP consistingof a 4C group of traces. For a certain rotation anglethat corresponds to the minimum energy on the off-diagonal components D′XY and D

′

Y X , the data appear

as if they were acquired in one of the symmetry planes.This means that the diagonal components, D′XX andD′Y Y , correspond to the fast and slow shear waves andcan be processed to estimate the splitting coefficient.

The 4C rotation dramatically improves the qualityof shear-wave reflection data and makes them suitablefor lithology discrimination (Alford, 1986). By combin-ing Alford rotation of VSP data with layer stripping,Winterstein and Meadows (1991) evaluate S-wave split-ting related to in-situ stress and fractures at Cymricand Railroad Gap oil fields (California, USA). WhereasAlford’s method assumes the principal anisotropy direc-tions (i.e., the azimuths of the symmetry planes) to beinvariant with depth, Winterstein and Meadows (1991)identify well-resolved, abrupt changes in the splittingcoefficient at several depth levels that could be im-portant for reservoir characterization. Thomsen et al.(1999) extend the layer-stripping technique to reflectedS-waves and discuss the analytic basis for separating thesplit shear modes on both 4C and 2C (single-source)data.

There have been numerous successful applicationsof shear-wave splitting for purposes of fracture charac-terization (e.g., Mueller, 1990; Crampin, 2003; Vascon-celos and Grechka, 2007). Traveltime and amplitude dif-


ferences between the fast and slow shear waves, as wellas their NMO ellipses, can help estimate fracture ori-entation, density and, in some cases, make inferencesabout fluid saturation. Also, Angerer et al. (2000) showthat shear-wave splitting is a more sensitive time-lapse(4D) indicator of pressure changes in response to CO2injection than P-wave velocities. Their synthetic seis-mograms based on the anisotropic poroelastic theory ofZatsepin and Crampin (1997) match stacked data beforeand after injection. Terrell et al. (2002) arrive at sim-ilar conclusions in their time-lapse study of CO2 floodat Weyburn Field in Canada. There is little doubt thatmoveout and amplitude inversion of multicomponent,multiazimuth data offers the best hope of estimatingthe anisotropy parameters of subsurface formations.

9.2 Mode-converted PS-waves

The majority of multicomponent surveys is acquiredwithout shear-wave sources, so the reflected wave-field is largely composed of compressional waves andmode-converted PS-waves. The most prominent P-to-S conversion typically happens at the reflector; suchPS events are sometimes called “C-waves” (Thomsen,1999). For horizontally layered, azimuthally isotropicmedia converted PS-waves are polarized in the incidence(sagittal) plane (i.e., they result from P-to-SV conver-sion). However, if the incident P-wave propagates out-side vertical symmetry planes of azimuthally anisotropicmedia, the reflected PS-wave splits into the fast (PS1)and slow (PS2) modes, neither of which is generally po-larized in the sagittal plane (e.g., J́ılek, 2002a).

An important processing step for mode conversionsin the presence of azimuthal anisotropy is rotation ofreceiver directions from an acquisition coordinate sys-tem to a source-centered, radial and transverse coordi-nate system (Gaiser, 1999). This procedure reveals az-imuthal traveltime variations of PS1- and PS2-waves onthe stacked radial components, as well as polarity rever-sals in the principal anisotropy directions on the stackedtransverse components (Li and MacBeth, 1999).

Similar to pure-mode SS reflections, the fast andslow PS-waves have to be separated for further pro-cessing. The feasibility of PS-wave splitting analysis isdemonstrated by Garotta and Granger (1988) who an-alyze the amplitude ratios of the transverse and radialcomponents and apply 2C rotation and layer stripping.Gaiser (1997) shows that Alford rotation and layer strip-ping (a method similar to that of Winterstein and Mead-ows, 1991), are applicable to PS-waves in reverse VSPgeometry. His technique operates with 4C data fromequation 5 where the two rows correspond to source-receiver azimuths 90◦ apart. The principle of Alford ro-tation is extended to wide-azimuth PS-wave surveys byDellinger et al. (2002) who replace stacking of PS1 andPS2 reflections with an appropriately designed tensormigration. Their results from Valhall Field are mixed,

which suggests that azimuthal and lateral velocity varia-tions may seriously complicate PS-wave processing. Re-cently there has been renewed interest in developing amore formal inversion approach to the PS-wave layer-stripping problem where the objective function is for-mulated in terms of the PS1-wave polarization azimuthand the traveltime difference between the split PS-waves(e.g., Bale et al., 2009; Haacke et al., 2009; Simmons,2009).

