Isotropic and anisotropic velocity model building for subsaltseismic imaging

Robert Keys1, Tim Matava2, Douglas Foster3, and Don Ashabranner1

ABSTRACT

The effectiveness of basin simulators for deriving subsaltvelocity models has been previously shown through the useof a correlation to relate effective stress to velocity. We buildon this and others’ work by using physical models to relateporosity to velocity. This process yields a physically realiz-able isotropic velocity model that is consistent with the geo-logic model and matches the tomographic velocity modelabove salt and in regions where the tomographic velocityestimate is accurate. We then use a geomechanical simulatorto model the stress distribution in and around allochthonoussalt in which material properties between salt and sedimentchange. Our stress model is the basis for an anisotropic veloc-ity model using Murnaghan’s theory for finite elastic defor-mation. This formulation, with bounds placed on the elasticcoefficients, leads to significant imaging improvements adja-cent to salt.

INTRODUCTION

Issues with seismic-reflection imaging in and around large veloc-ity contrasts such as those presented by allochthonous and autoch-thonous salt are well-known (Leveille et al., 2011). For example,tomographic methods (Woodward et al., 2008) are challenged be-neath the salt by a poor signal-to-noise ratio, and reflection energy isgenerally limited by offset or angle range. We have also observedthat velocities extrapolated into subsalt areas are frequently too fast.Past approaches to address these difficulties have involved improve-ments in seismic acquisition (e.g., wide azimuth, coil surveys, etc.)and imaging algorithms (e.g., reverse time migration [RTM]). Wepropose that significant improvement in subsalt images may be ob-tained through additional geologic control in the subsalt migration

velocity model. In this paper, we demonstrate how to use an inte-grated basin simulation of a geologic model to develop isotropicand anisotropic velocity volumes for more accurate subsalt imaging.We use the term “model” to describe any calculated representa-

tion of a physical property. We use the term “physical model” todescribe a model (e.g., porosity, velocity, or stress) derived usinga physical law or conservation principle. For example, a model builtusing conservation of mass and momentum is a physical model.Stress derived from a constitutive relationship is also a physicalmodel because the derivative of the strain energy function with re-spect to strain must be positive definite. The latter example impliesthe use of an entropy principle. We refer to a rock-physics model asphysical if it satisfies the Voigt-Reuss bounds that are required foran elastic material.The first reported use of migration velocity models derived from

a geologic interpretation for subsalt imaging is by Albertin et al.(2002). They use a basin model to generate a present-day effectivestress volume. Then they use an empirical relationship betweenvelocity and effective stress, derived fromwell data (Petmecky et al.,2008), to create a velocity volume. More recently, De Prisco et al.(2014) and Szydlik et al. (2015) use methods presented by Breviket al. (2014) to build isotropic and anisotropic subsalt velocity mod-els with porosity, permeability, and effective stress volumes calcu-lated from an integrated basin simulator. In each of these cases, thebasin simulator is a tool that uses a geologic interpretation to createa migration velocity volume for the subsalt region, where tomog-raphy is either significantly limited or cannot be applied.Conventional basin models estimate vertical stress, which can

provide only isotropic velocity models. Anisotropic velocity modelscan be calculated using a full 3D geomechanical stress simulator.Fredrich et al. (2003) and Nikolinakou et al. (2012) use geomechan-ical models to show stress halos and arching in sediments aroundsalt bodies. As the rugosity of the salt increases, the stress effects areaccentuated. Rapid variations of velocity can distort subsalt imagesif not properly accounted for, and stress can produce these types of

Manuscript received by the Editor 15 June 2016; revised manuscript received 1 November 2016; published online 30 March 2017.1ConocoPhillips Company, Houston, Texas, USA. E-mail: robert.g.keys@conocophillips.com; Don.Ashabranner@conocophillips.com.2GGIM, Houston, Texas, USA. E-mail: tmatava@sbcglobal.net.3University of Texas at Austin, Institute for Geophysics, Austin, Texas, USA. E-mail: djfoster@ig.utexas.edu.© 2017 Society of Exploration Geophysicists. All rights reserved.

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GEOPHYSICS, VOL. 82, NO. 3 (MAY-JUNE 2017); P. S247–S258, 16 FIGS.10.1190/GEO2016-0316.1

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velocity anomalies. These stress effects can propagate hundreds ofmeters to kilometers away from the salt. Bachrach and Sengupta(2008) are the first to show the impact of stress perturbations dueto the interaction of salt and sediment on the stress field and theresulting impact on the velocities. In Matava et al. (2016), we usea similar approach to estimate the full anisotropic velocity modelaround salt bodies. Here, we expand on this approach by providingadditional details on the methodology for building a migrationvelocity volume.We use third-order elasticity (TOE) to link stress (strain) to seis-

mic velocities. The theory originated with Murnaghan (1937), whodescribes the elasticity of finite deformations and derives an expan-sion of the strain energy function in terms of strain perturbations.We start with the basin model derived isotropic velocity, which in-cludes salt, as the initial reference velocity, and then we use the TOE

terms of Murnaghan’s expansion to estimate changes in elasticproperties due to the perturbation in stress from the initial isotropicstate. A difficulty with the TOE method is that three independentadditional coefficients are required. Laboratory measurements ofthese coefficients using cores show a high degree of variability anduncertainty. Here, we use the experimental observation (e.g., Nurand Simmons, 1969) that velocity increases with increasing effec-tive stress to provide bounds for these coefficients. We use thesebounds to choose the TOE coefficients. Using this methodology,we find that seismic images around and beneath the salt can be im-proved. We will illustrate this methodology with examples from theGulf of Mexico.

