GEOPHYSICS, VOL. 62, NO. 6 (NOVEMBER-DECEMBER 1997); P. 1839-1854,18 FIGS.

Velocity analysis using nonhyperbolic moveoutin transversely isotropic media

Tariq Alkhalifah*

ABSTRACT

P-wave reflections from horizontal interfaces in trans-versely isotropic (TI) media have nonhyperbolic move-out. It has been shown that such moveout as well as alltime-related processing in TI media with a vertical sym-metry axis (VTI media) depends on only two parameters,Vnmo and 17. These two parameters can be estimated fromthe dip-moveout behavior of P-wave surface seismicdata.

Alternatively, one could use the nonhyperbolic move-out for parameter estimation. The quality of resultingestimates depends largely on the departure of the move-out from hyperbolic and its sensitivity to the estimatedparameters. The size of the nonhyperbolic moveout inTI media is dependent primarily on the anisotropy pa-rameter 17. An "effective" version of this parameter pro-vides a useful measure of the nonhyperbolic moveouteven in v(z) isotropic media. Moreover, effective 17, rlea ,is used to show that the nonhyperbolic moveout asso-

INTRODUCTION

Often, the earth's subsurface is dominated by interfaces thatare horizontal or subhorizontal. Therefore, reflections fromsuch interfaces are the primary source of information in muchseismic data. Moveout from horizontal and subhorizontal re-flectors, for example, provides useful velocity information.Fortunately, reflection moveouts from horizontal interfacesgenerally are well represented by truncated Taylor series-typecharacterizations of moveout in transversely isotropic (TI) me-dia with a vertical symmetry axis (VTI) (Hake et al., 1984;Tsvankin and Thomsen, 1994). These representations are ac-curate to and beyond the large offsets often used in practice.

Alkhalifah and Tsvankin (1995) demonstrated that, for TImedia with vertical symmetry axis (VTI media), just two

ciated with typical TI media (e.g., shales, with 17 ^_ 0.1)is larger than that associated with typical v(z) isotropicmedia. The departure of the moveout from hyperbolicis increased when typical anisotropy is combined withvertical heterogeneity.

Larger offset-to-depth ratios (X/D) provide morenonhyperbolic information and, therefore, increased sta-bility and resolution in the inversion for 1leff. The X/Dvalues (e.g., X/D > 1.5) needed for obtaining stabilityand resolution are within conventional acquisition limits,especially for shallow targets.

Although estimation of 11 using nonhyperbolic move-outs is not as stable as using the dip-moveout method ofAlkhalifah and Tsvankin, particularly in the absence oflarge offsets, it does offer some flexibility. It can be ap-plied in the absence of dipping reflectors and also maybe used to estimate lateral 17 variations. Application ofthe nonhyperbolic inversion to data from offshore Africademonstrates its usefulness, especially in estimating lat-eral and vertical variations in rl.

parameters are sufficient for performing all time-related pro-cessing, such as normal moveout (NMO) correction (includingnonhyperbolic moveout correction, if necessary), dip-moveout(DMO) correction, and prestack and poststack time migra-tions. Taking Vh to be the P-wave velocity in the horizontaldirection, their two anisotropy parameters are

o.s( Vh E — 8

Vmo -1 - 1+26' (1)

and the short-spread NMO velocity for a horizontal reflectoris

Vnmo = Vpo 1 + 28. (2)

Manuscript received by the Editor August 29, 1996; revised manuscript received January 13, 1997.*Formerly Center for Wave Phenomena, Colorado School of Mines, Golden, Colorado 80401; presently Department of Geophysics, StanfordUniversity, Stanford, California 94305-2215. E-mail: tariq@sep.stanford.edu.© 1997 Society of Exploration Geophysicists. All rights reserved.

1839

Dow

nloa

ded

10/2

5/15

to 1

34.7

.152

.117

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

1840

Alkhalifah

Here, Vp is the P-wave vertical velocity and E and S are twoof Thomsen's (1986) dimensionless anisotropy parameters.

Moreover, Alkhalifah and Tsvankin (1995) showed that theparameters 17 and Vnmo are obtainable solely from surface seis-mic P-wave data, specifically from estimates of stacking veloc-ity for reflections from interfaces having two distinct dips (theDMO method). The two-parameter representation and inver-sion also hold in v(z) media. Alkhalifah (1997) used the DMOinversion method to invert for vertical variations in n. How-ever, the DMO inversion in Alkhalifah and Tsvankin (1995)and Alkhalifah (1997) works only when reflectors with at leasttwo distinct dips (e.g., a fault and a gently dipping reflector)are present, as long as one of the dips is not close to 90°.

Hake et al. (1984) derived the three-term Taylor series ex-pansion of the reflection moveout from horizontal reflectorsin VTI media. The presence of the third term in their expan-sion implies nonhyperbolic moveout. Tsvankin and Thomsen(1994) recast the three-term expansion more compactly as afunction of Thomsen's (1986) parameters. Moreover, usingan asymptotic fit, Tsvankin and Thomsen (1994) suggested acorrection factor that approximates the deleted higher-orderterms of the Taylor series expansion, thus stabilizing the move-out at long offsets. Tsvankin and Thomsen (1995) studied theproblem of inverting for Thomsen's (1986) anisotropy param-eters (Vp0, E, and S) using the nonhyperbolic moveout of re-flections from horizontal interfaces. They found that such aninversion using only P-wave data would be highly ill condi-tioned because of the trade-off between the vertical velocityand anisotropic coefficients, which cannot be overcome by us-ing even a spread length that is twice the depth. Their obser-vation is in agreement with the two-parameter dependency oftime-related processing (Alkhalifah and Tsvankin, 1995). Theyalso pointed out the ambiguity in resolving the second-order(A 2) and fourth-order (A 4 ) coefficients of the Taylor series ex-pansion using traveltime moveout for ray angles up to 45 0 .Although such ambiguity exists in the general sense of usingtraveltimes to invert for A2 and A4, it can be overcome some-what by first extracting A 2 , which is simply the reciprocal of theNMO velocity squared, from conventional velocity semblanceanalysis, and in turn using it in the inversion for A 4 or, in mycase, i1. In fact, as shown later, a 2-D semblance scan over bothparameters proves to be a reliable method. Neidell and Taner(1971) have stated the clear benefits of semblance analysis forparameter extraction.

Byun et al. (1989) applied a two-parameter velocity analysison synthetic vertical-seismic-profiling data using a "skewed"hyperbolic-moveout formula for horizontal reflectors. Al-though their velocity analysis approach showed promise, theirnonhyperbolic (or skewed hyperbolic) formula was a coarseapproximation of the actual moveout in TI media (Tsvankinand Thomsen, 1994). For example, their formula requiredknowledge of the vertical P- and S-wave velocities, whereasthe true moveout is very much independent of these two pa-rameters (Alkhalifah and Tsvankin, 1995).

Use of the deviation of moveout from a hyperbolic naturefor parameter estimation in general depends on the size ofthe deviation as well as on the sensitivity of the nonhyper-bolic moveout to the estimated parameters and the absenceof complicating factors, such as lateral velocity variation. Inthis paper, I compare the size of nonhyperbolic moveout forreflections from horizontal interfaces in VTI media with that

associated with typical vertically inhomogeneous isotropic me-dia. Then, I invert for estimates of r1 using the nonhyperbolicmoveout and discuss the sensitivity of the inversion to errors inthe measured parameters, namely, Vnmo and traveltime. I alsoapply semblance analyses over nonhyperbolic trajectories toestimate both Vnmo and . The study includes field-data appli-cations that illustrate the usefulness of this method.

