CWP-563

Migration velocity analysis and imaging for tilted

TI media

Laxmidhar Behera1,2 and Ilya Tsvankin2

1National Geophysical Research Institute, Hyderabad 500007, India2Colorado School of Mines, Department of Geophysics, Center for Wave Phenomena, Golden, CO 80401-1887, USA.

ABSTRACT

Tilted transversely isotropic (TTI) formations cause serious imaging distortionsin active tectonic areas (e.g., fold-and-thrust belts) and in subsalt exploration.Here, we introduce a methodology for P-wave prestack depth imaging in TTImedia that properly accounts for the tilt of the symmetry axis as well as forspatial velocity variations.For purposes of migration velocity analysis (MVA), the model is divided intoblocks with constant values of the anisotropy parameters ǫ and δ and linearlyvarying symmetry-direction velocity VP0 controlled by the vertical (kz) andlateral (kx) gradients. Since estimation of tilt from P-wave data is generally un-stable, the symmetry axis is kept orthogonal to the reflectors in all trial velocitymodels. It is also assumed that the velocity VP0 is either known at the top ofeach block or remains continuous in the vertical direction. The MVA algorithmestimates the velocity gradients kz and kx and the anisotropy parameters ǫand δ in the layer-stripping mode using a generalized version of the methodintroduced by Sarkar and Tsvankin for factorized VTI media.Synthetic tests for several TTI models typical in exploration (a syncline, up-tilted shale layers near a salt dome, and a bending shale layer) confirm thatif the symmetry-axis direction is fixed, the parameters kz , kx, ǫ, and δ canbe resolved from reflection data. It should be emphasized that estimation of ǫ(with known VP0) in TTI media requires using nonhyperbolic moveout for longoffsets reaching at least twice the reflector depth. We also demonstrate thatapplication of VTI processing algorithms to data from TTI media may lead tosignificant image distortions and errors in parameter estimation, even when tiltis moderate (e.g., 20-30◦). The ability of our MVA algorithm to separate theanisotropy parameters from the velocity gradients can be also used in lithologydiscrimination and geologic interpretation of seismic data in complex areas.

Key words: transverse isotropy, velocity analysis, tilted symmetry axis,prestack migration, depth imaging, P-waves

1 INTRODUCTION

Transverse isotropy with a tilted symmetry axis (TTI)is a model of ten used to describe dipping shale layers inactive tectonic areas such as the Canadian Foothills andnear salt bodies (e.g., Vestrum et al., 1999; Tsvankin,2005). TTI symmetry can also be created by systemsof parallel dipping fractures embedded in otherwise

isotropic rock (Dewangan and Tsvankin, 2006a, 2006b).Serious distortions caused by TTI layers in conventional(isotropic) seismic imaging are well documented in theliterature (e.g., Isaac and Lawton, 1999; Vestrum et al.,1999). To properly image targets overlaid by TTI for-mations, migration algorithms have to be able to handlethe tilt of the symmetry axis, which often varies later-ally (Vestrum et al., 1999; Kumar et al., 2004). Since

34 L. Behera & I. Tsvankin

the influence of tilt creates ambiguity in parameter esti-mation, a major problem in seismic processing for TTImedia is accurate velocity analysis and model building.

P-wave kinematic signatures for tilted transverseisotropy can be described by the symmetry-directionvelocity VP0, Thomsen (1986) parameters ǫ and δ, andtwo angles responsible for the orientation of the sym-metry axis. In 2D models treated here, the symmetrydirection is described by the angle ν (tilt) with the ver-tical. Estimation of this parameter set even for a sin-gle horizontal or dipping TTI layer generally requirescombining P-wave data with mode-converted PS-waves(Dewangan and Tsvankin, 2006a; 2006b). Inversion ofP-wave reflection traveltimes alone for the TTI param-eters is generally nonunique, unless NMO ellipses fromreflectors with two different dips are available (Grechkaand Tsvankin, 2000).

For TTI shale layers, the symmetry axis is usuallyorthogonal to the layer boundaries, which helps to re-duce the number of independent parameters and makevelocity analysis more stable. If the dip φ of the reflec-tor is equal to the tilt ν, the dip-line P-wave normal-moveout (NMO) velocity in a homogeneous TTI layeris described by the isotropic cosine-of-dip dependence(Tsvankin, 1995; 1997):

Vnmo(φ) =Vnmo(0)

cos φ=

Vnmo(0)√

1 − p2 V 2

P0

, (1)

where p = sin φ/VP0 is the ray parameter of the zero-offset ray; note that p can be determined from timeslopes on the zero-offset or stacked section. In somecases (e.g., for a bending layer), it may be possible todirectly estimate the zero-dip NMO velocity given by

Vnmo(0) = VP0

√1 + 2δ . (2)

Then equation 1 can be used to find the vertical velocityVP0, which can be substituted into equation 2 to obtainδ. Still, the parameter ǫ even in this simple model re-mains unconstrained by the P-wave NMO velocity fromthe bottom of the layer. Grechka et al. (2001) demon-strate on physical-modeling data that for relatively sim-ple models it may be possible to estimate ǫ using theNMO velocities for reflections from deeper interfaces.

