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Answer 1

D i ff e rential interference results from the fact that neighboringreflectors increasingly interfere as the incidence anglei n c reases. When the reflectors come from the top and bottomof a thin bed of interest, the interference is called off s e t -dependent tuning. This tuning will cause false amplitude vari-ations with offset (AVO), not associated with either individualre f l e c t o r. When viewed in moved-out gathers, these eff e c t sappear to be the result of a stretched wavelet at larger off s e t s .

The following remedies to this problem have been proposed:

Rupert and Chun (1975) brought short segments of datainto alignment by constant time shifts. AVO analysis couldthen be applied to the shifted data without wavelet stretch.Differential interference still remained from events outside

the segments.

Byun and Nelan (1997) processed moved-out gathers witha time-varying filter to transform the stretched wavelet into

the unstretched one. This procedure generated a moved-out gather without wavelet stretch but amplified ambient

noise, sometimes to unbearable levels.

Castoro et. al. (2001) removed wavelet stretch from moved-out data by transforming it in the frequency domain. Thismethod strictly applies only to a relatively short window ofdata, since filtering in the Fourier domain is time-invariant,

but wavelet stretch is not. As the window length isdecreased, edge effects become more severe.

Trickett (2003) proposed a method of stretch-free stacking,which when applied to partial-offset gathers could be usedfor AVO analysis, even when reflection events cross. Theapplicability of this method for AVO analysis is still beingevaluated.

In the remainder of my reply, I will describe a fifth method,which generates a stretch-free AVO gradient, as opposed to astretch-free gather. It does this by estimating the contribution

12 CSEG RECORDER December 2004

Continued on Page 13

Geophysicists who have used AVO analysis for confirmation/detection of anomalies in their prospects, have

often tried to understand the factors that affect the pre-stack seismic amplitudes and attempted to compensate

for such effects. Amongst others, differential interference and offset tuning are two important effects that the

p re-stack data needs to be compensated for. These issues form the question for the Expert Answers c o l u m n

this month.

The Experts answering this question are familiar names in the seismic world, Herbert Swan(ConocoPhillips, Alaska) Roy White (Consultant, U.K.) and Jon Downton (Veritas, Calgary). We thank them

for sending in their responses to our question. The order of the responses given below is the order in which

they were re c e i v e d .

Satinder Chopra

Q . Differential interference and offset-dependent tuning are two serious factors that hamper confident AVOa n a l y s i s. What causes them and how do we effectively tackle them today?

F i g u re 1. Waveform and amplitude spectrum of a bandpass wavelet.

F i g u re 2. Top plot (a): The bandpass wavelet, w ( t ), (black), and the leakagewavelet BL( t ) ( red). Bottom plot (b): The optimal filter for estimating the noise-

f ree intercept in the presence of white noise, h1( t ) (black) and the optimal filterfor estimating the stretch error in the gradient, h2( t ) ( re d ) .


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December 2004 CSEG RECORDER 13

to the gradient from diff e rential interference, and thensubtracting it. Optimal performance is achieved in the presenceof a known noise spectrum.

D i ff e rential interference manifests itself as l e a k a g e f rom thenormal-incidence reflectivity series, a(t), to the AVO gradient.For a wavelet w(t) and stacking velocity Vs(t), this leakage is

approximated by

1. BL(t) = -{a(t) * [t w(t)] } [1 + 2tVs(t)/Vs(t)] / 2,

where * denotes convolution, and the prime denotes differen-tiation (Swan 1991). Given an AVO intercept trace,A(t)=a(t)*w(t), we can estimate the component of the gradientdue to differential interference using

2. BL(t) = -[A(t) * h2(t)] [1 + 2tVs(t)/Vs(t)] / 2,

where h2(t) is a linear filter which estimates the gradient error

due to stretch, from the intercept. Thestacking velocity should be smoothedto avoid sudden changes in Vs(t). Inthe presence of an intercept noises p e c t rum Sn(), the Fourier trans-form of the optimal h2 filter, in theleast-squares sense, is given by

3. H2() = -[ |W() |2 + W * () W ()] /[ |W() |2 + Sn() ] .

Note that this expression is invariantto a wavelet phase shift, and is stableeven when the wavelet Fourier trans-form W() vanishes. The filter h2(t)can also be obtained in the timedomain by a Levinson re c u r s i o n(Swan 1997).

For the case of the bandpass waveletwhose waveform and spectrum areshown in Figure 1, the gradientstretch error with a constant stackingvelocity, BL(t ), is shown as the redcurve in Figure 2a. This error iscaused by the wavelet eff e c t i v e l y

being stretched at large offsets. Theerror is zero at the wavelet center. The

red curve of Figure 2b re p resents the optimal h2(t) f i l t e r,computed assuming 1% white noise, whose spectrum is given byequation (3). Also shown in Figure 2b is the optimal h1(t) filter,which estimates the noise-free intercept in the presence of noise.Its Fourier transform is given by

4. H1() = |W()|2/ [ |W()|2 + Sn()].

The red dashed curve of Figure 3 is the result of convolving w ( t )with h2( t ). It closely approximates the leakage wavelet BL(t). A l s oshown is the result of convolving w(t) with h1(t). It closelya p p roximates w ( t ). Although not terribly important in thisexample, this filter applies the same noise-reduction regimen tothe intercept as to the gradient. This will ensure their spectra willmatch, and hence optimize the coherency of their cro s s - p l o t s .

