correction for the inﬂuence of velocity lenses on

CWP-668

Correction for the influence of velocity lenses onnonhyperbolic moveout inversion for VTI media

Mamoru Takanashi1, 2 & Ilya Tsvankin11 Center for Wave Phenomena, Geophysics Department, Colorado School of Mines, Golden, Colorado 804012 Japan Oil, Gas and Metals National Corporation, Chiba, Japan

ABSTRACT

Nonhyperbolic moveout analysis plays an increasingly important role in velocitymodel building because it provides valuable information for anisotropic param-eter estimation. However, lateral heterogeneity associated with stratigraphiclenses such as channels and reefs can significantly distort the moveout parame-ters, even when the structure is relatively simple.Here, we discuss nonhyperbolic moveout inversion for 2D models that include alow-velocity isotropic lens embedded in a VTI (transversely isotropic with a ver-tical symmetry axis) medium. Synthetic tests demonstrate that a lens can causesubstantial, laterally varying errors in the normal-moveout velocity (Vnmo) andthe anellipticity parameter η. The area influenced by the lens can be identifiedusing the residual moveout after the nonhyperbolic moveout correction and thedependence of errors in Vnmo and η on spreadlength.To remove lens-induced traveltime distortions from prestack data, we proposean algorithm that involves estimation of the incidence angle of the ray passingthrough the lens for each recorded trace. Using the velocity-independent layer-stripping method of Dewangan and Tsvankin, we compute the lens-inducedtraveltime shift from the zero-offset time distortion (i.e., from “pull-up” or“push-down” anomalies).Synthetic tests demonstrate that this algorithm substantially reduces the errorsin the effective and interval parameters Vnmo and η. The corrected traces andreconstructed “background” values of Vnmo and η are suitable for anisotropictime imaging and producing a high-quality stack.

Key words: P-waves, anisotropy, transverse isotropy, velocity analysis, lat-eral heterogeneity, velocity lenses, nonhyperbolic moveout inversion, traveltimeshifts

1 INTRODUCTION

Kinematics of P-wave propagation in VTI (transverselyisotropic with a vertical symmetry axis) media are gov-erned by the vertical velocity V0 and the Thomsen pa-rameters ε and δ (Tsvankin & Thomsen, 1994). P-wavereflection traveltime in laterally homogeneous VTI me-dia above a horizontal or dipping reflector depends onlyon the normal moveout velocity Vnmo and the anel-lipticity parameter η (Alkhalifah & Tsvankin, 1995;Tsvankin, 2005):

Vnmo = V0

√1 + 2δ , (1)

η =ε − δ

1 + 2δ. (2)

The parameters Vnmo and η, which control all P-wavetime-processing steps, can be obtained from nonhyper-bolic moveout or dip-dependent NMO velocity. In par-ticular, the nonhyperbolic moveout equation introducedby Alkhalifah & Tsvankin (1995) and its extension forlayered media (Alkhalifah, 1997; Grechka & Tsvankin,1998; Tsvankin, 2005) have been widely used for esti-mating Vnmo and η and building anisotropic velocitymodels.

Nonhyperbolic moveout analysis is performed un-der the assumption that the overburden is laterally ho-mogeneous on the scale of spreadlength. However, even

384 M. Takanashi & I. Tsvankin

gentle structures often contain small-thickness lenses(such as channels and carbonate reefs), whose width issmaller than the spreadlength (Armstrong et al., 2001;Fujimoto et al., 2007; Takanashi et al., 2008; Jenner,2009; see Figure 1). For isotropic media, lateral hetero-geneity of this type has been recognized as one of thesources of the difference between the moveout and truemedium velocities (Al-Chalabi, 1979; Lynn & Claer-bout, 1982; Toldi, 1989; Blias, 2009). Such lens-inducederrors in Vnmo lead to misties between seismic and welldata (Fujimoto et al., 2007).

Although the moveout parameters (especially η)were shown to be sensitive to correlated traveltime er-rors (Grechka & Tsvankin, 1998), overburden hetero-geneity is seldom taken into account in nonhyperbolicmoveout inversion. Grechka (1998) shows analyticallythat a constant lateral velocity gradient does not distortthe estimates of Vnmo and η, if anisotropy and lateralheterogeneity are weak. The second and fourth hori-zontal velocity derivatives, however, can cause errors inVnmo and η. Still, Grechka’s (1998) results are limited toa single horizontal layer and cannot be directly appliedto models with thin lenses.