In addition to such well-documented applications asimaging beneath gas clouds and lithology discrimina-tion, mode-converted data provide valuable attributesfor fracture/stress characterization (Gaiser, 2000). Af-ter performing layer stripping of 3D ocean-bottom-cable(OBC) PS-wave data over Valhall Field, Olofsson etal. (2002) describe a dramatic “ring of anisotropy” inthe overburden where the PS1-wave is polarized trans-versely around the production platform. The correla-tion of this anisotropy pattern with sea-floor subsidencecaused by the reservoir collapse after years of productionsuggests that shear waves are highly sensitive to local,deformation-induced stresses. Sensitivity of the polar-ization direction of the PS1-wave to local stresses overanticlines has also been observed at Emilio Field in theAdriatic Sea (Gaiser et al., 2002), and at Pinedale Fieldin Wyoming, USA (Gaiser and Van Dok, 2005). As il-lustrated by Figure 8, the PS1-wave at Emilio Field ispolarized parallel to the crest of a doubly plunging anti-cline (thick black arrows), where anisotropy is generallyhigher.

Finally, it is important to note that the moveoutasymmetry of PS-waves (i.e., their traveltime generallydoes not stay the same when the source and receiver areinterchanged) helps constrain the parameters of tiltedTI media (Dewangan and Tsvankin, 2006b) and char-acterize dipping (non-vertical) fracture sets (Angerer etal., 2002).

9.3 Joint processing of PP and PS data

Conventional isotropic processing of high-quality mul-ticomponent offshore OBC surveys routinely producesdepth misties between PP and PS sections, in large partdue to the strong influence of anisotropy on PS-wavemoveout. The high sensitivity of mode conversions toanisotropy represents an asset for joint anisotropic in-version of PP and PS data (e.g., Grechka et al., 2002a;Foss et al., 2005). For example, the parameters VP0, ǫ,and δ influence the kinematics of both P- and SV-wavesin TI media, which underscores the importance of mul-ticomponent data in anisotropic velocity analysis.

Widespread use of converted waves, however, ishindered not just by the higher acquisition cost ofmulticomponent surveys, but also by difficulties inPS-wave processing. Such properties of mode conver-sions as moveout asymmetry, reflection point disper-sal, and polarity reversals present significant challenges


VTI Isotropic

7 118 9 10 12 13 7 8 9 10 11 12 13

4.0

t, s

3.5

3.0

2.5

2.0

1.5

Top Balder

Figure 9. Common-conversion-point stacks of PSV-wavesfor a 2D line above the Siri reservoir in the North Sea (af-ter Grechka et al., 2002b). Acquired PP- and PSV-waveswere processed using the PP+PS=SS method to compute thetraveltimes of the corresponding SS (SVSV) reflections. Thesection on the left was computed with a VTI velocity modelobtained from stacking-velocity tomography of the recordedPP-waves and constructed SS-waves. The section on the rightwas produced without taking anisotropy into account.

for velocity-analysis and imaging algorithms. Theseproblems motivated the development of the so-called“PP+PS=SS” method designed to construct primarySS (in general, both S1 and S2) reflections with thecorrect kinematics from PP and PS data (Grechka andTsvankin, 2002). The key idea of the method, which op-erates on PP and PS reflections acquired in split-spreadgeometry, is to match the time slopes (horizontal slow-nesses) of PP- and PS-waves on common-receiver gath-ers. This procedure helps identify PP and PS events re-flected at the same (albeit unknown) subsurface points,and the SS-wave traveltime can be obtained as a sim-ple linear combination of the PP and PS times. Toavoid time picking, Grechka and Dewangan (2003) de-vised the full-waveform (interferometric) version of thePP+PS=SS method based on a specially designed con-volution of PP and PS traces.