ISOTROPIC VELOCITY MODEL BUILDING

One of the main issues with subsalt tomogra-phy is the lack of energy in the full-offset rangeof the image gathers. Figures 1 and 2 illustratethis point. Figure 1 shows image gathers and thefully migrated image from a conventionally de-rived velocity model, and Figure 2 shows imagegathers using a velocity model derived from a ba-sin model. Comparison of the gathers shows thatboth images are plausible because events in bothsets of gathers are flat over the available offsetrange. However, the image derived from the ba-sin simulation velocity (Figure 2) appears moregeologic. The basin model image minimizes con-flicting dips; the basement and deep reflectors aremore coherent and they reveal an interpreteddeep salt pedestal. The energy affecting the saltpedestal is migrated to other parts of the seismicimage in Figure 2. The quality of the image is animportant factor for selecting the preferred veloc-ity model. Our experience is that a more accurategeologic interpretation leads to a better imagethan a poor geologic interpretation.

In the following, we describe our process forcreating a migration velocity model from a basinmodel. Construction of the basin model requiresa geologic interpretation of a 3D seismic volumethat has an associated migration velocity volumeobtained by conventional means. We expect thatthe initial migration velocity is accurate above oraway from the salt, and our objective is to im-prove the seismic image beneath the salt witha migration velocity that is consistent with a geo-logic interpretation that is refined by this process.The 3D integrated basin simulators such as

those described by Moeckel et al. (1997) or Hant-schel and Kauerauf (2009) are designed to for-ward model pressure, temperature, and porosityusing a geologic interpretation for sediment type,sedimentation rate, and basin architecture. Thesesimulators use a 1D vertical compaction law tocalculate a change in porosity with a change ineffective stress that has the following form:

dϕ ¼ −Cdσeff ¼ −Cðdσlith − dPporeÞ; (1)

Figure 1. (a) Subsalt image gathers using velocities extrapolated into the subsalt region.(b) The full-fold RTM-migrated section.

Figure 2. (a) Image gathers using the velocities derived using the basin modeling meth-ods presented in this paper. (b) The full-fold RTM-migrated section. Reflectors at locationsA are deep basement reflectors interpreted as a salt pedestal and appear more geologic thanin the original image. At location B, misplaced energy and migration swings are removed.

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where ϕ is the porosity, σ is the stress (effective and lithostatic),and Ppore is the pore pressure. Compressibility is defined as C ¼−ð∂ϕ∕∂σeffÞ and, in this case, is the irreversible change in porosityassociated with compaction. The compressibility of sediments in acompacting margin is typically one to two orders of magnitudegreater than compressibilities associated with reversible processes(e.g., reflection seismic wave propagation) in the same sediments.High values for sediment compressibility combined with low per-meability result in overpressure, in which pore fluids carry a portionof the sediment load.Conservation laws are used to derive equations for mass and mo-

mentum that describe the pore pressure and vertical stress evolutionin the basin. Porosity changes with burial depth due to sedimentloading constitute a source term for fluids in the mass and momen-tum laws. We assume a single-phase fluid with Terzaghi coupling(Terzaghi and Peck, 1948) between lithostatic stress and pore pres-sure (equation 1). Equation 1 and lithostatic loading histories throughgeologic time, combined with mass and momentum balances on theliquid allow calculation of the pore pressure and effective stress forthe volume. In addition, other parameters are calculated, such as bulkdensity and bulk modulus as well as the regional flow field in a basin.Hantschel and Kauerauf (2009, chapter 2) provide details of inte-grated basin modeling to quantify pressure and porosity in a petro-leum system.We use a geologic interpretation of a 3D seismic volume to model

porosity changes due to sediment type, sedimentation rate, and ba-sin architecture. Porosity is a fundamental variable because of therelationship between porosity, effective stress, and permeability. Al-lochthonous salt may be treated as either a constant age sedimentarylayer with bounding surfaces or as facies within a sedimentary layer.In the first case, salt may also be inflated or deflated either geomet-rically or by changes in vertical load through time. Our experienceis that the best seismic images are obtained when the salt is treatedas sediment but is inflated or deflated through time.Construction of a basin model to create a velocity volume is per-

formed similarly to basin model applications for oil and gas explo-ration; however, emphasis is placed on porosity, pore pressure, andlithostatic load. Wireline data and control points are used to cali-brate porosity. Control points are pseudowells, in which a verticalvelocity profile is inverted for porosity with a rock-physics modeland used as if it were an exploration calibration well. Pressure onthe fluid phase is calibrated with wireline data and pressure mea-surements where available. Coupling between the porosity and theeffective stress through the compressibility of the sediments (equa-tion 1), insures consistency between the geologic interpretation andthe velocity model for the sediment type, sedimentation rate, andbasin architecture. We typically require three to five sediment typesbetween the seabed and deepest source rocks in the basin to con-struct the basin model. The sediment types have a shale fraction thatwe use later in the velocity modeling workflow.An approximation for the differential effective medium (DEM)

rock-physics model is used to calculate the velocity from basinmodel porosity, pore pressure, and shale fraction. Porosity and porepressure are calculated by the basin simulator. The shale fraction isdetermined from the lithologies that are used to populate the basinmodel. Typically, we use a shale fraction of 0.7−0.9 for shales and0.2−0.4 for sands. The DEM approximation represents the rock as amixture of fluid-filled (brine) ellipsoidal stiff (sand) and compliant(shale) pores embedded in a sand/shale matrix. The Voigt-Reuss-Hill