Additional information about the anisotropy pa-rameters is provided by nonhyperbolic moveout. If thesymmetry axis is orthogonal to the bottom of a TTIlayer, the P-wave quartic moveout coefficient on the dipline of the reflector is proportional to the anellipticityparameter η ≡ (ǫ−δ)/(1+2δ) (Pech et al., 2003). There-fore, if δ has been found by inverting NMO velocities,nonhyperbolic moveout can be used to constrain ǫ.

These parameter-estimation issues have serious im-plications for migration velocity analysis (MVA) in TTImedia discussed here. An efficient MVA method for TImedia with a vertical symmetry axis (VTI) media wasintroduced by Sarkar and Tsvankin (2004) who dividedthe model into factorized VTI blocks. A medium is

called factorized if all ratios of the stiffness elements cij

are constant, which implies that the anisotropy param-eters are constant as well. The reference velocity (in ourcase, VP0) in factorized models represents an arbitraryfunction of the spatial coordinates (Cerveny, 1989). Fac-torized transverse isotropy with a linear velocity func-tion VP0(x, z) is the simplest model that accounts forboth anisotropy and heterogeneity.

Since anisotropy parameters are usually obtainedwith a relatively low spatial resolution, neglecting theirvariation within factorized blocks does not impair thequality of the velocity model. Sarkar and Tsvankin(2003, 2004, 2006) demonstrate on synthetic and fielddata that if the velocity VP0 is known at a single pointin each factorized block, the MVA algorithm accuratelyestimates the parameters ǫ and δ, as well as the vertical(kz) and lateral (kx) gradients of VP0. It should be em-phasized that stable recovery of the parameter ǫ (or η)requires using either long-spread data (with the max-imum offset-to-depth ratio of at least two) or dippingevents.

Here, we extend the MVA algorithm of Sarkar andTsvankin (2004) to tilted transversely isotropic mediawith the symmetry axis confined to the vertical in-cidence plane. The model is divided into TTI blockswith constant values of ǫ and δ and a linearly vary-ing symmetry-direction velocity VP0. These blocks, how-ever, are not strictly factorized because the symmetryaxis is kept orthogonal to reflectors, which may have ar-bitrary shape. As a result, the symmetry-axis directionand, therefore, the ratios of the stiffnesses cij may varywithin each block. (Note that the stiffness ratios remainconstant in the rotated coordinate system tied to thesymmetry axis.)

We begin by introducing the methodology of Kirch-hoff prestack depth migration and migration velocityanalysis for TTI media. In particular, we describe themodifications needed to account for tilt in ray trac-ing and MVA-based parameter estimation. Then themethod is tested on several typical TTI models that in-clude dipping anisotropic layers. The velocity-analysisand imaging results are compared with those obtainedby VTI algorithms to illustrate the need to account fortilt in anisotropic imaging. We also study the influenceof spreadlength on the errors in the medium parame-ters and show that parameter ǫ cannot be constrainedwithout including residual moveout for large (reachingat least two) offset-to-depth ratios.

2 METHODOLOGY

The migration velocity analysis algorithm of Sarkar andTsvankin (2004), designed for piecewise-factorized VTImedia, includes the same two main steps as conventionalMVA in isotropic media (e.g., Liu and Bleistein, 1995).The first step is Kirchhoff depth migration with a trialvelocity model that creates an image of the subsurface.

Imaging for tilted TI media 35

After picking reflectors on the migrated section, sem-blance scanning is used to evaluate the residual move-out of reflection events on image gathers (often called“common image gathers” in offset domain). The secondstep is a linearized parameter update designed to min-imize the residual moveout after the next applicationof migration. The two steps are iterated until events inimage gathers are sufficiently flat.

In order to estimate the vertical gradient of VP0,Sarkar and Tsvankin (2004) use two reflectors locatedat different depths in each factorized block. Also, to con-strain the parameter η (and, therefore, ǫ), the residualmoveout in image gathers is described by the followingnonhyperbolic equation:

z2

M(h) = z2

M(0) + r1h

2 + r2

h4

h2 + z2

M(0)

, (3)

where zM

is the migrated depth and h is the half-offset.The parameters r1 and r2, which quantify the magni-tude of residual moveout, are estimated by a 2D sem-blance scan.