NONHYPERBOLIC MOVEOUT IN LAYERED MEDIA

Hake et al. (1984) derived a three-term Taylor series expan-sion for the moveout of reflections from horizontal interfaces inhomogeneous VTI media. If one ignores the contribution of thevertical shear-wave velocity VSO, which is negligible (Tsvankinand Thomsen, 1994; Alkhalifah and Lamer, 1994; Tsvankin,1995; Alkhalifah, manuscript in revision), their equation canbe simplified, when expressed in terms of YI and Vnmo, to

z x2 21)X4( 3 )t2(x)=t+ 0 v2 t2 v 4 ( )nmo 0 nmo

Here, t is the total traveltime, to is the two-way zero-offsettraveltime, and X is the offset. The first two terms on the rightcorrespond to the hyperbolic portion of the moveout, whereasthe third term approximates the nonhyperbolic contribution.Note that the third term (fourth order in X) is proportional tothe anisotropy parameter r^, which therefore controls nonhy-perbolic moveout directly.

Tsvankin and Thomsen (1995) derived a correction factor forthe nonhyperbolic term of the Hake et al. (1984) equation thatincreases accuracy and stabilizes traveltime moveout at largeoffsets in VTI media. The more accurate moveout equation,when expressed in terms of and Vnmo , is

X z 277X4

72(X) = + mo t0 Vn no(1 + AX2)'(4)

where

277A=

tzV 4 \ 11nmo— 0 nmo v2 VZ

and the horizontal velocity Vh from equation (1) for homoge-neous media is

Vh = Vnmo 1 + 217.

Through simple manipulation, equation (4) reduces to(Alkhalifah and Tsvankin, 1995)

= 0x2 _ 21,X4

5 Z 2 r z z t2(x ) t + Vnmo VnmoLtO Vnmo +(1+277)X2 ] ( )

Equations (4) and (5) differ for v(z) media, for which V,, can bedefined in at least two different ways. Equation (5) is accuratefor large offset-to-depth ratios. Note that setting X = 0 in thedenominator of the fourth-order term reduces equation (5)to equation (3). The additional X factor in the denominatorproduces an expansion that approximates the influence of theterms (beyond the fourth order) that were omitted in the Taylorseries expansion and therefore increases the moveout accuracyat large offsets. The higher-order approximation is based on the

Dow

nloa

ded

10/2

5/15

to 1

34.7

.152

.117

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

OE

ICD02

V=3.0 km/s

V=2.0 km/s

&-0.1, E=0.1

V= 4.O km/s

5=0.15, E=0.2

ii

0.5

0

-0.5

0 -1

^ 44) 2E

0

-2

-4

3

Nonhyperbolic-Moveout Anisotropy Analysis

1841

fact that as X becomes very large (goes to infinity), while t o isfinite, equation (5) reduces to

t2 (x} _ x2Z . (6)h

Therefore, equation (5) is asymptotically exact, because theraypath in this case is horizontal.

Equations (3) and (4) can be used as well in layered media,with a small-offset approximation of the type made in Dix(1955). Hake et al. (1984) and Tsvankin and Thomsen (1994)provided key equations for moveout in layered VTI media,but in terms of conventional elastic coefficients and Thomsen's(1986) parameters, respectively. Here, I recast their expressionsin terms of the practical anisotropy parameters r7 and Vnmo.

First, as usual, the normal-moveout (NMO) velocity involvesan rms average of velocities in the previous layers. Specifically,

Appendix A I find that equations (3) and (4) continue to hold,with being replaced by

)7eff(to) = 1tOV41

ZO) f'V4..(_C)[j + 877( r)]dr — 1gl nmo( }

(8)

Here, o(r) is the instantaneous value of the anisotropy param-eter r, as a function of the vertical reflection time. In homo-geneous isotropic media [ii(r) = 0], expressions (7) and (8)inserted into equation (3) reduce to the familiar three-termexpansion given in Taner and Koehler (1969), as shown inAppendix B. Tsvankin and Thomsen (1994) suggested that Vhshould be computed in equation (4) using the rms relation

2 1 t0Uh (to) = to f vh(r) dr , (9)

1 f roV mo(t0) _ —

vnmo(r) dr , (7)

t0 0

where

Vh(T) = Vnmo(T) 1 + 2i(r)•where all lowercase variables v, including V nmo , correspond tointerval velocity values and the integration is over time for thevertical raypath. Therefore, v nmo is the interval NMO velocitygiven by

Vnmo(r) = v(r) /1 + 25(r),

and v(r) is the interval vertical velocity. On the other hand,uppercase variables, including Vnmo , correspond to quantitiesthat are averages for the entire vertical column from the surfaceto the reflector of interest. (Recall that here Vnmo refers to theNMO velocity for horizontal or near-horizontal reflectors).

Next, starting with Tsvankin and Thomsen's (1994) expan-sion for the coefficient of the fourth-order term in Hakeet al. (1984) equation for moveout in layered VTI media, in

Note that, in this case, equation (4) is described by threeeffective parameters (Vnmo . Vh , and rIeff). This will complicatevelocity analysis and inversion applications requiring, amongother things, three-parameter searches. On the other hand, aslightly different definition of Vh , given by

Vh(to) = Vnmo(t0) 1 + 2 lleff(r0) , (10)

will reduce the number of effective parameters to two W.and Oeff), simplifying the equation for later uses. In this case,equation (5) holds for layered media, with rl being replaced by

Tleff, which is computed using equation (8).The right side of Figure 1 shows the percent error in the

computed moveout corresponding to reflections from (a) the

-a_

0.5 1 1.5 2 2.5 3X/D

FIG. 1. (Left) Three-layer model, with the first layer being isotropic (S = 0 and E = 0). (Right) Percent time error in moveout(departure from the exact moveout) corresponding to reflections from (a) the bottom of the second layer and (b) the bottom ofthe third layer. The gray curve corresponds to using equation (5), based on the definition of Vh in equation (10). The black curvecorresponds to using equation (4), with Tsvankin and Thomsen's (1994) definition of V h . The dashed curve corresponds to usingequation (3), a modification of the equation of Hake et al. (1984). Here, Vnmo, O7eff, and V h are calculated using equations (7), (8),and (9), respectively.

Dow

nloa

ded

10/2

5/15

to 1

34.7

.152

.117

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

1842

bottom of the second layer and (b) the bottom of the thirdlayer in the model shown on the left side of Figure 1. It isclear that for X/D < 2, the moveout corresponding to i eff ,

calculated using equation (5) (gray curve), has a smaller error(better approximates the exact moveout) than does the move-out corresponding to equation (4), using Vh calculated fromequation (9) (black curve). Both approximations give betterresults than does moveout described by equation (3) (dashedcurve), modified from Hake et al. (1984). V,, mo and rteff for allthree approximate curves are the same and are calculated us-ing equations (7) and (8), respectively. Therefore, not only didthe modified Vh expression (10) simplify the problem by re-ducing the number of required effective parameters, but also itapparently provided a better approximation of the exact move-out. Although only one example is shown here, this conclusionholds for many other v(z) VTI models tested.

A definition of Vh that is more in line with the asymptoticapproximation used to produce equation (4) is to take Vh asthe maximum horizontal velocity among the overlying layers.Such a definition of Vh , however, is not practical for typicalseismic spreads because, although the definition is accurateasymptotically, it can overestimate Vh considerably at practicaloffsets (e.g., for models that include a thin layer with a high Vh ).