average (Mavko et al., 1996, p. 127) is used to determine the bulk andshear modulus of the sand/shale rock matrix based on the shale frac-tion. Shale fraction is also used to divide the pore space into stiff andcompliant pores. Using the DEM approximation avoids some con-vergence issues associated with the numerical DEM algorithm. Fur-thermore, the elastic moduli calculated from the DEM approximationcan be shown to satisfy the Voigt-Reuss bounds required for real elas-tic materials. A derivation of the DEM approximation and a demon-stration that its elastic moduli satisfy the Voigt-Reuss bounds areprovided in Appendix A.The DEM approximation for the bulk modulus K is the unique

solution of the equation

K ¼ K0

0B@�1 − Kf

K0

��1 − Kf

K

� ð1 − ϕÞ

1CA

p0

; (2)

whereK0 is the grain bulk modulus,Kf is the pore-fluid bulk modu-lus, and ϕ denotes porosity. The shear modulusG is calculated fromgrain shear modulus G0 and ϕ with

G ¼ G0ð1 − ϕÞq0 : (3)

Exponents p0 and q0, also referred to as polarization factors (Ber-ryman, 1992), depend on the shape or aspect ratio of dry ellipsoidalsand and shale pores. The polarization factors p0 and q0 are vol-ume-weighted combinations of the stiff and compliant polarizationfactors:

p0 ¼ Vssp0ðαssÞ þ Vshp0ðαshÞ; (4)

q0 ¼ Vssq0ðαssÞ þ Vshq0ðαshÞ; (5)

where αss and αsh are the stiff and compliant ellipsoidal pore-aspectratios, respectively, and Vss and Vsh denote sand fraction and shalefraction, respectively. Formulas for the pore-aspect ratio dependentpolarization factors are given by Keys and Xu (2002). Equations 2and 3 are the basis for elastic, isotropic moduli we use to calculatevelocity of the sediments.A diagram illustrating the workflow to calculate a velocity model

from a geologic interpretation of seismic data is shown in Figure 3.The geologic interpretation is used with an integrated basin simu-lator to forward model present-day pressure and porosity at wellsand control points distributed throughout the volume. The DEMmodel uses the porosity, shale fraction, and pressure volumes to cal-culate an isotropic velocity volume. The schematic workflow can beiterative for large changes in the velocity models. Newly migratedvolumes may be reinterpreted to create an updated geologic inter-pretation and a recalibrated basin model. We illustrate this workflowwith an application to a Gulf of Mexico imaging project.The workflow to develop an isotropic velocity volume is general;

however, particular details are important for specific applicationssuch as subsalt velocity model building. Velocities derived fromtomography in the supra-salt section and outboard of allochthonoussalt are an important component for imaging the seismic volume.The tomographic velocity volume is most reliable in supra-salt min-ibasins and south of the Sigsbee escarpment in the Gulf of Mexicowhere salt is absent. Isotropic velocity models from the basin model

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have little geologic control from well data in the suprasalt region,thus tomography provides useful information on sediment proper-ties in this section.For our Gulf of Mexico imaging projects, we use a nearby well to

calibrate the DEM rock-physics model. We found the calibratedmodel to be robust, and it was unnecessary to adjust calibrationparameters throughout the greater Gulf of Mexico. We used selectedvelocity profiles from the tomography velocity volume to calculatepseudowell porosity profiles with the DEM rock-physics model(Figure 4). This process allows us to construct a velocity modelfrom the basin model that matches the tomographic velocity at con-trol points where velocity estimates from tomography are reliable.The elastic properties associated with the basin model constructedvelocity also necessarily satisfy the Voigt-Reuss bounds, which isimportant when we consider the anisotropic velocity model later. Inpractice, several pseudowell control points (<10) are used through-out the model to supplement drilled wells for calibration. Controlpoints are updated as the tomographic velocity volume evolves.Furthermore, the velocity volume derived from the basin modelis used to initialize and constrain subsequent tomography revisions.In summary, methods used to build isotropic velocity models are

physically based and designed to yield consistent results betweentomography and basin modeling velocity volumes. Tomography is

used in the shallow supra-salt section and merged with the basinmodel velocity to yield an enhanced velocity volume particularlyfor the subsalt portion of the volume. Control points in the formof pseudowells allow a direct comparison of tomographic and basinmodeling results and aid in obtaining a consistent velocity volume.Relying on physical models instead of correlations yields calibratedmodels in shorter times. Reducing the cycle time means that thecause-effect on the images is fresh in the mind of those involved inthe project, which further reduces the number of iterations to producea final image (typically less than 10 in most extensional settings).Improvements in an isotropic image for the subsalt Wilcox Play

in the Gulf of Mexico are shown in Figure 5. This image qualita-tively shows better delineation of subsalt reflectors in the target areaand an improved image of the basement structure. An additionalimpact of this reprocessing program is shown in Figure 6, whichdisplays changes in the structure of the reservoir against allochtho-nous salt. These changes result from initial velocities that were toofast and pushed basement horizons too deep. Reprocessing slowedthese velocities, decreased the depth to basement, and reducedthe dip of the horizon into the three-way trap against salt. Thesechanges in trap geometry modify predrill estimates of mean resour-ces. In our example (Figure 6), the changes increase the mean re-sources by approximately 50%. When the velocity is lowered, the

gross rock volumes and mean resources increase.It was our experience that reimaging projects al-ways required us to decrease initial velocitiesat depth.These imaging results are important from a

petroleum system perspective. The section shownin Figure 5 includes the entire petroleum systemfrom source to trap. This image allows interpreta-tion of not only the trap but also away from thetrap where fluids, presumed to have been expelledfrom a source rock, migrate laterally and verticallyto a reservoir. Improved images in reflector con-tinuity and depth increase our confidence in thepetroleum system that is presented as reducedprospect risk. Indeed, these improved images andthe significantly different structural maps implylarge changes to the petroleum system for the areain terms of fluid quality prediction, volume of hy-drocarbons available to trap, and equity negotia-tions (Figure 6). Therefore, it makes sense toconduct this type of reimaging program early inthe exploration play process rather than later afterland is acquired and equity negotiations occur.