To make the modeling and migration algorithms ofSarkar and Tsvankin (2004) suitable for a medium com-posed of TTI blocks or layers, we used ray-tracing soft-ware that can handle an arbitrary tilt of the symmetryaxis (Seismic Unix codes “unif2aniso” and “sukdsyn2d”;see Alkhalifah, 1995). The model is assumed to be 2D,with the vertical incidence plane containing the sym-metry axis and, therefore, the reflector normals. At theparameter-estimation step, we keep the symmetry axisorthogonal to the reflectors picked on the trial imageand update the parameters ǫ and δ and the gradientskz and kx of the symmetry-direction velocity VP0 (ǫ,δ, kz and kx are constant within each block). As in theVTI case, the velocity VP0 either has to be known at onepoint in each factorized block or assumed to be contin-uous in the vertical direction. The MVA is applied in alayer-stripping mode starting at the top of the model,with at least two reflectors at different depths analyzedfor each factorized block.

For TTI media with a positive vertical gradientin VP0, reflections from steeply dipping interfaces of-ten arrive at the surface as turning rays (Tsvankin,1997; 2005). Hence, our algorithm properly accounts forturning-ray reflections in the computation of the trav-eltime field used by the migration operator.

3 TESTS ON SYNTHETIC DATA

Here, we generate synthetic seismograms and test ourMVA/imaging algorithm on three common geologicalmodels that often include TTI layers: a syncline, a saltdome flanked by uptilted shale layers, and a bendingshale layer. To conform to our model assumptions, thesymmetry axis is kept orthogonal to the bottom of theTTI layers.

0

2

4

Dep

th (

km)

0 2 4 6Distance (km)

Figure 1. Model with a TTI syncline sandwiched betweentwo isotropic layers. The bold lines mark the layer bound-aries; the additional reflectors used in MVA are shown by thethinner lines. The parameters of the TTI layer are VP0 = 2.3km/s, kz = 0.6 s−1, kx = 0.1 s−1, ǫ = 0.1, and δ = −0.1(η = 0.25). The symmetry axis (marked by the arrows) isorthogonal to the layer’s bottom; the dips are 30◦. The toplayer has VP0 = 1.5 km/s, kz = 1.0 s−1, kx = ǫ = δ = 0;for the bottom layer, VP0 = 2.7 km/s, kz = 0.3 s−1, andkx = ǫ = δ = 0. The velocity VP0 is specified at the top ofeach layer at the 1 km coordinate.

3.1 Syncline model

The first model includes a TTI syncline with dips of 30◦

sandwiched between two isotropic media (Figure 1). Theisotropic layers are vertically heterogeneous but have nolateral velocity gradient, while the TTI layer is bothvertically and laterally heterogeneous. As required bythe MVA algorithm, each layer contains two reflectinginterfaces, with every second reflector representing theboundary between layers. Synthetic data generated byanisotropic ray tracing (see above) consist of 260 shotgathers with shot and receiver intervals of 25 m and 40traces per gather.

The section obtained after anisotropic prestackdepth migration using the ray-tracing algorithm men-tioned above with the true model parameters for alllayers is shown in Figure 2a. The influence of the tiltof the symmetry axis in the TTI layer is taken into ac-count during the computation of the traveltime tableby anisotropic ray tracing. Then the same traveltimetable is used for Kirchhoff prestack depth migration. Asexpected, all reflectors are well focused and accuratelypositioned, and all image gathers in Figure 2b are per-fectly flat.

3.1.1 TTI velocity analysis and migration

We applied our MVA algorithm for TTI media to 10image gathers located at horizontal coordinates rangingfrom 1.5 km to 5.0 km (Figure 3). Since it is essential to

36 L. Behera & I. Tsvankin

0

2

4

Dep

th (

km)

0 2 4 6(a)

(b)

Distance (km)

0

2

4

Dep

th (

km)

0 100 200Midpoint

Figure 2. (a) True image of the model from Figure 1 ob-

tained by anisotropic prestack depth migration with the cor-rect medium parameters. (b) The corresponding image gath-ers in the offset domain at 0.4 km intervals (displayed withthe corresponding midpoints along the x-axis) for each re-flector.

use nonhyperbolic moveout in parameter estimation forTTI media (see the introduction), the maximum offset-to-depth ratio for the bottom of the TTI layer is closeto two. The medium parameters are estimated in thelayer-stripping fashion starting at the surface.

The initial velocity model used in the first iterationof MVA is homogeneous and isotropic. For the first (top)layer, the velocity VP0 is assumed to be known at a sin-gle surface location [VP0(x = 1 km, z = 0) = 1500 m/s].This assumption is needed only if this layer is treatedas anisotropic. We also assign the correct value to thesymmetry-direction velocity VP0 at the top of the sec-ond and third layers. As discussed above, the symmetryaxis in the second layer is kept orthogonal to the layer’sbottom.