Isotropic media are simply a subset of VTI media in whiche and S equal zero [(r) = 0]. Therefore, equations (3) and (5)can be used to approximate moveout in isotropic layered me-dia. Thus, although the anisotropy parameter ?1(r) equals zerothroughout, because the medium is inhomogeneous, 77eff(tO),as given by equation (8), is nonzero. In fact, equation (3) re-duces to the familiar three-term expression given by Taner andKoehler (1969) for isotropic media (Appendix B). Therefore,the value of i7eff also can be used to describe the departure fromhyperbolic moveout caused by the inhomogeneity in isotropiclayered media above a certain reflector. For v(z) isotropic me-dia, equation (4) would yield hyperbolic moveout if Tsvankinand Thomsen's (1994) definition of Vh [equation (9)] were used.Thus, nonhyperbolic moveout associated with vertical inhomo-geneity would be ignored. Better estimates of the moveout areachieved by using the new definition of Vh [equation (10)].For such isotropic media, however, the nonhyperbolic move-out given by equation (5) would be slightly less accurate thanthat given by equation (3) (Appendix B). The reason for thereduced accuracy is that the correction factor introduced inTsvankin and Thomsen (1994) is based on the anisotropy as-sumption only. Therefore, although it produces a highly accu-rate moveout description for homogeneous VTI media, equa-tion (5) results in increased error when vertical inhomogeneityis introduced into the model (Appendix B). Fortunately, the er-rors arising from using equation (5) for all models shown (usingexamples with strong vertical inhomogeneity) nevertheless areless than 0.5% for X/D < 2, rather independent of the strengthof anisotropy.

Alkhalifah

From Figure 1, one cannot distinguish between the amountof nonhyperbolic moveout attributable to anisotropy and thatattributable to inhomogeneity. If this medium, with its largevertical inhomogeneity, were strictly isotropic (E = 0 and S = 0in each layer), then 1leff, calculated using equation (8), wouldequal 0.06. In contrast, the presence of anisotropy resulted inrleff = 0.19. The difference between the two rneff values, how-ever, is not directly attributable to anisotropy because the re-lationship between these factors is nonlinear.

The value of 0.06 for ?Jeff in this three-layer example re-sults from a strong vertical inhomogeneity. I find 77eff valuesassociated with more typical v(z) (average gradient of 0.6 s -1 )isotropic media to be much smaller than 0.1, a common valuefor typical TI media. Thus, nonhyperbolic moveout is less se-vere for typical v(z) isotropic media than for common homo-geneous VTI media. For example, one can approximate thevelocity increase with depth in an isotropic medium [i (r) = 0]by using a constant velocity gradient a with vo as the velocityat the surface; that is,

v(z) = vo + az.

For such a medium, velocity can be expressed in terms of two-way vertical traveltime to as

v(to) = voeo.5ato

Through straightforward derivation using equations (7) and(9), for such a constant-gradient medium,

1 0.5atonett(to)

= 8 tanh(0.5ato)1 (11)

Here, tanh is the hyperbolic tangent function. Note that 1leff

is independent of vo. Therefore, any linear velocity functionwith the same velocity gradient will lead to the same degree ofnonhyperbolic moveout, independent of vo .

Figure 2 shows 7leff values as a function of vertical timefor three values of velocity gradient a. All three curves showmodest values of 1leff when compared with for typical ho-mogeneous TI media [e.g., Taylor sandstone, where rj =0.156 (Thomsen, 1986)]. This result supports the contentionthat anisotropy typically introduces a larger departure from

PROPERTIES OF NONHYPERBOLIC MOVEOUT

From equations (3) and (5), the value of rieff for a given V,,mo

and to directly describes the degree of nonhyperbolic moveoutin both anisotropic and isotropic layered media. For 1leff = 0,the fourth-order term in equation (5) vanishes, and the move-out is hyperbolic, as is the case in homogeneous isotropic orelliptically isotropic media.

FIG. 2. Values of nef in an isotropic medium with a constantvelocity gradient, as a function of zero-offset time to , for threevalues of velocity gradient a.

Dow

nloa

ded

10/2

5/15

to 1

34.7

.152

.117

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

Nonhyperbolic-Moveout Anisotropy Analysis

1843

hyperbolic moveout than does velocity layering. From Figure 2,it might seem that the nonhyperbolic moveout at later times islarge in v(z) media, but as to increases (to > 2 s), X/D decreases,since the maximum offset usually remains constant. Therefore,the significance of nonhyperbolic moveout becomes smaller(Al-Chalabi, 1974). In other words, although i7eff increases withto in Figure 2, the decrease in X/D reduces its influence. There-fore, nonhyperbolic moveout attributable to smooth verticalinhomogeneity is small at all times. In contrast, for homoge-neous TI media, is constant, so nonhyperbolic moveoutclearly is largest at early times, when X/D is large.

Following steps similar to those used in deriving equa-tion (11), one can derive an analytical expression of r) eff forfactorized TI (FTI) media [i.e., TI media for which anisotropyparameters r, S, and E are independent of position (Cerveny,1989; Alkhalifah and Lamer, 1994)1 with a linear velocity vari-ation. In FTI media,

vnmo(r) = v(r)/1 + 28 (12)

and

Vh(T) = Vnmo(t)/1 + 217T1,

where rq 7- is a constant 7 value for FTI media. For such a me-dium, with a constant vertical gradient in the vertical P-wavevelocity, rieff is given by

1 0.5a t0tleff = g [(1 + 8ulTr) tanh(0.5ato) — 11.

(13)

The difference between the Tieff value for FTI media [equa-tion (13)] and that for isotropic media [equation (12)] is givenby

0.5at0Aileff = 11TI (14)

tanh(0.5ato)

On the basis of equation (3), Orjeff describes the differencein the moveout curves for the two media. Moreover, since themoveout curve described by equation (3) is a reasonable ap-proximation of the zero-offset diffraction curve, even in layeredmedia, the difference between the moveout curves associatedwith VTI media and isotropic media can provide some insightinto errors that can result from using isotropic migration fordata from VTI media.

ESTIMATING ANISOTROPY USINGNONHYPERBOLIC MOVEOUT

If the maximum offset is large enough (offsets in marinesurveys often exceed 4 km) relative to reflector depth and theresolution of the data is high, it is possible to estimate thedegree of nonhyperbolic moveout attributable to anisotropy.

For X/D < 1, in layered VTI media the moveout is approx-imately hyperbolic and is given by

XZth = to +

Vnmo

Subtracting equation (5) from this equation and replacing r^with Tieff results in

Ot t — t 2 t2 =2_effX415

hVnmo [t0 Vnmo + (1 + 2Teff)X2] ( )

the amount of time-squared deviation attributable to thenonhyperbolic moveout. A straightforward manipulation of

equation (15) results in2 / 2 2 2)^t Unmo N nmo + X

tleff = 2X2 (X2 — At 2 v2 0 )

(16)

with an accuracy governed by that of equation (5). Note thatthis expression is singular for X = 0. It is clear that no Teffinformation can be extracted from small offsets. The stabilityin estimating T7eff is expected to increase with offset.

To estimate 7leff using equation (16), one must first obtainVnmo , the short-spread NMO velocity corresponding to a hor-izontal reflector. This velocity can be obtained using conven-tional velocity analysis based on a moveout spread that sat-isfies X/D < 1. Assuming that an accurate Vnmo is obtained,then At' can be measured from the reflection moveout in theseismic data. One way to measure At 2 for use in equation (16)is to apply an NMO correction using Vnmo and to compute

Ot t = to — teo. , (17)

where tcor corresponds to the moveout traveltime after NMOcorrection. It is clear that if the true moveout is hyperbolic,then tt equals to and therefore At2 = 0.

If one assumes no lateral velocity variation, the accuracyof the derived Tieff depends primarily on the accuracy of themeasured Vnmo and At2 , which in turn depends on the accuracyof Vnmo . Therefore, the sensitivity measure must combine theinfluences of errors in Vnmo on both At2 and Tieff. Figure 3 showsthe sensitivity of Tieff to errors in the measured Vnmo (i.e., fromvelocity analysis), calculated using equation (16). This examplecorresponds to a homogeneous medium with = 0.1, Vnmo =2.0 km/s, and to = 2.0 s.