ANISOTROPIC VELOCITY MODELBUILDING

Seismic velocities are influenced by stress,and the presence of salt can produce stress-in-duced anisotropy in the velocity field. Geome-chanical models have demonstrated the impactof stress halos and arching around rugose saltbodies (Fredrich et al., 2003; Nikolinakou et al.,2012) and on velocity anisotropy (Bachrach andSengupta, 2008). Their model results suggestthese stress arches and halos extend hundreds ofmeters in the sediment away from the salt body.

Figure 3. The workflow we developed for using a geologic interpretation of seismicdata to create a calibrated velocity model to migrate seismic data. The basin modelis calibrated in the normal way for pressure and porosity to wells in the area. Pressureand porosity volumes, along with a volume for the shale fraction and a calibrated DEMmodel, are used to calculate an isotropic velocity volume. A geomechanics simulator isused to calculate stress in the volume, and the TOE method is used to estimate velocity,including anisotropy in the volume. Adding a temperature model to the calibration,along with a source-rock model, means that a physical model for the petroleum systemcan be developed during the imaging part of the exploration program.

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Poroelastoplastic modeling (Nikolinakou et al., 2012) shows thatstress-induced effects can extend kilometers away from salt bodies.We use a geomechanical simulation of our basin model to predictstress perturbations from an isotropic reference model and to esti-mate velocity anisotropy and anisotropic imaging parameters.Specifically, we use the theory of finite elastic deformation de-

veloped by Murnaghan (1937) to calculate the impact of stress onvelocity. Murnaghan (1937) derives an expansion of the strain en-ergy density with respect to strain perturbations. Neglecting pertur-bations in strain energy of second order and beyond, this expansionyields the following approximation for the stress-dependent elasticstiffness:

Cijkl ¼ C0ijkl þ C0

ijklmnemn; (6)

where Cijkl is the stress-dependent stiffness tensor, C0ijkl is the refer-

ence isotropic stiffness tensor, and emn is the strain tensor that isrelated to the stress σij through Hooke’s law

σij ¼ C0ijklekl: (7)

The sixth-order tensor C0ijklmn in equation 6 contains the TOE

coefficients. For an isotropic reference medium, there are five non-zero TOE coefficients, of which three are independent (Prioul et al.,2004). Common practice uses core measurements to estimate therequired coefficients, but these measurements have considerable

variability and significant uncertainty due to the complexity ofthe laboratory measurements.Although there is large uncertainty in the TOE coefficients, we

can still derive useful bounds for them. Experimental observationsshow that P- and S-wave velocities increase in the direction of anapplied force (Nur and Simmons, 1969). Assuming the sign con-vention that compression is positive (and tension is negative), theseobservations imply that independent TOE coefficients, in Voigt no-tation, satisfy the bounds

−1

2C0111 < C0

112 < C0111; (8)

0 < C0155 < C0

144; (9)

along with the symmetry conditions of C0144 ¼ ðC0

112 − C0123Þ∕2 and

C0155 ¼ ðC0

111 − C0112Þ∕4 (see Appendix B). We constrain the TOE

coefficient C0111 to be proportional to the reference stiffness coef-

ficient C011 and choose the remaining coefficients to satisfy the sym-

metry conditions and bounds in equations 8 and 9. We choose theTOE coefficients C0

112 and C0155 to be at or near the midpoint of their

bounded ranges. The scale factor to calculate C0111 impacts the mag-

nitude of anisotropy but not the direction, which is determined bystress. We chose the scale factor to match the magnitude of the δshown in Figure 7. Using equations 6 and 7 with these TOE coeffi-cients and the stress perturbations from the basin simulator, we obtaina stress-dependent elastic stiffness tensor that produces an anisotropicvelocity model for which P- and S-wave velocities increase in thedirection of increasing stress. We use this stress-dependent stiffness

Figure 5. A comparison between the initial image and the final im-age after reprocessing. The final image followed a workflow outlinedin the text, which combined tomographic velocities and basin mod-eling velocities to produce a final image. Deep reflectors, interpretedas basement, are more continuous and flatter in the final image. Re-flectors beneath the salt are more continuous, but they are also shal-lower and have less dip than the original image into the truncationwith salt.

Figure 4. A DEM-calculated porosity profile (black dots) from thetomographic velocity model compared with the basin model porosity(red line) at a particular control point. The horizontal lines are strati-graphic tops included in the basin model. The basin model porosityprofile matches the general trend of the porosity profile derived fromthe velocity model; however, the derived porosity profile shows lowporosity (fast sediments) at approximately 4 km. Similarly, the basinmodel suggests too low porosity at approximately 9 km. Resolvingthese issues with physical models (rather than correlations) makesmore consistent the geologic and geophysical models.