The inverted parameters are close to the true valuesin all three layers, and the migrated image (Figure 3a)is practically indistinguishable from the benchmark sec-tion in Figure 2. According to the analysis in the intro-duction, the NMO velocities from the horizontal anddipping reflectors in the TTI layer constrain the param-eters VP0 and δ (see equations 1 and 2). Although this

result is obtained for a homogeneous TTI medium, itshould remain valid for our heterogeneous model be-cause we estimate the velocity gradients by using imagegathers at different depths and lateral positions. Also,we make the parameter estimation more stable by as-suming that the velocity VP0 is known at the top ofthe layer. To resolve ǫ, it is necessary to use nonhyper-bolic moveout on long spreads, which is controlled bythe anellipticity parameter η ≈ ǫ− δ. Evidently, the rel-atively large offset-to-depth ratios (up to two) we usedin MVA are sufficient to provide a tight constraint onǫ. The influence of spreadlength on the stability of pa-rameter estimation is analyzed in more detail below.

We also computed the error bars for each parameterby setting the standard deviation in the picked migrateddepths on selected image gathers to ± 5 m. The pickingerrors are then substituted in the inversion operator tofind the corresponding standard deviations of the modelparameters (Sarkar and Tsvankin, 2004). This proce-dure yields relatively small errors of up to ±0.02 in allestimated velocity gradients and anisotropy parameters.

The improvements achieved by the MVA algorithmin reducing the residual moveout in image gathers is il-lustrated in Figures 3b-d. After six iterations of MVAusing the first two reflectors, image gathers of eventsin the first layer are flat, but there is substantial resid-ual moveout in the two deeper layers (Figure 3c). Uponcompletion of the parameter estimation for all three lay-ers, image gathers are flat throughout the model (Fig-ure 3d).

3.1.2 Sensitivity to spreadlength

To quantify the dependence of errors in the mediumparameters on the maximum offset-to-depth ratio, werepeated MVA for the TTI syncline model with a rangeof spreadlengths. As before, the errors were computedby the algorithm of Sarkar and Tsvankin (2004) for a±5 m picking error in migrated depths. The increasein the offset-to-depth ratio (computed for the midpointof the dipping reflectors) makes estimation of all pa-rameters much more stable (Figure 4). The parameterskz, kx, and δ are constrained by NMO velocity, whichcan be measured on conventional-length spreads closeto the reflector depth. For that reason, the error curvesfor kz, kx, and δ show a similar trend and flatten outfor offset-to-depth ratios between unity and 1.5.

In contrast, the error in the parameter ǫ continuesto decrease until the maximum offset reaches two timesthe reflector depth, which is in good agreement with theanalytic results discussed above. Indeed, ǫ in our modelis determined from the quartic moveout coefficient (i.e.,from nonhyperbolic moveout), which is poorly resolvedif the maximum offset-to-depth ratio is smaller than two(Tsvankin, 2005). Thus, the results in Figure 4 suggestthat for stable estimation of all relevant parameters of

Imaging for tilted TI media 37

0

2

4

Dep

th (

km)

0 2 4 6(a) (b)

(c) (d)

Distance (km)0 100 200

Midpoint

0

2

4

Dep

th (

km)

0 100 200Midpoint

0 100 200Midpoint

Figure 3. (a) Final image of the syncline model obtained after MVA and prestack depth migration for TTI media. Theestimated parameters of the first (subsurface) layer are kz = 0.99 s−1 and kx = ǫ = δ = 0. For the second layer, kz = 0.59 s−1,kx = 0.09 s−1, ǫ = 0.09, and δ = −0.11. For the third layer, kz = 0.29 s−1 and kx = ǫ = δ = 0. The error for each parametervaries from ±0.01 to ±0.02, if the depth picking error is assumed to be ±5 m. Image gathers obtained (b) with the initial modelparameters before MVA; (c) after six iterations of MVA for the first two layers; (d) after MVA for all three layers.

TTI media, it is highly desirable to employ offsets twiceas large (or more) as the reflector depth.

3.1.3 Influence of noise

To assess the stability of our MVA and migration algo-rithms, we added random uncorrelated Gaussian noiseto the synthetic data set for the model from Figure 1.The signal-to-noise ratio (S/N), measured as the ratioof the peak signal amplitude to the root-mean-square(rms) amplitude of the background noise, is close to two;the frequency bands of the signal and noise are identical(Figure 5a). The semblance maxima for most events inthe presence of noise become less focused (Figure 5b),which enhances the tradeoff between the moveout pa-

rameters r1 and r2 in equation 3 (Tsvankin, 2005). How-ever, since any pair of values (r1, r2) within the inner-most semblance contour provides nearly the same vari-ance of migrated depths, this tradeoff does not hamperthe convergence of the MVA algorithm.

On the whole, despite the low S/N ratio, randomnoise does not significantly distort the MVA results(only the error in ǫ is non-negligible). Although the im-aged reflectors are not as well focused as those on thenoise-free section, they are clearly visible and correctlypositioned (Figure 5c).