As expected, errors are smaller when longer offsets are usedin estimating T7eff , again as long as velocity does not vary lat-erally. Therefore, any inversion technique (i.e., least squares)based on the nonhyperbolic method to obtain Tieff from mea-surements at different offsets should benefit from weightingfactors that favor the far offsets. Furthermore, for a fixed off-set, errors clearly increase with an increase in either to or Vnmo ,since an increase in either implies a reduced ratio X/D.

Note in Figure 3 that even at zero error in Vnmo , the in-verted Tieff is not exactly 0.1. The error can be attributed tothe difference between the forward time calculation, which

FIG. 3. Calculated values as a function of error in the NMOvelocity for offset-to-depth ratios X/D = 1.5 (dashed blackcurve), X/D = 2 (solid gray curve), and X/D = 2.5 (solidblack curve). Here, to = 2.0 s, and Ti for the model is 0.1.

Dow

nloa

ded

10/2

5/15

to 1

34.7

.152

.117

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

1844

involves exact ray tracing, and the inversion process based onequation (16), which is an approximation. This difference in thehomogeneous model given above is largest for about X/D = 2.

In any case, the errors caused by equation (16), at the correctVnmo , are small for typical anisotropies. Therefore, I rely on thisanalytical representation to accomplish most of the inversionsin this paper.

The approach described above for estimating Vnmo and isintroduced to develop insights into the nonhyperbolic inver-sion problem. A more practical approach, based on the 2-Dsemblance analysis method, is discussed next.

SEMBLANCE ANALYSIS BASED ONHYPERBOLIC MOVEOUT

Semblance analysis is less sensitive to traveltime errors thanis traveltime inversion and generally produces more stable re-sults. The semblance coefficient is defined as the ratio of theoutput energy over a window of a stack of traces to the inputenergy in the unstacked traces. In mathematical terms, S k , thesemblance coefficient for M traces, is

k+n/2 M 2L2=k—N/2 L Y-i=lfij(i.k)]

Sk k+N/2 M 2M V e =k—N/2 Ei= 1 fiJ(i.t)

where f, is the recorded data in trace i at the time sample jand j is a function of the zero-offset time sample k and thetrace (offset) i. The window size N + 1, usually set at abouthalf the dominant period of the wavelet, is used to smooth thesemblance spectrum estimates. The semblance coefficient hasa maximum value of unity (when all traces are identical) and aminimum value of zero. Semblance summation in this form isbiased against randomness and sudden variations in amplitudeand polarity. Also, unlike simple summation, as in conventionalstacking, it is insensitive to the overall trace amplitude. Specif-ically, events with identical moveout but differing in amplitudeby a scaling factor produce the same semblance response.

Estimating V,,mo through semblance velocity analysis is basedon summing data over hyperbolic trajectories controlled bythe trial moveout velocity, which defines j(i, £). Therefore, the

Alkhalifah

velocity panel that shows the highest amplitude (stack power)for a specific time, through summation or some semblance mea-sure, defines the stacking velocity. In homogeneous isotropicmedia, in which the moveout is hyperbolic, the stacking velocityis identical to the NMO velocity of the medium. In anisotropicas well as inhomogeneous isotropic media (Al-Chalabi, 1974),the moveout is no longer hyperbolic, and the nonhyperbolicportion of the moveout can distort estimates of stacking ve-locity so that they differ from the NMO velocity, with the dif-ference being proportional to the size of the nonhyperbolicmoveout. As demonstrated earlier, nonhyperbolic moveoutsare larger for typical anisotropy than for typical vertical in-homogeneity. Therefore, the difference between the stackingvelocity and the NMO velocity is expected to be larger inanisotropic media. For i > 0, which is the typical situation,moveouts at far offsets deviate from the hyperbolic trajectoryat shorter times, resulting in higher stacking-velocity estimatesfrom velocity analysis. The size of the deviation of stacking-velocity estimates depends primarily on the range of offsetsused in the velocity analysis.

Figure 4 shows velocity analysis panels for various offset-to-depth ratios (X /D) used in the analysis. Here, the reflection isat to = 2.0 s, rj for the model is 0.1, and V.,, = 2.0 km/s. Ran-dom noise with an rms signal-to-noise ratio of 3 was added to allsynthetic examples used in this paper. The synthetic data usedin the semblance analysis are shown in Figure 5a. To illustratethe size of the nonhyperbolic moveout, the same synthetic dataafter NMO correction using the medium NMO velocity areshown in Figure 5b. Estimated stacking velocities in Figure 4increase with increasing X/D; for X/D = 2, even the reflec-tion time is distorted from the actual zero-offset time of 2 s.While this spread-length bias increases with increasing X /D,the ability to resolve the estimated velocity also increases withincreasing offset used in the analysis. For X/D = 1, which istypically used in conventional velocity analysis, the stacking ve-locity from Figure 4 is estimated to be 2.03 km/s, 1.5% higherthan Vnmo . If a smaller X/D is used (i.e., X/D = 0.5), the errorin estimating Vnmo becomes less than 1%, although the reso-lution is poorer. Theoretically, as offset approaches zero, thestacking velocity should approach Vnmo . Practically, however,

FIG. 4. Velocity analysis panels for various offset-to-depth ratios X/D used in the analysis. Here, the reflection is at to = 2.0 s (depthD = 2 km), rj for the model is 0.1, and Vnmo = 2.0 km/s. The peak frequency of the Ricker wavelet used in the analysis here andthroughout the paper is 40 Hz.

Dow

nloa

ded

10/2

5/15

to 1

34.7

.152

.117

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

Nonhyperbolic-Moveout Anisotropy Analysis

1845

as the range of offsets used decreases, velocity analysis suffersfrom reduced resolution. The trade-off between resolution andaccuracy in stacking velocity depends mainly on the peak fre-quency of the wavelet. Once a choice is made regarding thistrade-off, one can use Figure 3 to relate possible Vnmo errors tothe accuracy expected in inverting for rj from traveltime picks.

Figure 6 shows semblance results from summing again overhyperbolic trajectories, controlled by the stacking velocity, fora single reflection event with a zero-offset time of 2 s. The gen-eral model is the same as in Figure 4, with (a) rl = 0 (isotropicmodel) and (b) rl = 0.1. In both cases, Vnmo = 2.0 km/s. The

vertical axis in Figure 6 corresponds to the maximum offsetused in the semblance analysis. For smaller maximum offsets,the resolution is poor and the velocity is unresolvable. As themaximum offset increases, so does the resolution. Neverthe-less, a clear shift of the best-fit stacking velocity occurs in theTI model, a direct influence of nonhyperbolic moveout. Theshift is dramatic as one approaches X/D = 2 (at an offset of4.0 km). Also, as the maximum offset increases, the semblancepower decreases, because the best-fit hyperbolic moveout failsto simulate the true nonhyperbolic moveout. As the amplitudedecreases, the contribution of noise to the analysis increases.

FIG. 5. Synthetic data generated for a model with one horizontal reflector at a depth of 2 km and with mediumparameters Vnmo = 2.0 km/s and r = 0.1. (a) Data before NMO correction. (b) Data after NMO correction usingan NMO velocity of 2.0 km/s.

FIG. 6. Velocity analysis panels for various maximum offsets used in the analysis. Here, the reflection is at t o = 2.0 s,Vnmo = 2.0 km/s (depth D = 2 km), and rr = 0 (a) or 0.1 (b).

Dow

nloa

ded

10/2

5/15

to 1

34.7

.152

.117

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

1846

In the next section, I demonstrate, through a semblance anal-ysis over nonhyperbolic moveout, how to reduce the errors inestimating Vnmo that arise in long-offset data.