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tensor to calculate the Thomsen delta and epsilonanisotropic imaging parameters.Equations 8 and 9 describe the stress-depen-

dent stiffness tensor in a local-coordinate systemdefined by the principal stress directions. Toapply equations 8 and 9 at a given point, we mustfirst determine the stress perturbation at thatpoint. We take the reference stress to be the stresstensor for a homogeneous isotropic body sub-jected to a force equal to the lithostatic load inthe direction of the main principal stress. Thestress components in the directions perpendicularto the main principal stress are the lithostatic loadscaled by ν∕ð1 − νÞ, where ν is the Poisson’sratio. The reference stress represents the stressobserved in the absence of any inhomogeneitiesin the medium such as salt. The stress field pro-duced by the geomechanical simulator is im-pacted by the presence of inhomogeneities, andthe difference between the geomechanical simu-lator stress field and the reference stress is thestress perturbation. Applying equations 6 and 7to the stress perturbation yields the stress-depen-dent elastic stiffness tensor in the local-coordi-nate system. Herwanger and Horne (2009)show how the stress-dependent elastic stiffness

tensor can be rotated from the local principal stress-coordinate sys-tem to the global-coordinate system, and then velocities in theglobal-coordinate system can be computed from the rotated stresstensor.Typically, for subsalt imaging projects, δ is determined from

a vertical seismic profile (VSP) or check-shot well data; however,wells may be located quite a distance from the area of interest. Al-khalifah (1997) discusses issues related to estimating anisotropicparameters. In theory, the ϵ parameter may be determined by flat-tening image gathers near the wells. In practice, ϵ is chosen to beproportional to δ. Once determined, these 1D, depth-dependentanisotropic parameters are extended laterally from the seabed overlarge distances and are fixed even though velocities are updated dur-ing model building.In our process, δ and ϵ are updated by stress modeling and change

as the 3D stress varies laterally. Rasolofosaon (1998) shows that ifthe second-order strain perturbations are neglected, then the stress-induced anisotropy is elliptical and δ is equal to ϵ. Figure 7 showsa standard model for δ. Figure 8 shows the stress estimate for δ.Because our anisotropy is stress induced, it is elliptical and weuse ϵ ¼ δ.The seismic images for these two anisotropic models are shown

in Figures 9 and 10, respectively. Both of these images are producedby a 3D tilted transverse isotropic (TTI) prestack RTM algorithm.The stress-induced ϵ and δ parameters are calculated from the stiff-ness tensor in the principal stress-coordinate system. The TTI ori-entation is the same for both calculations for comparison purposes.Overall, the stress-induced anisotropic model improves the coher-ency of reflectors in the deeper section. Notice that the positioningof events is shifted between these two sections. This occurs becauseon the conventional sections, tomographic updates of velocity op-timally flatten the gathers. When a different δ and ϵ are used, thegathers will not be completely flat, so an additional iteration of

Figure 6. A base map showing the interpreted Wilcox reflector for the initial image andthe final reprocessed image. In this case, the Wilcox is subcropped by allochthonous salton the bottom left half of the figure. Reprocessing decreased the velocity in the subsaltsection. Slower velocities decreased significantly the depth of the Wilcox horizon at thesubcrop to salt as well as in the center of the basin. In addition, the area of closure at thesubcrop changed significantly. The lower dip of the Wilcox into salt, which is evident inthe reprocessed image, results in higher mean resources assigned to the prospect. Themean resources increased approximately 50%, which we found typical for this work.

Figure 7. Standard model for the Thomsen’s anisotropic parameterδ. The blue indicates no anisotropy, δ ¼ 0. This figure shows themodel for δ adjusted for water bottom depth; δ varies only in thedepth direction. The value of ϵ for this application is twice the valueof δ.

Figure 8. Stress-induced anisotropic model for the Thomsen’s aniso-tropic parameter δ. The color blue indicates no anisotropy, δ ¼ 0.Because of stress variations, the model for δ varies laterally. The blueareas are salt bodies that are isotropic. For stress-induced anisotropy,the ϵ parameter is, to first order, equal to δ.

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tomography is required. Prior to the tomographic update, the loca-tion of salt bodies is reinterpreted.Additional depth images are shown in Figures 11, 12, 13, and 14.

In Figures 11 and 13, the 1D model for δ and ϵ is used, and in Fig-ures 12 and 14, the stress-induced anisotropic model is used.Each of these examples show improved imaging in the deeper

section below the salt bodies using the stress-induced parameters.The base of salt reflection is also more visible. These image im-provements enable more detailed mapping of subsalt structures.

Standard estimates of δ and ϵ tend to vary only with respect todepth. The stress-induced anisotropic parameters vary laterally andcorrelate to stress variations around salt bodies. If velocities varylaterally, the anisotropic parameters should vary laterally as well.Seismic images derived from the stress-induced anisotropic param-

Figure 9. Seismic image using the 1D anisotropic model. This is aprestack depth-migrated image using the δ shown in Figure 7. Thevalue of ϵ is twice the value of δ. This image is produced by a 3Dprestack RTM algorithm.

Figure 10. Seismic image using the stress-dependent anisotropicmodel. This is a prestack depth-migrated image using stress depen-dent ϵ and δ shown in Figure 8. This image is produced by a 3Dprestack RTM algorithm.

Figure 11. Seismic image using the 1D anisotropic model. This is aprestack depth-migrated image using the 1D model for δ and ϵ. Thisimage is produced by a 3D prestack RTM algorithm.

Figure 12. Seismic image using the stress-dependent anisotropicmodel This is a prestack depth-migrated image using the stress de-pendent ϵ and δ. This image is produced by a 3D prestack RTMalgorithm.