38 L. Behera & I. Tsvankin

0.00

0.05

0.10

0.15

0.20

0.25

Err

or in

kz

(s−

1 )

0.8 1.2 1.6 2.0 2.40.00

0.02

0.04

0.06

0.08

0.10

Err

or in

kx

(s−

1 )

0.8 1.2 1.6 2.0 2.4

0.00

0.02

0.04

0.06

0.08

0.10

Err

or in

ε

0.8 1.2 1.6 2.0 2.4

Offset/depth

0.00

0.02

0.04

0.06

0.08

0.10

Err

or in

δ

0.8 1.2 1.6 2.0 2.4

Offset/depth

Figure 4. Influence of the maximum offset-to-depth ratio on the absolute errors in the parameters kz, kx, ǫ, and δ. The errorsare estimated from the MVA results for the left part of the syncline model in Figure 1.

3.1.4 Errors of VTI-based processing

Since most anisotropic imaging algorithms used in in-dustry are designed for VTI media, it is important toevaluate the influence of the tilted symmetry axis onthe quality of the migrated image. The section in Fig-ure 6 is obtained by setting the tilt in the synclineto zero, which makes the second layer VTI. Althoughthe velocity model includes the correct values of ǫ andδ, the dipping reflectors in the TTI layer are stronglymisfocused and somewhat shifted spatially. The imagegathers for these dipping interfaces exhibit significantresidual moveout (Figure 6b), which indicates that thevelocity field is highly inaccurate.

To emulate a complete VTI processing sequence ap-plied to this model, we repeated migration velocity anal-ysis, but without allowance for a tilted symmetry axis(i.e., the MVA code treated the second layer as VTI).After several iterations of parameter updating, the im-age gathers are largely flattened, and the image qualityis only marginally inferior to that achieved for the true

model (Figure 7). The parameters kz, ǫ, and δ of thesecond layer, however, are distorted. These errors areintroduced by the MVA algorithm, which has to flattenimage gathers in the TTI layer with the incorrect tilt ofthe symmetry axis. It is interesting to note that despitethe distortions in ǫ and δ, the best-fit VTI model hasan accurate value of the anellipticity parameter η.

The ability of the VTI-based algorithm to compen-sate for the influence of tilt decreases for a larger relativethickness of the TTI syncline (Figure 8). Because of themore significant contribution of the interval traveltimein the TTI layer, the dipping reflectors in Figure 8 aremisfocused and shifted in the vertical direction. Suchartifacts generated by VTI imaging can serve as an in-dication that the medium immediately above the dis-torted reflectors may have a tilted symmetry axis. Thequality of the image produced by VTI processing alsodecreases for strongly anisotropic TTI models and largermagnitudes of the parameters ǫ, δ, and η.

Imaging for tilted TI media 39

0

2

4

Tim

e (s

)

0 500 1000(a) (b)

(c)

r

r

Offset (m)

-0.6

-0.4

-0.2

0

0.2

0.4

-0.4 -0.2 0 0.2 0.4

*

0

0.2

0.4

0.6

0.8

2

1

0

2

4

Dep

th (

km)

0 2 4 6Distance (km)

Figure 5. Influence of Gaussian noise on MVA and migration for the model from Figure 1. (a) One of the noise-contaminatedshot gathers (the lateral coordinate is close to 2 km). (b) The semblance scan for the bottom of the TTI layer (the lateralcoordinate is 1.9 km) computed as a function of the moveout coefficients r1 and r2 (equation 3). The maximum semblance ismarked by the star. (c) The image obtained for the noise-contaminated data set. The estimated parameters of the first layerare kz = 1.06 s−1, kx = 0.01 s−1, ǫ = −0.01, and δ = 0. For the second layer, kz = 0.56 s−1, kx = 0.11 s−1, ǫ = 0.14, andδ = −0.08. For the third layer, kz = 0.35 s−1, kx = 0.01 s−1, ǫ = 0.02, and δ = −0.02. The errors for each parameter vary from±0.03 to ±0.05 under the assumption that the picking error for the noisy data is ±20 m.

3.2 Salt-dome model

The next test is performed for a simplified salt model,which can be considered typical for subsalt explorationplays. The model includes an isotropic salt dome withsteep flanks overlaid by a TI shale formation. The sym-metry axis in the shale is vertical directly above thedome and tilted (orthogonal to the bedding) in the dip-ping layers on both sides of the salt body (Figure 9).TTI migration, applied in the same way as for the syn-

cline model using the correct medium parameters, pro-duces a sharp, accurate image (Figure 10a) with flatimage gathers even at the steep flanks of the salt (Fig-ure 10b).

3.2.1 Processing using TTI and VTI models

When tilt is properly taken into account by the MVA al-gorithm, both the dipping reflectors and the salt dome

40 L. Behera & I. Tsvankin

0

2

4

Dep

th (

km)

0 2 4 6(a)

(b)

Distance (km)

0

2

4

Dep

th (

km)

0 100 200Midpoint

Figure 6. (a) Image of the model from Figure 1 obtained

without taking the symmetry-axis tilt in the second layerinto account; the rest of the model parameters are correct.(b) The corresponding image gathers.

are well focused and properly positioned (Figure 11).As before, the parameters are estimated in the layer-stripping mode using the correct values of the velocityVP0 at the top of each layer. For purposes of MVA, theshale formation was divided into two blocks along thevertical axis of the salt dome. Errors in both anisotropyparameters and velocity gradients are relatively small,although ǫ is not as well constrained as δ. The largererror in ǫ is expected because, as discussed above,this parameter does not influence NMO velocity andis constrained only by nonhyperbolic moveout on longspreads.