Recently, isotropic traveltime tomography has beenused to estimate the velocity inside the lens and re-move the lens-induced velocity errors (Fujimoto et al.,2007; Fruehn et al., 2008). These case studies showthe importance of integrating seismic and geologic in-formation and understanding the relationship betweenthe overburden heterogeneity and velocity errors. Inprinciple, laterally varying anisotropy parameters canbe estimated from anisotropic reflection tomography(e.g. Woodward et al., 2008). However, if the lens lo-cation is unknown, lens-induced traveltimes shifts canhinder accurate parameter estimation on the scale ofspreadlength.

Here, we study the influence of velocity lenses onnonhyperbolic moveout inversion for 2D VTI models. Toanalyze lens-induced distortions of reflection data, weperform finite-difference modeling and apply moveoutinversion using the Alkhalifah-Tsvankin (1995) nonhy-perbolic equation. We show that even a relatively thinvelocity lens may cause pronounced errors in the move-out parameters Vnmo and η and describe several crite-ria that can help identify range of common-midpoint(CMP) locations, for which reflected rays cross thelens. To remove lens-induced traveltime shifts, we pro-pose a correction algorithm designed for gently dip-ping anisotropic layers. Synthetic tests demonstratethat this algorithm suppresses lens-related distortionson the stacked section and substantially reduces errorsin the effective and interval parameters Vnmo and η.

Figure 1. Time-migrated section from the central North Sea(after Armstrong et al., 2001). Amplitude anomalies at thebottom of the channel-like structures (arrows) and pull-upanomalies below the structures (inside the rectangles) indi-cate the presence of lateral heterogeneity associated with thechannel fills. Pull-up and push-down anomalies caused byhigh and low velocities, respectively, in channels or carbon-ate reefs are also observed in other hydrocarbon-producingregions, such as the Middle East and Northwest Australia.

2 DISTORTIONS CAUSED BY VELOCITYLENSES

To generate synthetic data, we perform finite-differencesimulations (using Seismic Unix code suea2df ; Juhlin,1995) and ray tracing for 2D models that include a low-velocity isotropic lens inside a VTI layer. The parame-ters Vnmo and η are estimated from nonhyperbolic move-out inversion based on the Alkhalifah-Tsvankin (1995)equation:

t2 = t20 +x2

V 2nmo

− 2η x4

V 2nmo[t

20 V 2

nmo + (1 + 2η)x2], (3)

where t is the P-wave traveltime as a function of theoffset x and t0 is the zero-offset time. Equation 3 canbe applied to layered VTI media with the effective pa-rameters given by (Tsvankin, 2005):

V 2nmo(N) =

1

t0(N)

NXi=1

(V (i)nmo)

2 t(i)0 , (4)

η(N) =1

8

1

V 4nmo(N) t0(N)

"NX

i=1

(V (i)nmo)

4(1 + 8η(i)) t(i)0

#

−1

ff, (5)

where t(i)0 , V

(i)nmo, and η(i) are the interval values, and N

is the number of layers.Although the Alkhalifah-Tsvankin equation pro-

vides a good approximation for P-wave moveout in VTImedia, the estimates of η are sensitive to correlated trav-eltime errors because of the tradeoff between η and Vnmo

Correction for lenses in nonhyperbolic inversion 385

Figure 2. Single VTI layer with an isotropic lens. Thelens velocity is 3 km/s; the background parameters areV0 = 4 km/s, δ = 0.07 and ε = 0.16. Points A, B and Ccorrespond to CMP locations discussed in the text. The testis performed for a spreadlength of 4 km; the target depth is2 km.

(Grechka & Tsvankin, 1998). In our model, such errorsare caused by an isotropic velocity lens in the overbur-den.

2.1 Single-layer model

First, we consider a rectangular lens embedded in a ho-mogeneous VTI layer (Figure 2). The section in Fig-ure 3 is computed by a finite-difference algorithm forcommon-midpoint (CMP) gathers outside the lens (lo-cation A) and at the center of the lens (location B).Whereas the lens does not distort traveltimes at loca-tion A, it causes a near-offset time delay of 17 ms andwaveform distortions (related to the influence of the sideand edges of the lens) in the mid-offset range at locationB.