SEMBLANCE ANALYSIS BASED ONNONHYPERBOLIC MOVEOUT

To use the semblance coefficient with a nonhyperbolic move-out trajectory, I simply describe j (i, £) using the nonhyperbolicmoveout equation (5) instead of the hyperbolic one (May andStratley, 1979). However, in this case, the moveout dependson two parameters rather than one, thus expanding the di-mensionality of the search. The nonhyperbolic scan below isapplied over Vh and Vnmo , rather than rl and Vnmo , so that bothaxes have the same units to simplify comparison of resolutionand accuracy.

Figure 7 shows the semblance coefficient as a function ofVn,,,o and Vh for a model with a horizontal reflector at a depthof 2.0 km beneath a homogeneous TI medium with Vnmo =2.0 km/s and rl = 0.1. The zero-offset reflection time of theRicker wavelet was 2 s, and the scan was done by settingto = 2.0 s. A 3-D scan would require a search over zero-offsettime as well. Figure 7a corresponds to a maximum X/D of 1.5used in the semblance summation, Figure 7b corresponds to amaximum X/D of 2.0, and Figure 7c corresponds to a maxi-mum X/D of 2.5. As expected, resolving power (reciprocallyrelated to the overall size of the elongated, nearly ellipsoidaldarkened region) increases with larger maximum offsets. Infact, because this elongated region tilts further from the verticalas X/D increases, the ability to resolve Vh increases consider-ably when larger offsets are included. The maximum semblanceresponse for any of the three maximum X/D values could bepicked at Vnmo = 2.0 km/s and Vh = 2.18 km/s. (The confidencein this pick increases with increasing offsets used in the analy-sis.) These values of Vnmo and Vh result in i = 0.095, which isclose to the actual value of 0.1. The slightly low estimate forrl arises from using nonhyperbolic equation (5), which is anapproximation (although a good one) of the actual moveout.

A practical approach that can reduce the cost of a 3-D scanover Vnmo, Vh , and to is an iterative 2-D technique, in which onescans once over Vnmo using small offsets, fixes the interpretedvalues of Vnmo , and then does another scan, this time over Vh

using the whole spread. The results of the Vh analysis then can

Alkhalifah

be used to scan again over Vnmo and so on until a convergencecriterion is met. As has been suggested in May and Stratley(1979), convergence is guaranteed through the use of an or-thonormal basis (i.e., the Legendre polynomials) to representthe moveout polynomial [equation (3)].

The NMO velocity obtained from semblance analysis usu-ally is more accurate than that extracted by fitting a hyper-bolic curve, in a least-squares sense, to the moveout of a re-flection. One should expect a similar improvement in accuracywhen nonhyperbolic moveouts are used in the semblance anal-ysis. In an analysis based on least-squares fitting of traveltimes,Tsvankin and Thomsen (1995) concluded that the second- andfourth-order coefficients of the Taylor series expansion of theTI moveout are not resolvable from traveltime moveout curvesin VTI media. The reason that the semblance approach reducesthis ambiguity in resolving the anisotropy parameters discussedin Tsvankin and Thomsen (1995) basically is described by theconcept of objective function (a function of the unknown pa-rameters formed so that a maximum or a minimum value ofthe function corresponds to a solution of the problem).

The objective function for the semblance approach, given bythe semblance responses in Figure 7 for a 40-Hz peak frequen-cy, has a more stable maximum (closer to the actual value) thandoes the objective function calculated on the basis of a least-squares traveltime fitting of the moveout over the same rangeof offsets (Figure 8). In particular, the least-squares methodis more sensitive to the shortcomings of the moveout approx-imation than is the semblance approach. This is obvious byobserving the amount of shift of the maximum (or the mini-mum) from the true values for the model. The velocity anal-ysis objective function also is more stable and less sensitiveto noise and traveltime errors than is the least-squares trav-eltime fitting approach. Figure 9 shows (a) semblance analysisand (b) least-squares traveltime fitting objective functions aftersubjecting the synthetic data of Figure 7 to random traveltimeshifts between 0 and 0.5% (=10 ms), as might happen aftera poor static correction. The mean of these traveltime shiftsis 0.25% (=5 ms). Clearly, the objective function of the least-squares fitting approach is influenced much more by the errors(shifted from the true value) than is that of the semblanceapproach. The resolution of the objective function of the sem-blance approach also depends largely on the peak frequency

FIG. 7. Nonhyperbolic velocity analysis using (a) X/D = 1.5, (b) X/D = 2.0, and (c) X/D = 2.5. Here, the reflection is at to = 2.0 s,it for the model is 0.1, and Vnmo = 2.0 km/s. The gray curves are contour lines for based on equation (1).

Dow

nloa

ded

10/2

5/15

to 1

34.7

.152

.117

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

Nonhyperbolic-Moveout Anisotropy Analysis

1847

of the wavelet. The dominant frequency of 40 Hz used here isquite representative of frequencies in seismic data.

ANISOTROPY AND VERTICAL HETEROGENEITY

In typical TI media, is positive, so the fourth-order term inequation (5) is negative. Similarly, in typical isotropic media,in which velocity varies with depth, the fourth-order term isagain negative, producing a similar moveout behavior. Specif-ically, both tz — X2 curves are convex upward (Hake et al.,1984). Although Oeff in heterogeneous isotropic media is usu-ally smaller than that in homogeneous TI media, the similarityin moveout behavior will raise problems in deciding how muchof the inverted rff to attribute to anisotropy and how muchto attribute to inhomogeneity. Nevertheless, it is expected thatthe dominant portion of the nonhyperbolic moveout often canbe attributed to anisotropy, and if inhomogeneity is resolvablethrough other techniques (i.e., conventional velocity analysisfor a sequence of reflections), then the relative contributionsfrom anisotropy and vertical heterogeneity can be assessed.

The relative sensitivity of dip-moveout processing to aniso-tropy and vertical inhomogeneity differs from that of nonhy-

perbolic moveout. For example, primarily for moderate to lowdips, a dip-moveout impulse response for typical anisotropy(i > 0) can be approximated as a stretched version of that forhomogeneous isotropy, whereas the v(z) isotropic impulse re-sponse is a squeezed version of the homogeneous isotropic one(Alkhalifah, 1996). Therefore, the presence of both anisotropyand inhomogeneity in a medium leads to DMO actions that areopposite one another, whereas the actions of both anisotropyand inhomogeneity increase the nonhyperbolic moveout of re-flections from horizontal interfaces.

Suppose that only one reflection is strong enough to showmeasurable nonhyperbolic moveout in a v(z) VTI medium. Areasonable approach might be to consider the medium FTI andtherefore obtain a constant 1) TI . Using equation (8) and settingr^(r) to be constant (=OTI ) result in

1 m11T1 = g ( 1 + 817eff) 1 (' to

V 4 /

o( to)— 1 (18)

to JO vnmo(r) dr

which enables one to deduce an average corresponding solelyto anisotropy. Equation (18) is important because, as shownlater in the first field-data example, especially at early recording

FIG. 8. Root-mean-square sum of the difference between the actual moveout and the moveout given by equation (4) as a functionof Vh and Vnmo using (a) X/D = 1.5, (b) X/D = 2.0, and (c) X/D = 2.5. This is the same model as that was used in Figure (7), inwhich the reflection is at to = 2.0 s, rl for the model is 0.1, and Vnmo = 2.0 km/s.

FIG. 9. Objective functions for (a) semblance analysis and (b) least-squares fitting for X/D = 2, as in Figures 7band 8b, respectively, after adding random traveltime shifts between 0 and 0.5% (=10 ms) to the data. The arrowin (b) points to the peak that is severely shifted because of the added errors.

Dow

nloa

ded

10/2

5/15

to 1

34.7

.152

.117

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

1848

times (to <2 s), often only one reflection will be strong enoughto show measurable nonhyperbolic moveout. (Lateral velocityvariation, of course, would complicate this interpretation.)