Figure 12 shows the processing results obtained fora velocity model that does not include tilt and treatsthe shale as VTI (the parameters ǫ, δ, kz, and kx arecorrect). The substantial residual moveout in the im-age gathers for the dipping reflectors and pronouncedimage distortions indicate that the influence of tilt forthis model is more significant than that for the synclinemodel discussed above.

Similar to the previous example, the VTI imag-ing result can be improved by deriving the best-fit VTImodel from migration velocity analysis (Figure 13). Al-

0

2

4

Dep

th (

km)

0 2 4 6(a)

(b)

Distance (km)

0

2

4

Dep

th (

km)

0 100 200Midpoint

Figure 7. (a) Image of the model from Figure 1 obtained

after applying MVA under the assumption that the secondlayer is VTI. The estimated parameters of the second layerused in the migration are kz = 0.53 s−1, kx = 0.12 s−1,ǫ = 0.15, and δ = −0.06 (η = 0.24). (b) The correspondingimage gathers.

0

2

4

Dep

th (

km)

0 2 4 6Distance (km)

Figure 8. Same as Figure 7a (i.e., the image obtained for thebest-fit VTI model), but the thickness of the TTI layer fromFigure 1 is increased by 200 m. The estimated parametersare kz = 0.52 s−1, kx = 0.12 s−1, ǫ = 0.13, and δ = −0.08(η = 0.26).

Imaging for tilted TI media 41

0

2

4

Dep

th (

km)

0 2 4 6Distance (km)

Isotropic

Shale Shale

Salt

Figure 9. Simplified salt model that includes a salt domeoverlaid by a TI shale formation. The symmetry axis in theshale (marked by the arrows) is vertical on top of the salt andorthogonal to the bedding in the uptilted layers, which aredipping at 30◦. The parameters of the shale are VP0 = 2.6km/s, kz = 0.6 s−1, kx = 0.2 s−1, ǫ = 0.3, and δ = 0.15. Thesubsurface horizontal layer is isotropic with VP0 = 1.5 km/s,kz = 1.0 s−1, and kx = ǫ = δ = 0; for the salt dome,VP0 = 4.5 km/s, kz = kx = 0.1 s−1, ǫ = δ = 0.

0

2

4

Dep

th (

km)

0 2 4 6(a)

(b)

Distance (km)

0

2

4

Dep

th (

km)

0 100 200Midpoint

Figure 10. (a) True image of the model from Figure 9 ob-tained by anisotropic prestack depth migration with the cor-rect parameters. (b) The corresponding image gathers.

0

2

4

Dep

th (

km)

0 2 4 6(a)

(b)

Distance (km)

0

2

4

Dep

th (

km)

0 100 200Midpoint

Figure 11. (a) Final image of the salt model obtained after

MVA and prestack depth migration for TTI media. The esti-mated parameters of the first subsurface layer are kz = 0.97s−1 and kx = ǫ = δ = 0. For the shale layer to the leftof the salt, kz = 0.58 s−1, kx = 0.19 s−1, ǫ = 0.34, andδ = 0.14. To the right of the salt, kz = 0.59 s−1, kx = 0.18s−1, ǫ = 0.32, and δ = 0.15. For the salt, kz = 0.09 s−1,kx = 0.1 s−1, and ǫ = δ = 0. The parameters of the left andright blocks below the shale layer on either side of salt domeare kz = 0.11, 0.10 s−1, kx = 0.09, 0.12 s−1, ǫ = δ ≈ 0.0respectively. The error for each parameter varies from ±0.01to ±0.03, if the depth picking error is assumed to be ±5 m.(b) The corresponding image gathers.

though the reflectors inside and below the shale for-mation are better focused than those in Figure 12, theflanks of the salt body are somewhat shifted laterallyand look blurry (Figure 13a). Also, the image gathers,especially those near the salt, are not completely flat-tened (Figure 13b). Note that the improved image of thedipping reflectors in Figure 13a is achieved by distort-ing the parameter δ and, to a lesser extent, the velocitygradients in the shale. Clearly, ignoring the tilt of thesymmetry axis in dipping TI layers in the overburdenmay cause serious problems in imaging salt bodies and,therefore, subsalt reservoirs.

42 L. Behera & I. Tsvankin

0

2

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Dep

th (

km)

0 2 4 6(a)

(b)

Distance (km)

0

2

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Dep

th (

km)

0 100 200Midpoint

Figure 12. (a) Image of the model from Figure 9 obtained

without taking the symmetry-axis tilt into account; the restof the model parameters are correct. (b) The correspondingimage gathers.