Using equation 3, we find the best-fit Vnmo and η forthe target reflector from a 2D semblance scan; for com-parison, we also perform conventional hyperbolic move-out inversion (Figure 4). The NMO velocity estimatedfrom the nonhyperbolic equation at location A is closeto the analytic value. At location B, however, Vnmo isabout 10% greater, although the exact effective NMOvelocity should decrease by 2% due to the low veloc-ity inside the lens. At a CMP location near the edgeof the lens (location C), Vnmo is 7% smaller than theexact value. Interestingly, nonhyperbolic moveout inver-sion produces an error in Vnmo, which is two times largerthan that obtained from hyperbolic moveout analysis.

The reason for the lens-induced distortion in Vnmo

is described in Al-Chalabi (1979) and Biondi (2006)(Figures 5a,b). Near-offset rays at location B pass twicethrough the lens, while far-offset rays miss the lens com-pletely. Since the lens has a lower velocity, this leads to asmaller traveltime difference between the near- and far-offset traces and, therefore, a higher NMO velocity. Incontrast, for location C, the lens is missed by near-offset

(a)

(b)

Figure 3. Comparison of CMP gathers for the model fromFigure 2 computed (a) outside the lens (location A) and(b) above the center of the lens (location B) with a finite-difference algorithm. The traveltime shifts at near offsets (ar-rows) at location B cause significant errors in the parametersVnmo and η. The lens-related waveform distortion at locationB is contoured by the ellipse.

rays and the traveltime difference between the near andfar offsets becomes larger, which reduces Vnmo (Figure5b).

The nonhyperbolic inversion gives a closer approxi-mation to the actual traveltime due to the contributionof the additional parameter η. Hence, the best-fit non-hyperbolic moveout curve at location B reproduces theincrease in the near-offset traveltime, which causes apronounced deviation of the estimated Vnmo from theexact value (Figure 5c). Note that the hyperbolic cor-rection distorts the velocity Vnmo outside the lens dueto the influence of nonhyperbolic moveout.

The laterally varying η-curve resembles the re-versed version of the Vnmo-curve. While in the absence


(a)

(b)

Figure 4. Lateral variation of the inverted (a) Vnmo and (b)η (bold solid lines) along the line using a spreadlength of 4 km(the spreadlength-to-depth ratio X/D = 2). The dashedline on plot (a) is the NMO velocity obtained from hyper-bolic moveout analysis for the same spreadlength. The exacteffective Vnmo (equation 4) and η (equation 5) are markedby thin solid lines.

of the lens the effective η at location B should almostcoincide with the background η = 0.08, the estimatedη = –0.07 is much smaller. The understated value of ηis explained by the need to compensate for the over-stated estimate of Vnmo in reproducing traveltimes atmoderate and large offsets (Grechka & Tsvankin, 1998;Tsvankin, 2005). The magnitude of the variation (thedifference between the largest and smallest values) in ηalong the line is close to 0.3.

2.2 Dependence of distortions on the lensparameters

Using ray-traced synthetic data, we investigate the de-pendence of the inverted moveout parameters on the ve-locity, width, and depth of the lens. The replacement offinite differences with ray tracing does not significantlychange the inversion results.

As expected, the magnitude of the errors in Vnmo

(a)

(b)

(c)

Figure 5. (a) Schematic picture of near- and far-offset ray-paths from a horizontal reflector beneath a low-velocity lensat three CMP locations (modified from Biondi, 2006). (b)The influence of the lens on the moveout curves. The ray-paths and moveout curves at locations A, B and C are shownby dotted, solid and dashed lines, respectively. (c) Schematicestimated moveout curves (dotted lines) obtained from hy-perbolic (left) and nonhyperbolic (right) inversion at locationB. The actual moveouts at locations A and B are shown bythin and bold solid lines, respectively.

and η is proportional to the velocity contrast betweenthe lens and the background (Figure 6a). When thespreadlength is fixed, the time distortions depend onthe ratio W/L′, where W is the width of the lens andL′ is the maximum horizontal distance between the in-cident and reflected rays at the lens depth (Figure 5a).Note that L′ decreases with increasing lens depth. Forthe model used in the test, the distortions in Vnmo andη are largest when the width of the lens is 0.5 km (orW/L′ = 0.25) (Figure 6b). On the other hand, the errorin Vnmo estimated from hyperbolic moveout inversionhas a flat maximum for the width ranging from 0.5 kmto 1.5 km.