ADVANTAGES OF THE NONHYPERBOLICINVERSION METHOD

Alkhalifah and Tsvankin (1995) have developed a proce-dure for estimating q and Vnmo in layered TI media using theshort-spread moveout behavior for dipping reflectors (DMOmethod). Even vertical variations of i can be estimated usingthe DMO method (Alkhalifah, 1997). That DMO-based pro-cedure is probably more stable in inverting for the anisotropyparameters than is the method described above, especially inthe absence of large offsets and at later times (deeper targets),at which X/D is small.

Unlike the DMO method, however, the nonhyperbolicmoveout method discussed here does not require dipping re-flectors and therefore is more flexible and can be applied to abroader range of field data. Moreover, it provides more oppor-tunity to obtain lateral variations in . For example, statisticalestimation of lateral variations in i7 can be made from data atmany common-midpoint (CMP) locations.

Given the trade-off between Vnmo and r7 in equation (1), theerrors in estimating Vh using the nonhyperbolic moveout gener-ally are small. The horizontal velocity Vh is the necessary quan-tity for migration of a vertical reflector to its true position. Us-ing the nonhyperbolic inversion, it usually is estimated (in thepresence of large offsets) with a higher accuracy than is Vnmo .

Therefore, one can better construct the time-migration impulseresponse with the nonhyperbolic moveout method than withisotropic methods.

One area in which r, measurements from nonhyperbolicmoveout can play a major role is in the presence of very steep(nearly vertical) reflectors, such as flanks of salt domes in theGulf of Mexico where, in addition, reflections from interfaces

Alkhalifah

with intermediate dips may not be available. Alkhalifah andTsvankin (1995) showed that the DMO method fails to yieldaccurate values of ri for such steep dips, primarily because themoveout for such reflections in TI media is not distinguishablefrom that in isotropic media or in any other anisotropic model.Therefore, the moveout for such dips becomes somewhat in-dependent of the anisotropy parameter rt. The nonhyperbolicmoveout for events from subhorizontal reflectors, which canprovide more accurate values of Vh , potentially can provide iiinformation for improved migration of data from steep reflec-tors.

This nonhyperbolic method, however, is based on the as-sumption of lateral homogeneity, with some tolerance, as isusually the case with v(z) algorithms, for mild lateral inhomo-geneity (i.e., smooth lateral variations). Therefore, strong lat-eral inhomogeneities will cause problems for the method andrequire a much more advanced treatment, which is beyondthe scope of this paper. In media with strong lateral inhomo-geneities, 77eff still is possibly measurable and can be used toaid in making nonhyperbolic moveout corrections, but it hasno simple interpretation in terms of medium properties.

FIELD-DATA EXAMPLES

Figure 10 shows a seismic line from offshore Africa. The linewas processed using conventional NMO and DMO algorithmswithout taking anisotropy into account. Velocity analysis showsa general vertical velocity increase with depth that can be sim-plistically modeled with a constant gradient of about 0.7 s -1 .

As Alkhalifah and Tsvankin (1995) demonstrated, these dataare influenced by the presence of anisotropy. Moreover, us-ing the DMO method for estimating it and Vnmo , Alkhalifah(1996) showed that the anisotropy is strongest above to = 2 sin a massive shale formation.

Figure 11 shows CMP gathers after applying isotropic homo-geneous DMO and NMO corrections. The NMO correction is

FIG. 10. CMP-stacked seismic line (offshore Africa) after application of NMO correction along with isotropichomogeneous DMO. The distance between CMPs is 12.5 m.

Dow

nloa

ded

10/2

5/15

to 1

34.7

.152

.117

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

Nonhyperbolic-Moveout Anisotropy Analysis

1849

based on the velocities obtained from conventional velocityanalysis, using a spread given by X/D < 1. DMO was appliedto reduce even the small distortion of the stacking velocitycaused by the mild dip (^-6°) of the reflector at about t o = 1.8 s.The two subparallel events prior to to = 2.0 s show a signifi-cant departure from hyperbolic moveout. If the deviations inFigure 11 were caused by NMO velocity overcorrection (usinglower-than-true velocities), then these moveout curves wouldhave departures from to proportional to X2 . The fact that thesecurves are practically straight for X/D < 1 implies that they arecontrolled by higher-order terms of the Taylor series expansion

(e.g., X4 ).A detailed portion of Figure 11 (Figure 12) helps in pick-

ing reflection times and therefore in measuring At e . Some of

the moveouts (e.g., the reflection at t o = 1.86 s and CMP loca-tion 700) have a slight initial plunge before the larger offsetsat which the nonhyperbolic behavior dominates the moveout.This initial plunge results from using a V„ mo value in the NMOcorrection that is higher than the true value. As suggested inFigure 4, the higher velocity is probably a result of spread-length bias, which arises from the attempt to fit nonhyperbolicmoveout with hyperbolic curves. Analysis over nonhyperbolicmoveout should overcome such a problem as well as providean estimate of the nonhyperbolic portion of the spread.

Figure 13 shows the semblance response using nonhyper-bolic moveouts as a function of Vn,,,o and Vh (similar to Figure 7)for the same reflection events as those shown in Figure 12 atCMP locations 700, 800, and 900. Note that, among the three

FIG.11. CMP gathers at locations 700, 800, and 900 after NMO correction and isotropic homogeneous DMO. The NMO correctionis based on velocities obtained from conventional velocity analysis with X/D < 1.

FIG. 12. Detail of Figure 11. The black curve is the approximate location of the zero-crossing of the reflection wavelet. It indicatesthe general shape of the reflection moveout.

Dow

nloa

ded

10/2

5/15

to 1

34.7

.152

.117

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

1850

locations, Vnmo decreases monotonically from 2.15 kmls at CMPlocation 700 to 2.07 km/s at CMP location 900. This decreasecorresponds mainly to the decrease in the zero-offset times ofthese reflections, which reflects the general velocity increasewith depth obtained by Alkhalifah (1996). On the other hand,the Vh or rj has the highest value at the middle location, CMPlocation, 800. At this CMP location, neff is 0.28, whereas at CMPlocations 700 and 900, rleff is 0.2 and 0.16, respectively. Thesevalues of rj include the combined influence of anisotropy and in-homogeneity. Using equation (18) in an attempt to remove theinfluence of vertical inhomogeneity, I estimate liTI to be 0.15,0.22, and 0.13 for CMP locations 700, 800, and 900, respectively.I attribute these results primarily to anisotropy. These valuesare, on average, higher than those obtained by Alkhalifah andTsvankin (1995) and Alkhalifah (1996) with the DMO-basedapproach. However, their measurements correspond primarilyto the region near CMP location 900, which has an Ti value herethat is more in agreement with their calculations. Finally, con-sidering the overall small lateral velocity variation, the result-ing estimate of 17 suggests the presence of anisotropy. Furtheranalysis of the relative importance of anisotropy and lateralvelocity variation, however, is necessary.

Figure 14 shows a schematic plot of the raypath from thesource down to the reflection point and back up to the receiverfor the maximum offset used in the semblance analysis at CMPlocation 800 in Figure 13. The ray bending is caused by the ver-tical increase in velocity with depth. Unlike in NMO velocityanalysis, the estimates rely on information from large offsets.Therefore, the subsurface influence on 17 estimates is not later-ally local (i.e., near the CMP location of measurement). Thus,although the use of large offsets can help improve the resolu-tion of the inversion, it can hamper the lateral resolution ofestimating il.

To obtain a better understanding of the lateral variations inq in the field example given above, one should compare CMPlocations that are at least 3 km apart (the distance correspond-ing to the maximum offset used in the analysis). Moreover,better results would be obtained if Ti estimates were made atmany more CMP locations and then averaged (smoothed) over3-km intervals with, for example, a Gaussian window.