3.3 Bending-layer model

Complex tectonic processes in fold-and-thrust belts,such as the Canadian Rocky Mountain Foothills, some-times produce bending shale layers with variable dip.Here, we process synthetic data generated for a TTIthrust sheet (Figure 14) fashioned after the physicalmodel of Leslie and Lawton (1996). This physical-modeling data set was used by Grechka et al. (2001)for anisotropic parameter estimation. The algorithm ofGrechka et al. (2001), however, operates only with NMOvelocities measured on conventional spreads and relieson several simplifying assumptions about the model.

The benchmark image computed for the true modelis shown in Figure 15a. Apart from relatively low am-plitudes of steeply dipping interfaces due to insufficientaperture, the reflectors are well focused and positioned,and all image gathers in Figure 15b are flat. To apply theMVA algorithm, we divided the thrust sheet into fourblocks with different dips and carried out the parame-ter estimation separately for each block. As was the casefor the previous two models, the TTI algorithm yieldsnot only accurate parameter estimates, but also a high-quality image (Figure 16). If the medium parameters

0

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0 2 4 6(a)

(b)

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0

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Figure 13. (a) Image of the model from Figure 9 obtained

after applying MVA under the assumption that the shaleformation is VTI. The estimated parameters for the shalelayer to the left of the salt are kz = 0.62 s−1, kx = 0.17 s−1,ǫ = 0.32, and δ = 0.12. To the right of the salt, kz = 0.62 s−1,kx = 0.18 s−1, ǫ = 0.30, and δ = 0.11. (b) The correspondingimage gathers.

(except for the tilt) are assumed to be the same in theblocks with different dips, there is no need to specifythe symmetry-direction velocity VP0 in the TTI layer.

When MVA does not take tilt into account, allboundaries in the thrust sheet are poorly focused, withnoticeable artifacts at the points where the interfaceschange dip (Figure 17a). It is interesting that the errorsin the medium parameters produced by the VTI algo-rithm are relatively minor. Apparently, image gathersfor the thrust sheet (Figure 17b) could not be flattenedby distorting the anisotropy parameters or velocity gra-dients, as long as the symmetry axis is vertical.

4 DISCUSSION AND CONCLUSIONS

The combination of tilted transverse isotropy and struc-tural complexity in many important exploration playsmakes it imperative to apply advanced migration veloc-ity analysis (MVA) methods and prestack depth imag-ing. Here, we presented an MVA methodology for P-

Imaging for tilted TI media 43

0

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th (

km)

0 3 6 9Distance (km)

650

550

300

Figure 14. TTI thrust sheet with variable dip for differ-ent blocks (0◦, 30◦, 55◦, and 65◦) and the symmetry axis(marked by the arrows) orthogonal to the boundaries. Ex-cept for the symmetry-axis direction, the parameters of allTTI blocks are the same: VP0 = 2.3 km/s, kz = 0.6 s−1,kx = 0.1 s−1, ǫ = 0.1, and δ = −0.1. The subsurface layeris thin (60 m) and has VP0 = 1.5 km/s, kz = 1.0 s−1, andkx = ǫ = δ = 0 (i.e., it represents the weathering layer); thevelocity VP0 across the bottom of the layer is continuous.The medium around the TTI sheet is isotropic with kz = 1.0s−1, and kx = ǫ = δ = 0. The horizontal layer at the bot-tom of the model has VP0 = 3.5 km/s, kz = 0.3 s−1, andkx = ǫ = δ = 0.

waves in heterogeneous TTI media based on dividing themodel into “quasi-factorized” blocks. The anisotropyparameters ǫ and δ in each block are constant, whilethe symmetry-direction velocity VP0 represents a linearfunction of the spatial coordinates and is described bythe vertical gradient kz and lateral gradient kx. To re-duce the uncertainty in parameter estimation, the sym-metry axis in each block or layer is taken to be orthog-onal to the reflector at the bottom of the block. Sincereflectors may have arbitrary shape, the symmetry-axisorientation generally varies in space, which means thatblocks are not fully factorized. (In factorized TI media,the symmetry-axis direction is fixed.)

Our algorithm represents an extension to TTI me-dia of the MVA methodology developed by Sarkar andTsvankin for vertical transverse isotropy. MVA is com-bined with Kirchhoff prestack depth migration basedon anisotropic ray tracing for heterogeneous TI mediawith arbitrary tilt. Parameter estimation is performedin the layer-stripping mode starting at the surface, withthe symmetry-direction velocity VP0 either specified ata single point in each block or assumed to be continuousin the vertical direction. To estimate the vertical gradi-ent kz, we use image gathers for at least two reflectorsat different depths within each block.