(a)

(b)

(c)

Figure 6. Dependence of the magnitude of the variation in Vnmo (left) and η (right) on (a) the velocity contrast defined as(Vlens − Vback)/Vback, where Vlens is the lens velocity and Vback is the background velocity, (b) the width and (c) the depth ofthe lens. Vnmo is obtained from nonhyperbolic (solid lines) and hyperbolic (dashed lines) moveout inversion. The spreadlengthis 4 km (offset-to-depth ratio X/D = 2).

Since W/L′ at the surface is 0.25 (W = 1 km,spreadlength is 4 km) in this model, a shallower lenscauses larger errors in Vnmo and η (Figure 6c). For adepth of 0.25 km (W/L′ = 0.29), the errors are close tothe largest distortions for the test in Figure 6b.

2.3 Identifying lens-induced distortions

Identifying the range of CMP locations influenced bythe lens is critical for avoiding the use of distorted pa-rameters. It is clear from the above results that largevariations of Vnmo and η on the scale of spreadlengthare strong indications of the lens. Using the single-layerlens model, we suggest two additional indicators of the


(a)

(b)

Figure 7. Semblance value for moveout-corrected gathersbefore (dashed line) and after (solid line) applying trim stat-ics. The data were contaminated by random noise with thesignal-to-noise ratio equal to (a) 10 and (b) 5.

lens – residual moveout after application of nonhyper-bolic moveout correction and the dependence of Vnmo

and η on spreadlength.The moveout curve distorted by the lens cannot

be completely flattened by the nonhyperbolic move-out equation. To estimate the magnitude of the resid-ual moveout, one can use so-called trim statics (Ursen-bach & Bancroft, 2001). Trim statics involves cross-correlation between a near-offset trace and all offsettraces, which helps evaluate the statics shifts neededto eliminate the residual moveout. Due to the presenceof residual moveout in the area influenced by the lens,application of trim statics increases the semblance (Fig-ure 7). Still, the semblance value after trim statics atlocation B is lower than that at location A because ofthe lens-induced waveform distortions.

Trim statics, however, may not perform well whenthe data contain random or coherent noise (Ursenbach& Bancroft, 2001). If the signal-to-noise (S/N) ratiois less than five, trim statics increases the semblanceby aligning noise components in the statics-correctedgather (Figure 7). Thus, trim statics can be used to de-lineate the area influenced by the lens only for relativelyhigh S/N ratios.

(a)

(b)

Figure 8. Dependence of (a) Vnmo and (b) η on thespreadlength-to-depth ratio (X/D).

Another possible lens indicator is the variation ofthe moveout parameters with spreadlength. As shownin Figure 8, the shape of the Vnmo- and η- curvesis highly sensitive to the spreadlength-to-depth ratio(X/D). In contrast, the estimated moveout parametersat location A outside the lens are weakly dependent onspreadlength.

2.4 Layered model

The conclusions drawn above remain valid for a morerealistic, layered model containing a parabola-shapedlens, which causes a maximum time distortion (or push-down anomaly) of 18 ms (Figure 9). We generate syn-thetic data with finite-differences and apply nonhyper-bolic moveout inversion for the two interfaces (A andB) below the lens (Figure 10).

For a spreadlength of 4 km, the maximum distor-tion (the maximum deviation from the exact value) inVnmo reaches approximately 9% for interface A and 11%for interface B (Figure 10), while the distortion in η


(a)

(b)

Figure 10. Lateral variation of estimated Vnmo (left) and η (right) for the model from Figure 9 for (a) interface A and (b)interface B. The dashed lines correspond to a spreadlength of 4 km, solid lines [only on plot (b)] to a spreadlength 6 km. Thethin solid lines mark the exact parameters.

Figure 9. Layered model with a parabola-shaped lens.The first layer is isotropic and vertically heterogeneous; V0

changes from 1.5 km/s at the surface to 2.5 km/s at the1 km depth. The second layer is homogeneous VTI withV0 = 3.5 km/s, δ = 0.07 and ε = 0.16 and contains anisotropic lens with V0 = 2.7 km/s. The maximum thicknessof the lens is 100 m. The third layer is homogeneous VTIwith V0 = 4.2 km/s, δ = 0.05 and ε = 0.1.

reaches 0.15 and 0.33, respectively. The larger errors forinterface B are related to its lower ratio X/D.