Figure 13. This is a prestack depth-migrated image using the 1Dmodel for ϵ and δ. This image is produced by a 3D prestackRTM algorithm.

Figure 14. Seismic image using the stress-dependent anisotropicmodel. This is a prestack depth-migrated image using the stress-de-pendent ϵ and δ. This image is produced by a 3D prestack RTMalgorithm.

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eters show improved coherency deeper in the section than in theconventional model. Subsalt reflectors and base-of-salt reflectionsare better imaged with the stress-induced anisotropic parameters.The improvement of the images is evidence that a laterally varyinganisotropic model is more accurate than the standard 1D model for ϵand δ.

CONCLUSION

In summary, we have shown the improvements in subsalt seismicimages when tomographic velocity volumes are modified with geo-logically derived velocity models. We build on the previous work byshowing how physical-based models, using a constitutive relation-ship that relates porosity to effective stress, can be used to constrainthe isotropic material properties of the sediments. A DEM model,which converts a basin model porosity into an isotropic velocitymodel, is the basis for this new isotropic velocity model. This newisotropic model is physically realizable in the sense that it satisfiesthe Voigt and Reuss bounds for isotropic elastic materials. Key tothis workflow is a geologic interpretation of the reimaged seismicdata. Because we use physically based models for the isotropic por-tion of imaging, we are able to extend these imaging improvementswith anisotropic stress fields available from the geomechanical sim-ulator to develop anisotropic velocity models.We use the finite elastic deformation theory developed by Mur-

naghan to modify the isotropic elastic stiffness coefficients forstress. Although there is considerable uncertainty in the TOE stiff-ness coefficients of this theory, by choosing coefficients within thebounds (equations 8 and 9), we are certain that propagation velocityincreases in the direction of increasing stress, which is consistentwith experimental observations.Images produced from anisotropic velocity models represent a

second-order effect on the isotropic velocity volume, but improve-ments in image quality are still significant. Edges of salt becomebetter defined and deep-sediment reflectors, often considered base-ment, are better resolved, which increase confidence in the geologicinterpretation of the seismic data.We have shown that physically based models are able to produce

geologically and geophysically consistent seismic volumes. A by-product of these reimaging programs is a petroleum system view ofprospects and plays that is developed at the same time as the reimag-ing program, which is typically early in an exploration program. Thismeans that the petroleum systems analysis of the play, including trapsize and position, mobility and fluid quality, are developed earlywhen flexibility usually remains in the direction of the program.

ACKNOWLEDGMENTS

A host of geoscientists contributed their time and effort on im-aging projects using methods presented in this paper. Their contri-butions included much ingenuity and insight. For their efforts, wethank J. Chang, M. Clavaud, B. Foster, E. Frugier, J. Law, Z. Li, G.Neupane, J. Walters, J. Western, and Y. Zeng. We thank J. Berrymanfor his time and insight and Y. Zhang for reviewing and improvingthis paper. We also thank the associate editor, T. Alkhalifah, M.Woodward, and two additional anonymous reviewers who mademany helpful suggestions. Finally, we thank the ConocoPhillipsCompany for supporting this work and PGS and CGG seismic com-panies for permission to publish the seismic images.

APPENDIX A

AN APPROXIMATION FOR THE DIFFERENTIALEFFECTIVE MEDIUM ROCK-PHYSICS MODEL

Keys and Xu (2002) derive an approximate solution of the DEMequations for the case of dry-rock bulk and shear modulus. In thefollowing, we extend their results to obtain estimates for fluid-saturated bulk and shear modulus with noninteracting pores. TheDEM method derives bulk and shear modulus by incrementallyadding pore space to a rock matrix. Berryman (1992) shows that,in the limit as the incremental pore space goes to zero, the DEMequations converge to the set of differential equations:

ð1 − ϕÞ dKðϕÞdϕ

¼ ðKf − KðϕÞÞp; (A-1)

ð1 − ϕÞ dGðϕÞdϕ

¼ ðG − GðϕÞÞq; (A-2)

with

Kðϕ ¼ 0Þ ¼ K0 and Gðϕ ¼ 0Þ ¼ G0; (A-3)

where ϕ is the porosity, K is the bulk modulus of the fluid-saturatedrock, Kf is the bulk modulus of the pore fluid, and K0 is the grainbulk modulus. Likewise, G is the shear modulus of the fluid-satu-rated rock, G is the shear modulus of the pore inclusion (G ¼ 0 for afluid-filled inclusion), and G0 is the grain shear modulus. Berryman(1992) refers to the parameters p and q as polarization factors; pand q depend on the bulk and shear modulus of the saturated rock,the grain properties, and pore-aspect ratios of the ellipsoidal inclu-sions that represent the pore space. Formulas for p and q are givenby Keys and Xu (2002). Because of the polarization factors, equa-tions A-1 and A-2 are coupled, nonlinear differential equations thatmust be solved numerically. However, with a good estimate for thepolarization factors, a reasonable approximate solution for theseequations can be obtained.We observed that for ellipsoidal pores with small aspect ratios,

1∕p varies almost linearly with respect to Kf∕K. In particular, weapproximate p with

1

p¼ 1

p1

Kf

Kþ 1

p0

�1 −

Kf

K

�; (A-4)

where p0 ¼ pðKf∕K ¼ 0Þ and p1 ¼ pðKf∕K ¼ 1Þ. That is, p0 isthe polarization factor for dry rock and p1 is the value of the polari-zation factor in the extreme case that Kf ¼ K. For the latter case,p1 ¼ 1. If the medium is comprised of materials with different poreaspect ratios, the polarization factors are the weighted average (byvolume) of the polarization factors for each of the individual materials.Inserting equation A-4 into equation A-1 and integrating over ϕ

gives the result

K ¼ K0

264�1 − Kf

K0

��1 − Kf

K

� ð1 − ϕÞ

375p0

: (A-5)