If the velocity VP0 is known, the parameter δ inTTI media with the symmetry axis orthogonal to the re-flector can be constrained using the NMO velocities for

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Figure 15. (a) True image of the model from Figure 14

obtained by anisotropic prestack depth migration with thecorrect parameters. (b) The corresponding image gathers.

either horizontal or dipping events. In contrast to VTImedia, where the parameters η and ǫ (with known VP0)for some models can be found from dip-dependent NMOvelocity, estimation of these parameters in TTI mediagenerally requires using nonhyperbolic moveout. In ourMVA algorithm, the residual moveout in long-spreadimage gathers is evaluated using the two-parameter non-hyperbolic equation described by Sarkar and Tsvankin.We found that sufficiently stable estimation of the pa-rameter ǫ in TTI media with a fixed symmetry-axisorientation can be achieved if the offset-to-depth ratioreaches at least two.

The MVA and migration algorithms were testedon several typical TTI models including a syncline, asalt dome with dipping TTI layers on both sides, anda bending TTI layer (thrust sheet). For all three mod-els we were able to accurately reconstruct the velocitygradients kz and kx throughout the medium and theanisotropy parameters ǫ and δ in the TTI blocks. Themigrated sections computed with the estimated veloc-ity model are practically indistinguishable from the trueimages, with good focusing and positioning of reflectorsbeneath the TTI formations.

To assess the influence of tilt on image quality, we

44 L. Behera & I. Tsvankin

0

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(b)

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Figure 16. (a) Final image of the thrust model obtained

after MVA and prestack depth migration for TTI media. Theparameters were estimated for each block of the thrust sheet(i.e., for each dip) separately. For the horizontal TTI block(dip=0◦), kz = 0.62 s−1, kx = 0.11 s−1, ǫ = 0.11, andδ = −0.09; for the 30◦ dip, kz = 0.59 s−1, kx = 0.12 s−1,ǫ = 0.09, and δ = −0.11; for the 55◦ dip, kz = 0.58 s−1,kx = 0.09 s−1, ǫ = 0.11, and δ = −0.08; for the 65◦ dip,kz = 0.62 s−1, kx = 0.12 s−1, ǫ = 0.09, and δ = −0.1. Forthe horizontal layer beneath the TTI sheet, kz = 0.29 s−1,kx = ǫ = 0, and δ = 0.01. The error for each parameter variesfrom ±0.01 to ±0.03, if the depth picking error is assumedto be ±5 m. (b) The corresponding image gathers.

migrated the data with the VTI model (i.e., with zerotilt) that has the correct values of VP0, kz, kx, ǫ andδ. Although the tilt in our first two models (the syn-cline and salt dome) is moderate (30◦), setting it to zeroresults in significant misfocusing and mispositioning ofreflectors. The inaccuracy of the VTI velocity field alsomanifests itself through substantial residual moveout inimage gathers.

In order to emulate a complete VTI processing se-quence applied to TTI media, we per formed MVA with-out allowance for a tilted symmetry axis to obtain the“best-fit” VTI model. The MVA algorithm can achievepartial flattening of image gathers with the incorrecttilt, but at the expense of distorting the medium param-eters, especially ǫ and δ (although the value of η remains

0

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Figure 17. (a) Image of the model from Figure 14 ob-

tained after applying MVA under the assumption that thethrust sheet is VTI. The parameters of the horizontal block(dip=0◦) are kz = 0.59 s−1, kx = 0.09 s−1, ǫ = 0.09, andδ = −0.11; for the 30◦ dip, kz = 0.62 s−1, kx = 0.11 s−1,ǫ = 0.11, and δ = −0.09; for the 55◦ dip, kz = 0.61 s−1,kx = 0.1 s−1, ǫ = 0.13, and δ = −0.12; and for the 65◦ dip,kz = 0.63 s−1, kx = 0.13 s−1, ǫ = 0.11, and δ = −0.08. (b)The corresponding image gathers.

accurate). Such artificial adjustments in ǫ and δ improveimage quality, although migrated sections typically areinferior to those generated with the TTI model. Also,the ability of the VTI-based algorithm to compensatefor the influence of tilt decreases for more complicatedmodels and TTI layers with relatively large thickness orstrong anisotropy.

On the whole, the MVA methodology introducedhere provides a practical tool for building TTI velocitymodels with minimal a priori information. Combinedwith prestack depth migration, this MVA algorithm canbe efficiently used to image targets beneath TTI forma-tions in structurally complex environments.

5 ACKNOWLEDGMENTS

We are grateful to the A(nisotropy)-Team of the Cen-ter for Wave Phenomena (CWP), Colorado School of

Imaging for tilted TI media 45

Mines (CSM), and other colleagues at CWP for fruit-ful discussions. L. B. thanks the Department of Scienceand Technology (DST), Govt. of India, for awarding himBOYSCAST Fellowship and V. P. Dimri, Director ofthe National Geophysical Research Institute (NGRI),Council of Scientific and Industrial Research (CSIR),for granting him permission to pursue postdoctoral re-search in CWP. This work was partially supported bythe Consortium Project on Seismic Inverse Methods forComplex Structures at CWP.

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