When we use a spreadlength of 6 km (X/D = 2),the distortions in Vnmo and η for interface B decreaseto 5% and 0.08, respectively. As is the case for a ho-mogeneous background medium, the moveout-correctedgather exhibits residual moveout in the area influencedby the lens (Figure 11). Thus, the presence of residualmoveout and the dependence of the moveout parame-ters on the spreadlength can serve as lens indicators forlayered media as well.

3 CORRECTION ALGORITHM

It is clear from the modeling results that even a thin lenscan cause significant errors in the parameters Vnmo andη. Another serious lens-induced distortion is the push-down anomaly on the stacked time section (Figure 12a).Although the time anomaly becomes smaller if the stackis produced using the background moveout parametersestimated away from the lens, the stacked event thenhas a smaller power because of a larger residual moveout


Figure 11. Moveout-corrected gathers computed using thebest-fit parameters Vnmo and η. Residual moveout is ob-served inside the area marked by the dashed line.

(a)

(b)

Figure 12. Stacked section generated using (a) the best-fitmoveout parameters and (b) the background parameters.

(Figure 12b). Clearly, it is desirable to produce an accu-rate stacked section without reducing the stack power.Here, we introduce two methods for correcting P-wavedata from layered VTI media for the influence of thelens. One of them is designed to mitigate the distor-tions on the stacked section using trim statics. The other

method makes it possible to remove the traveltime dis-tortions from each recorded trace and, therefore, obtainboth accurate moveout parameters and a high-qualitystack.

3.1 Trim statics

By eliminating residual moveout, trim statics makes alltraces kinematically equivalent to the zero-offset trace(Figure 13a). Thus, trim statics increases stack powerand generates a stack that kinematically reproducesthe zero-offset section (compare Figure 13b with Fig-ure 12a).

To remove the zero-offset time distortion, we as-sume that the zero-offset raypath is not influenced bythe lens and remains vertical for all horizontal inter-faces. Then the distortion of t0 should be the same atinterfaces A and B. This assumption allows us to usethe estimated push-down at interface A for correctingthe time distortions for both interfaces. The resultingstacked section is kinematically correct and has a highstack power (Figure 13c). However, as discussed above,trim statics works only for high S/N ratios and cannotbe used to estimate the background values of Vnmo andη.

3.2 Prestack traveltime shifts

3.2.1 Method

The correction algorithm discussed here is designed fora horizontally layered overburden containing the lens,but the target reflector can be dipping or curved. Unlikethe statics correction, this technique involves computa-tion of traveltime shifts as functions of offset and targetdepth (Figure 14a). As the input data we use the zero-offset time shifts (“pull-up” or “push-down” anomalies,∆t0) for the horizontal reflector immediately below thelens. The lens-related perturbation of the raypath isassumed to be negligible, so that the ray in the layercontaining the lens can be considered straight. Thenthe ray crossing the lens can be reconstructed using thevelocity-independent layer-stripping method (VILS) ofDewangan & Tsvankin (2006).

VILS builds the interval traveltime-offset functionby performing kinematic downward continuation of thewavefield without knowledge of the velocity model. Eachlayer in the overburden is supposed to be laterally ho-mogeneous with a horizontal symmetry plane, so thatthe raypath of any reflection event is symmetric withrespect to the reflection point. The bottom of the tar-get layer, however, can be curved and the layer itselfcan be heterogeneous. Wang & Tsvankin (2009) showthat VILS provides more robust estimates of the intervalmoveout parameters in VTI and orthorhombic modelsthan Dix-type equations.

VILS can be applied to our model under the as-


(a)

(b) (c)

Figure 13. (a) Moveout-corrected gather after application of trim statics; (b) the stacked section after trim statics, and (c)the stacked section from plot (b) after removing the push-down anomaly at interface A.

sumption that the raypath in the overburden is notdistorted by the lens. The idea of VILS is to identifyreflections from the top and bottom of a certain layerthat share the same upgoing and downgoing ray seg-ments. This is accomplished by matching time slopeson common-receiver and common-source gathers. Ap-plication of VILS to the reflections from the target andtop of the layer containing the lens yields the horizontalcoordinates xT1 and xR1 (Figure 14a). Likewise, the co-ordinates xR2 and xT2 are estimated by combining thetarget event with the reflection from the bottom of thelayer containing the lens.