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A similar approach yields an approximationfor the shear modulus G, but because G ¼ 0, theexpression for G reduces to the dry rock approxi-mation derived by Keys and Xu (2002)

G ¼ G0ð1 − ϕÞq0 : (A-6)

Equation A-5 has a unique solution for K be-tween Kf and K0. Furthermore, we show thatthe unique solution of A-5 satisfies the Voigtand Reuss bounds required for an isotropic elasticmaterial. First, however, we compare our approxi-mate solution with the solution of the DEM equa-tions. Figure A-1 shows the bulk modulus derivedfrom the DEM equations (blue) and the bulkmodulus estimated (red) from equation A-5 forvarying pore-aspect ratios and fluid bulk modulus.We have found that pore-aspect ratios between0.01 and 0.16 are sufficient to model typicalsand/shale mixtures. The fluid bulk moduli in Fig-ure A-1 span the range of moduli consistent withgas, oil, and brine. Figure A-1 shows that there isclose agreement between the DEM solution forbulk modulus and the approximation obtainedfrom equation A-5. The accuracy of the dry-rockapproximation for shear modulus (equation A-6)was discussed by Keys and Xu (2002).To show that equation A-5 has a unique sol-

ution for bulk modulus K between Kf and K0

that satisfies the Voigt and Reuss bounds, letFðkÞ denote the auxiliary function:

FðkÞ ¼ k − K0

264�1 − Kf

K0

��1 − Kf

k

� ð1 − ϕÞ

375p0

:

(A-7)

Using equation A-7, then limk→KfFðkÞ ¼

−∞. Thus, FðkÞ is negative as k approachesKf . In addition, FðK0Þ ¼ K0½1 − ð1 − ϕÞp0 � >0 provided that p0 > 0. Figure A-2 shows thepolarization factors p0 and q0 for pore-aspect ra-tios between zero and one and Poisson’s ratiosgreater than zero. Both polarization factors arepositive for these pore-aspect ratios and Poisson’sratios. Because FðkÞ is real and continuous forKf < k < K0, then FðkÞ must be zero for somevalue of k ¼ KD between Kf and K0. Further-more, because

dFðkÞdk

> 0; (A-8)

then FðkÞ is monotonically increasing; conse-quently, KD can be the only zero of FðkÞ betweenKf and K0. The solution of equation A-5 is there-fore unique. This argument also shows that equa-

Figure A-1. Comparison of the DEM solution for bulk modulus and the DEM approxi-mation (equation A-5). The DEM solution for bulk modulus (blue) is compared with theapproximation for the bulk modulus given by equation A-5 for pore-aspect ratios rang-ing from 0.01 to 0.16, porosities from 0% to 40%, and pore-fluid bulk moduli varyingfrom 0.01 (gas) to 2.8 GPa (brine). The range of aspect ratios is sufficient to representtypical sand/shale (stiff/compliant) mixtures.

Figure A-2. Polarization factors p and q as a function of pore-aspect ratio. Polarizationfactors (a) p0 and (b) q0 are shown for pore-aspect ratios between 0 and 1 (sphericalpores) and Poisson’s ratios between 0 and 0.4. For these variations, both polarizationfactors are greater than 1.0.

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tion A-5 can be solved with the bisection algorithm, using Kf andK0

to bracket the solution.Let KR denote the Reuss bound for bulk modulus:

1

KR¼ ϕ

Kfþ 1 − ϕ

K0

; (A-9)

then

FðKRÞ¼KR−K0

264�1−Kf

K0

��1−Kf

KR

�ð1−ϕÞ

375p0

;

¼KR−K0

264

�1−Kf

K0

�

1−Kf

�ϕKfþ 1−ϕ

K0

�ð1−ϕÞ

375p0

;

¼KR−K0

264

�1−Kf

K0

�

1−ϕ−Kf

�1−ϕK0

�ð1−ϕÞ

375p0

¼KR−K0<0:

(A-10)

The Voigt bound KV for bulk modulus is

KV ¼ ϕKf þ ð1 − ϕÞK0; (A-11)

FðKVÞ ¼ KV − K0

264�1 − Kf

K0

��1 − Kf

KV

� ð1 − ϕÞ

375p0

¼ KV

�1 −

�KV

K0

�p0−1

�: (A-12)

Figure A-2 shows that p0 is greater than one, which means theright side of equation A-12 is positive. Consequently,

FðKRÞ < 0 ¼ FðKDÞ ¼ 0 < FðKVÞ: (A-13)

Because FðkÞ monotonically increases, it follows immediatelyfrom equation A-13 that the solution of equation A-3, KD, mustsatisfy the Voigt-Reuss bounds:

KR < KD < KV: (A-14)

A similar argument holds for the shear modulus.