Although the PP+PS=SS method should be pre-ceded by PP-PS event registration, it does not requireinformation about the velocity field and is valid for arbi-trarily anisotropic, heterogeneous media. The moveoutsof the recorded PP-waves and computed SS-waves canbe combined in anisotropic velocity analysis using, forexample, 3D stacking-velocity tomography (Grechka etal., 2002a). The case study from the North Sea in Fig-ure 9 demonstrates that this methodology greatly im-proves the quality of PS-wave stacked sections (Grechkaet al., 2002b). Application of the PP+PS=SS methodfollowed by VTI processing provided a much better im-age of the reservoir top (top Balder, the deepest ar-row on the left) and a crisp picture of faulting in theshallow layers. Accounting for anisotropy also boosted

higher frequencies in the stack and, therefore, increasedtemporal resolution.

10 FRACTURE CHARACTERIZATION

By some estimates, fractured reservoirs contain aboutone-third of the world’s hydrocarbon reserves. Sincealigned fractures create velocity and attenuationanisotropy on the scale of seismic wavelength, seis-mic fracture characterization is largely based onthe anisotropic processing/inversion methods discussedabove.

In the past few years, significant progress has beenachieved both in effective media theories and seismiccharacterization of multiple fracture sets. The theoreti-cal advances are mainly attributed to increased comput-ing power, which made it possible to construct so-calleddigital rocks and examine how such realistic features ascrack intersections, shape irregularities, microcorruga-tion, and partial contacts of the fracture faces influencethe effective elastic properties. It has been shown thatmultiple sets of irregular, possibly intersecting fracturesthat have random shape irregularities are well approx-imated by isolated, penny-shaped cracks (Grechka andKachanov, 2006, and references therein).

Another important result, known from theoreticalstudies of Kachanov (1980, 1993) and confirmed nu-merically by Grechka et al. (2006), is that multiple,arbitrarily oriented sets of fractures embedded in anotherwise isotropic host rock yield an effective mediumof approximately orthorhombic symmetry. This state-ment is valid for both dry and liquid-filled fractures.The former are close to so-called scalar cracks (in ter-minology of Schoenberg and Sayers, 1995), which al-ways yield effective orthotropy (i.e., orthorhombic sym-metry) in the non-interaction approximation. The lattercontribute mainly to shear-wave anisotropy (i.e., to pa-rameters analogous to the splitting coefficient γ(S)) andalso do not produce any substantial deviations from or-thorhombic symmetry.

The closeness of the effective elasticity of crackedsolids to orthotropy implies that multiple systems offractures appear to long (compared to the fracture sizes)seismic waves as three orthogonal or principal sets. Forinstance, N sets of dry fractures that have the individualcrack densities e(k) and the normals n(k) (k = 1, . . . , N)to the fracture faces are equivalent to three “principal”sets, whose densities and orientations are found as theeigenvalues and eigenvectors of the crack-density tensor(Kachanov, 1980, 1993):

αij =N

X

k=1

e(k) n(k)i n

(k)j , (i, j = 1, 2, 3) . (7)

Vasconcelos and Grechka (2007) employ this the-ory to characterize multiple vertical fracture sets fromwide-azimuth, multicomponent seismic reflection data


recorded at Rulison Field. The fracture orientations ob-tained from seismic data are consistent with the FMI(Formation MicroImager) log acquired in the studyarea. In addition, Vasconcelos and Grechka (2007) con-struct an orthorhombic velocity model of the Ruli-son reservoir by jointly inverting P- and S-wave NMOellipses. This inversion is possible primarily becausecrack-induced orthotropy is governed by fewer inde-pendent parameters than general orthorhombic media,making estimation of these parameters better posed andea sier to implement. Still, comparison of the spatiallyvarying crack densities with the estimated ultimate re-covery (EUR) of the available wells shows little correla-tion. This problem, typical for a number of other tight-gas fields in North America, should motivate furtherdevelopment of robust seismic technologies capable ofdetecting accumulations of hydrocarbons in fracturedformations.

11 THE ROAD AHEAD

Progress in geophysics is usually driven by data; when-ever we acquire a new type of data, we can expect todiscover unexpected features that cannot be handledby existing methodologies. In hindsight, these surprisesshould have been foreseen (and maybe were foreseen bya few savants), but they always do surprise most of us.