Under the straight-ray assumption, we find the hor-izontal coordinates of the crossing points and the rayangles (Figures 14a,b):

xT L = xT1 +z′

T L(xT2 − xT1)

z, (6)

xRL = xR1 −z′

RL(xR1 − xR2)

z, (7)

cos θT L =zp

(xT2 − xT1)2 + z2

, (8)

cos θRL =zp

(xR1 − xR2)2 + z2

, (9)

where z is the thickness of the layer with the lens, andz′

T Land z

′

RLare the distances from the lens to the top

of the layer at locations xT L and xRL , respectively.

If the lens produces a sufficiently strong reflectionand the layer is vertically homogeneous, the ratio z′/zcan be estimated from the corresponding zero-offsettraveltimes (t′/t). In the layered model, we can clearlyidentify the lens reflection at t = 1.15 s on the stackedsection (Figure 15a). This indicates that the horizontalcoordinates and the ray angles can be estimated withoutcomplete information about the velocity and anisotropyparameters. Then the total lens-related traveltime shiftfor the target event (∆tta) can be computed as

∆tta =1

2

„∆t0 (xT L , 0)

cos θT L

+∆t0 (xRL , 0)

cos θRL

«, (10)

where ∆t0(xT L , 0) and ∆t0(xRL , 0) are the zero-offsettime distortions below the lens at locations xT L and


(a)

(b)

Figure 14. (a) Ray diagram of the correction algorithm.The horizontal coordinates xT1 , xT2 and xR1 , xR2 are deter-mined from the velocity-independent layer-stripping method.(b) Upgoing ray segment crossing the lens. Using the valuesof z′

RLand z, we can compute the horizontal location of the

crossing point (xRL ) and the ray angle (θRL ).

xRL , respectively. Both ∆t0(xT L , 0) and ∆t0(xRL , 0)can be estimated from the near-offset stack. The rayangles θT L and θRL do not have to be the same, whichmakes the algorithm suitable for dipping or curved tar-get reflectors.

After the correction, the kinematics of the prestackdata should be close to the reflection traveltime de-scribed by the background values of Vnmo and η. The in-terval parameters V

(i)nmo and η(i) can be computed using

the layer-stripped data corrected for the lens-inducedtime shifts. The removal of the time distortions alsohelps generate an accurate stacked section.

3.2.2 Synthetic test

The prestack correction algorithm is tested here on thelayered model from Figure 9. First, we need to estimatethe three required input quantities: ∆t0, the ratio z′/z,and the thickness z of the layer containing the lens. Thevalues of ∆t0 (Figure 15b) and z′/z (t′/t) are obtainedfrom the near-offset stacked section (Figure 15a). Forpurposes of this test, the thickness of the layer contain-ing the lens is assumed to be known.

Application of traveltime shifts computed fromequation 10 eliminates the time-varying push-down

(a)

(b)

Figure 15. (a) Near-offset stacked section obtained for theoffset range from 0 to 200 m, and (b) the magnitude of thepush-down anomaly (solid line) estimated by picking themaximum amplitude along interface A. The dotted line in(b) is the exact ∆t0.

anomaly and increased the S/N ratio of the stackedsection (Figure 16a). Also, the correction significantlyreduces the residual moveout in the moveout-correctedgathers (Figure 16b) and the errors in the effectiveparameters Vnmo and η (Figure 17a). For interface B,the distortion in Vnmo decreases from 5% to less than1%, and in η from 0.08 to 0.02. Figure 17b shows thatthe correction algorithm also produces much more ac-curate interval parameters Vnmo and η estimated fromthe layer-stripped data. The remaining errors are largelycaused by the straight-ray assumption for the layer con-taining the lens.

It is important to evaluate the sensitivity of the pa-rameter estimation to errors in the input data. Exten-sive testing shows that when the error in ∆t0 is smallerthan 25%, the moveout-corrected gather is almost flat.To test the sensitivity to the ratio z′/z, we move thelens down by 100 m and 200 m, which corresponds to10% and 20% errors in z′/z. Although distortions in themoveout-corrected gather become noticeable when theerror reaches 20%, the magnitude of the residual move-out is still much smaller than that before the correction.