APPENDIX B

BOUNDS FOR THE THIRD-ORDER ELASTICCOEFFICIENTS

The observation that P- and S-wave velocities increase in thedirection of increasing stress can be used to bound the TOE coef-ficients. Let σij be a stress perturbation from an initial reference

state of stress with principal components σ1, σ2, and σ3 in the di-rection of each of the coordinate axes. Assume that the stress σij issufficiently small such that the second-order strain perturbations arenegligible. Then, in Voigt notation, the principal strain componentse1, e2, and e3 are given by

σi ¼ C0ijej; (B-1)

where C0ij denotes the second-order stiffness tensor for the isotropic

reference medium. Because second-order strains are negligible, thestrain-dependent stiffness tensor is

Cij ¼ C0ij þ C0

ijkek; (B-2)

where C0ijk is the third-order stiffness tensor for the isotropic refer-

ence medium whose elements are the TOE coefficients.Now, let σij be a stress perturbation from the reference state with

principal components σ1, σ2, and σ3 in the direction of each of thecoordinate axes. Assume that the stress σij is also sufficiently smallsuch that the second-order strain perturbations can be neglected.Then, the corresponding strain and stiffness tensors satisfy the equa-tions:

σi ¼ C0ijej; (B-3)

Cij ¼ C0ij þ C0

ijkek: (B-4)

Define δei ¼ ei − ei to be the change in strain, let δσi ¼ σi − σibe change in stress, and let δCij ¼ Cij − Cij denote the change inelastic stiffness. From equations B-1 to B-4, it follows that:

δσi ¼ C0ijδej; (B-5)

δCij ¼ C0ijkδek: (B-6)

From the Kelvin-Christoffel equations (Herwanger and Horne,2009), δC11 ¼ δðρV2

11Þ, δC22 ¼ δðρV222Þ, and δC33 ¼ δðρV2

33Þ,where V11, V22, and V33 denote the P-wave velocities in the threeprincipal stress directions. Similarly, δC44 ¼ δðρV2

23Þ, δC55 ¼δðρV2

13Þ, and δC66 ¼ δðρV212Þ, where V23, V13, and V12 denote

the S-wave velocities in the three principal stress directions. Thus,equation B-6 relates the change in P- and S-wave velocities to thechange in strain. We first focus on the P-wave component of equa-tion B-6.Through compaction, an increase in stress causes an increase in

density. Observing that for a given principal stress direction, an in-crease in stress causes an increase in velocity in the same direction,it follows that an increase in stress must cause an increase in theelastic stiffness coefficient in the associated principal direction.We have assumed that the elastic deformation is reversible, so adecrease in stress must reduce the corresponding elastic stiffness.Consequently, for the given direction of principal stress, the changein the elastic stiffness coefficient has the same sign as the change in

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stress. As a sign convention, we take compression to be positive andtension is negative. Then, our observation that P-wave velocity in-creases in the direction of applied stress leads to the condition that

0 < δσ1δC11 þ δσ2δC22 þ δσ3δC33: (B-7)

From equations B-5 and B-6, the change in P-wave velocity in theprincipal stress directions can be expressed as

δC¼

264δC11

δC22

δC33

375¼

264C0111 C0

112 C0112

C0112 C0

111 C0112

C0112 C0

112 C0111

375264C011 C0

12 C012

C012 C0

11 C012

C012 C0

12 C011

375−1

×

264δσ1

δσ2

δσ3

375¼

264a b b

b a b

b b a

375264δσ1

δσ2

δσ3

375¼ Aδσ; (B-8)

where

a ¼ ðC011 þ C0

12ÞC0111 − 2C0

12C0112

ðC011 − C0

12ÞðC011 þ 2C0

12Þ

and b ¼ C011C

0112 − C0

12C0111

ðC011 − C0

12ÞðC011 þ 2C0

12Þ: (B-9)

Then, applying equation B-7

0 < δσTδC ¼ δσTAδσ: (B-10)

The constraint B-10 must hold for arbitrary δσ, which is possible ifand only if A is positive definite. Furthermore, A is positive definiteif and only if its eigenvalues are positive. The eigenvalues of A are

a − b ¼ C0111 − C0

112

C011 − C0

12

and aþ 2b ¼ C0111 þ 2C0

112

C011 þ C0

12

:

(B-11)

Because the eigenvalues are positive, C0111 − C0

112 > 0 andC0111 þ 2C0

112 > 0, or

−1

2C0111 < C0

112 < C0111: (B-12)

In a similar manner, S-wave components of equations B-5 and B-6 can be written as

24 δC44

δC55

δC66

35 ¼

24 a b bb a bb b a

3524 δσ1δσ2δσ3

35; (B-13)

where

a ¼ ðC011 þ C0

12ÞC0144 − 2C0

12C0155

ðC011 − C0

12ÞðC011 þ 2C0

12Þ

and b ¼ C011C

0155 − C0

12C0144

ðC011 − C0

12ÞðC011 þ 2C0

12Þ: (B-14)

Because S-wave velocity should also increase in the direction ofincreasing stress, it follows that

−1

2C0144 < C0

155 < C0144: (B-15)

However, the third-order coefficients C0144 and C

0155 are related to

the third-order coefficients C0111, C

0112, C

0123, which imposes addi-

tional constraints. In particular, because

C0155 ¼

1

4ðC0

111 − C0112Þ; (B-16)

the constraint B-12 yields the bound

0 < C0155 < C0

144: (B-17)

In summary, to satisfy the requirement that velocity increases inthe direction of increased effective stress, the TOE stiffness coef-ficients must satisfy the constraints

−1

2C0111 < C0

112 < C0111 and 0 < C0

155 < C0144: (B-18)

Note that there is no P-wave anisotropy at the upper bound forC0112 ¼ C0

111. In fact, if C0144 is zero, then the strained elastic stiff-

ness is isotropic regardless of the applied stress perturbations. Maxi-mum P-wave anisotropy occurs at the lower bound for C0

112.

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