Today, the industry routinely acquires high-qualitywide-azimuth 3D marine data with the goal of better il-luminating subsalt targets. When processing such data,we are discovering that azimuthally variable seismic ve-locity is often required to flatten the wide-azimuth im-age gathers. This will surely lead us to further developmethods dealing with azimuthal anisotropy, which havebeen applied primarily to land data sets. Also, it is al-ready clear that horizontal transverse isotropy (HTI) isnot an appropriate model for most formations with ver-tical cracks, and TTI is probably an oversimplified sym-metry for dipping beds. Future developments will in-clude extension of velocity-analysis and migration algo-rithms to more realistic orthorhombic models. Althougha solid foundation for parameter estimation and imagingin orthorhombic media has already been built, findingrobust and cost-effective processing solutions is a seri-ous challenge, especially for tilted orthotropy. Also, itwould not be practical to operate with a parameter setthat is not constrained by available seismic data. An-other direction of future research with a high potentialpayoff in velocity analysis is anisotropic full-waveforminversion of reflection data, which should become feasi-ble with continuing increase in computing power.

An interesting feature of anisotropy is that, al-though usually it is weak (i.e., the dimensionlessanisotropy parameters typically are much smaller thanunity), in many contexts it has a strong influence on seis-mic data. In particular, the contribution of anisotropyto reflection coefficients is comparable to the isotropic

“fluid” and “lithology” factors, which is particularlynoticeable in the azimuthally varying P-wave AVO re-sponse. Anisotropy in the overburden also causes pro-nounced distortions in the geometric-spreading factorfor reflected waves. Hence, it is easy to predict that moreemphasis will be placed on understanding and utiliz-ing amplitude signatures in anisotropic media, likely in-cluding attenuation measurements. Experimental dataindicate that attenuation anisotropy, especially thatproduced by fluid-saturated fractures, may be ordersof magnitude higher than velocity anisotropy. There-fore, azimuthally varying (and, possibly, frequency-dependent) attenuation coefficients may provide sensi-tive reservoir-characterization attributes.

Anisotropic phenomena are especially noticeable inshear and mode-converted wavefields; it is usually im-possible to deal with shear data without consideringanisotropy. In so doing, completely new concepts (un-known in isotropy) arise, such as shear-wave splitting.For example, acquisition of high-quality PS-wave data inrecent years revealed strong conversion of energy (P-to-S) at near-normal incidence, which is prohibited by thestandard model of plane-wave reflection from a planarboundary between isotropic or VTI halfspaces. Someof candidate explanations of these anomalous PS ar-rivals involve anisotropy (e.g., tilted TI on either sideof the reflector). Whatever the eventual solution to thisproblem, it will likely entail a revision of conventionalAVO models and algorithms for both PP- and PS-waves.Also, wide-azimuth, multicomponent data will play amajor role in robust parameter estimation for realisticorthorhombic and, in some cases, lower-symmetry me-dia. Note that the split shear-wave primary reflections(S1 and S2) with the correct kinematics can be gener-ated from wide-azimuth PP and PS data using the 3Dversion of the PP+PS=SS method.

Whereas anisotropic P-wave imaging essentiallyamounts to looking “past” anisotropy at explorationtargets, progress in processing/inversion techniques isputting more emphasis on employing anisotropy pa-rameters as attributes in reservoir characterization andlithology discrimination. One of interesting emergingapplications of anisotropic attributes is in time-lapseseismic for compacting reservoirs because the shear-wave splitting coefficient, traveltime shifts and othercompaction-related signatures are strongly influencedby stress-induced anisotropy. Physical characterizationof the subsurface in terms of lithology, fluids, fractures,pore pressure, and permeability will require improvedrock-physics and geomechanics methods operating withanisotropic models.

12 ACKNOWLEDGMENTS

We are grateful to our numerous colleagues for their con-tributions discussed in the paper and for fruitful discus-sions that improved our understanding of the subject.


We appreciate the reviews by K. Helbig, A. Rüger andA. Stovas, who made a number of helpful suggestions.I.T. acknowledges the support of the Center for WavePhenomena at Colorado School of Mines.

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