(a)

(b)

Figure 16. (a) Stacked section and (b) moveout-correctedgathers obtained after applying prestack traveltime shiftsthat compensate for the influence of the lens.

Finally, a thickness error up to 20% proves to have lit-tle impact on the output of the correction algorithm.An accurate stacked section can be generated even forsomewhat larger errors in these input quantities.

4 DISCUSSION

The correction algorithm requires knowledge of the zero-offset time anomaly ∆t0, the ratio z′/z and the thicknessz of the lens-containing layer. In the synthetic test, ∆t0was accurately estimated from the push-down anomalyon the near-offset stacked section, and the ratio z′/z wasobtained from the corresponding time ratio t′/t usingthe reflection from the lens (Figure 15). Since depthuncertainty seldom exceeds 20% in practice, errors inz are not expected to have a significant impact on thecorrection results.

The suggested approach should be applicable tomany field data sets. For example, the time section fromthe central North Sea in Figure 1 contains channel-likestructures and pull-up anomalies (marked area in Figure1), which indicate the presence of high-velocity chan-

nel fills (Armstrong et al., 2001). The lens reflectionsare sufficiently strong for estimating the ratio t′/t (and,therefore, z′/z), and the pull-up time anomaly can beaccurately measured as well.

Our algorithm can also be applied to layered mediawith multiple lenses, if it is possible to estimate the val-ues of ∆t0 and z′/z for each lens separately. Then thetotal traveltime shifts are obtained by summing the in-dividual lens-induced time distortions. However, the al-gorithm will produce distorted time shifts when a layercontains multiple lenses or the lens reflections cannotbe identified. Also, the algorithm assumes a laterallyhomogeneous overburden and straight rays in the layerscontaining the lenses. Therefore, the correction may be-come inaccurate when the overburden includes dippinginterfaces or has a strong velocity contrast between thelens and the background.

5 CONCLUSIONS

We demonstrated that a relatively thin velocity lensmay cause significant, laterally varying distortions inthe moveout parameters Vnmo and η estimated fromnonhyperbolic moveout analysis. The magnitude of thedistortion depends on the width and depth of the lensand is proportional to the velocity contrast between thelens and the background. The error in Vnmo is largerafter nonhyperbolic moveout inversion compared withthe conventional hyperbolic algorithm applied for thesame spreadlength, particularly when the lens is narrowor is located in a shallow layer. Hence, although non-hyperbolic moveout analysis produces smaller residualmoveout and higher stacking power than the hyperbolicequation, it does not guarantee a more accurate estima-tion of NMO velocity in the presence of lateral hetero-geneity.

Identifying the area influenced by the lens is crit-ical for avoiding use of distorted moveout parameters.We showed that the residual moveout can serve as a lensindicator because the lens-induced distortion cannot becompletely removed by nonhyperbolic moveout inver-sion. The presence of residual moveout can be identifiedfrom the increase in semblance after application of trimstatics, provided the signal-to-noise ratio is sufficientlyhigh. A lens also manifests itself by making the moveoutparameters strongly dependent on spreadlength and thelateral coordinate.

To correct for lens-induced traveltime shifts onprestack data, we developed an algorithm based onvelocity-independent layer stripping (VILS). Synthetictests confirmed that the algorithm successfully removeslens-induced distortions on the stacked section and sub-stantially reduces the errors in the effective and intervalparameters Vnmo and η. The correction requires esti-mates of the zero-offset time distortion ∆t0, the thick-ness z of the layer containing the lens and the ratio z′/z,where z′ is the distance between the lens and the top of


(a)

(b)

Figure 17. (a) Inverted effective parameters Vnmo (left) and η (right) for interface B before (dashed line) and after (thicksolid line) applying the correction algorithm (X/D = 2). (b) The interval parameters Vnmo and η in the third layer (2-3 km)estimated before (dashed line) and after (thick solid line) applying the correction algorithm. The spreadlength (before applyingVILS) is 6 km. Thin solid lines mark the exact parameters.

the lens-containing layer. The parameters ∆t0 and z′/zcan be obtained from reflection data, while z cannot befound without additional (e.g., borehole) information.However, errors up to 20% in the ∆t0 and z, as wellas a 10% error in the ratio z′/z, do not significantlyhamper the performance of the algorithm.