Figure 15 shows another portion of the data set from off-shore Africa that is dominated by horizontal (or subhorizontal)

Alkhalifah

events. The large number of strong horizontal reflectors shouldprovide an excellent setting for applying the layer-stripping ap-proach discussed earlier. This data set also includes offsets upto 4.3 km, which will help in boosting the resolution of thesemblance analysis at later times. Nevertheless, the measure-ments at later times still suffer from lower resolution becauseof smaller X/D as well as increased layer-stripping errors thatpropagate from the top to the bottom of the section. Figure 16shows four sample nonhyperbolic semblance responses calcu-lated at 1.24,1.86, 2.28, and 2.99 sat CMP location 300. Pickingthe Vnm,, and Vh values corresponding to the maximum sem-blance responses for these times, as well as other ones, andinserting them into equations (7) and (8) yields the velocityand Ti curves shown in Figure 17 (black curves). Because theseare marine data, Vnmo and Ti are set to equal 1.5 km/s and zero,respectively, at the surface. The curves of interval Vnmo and Tihave been convolved with a simple smoothing window of length0.2 s. The gray curves, on the other hand, describe the upperand lower limits of possible parameter values corresponding tothe uncertainties in picking Vnmo and Vh (e.g., picking within theblack region in Figure 16). The range of possible values is cal-culated by evaluating the derivatives of the interval values withrespect to the measured effective ones and multiplying thesederivatives by the uncertainty in the measurement of V,,mo andVh . As above, I ignore the influence of lateral velocity varia-tion on the results. However, the lateral velocity variation inthis region is mild (<2%).

FIG. 14. Schematic plot of the raypath for CMP location 800 fora ray traveling from the source down to the reflection point andback up to the receiver, based on the maximum-offset raypathused in the analysis.

FIG. 13. Nonhyperbolic velocity analysis for CMP locations 700, 800, and 900. Here, to, which varies from one CMP to another, hasbeen extracted from Figure 12. The gray curves correspond to contour lines describing Ti

Dow

nloa

ded

10/2

5/15

to 1

34.7

.152

.117

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

Nonhyperbolic-Moveout Anisotropy Analysis

1851

Note that in Figure 16 the semblance resolution, especiallyfor Vh , decreases with increasing zero-offset time because ofthe reduction in X/D. This will degrade the accuracy of pick-ing, resulting in errors in the interval values that increase withvertical time. Also, as with any other layer-stripping applica-tion, the interval values at later times have errors accumulatedfrom measurements at earlier times. So, in Figure 17, rj valuesbeyond to = 2.0 s are not that reliable. On the other hand, theincrease of rl up to to = 1.8 s is reasonable because this increase

is maintained even when the measured values at to = 1.86 s inFigure 16 are perturbed within the range of acceptable picks(the black region). For example, although Vh was evaluated atthe maximum semblance to equal 2.05 km/s, one can assumea margin of error, corresponding to the black region, of about+0.04 km/s, which corresponds to about 2% error. The limitsof this margin are given by the gray curves in Figure 17. Withinthis margin of error, the ij curve in Figure 17 always increasesup to the maximum value at 1.8 s, but the particular form of

FIG. 15. Time-migrated seismic line (offshore Africa). This line is near the line shown in Figure 10, but indeeper waters.

FIG. 16. Nonhyperbolic velocity analysis for CMP location 300 from Figure 15 at different to values. The graycurves correspond to contour lines describing r?eff.

Dow

nloa

ded

10/2

5/15

to 1

34.7

.152

.117

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

1852

Alkhalifah

the increase depends on the Vh value. Therefore, the increasein it up to 1.8 s is more or less an accurate representation of theanisotropy variation beneath CMP location 300. The reliabilityof the results at these times was enhanced further by observ-ing similar results from neighboring CMP locations (i.e., CMPlocations 200 and 250).

One major problem in estimating interval values using layerstripping (i.e., the Dix equation) is that close measurementsin time can result in erroneous interval values. An alternativeto layer stripping is nonlinear inversion. Thus, one can fit aninterval velocity and r7 model to the measured Vnmo and Vh withsome regularization included. The inversion approach has theadvantage of mixing measurements from neighboring CMPs inan effort to stabilize the results. Such mixing of measurementsalso helps smooth the interval estimates laterally.

CONCLUSIONS

The nonhyperbolic moveout behavior of reflections fromhorizontal interfaces is an important source of velocity infor-mation for processing, especially in anisotropic media. In mostanisotropic media, the nonhyperbolic moveout is relativelylarge, larger, in fact, than that in typical vertically heteroge-neous isotropic media. Therefore, it usually is observable andmeasurable and thus can be used to invert for medium param-eters. Although estimates of n derived from the nonhyperbolicmoveout method alone might not be reliable enough to usedirectly in lithology interpretation, such results can play a ma-jor role in processing and supporting estimates of TI and V o

obtained from the DMO method (Alkhalifah and Tsvankin,1995), as well as in providing 77 values in areas where the dip-ping features required by the DMO method are absent.

The effective T7 values obtained from the NMO methodcan be used as an indicator of the relative contributions ofanisotropy and vertical heterogeneity to nonhyperbolic move-out. This indicator demonstrates that nonhyperbolic moveoutassociated with typical TI media (i7 ^_ 0.1) is greater than thatassociated with typical v(z) isotropic media. Therefore, apply-ing the nonhyperbolic moveout correction prior to stacking is

Vnmo2000 4000

11

CDEH

aDEH

FIG. 17. Interval values Vnmo and TI as a function of vertical time(black curves) at CMP location 300. The margin of possiblevalues attributable to picking uncertainties lies between thetwo gray curves.

more important in such TI media than in v(z) media. In anycase, both anisotropy and vertical heterogeneity can be takeninto account in inversion of nonhyperbolic data when velocityanalysis is done over a range of reflector times. The importanceof anisotropy versus that of lateral velocity variation remainsan area for future study.

The process of extracting T7 values from the nonhyperbolicmoveout behavior of reflections is sensitive to errors in themeasured Vnmo , with the sensitivity decreasing as offsets usedin the inversion increase. Sensitivity also decreases at smallervertical times corresponding to smaller depths and thus largeroffset-to-depth ratios. In TI media, with positive n, nonhyper-bolic moveout tends to overestimate the value of Vnmo ob-tained using velocity analysis, depending mainly on the offset-to-depth ratio X/D used in the velocity analysis. For a typicalX/D < 1.0, such increases do not exceed 2%. Depending onthe ratio X/D used in the inversion for , this overestimation ofVVo results in an estimate of q that is low, by no more than 0.04for a model with r = 0.1. However, estimates of 11 and Vnmo

can be used iteratively to improve further estimates of oneanother. At increased cost, semblance analysis over nonhyper-bolic trajectories can reduce such errors and thereby providebetter estimates of V,,mo and rj.

The nature of seismic data, which are dominated by hori-zontal and subhorizontal reflections, makes this method morewidely applicable than the DMO method of Alkhalifah andTsvankin (1995), which relies on the presence of reflectionswith different dips. The nonhyperbolic moveout method inhomogeneous media needs only one reflector, preferably ahorizontal one, for the inversion to work. Therefore, it canbe applied almost anywhere, which helps in estimating lateralvariations in 17. Use of larger offsets in the nonhyperbolic inver-sion, however, which are necessary to stabilize the inversion,can degrade the lateral resolution of 17.

One area in which measurements from nonhyperbolicmoveout can play a major role is in the presence of very steep(near vertical) reflectors, such as flanks of salt domes in theGulf of Mexico where, in addition, reflectors from interfaceswith intermediate dips may not be available. The DMO methodof Alkhalifah and Tsvankin (1995) fails for such steep dips,primarily because the moveout for such reflections in TI me-dia is not distinguishable from that in isotropic media. There-fore, nonhyperbolic moveout from horizontal events may bethe only information in surface seismic data that can provide r^estimates necessary for better migration of data from such dips.

Using the nonhyperbolic moveout method on data from off-shore Africa helped to estimate ri both vertically and laterally.However, the accuracy of the inversion decreased dramaticallywith depth because of the reduced X/D and layer-strippingerrors.

ACKNOWLEDGMENT

I am grateful to Ken Lamer and John Toldi for helpful discus-sions. I thank John Toldi and Chris Dale of Chevron OverseasPetroleum, Inc., for providing the field data. Special thanks arealso due to the Center for Wave Phenomena, Colorado Schoolof Mines, for technical support and to KACST, Saudi Arabia,for financial support. Financial support for this work also wasprovided in part by the United States Department of Energy(DOE) (this support does not constitute an endorsement byDOE of the views expressed in this paper).

Dow

nloa

ded

10/2

5/15

to 1

34.7

.152

.117

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

Nonhyperbolic-Moveout Anisotropy Analysis

1853

REFERENCES

Al-Chalabi, M., 1974, An analysis of stacking, rms, average, andinterval velocities of horizontally layered ground: Geophys. Prosp.,22, 458-475.

Alkhalifah, T., 1996, Transformation to zero offset in transverselyisotropic media: Geophysics, 61, 947-963.

1997, Seismic data processing in vertically inhomogeneous TImedia: Geophysics, 62, 662-675.

Alkhalifah, T., and Lamer, K., 1994, Migration errors in transverselyisotropic media: Geophysics, 59, 1405-1418.

Alkhalifah, T., and Tsvankin, I.,1995, Velocity analysis for transverselyisotropic media: Geophysics, 60,1550-1566.

Byun, B. S., Corrigan, D., and Gaiser, J. E., 1989, Anisotropic ve-locity analysis for lithology discrimination: Geophysics, 54, 1564-1574.

Cerveny, V., 1989, Ray tracing in factorized anisotropic inhomoge-neous media: Geophys. J. Internat., 94, 575-580.

Dix, C. H., 1955, Seismic velocities from surface measurements:Geophysics, 20, 68-86.

Hake, H., Helbig, K., and Mesdag, C. S., 1984, Three-term Taylorseries for t 2 - x= curves over layered transversely isotropic ground:Geophys. Prosp., 32, 828-850.

May, B. T., and Stratley, D. K., 1979, Higher-order moveout spectra:Geophysics, 44, 1193-1207.

Neidell, N. S., and Taner, M. T., 1971, Semblance and other coherencymeasures for multichannel data: Geophysics, 36, 482-497.

Shah, P. M., and Levin, F K., 1973, Gross properties of time-distancecurves: Geophysics, 28, 643-656.

Taner, M. T., and Koehler, F. 1969, Velocity spectradigital computerderivation and applications of velocity functions: Geophysics, 34,859-881.

Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, 1954-1966.

Tsvankin, I., 1995, P-wave signatures and notation for transverselyisotropic media: An overview: Geophysics, 61, 467-483.

Tsvankin, I., and Thomsen, L., 1994, Nonhyperbolic reflectionmoveout in anisotropic media: Geophysics, 59, 1290-1304.

1995, Inversion of reflection traveltimes for transverse isotropy:Geophysics, 60, 1095-1107.

APPENDIX A

EFFECTIVE 71 IN LAYERED MEDIA

For multilayered TI media, the exact quartic coefficient A4of the Taylor series expansion (Hake et al., 1984; Tsvankin andThomsen, 1994) is given by

A4 = (E i [vnmo] 2 At (') ) 2 - t0Et []4 (t)

4 ( [v(t) ]

2

Atli)I

4{ nmo

r 1 8

to ^i`44i L vntn0 ] [At (') ] 3+ 2 4 (A 1)

(ji [vnmo] Atl` ))

which includes (in the first term) ray bending attributable tothe layered structure. Here, Lt'> and unmo are the two-wayzero-offset time and the NMO velocity for a given layer i, re-spectively. A4i is the quartic coefficient A 4 in layer i. When thecontribution of the vertical shear-wave velocity is ignored, it isgiven by

2^(i)

A4i = -, (A-2)

[At ( ` ) ]2 [vnmo]

4

where t7 ( ' ) is the t7 value for a given layer i. On the other hand,again ignoring the contribution of the vertical shear-wave ve-locity, A 4 is given by

A4 = - 2 ^ f(A-3)t0 Vnmo

Inserting A4, into equation (A-1) and using integration in-stead of summation gives

A4 ( t0) _ tO Vnmo( to) - tO J0' t

0ynmo\/ t) dt

44t0 V

8nmo

8t0 f0 7)\t)ynmo( t)dr- (A-4)

4 84t0 VnmoSubstituting equation (A-3) into equation (A-4), with straight-forward manipulation, results in

( to

^eff(t0) = g j toval (to)

nmo (t)[l + 8^7(t)J dt - i J.nmo

(A-5)

APPENDIX B

NONHYPERBOLIC MOVEOUT IN ISOTROPIC LAYERED MEDIA

Typically, reflections in v(z) isotropic media are approxi-mated by hyperbolic moveout characterized by rms velocity.If the subsurface velocity has a large variation with depth, thehyperbolic moveout assumption is less accurate and an addi-tional term in the Taylor series expansion is needed to bettersimulate the moveout. The three-term expansion (Taner andKoehler, 1969; Shah and Levin, 1973; Al-Chalabi, 1974) forisotropic media is given by

41- 44 X4

x2 Vnt = to + 2 + 2 4 o, (B-1)

Vnmo 4t0 Vnmo

where Vnmo is calculated using equation (7) and V4 is calculatedusing

fr

V4 (ta) o o

v4 (t) dt. (B -2)

The interval quantities vnmo(r) and v(r), in this case, areequal. The first two terms in equation (B-1) correspond to hy-perbolic moveout, and the additional term is the nonhyperbolicterm, which provides better moveout description at larger off-sets. Actually, equation (3), with OOeff given by equation (8),reduces to equation (B-1) for isotropic media.

The left portion of Figure B-1 shows a v(z) isotropic modelconsisting of three layers. The right portion shows the percent

Dow

nloa

ded

10/2

5/15

to 1

34.7

.152

.117

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/

0.5

0

—0.5

p —1

0-1.5

(1) 1.51

E 0.5F- C

—0.5

—1—1 . ^

0.5 1 1.5 2 2.5 3

. t.,o.0

CAA VP-3.0 km/s

V=2.0 km/s

I

V=4.0 km/sci)

1854

Alkhalifah

31 IFYI. B-1. (Left) Model with three isotropic layers. (Right) Percent time error in moveout corresponding to reflections from (a)the bottom of the second layer and (b) the bottom of the third layer. The gray curve corresponds to equation (5). The black curvecorresponds to equation (3). The dashed curve corresponds to the hyperbolic moveout. V„mo for all three curves is calculated usingequation (7).

error in the computed moveout corresponding to reflectionsfrom (a) the bottom of the second layer and (b) the bot-tom of the third layer. The time errors resulting from usingequation (B-1) [or equation (3)], given by the black curve,are smaller overall than those resulting from using equation(5), given by the gray curve. Both results, however, are betterthan those obtained using a hyperbolic moveout, given by the

dashed curve. Nevertheless, the difference between the twononhyperbolic approximations is small, even for such a strongvertical inhomogeneity. This difference reduces dramaticallyfor a smoother heterogeneity. The additional X factor in thedenominator of the fourth-order term in equation (5), althoughextremely important for VTI models, generally is ineffectivefor isotropic models.

Dow

nloa

ded

10/2

5/15

to 1

34.7

.152

.117

. Red

istr

ibut

ion

subj

ect t

o SE

G li

cens

e or

cop

yrig

ht; s

ee T

erm

s of

Use

at h

ttp://

libra

ry.s

eg.o

rg/