Although we presented the correction method fora 2D model that contains a single lens, it can be ex-tended to wide-azimuth data from layered media withwell-separated multiple lenses. Potentially, the 3D ver-sion of the algorithm can be used to correct for the influ-ence of small-scale lateral heterogeneities on azimuthalmoveout inversion.

6 ACKNOWLEDGMENTS

We are grateful to members of the A(nisotropy)-Teamof the Center for Wave Phenomena (CWP), ColoradoSchool of Mines (CSM), for helpful discussions. Thiswork was supported by Japan Oil, Gas and MetalsNational Corporation (JOGMEC) and the Consortium

Project on Seismic Inverse Methods for Complex Struc-tures at CWP.

REFERENCES

Al-Chalabi, 1979, Velocity determination from seismicreflection data , Applied Science Publishers.

Alkhalifah, T., and Tsvankin, I., 1995, Velocity anal-ysis for transversely isotropic media: Geophysics, 60,1550–1566.

Alkhalifah, T., 1997, Velocity analysis using nonhyper-bolic moveout in transversely isotropic media: Geo-physics, 62, 1839–1854.

Armstrong, T., McAteer, J., and Connolly, P., 2001,Removal of overburden velocity anomaly effects fordepth conversion: Geophysical Prospecting, 49, 79–99.

Biondi, B. L., 2006, 3D seismic imaging , Society ofExploration Geophysicists.

Blias, E., 2009, Stacking velocities in the presence of


overburden velocity anomalies: Geophysical Prospect-ing, 57, 323–341.

Dewangan, P., and Tsvankin, I., 2006, Velocity-independent layer stripping of PP and PS reflectiontraveltimes: Geophysics, 71, no. 4, U59–U65.

Fruehn, J., Jones, I. F., Valler, V., Sangvai, P., Biswal,A., and Mathur, M., 2008, Resolving near-seabed ve-locity anomalies: Deep water offshore eastern India:Geophysics, 73, no. 5, VE235–VE241.

Fujimoto, M., Takanashi, M., Szczepaniak, M., andYoshida, T., 2007, Application of prestack depth mi-gration across the Ichthys field, Browse basin: ASEGAnnual Meeting, Expanded Abstracts.

Grechka, V., and Tsvankin, I., 1998, Feasibility of non-hyperbolic moveout inversion in transversely isotropicmedia: Geophysics, 63, 957–969.

Grechka, V., 1998, Transverse isotropy versus lateralheterogeneity in the inversion of P-wave reflectiontraveltimes: Geophysics, 63, 204–212.

Jenner, E., 2009, Data example and modelling studyof P-wave azimuthal anisotropy potentially caused byisotropic velocity heterogeneity: First Break, 27, 45–50.

Juhlin, C., 1995, Finite-difference elastic wave propa-gation in 2D heterogeneous transversely isotropic me-dia: Geophysical Prospecting, 43, 843–858.

Lynn, W. S., and Claerbout, J. F., 1982, Velocity es-timation in laterally varying media: Geophysics, 47,884–897.

Takanashi, M., Kaneko, M., Monzawa, N., Imahori,S., Ishibashi, T., and Sakai, A., 2008, Removal ofoverburden channel effects through channel velocitymodeling and prestack depth migration for an oilfield offshore Abu-Dhabi: Abu Dhabi InternationalPetroleum Exhibition and Conference, SPE 117982.

Toldi, J., 1989, Velocity analysis without picking: Geo-physics, 54, 191–199.

Tsvankin, I., and Thomsen, L., 1994, Nonhyperbolicreflection moveout in anisotropic media: Geophysics,59, 1290–1304.

Tsvankin, I., 2005, Seismic signatures and analysis ofreflection data in anisotropic media , 2nd ed, ElsevierScience Publishing Company, Inc.

Ursenbach, C. P., and Bancroft, J. C., 2001, Playingwith fire: Noise alignment in trim and residual statics:71th Annual Internatinal Meeting, SEG, ExpandedAbstracts, 1973-1976.

Wang, X., and Tsvankin, I., 2009, Estimation of inter-val anisotropy parameters using velocity-independentlayer stripping: Geophysics, 74, no. 5, WB117–WB127.

Woodward, M. J., Nichols, D., Zdraveva, O., Whitfield,P., and Johns, T., 2008, A decade of tomography:Geophysics, 73, no. 5, VE5–VE11.

correction for the inﬂuence of velocity lenses on

Documents