Figure 4 shows a synthetic CMP gather formed from a 50 ftsection of 2.3 g/cm3 material embedded into a constant2.5 g/cm3 substrate. Neither the acoustic velocity (10 kft/s) nor

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F i g u re 3. The actual and estimated noise-free wavelets (black) and the actual andestimated gradient leakage wavelets (re d ) .

F i g u re 5. An AVO crossplot without gradient stretch correction. Various false AVO anomalies are appare n t .

F i g u re 4. A synthetic CMP gather that illustrates offset-dependent tuning.


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the shear velocity (5 kft/s) variesthrough this model. Such a density-only contrast is expected to produce a

background AVO response. Using 28Hz as the wavelet center frequency,this bed is below the tuning thicknessof 89 ft. Wavelet stretch is noticeableout to the farthest offset, which corre-

sponds to an angle of about 45.

AVO intercept A(t) and gradient B(t)were computed from this gather via aleast-squares fit at each time to theequation

5. S(t, ) = A(t) + B(t)sin2 + C(t)sin2tan2,

where S(t, ) is the synthetic gatherand is the incidence angle. Figure 5shows a cross-plot of this gradientversus intercept, both filtered by h1(t).The color of the dots corresponds to

time. The A- B plane is subdividedinto regions that correspond tocommonly used AVO classifications(Castagna and Swan, 1997), as shown.The left side of this figure shows the

i n t e rcept trace, repeated five times. The background colorsmatch those of the AVO classifications. Although the central topand base reflectors correctly indicate background (gray) reflec-tors, there are prominent false AVO anomalies as far away as 30ms (150 ft) from the central lobes.

After the portion of the gradient BL(t) due to differential inter-ference is subtracted from the gradient obtained from equation(5) and cross-plotted with the intercept, the result is shown in

Figure 6. Now the two traces are much more tightly coupled, andthe false AVO anomalies are removed. The hodogram barelygrazes the class 1 top polygon, but other than that, correctlyremains in background territory.

If the thickness of the low-density zone is varied from 100 ft to 10ft, the results are shown in Figures 7 and 8. Figure 7 shows theintercept trace and apparent AVO classification as a function ofwedge thickness, when differential interference is not removed.Spurious AVO anomalies are at their worst at around half thetuning thickness (/8). Figure 8 shows the improved result whendifferential interference is removed.

References

Byun, Bok S. and Nelan, E. Stuart, 1997,Method and system for correcting seismic tracesfor normal move-out correction, U. S. Patent 5,684,754.

Castagna, John P. and Swan, Herbert W., 1997, Principles of AVO crossplotting, TheLeading Edge, 16, No. 4, pg. 337-342.

Castoro, Alessandro, White, Roy E. and Thomas, Rhodri D., 2001, Thin-bed AVO:Compensating for the effects of NMO on reflectivity sequences, Geophysics, 66, No. 6, pg.1714-1720.

Rupert, G. B. and Chun, J. H., 1975, The block move sum normal moveout correction,Geophysics, 40, No. 1, pg. 17-24.

Trickett, Stewart R., 2003, Stretch-free stacking, 73rd Ann. Internat. Mtg. Soc.Exploration Geophysicists, pg. 2008-2011.

Swan, Herbert W., 1991,Amplitude-versus-offset measurement errors in a finely layeredmedium, Geophysics, 56, No. 1, pg. 41-49.

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F i g u re 8. With offset-dependent tuning removed, the false AVO anomalies disappear.

F i g u re 7. A wedge plot without gradient stretch removal that shows coherent falseAVO anomalies above and below the target event.

F i g u re 6. The same crossplot after gradient stretch removal. A much more accurate picture emerg e s .


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________________1997, Removal of offset-dependent tuning in AVO analysis, 67th Ann.Internal Mtg. Soc Exploration Geophysicists, pg. 175-178.

Herbert Swan

ConocoPhillips, Alaska

Answer 2

Differential interference is a universal affliction of reflection seis-mology. The separation of reflectors in depth is generally much

less than the dominant wavelength of the waveforms that returnto the recorders. In general too, reflector spacing varies laterally.The consequence is that the primary reflection signal consists ofa multitude of interfering reflection pulses, or seismic wavelets,that produce images of the subsurface that are dominated bydifferential interference. Although occasionally a reflection may

be considered for practical purposes an isolated reflection, it isdifferential interference that is the norm. In the offset domain,differential interference is again the norm for the simple reasonthat normal moveout curves are rarely parallel. So the net wave-form from two or more neighboring reflectors varies with offset.

One could also cite differential interference from multiple reflec-tions. Although that has serious consequences for AVO analysis,

it isnt really what the question is about. For AVO analysis, onehas to start in the offset domain in order to explain the effect ofdifferential interference and offset-dependent tuning on an AVOresponse. To do that I first consider the archetypal example of theAVO response of a thinning bed encased in a uniform shale. Thatleads into the impact of NMO stretch, tuning and thin beds onAVO inversion. I conclude with some remarks about AVO inver-sion and layered inversions that may provoke further comment.

AVO response of a thin bed

Figure 1 shows a rock physics model of a sandstone sandwichedwithin a shale, based on a reservoir in the central North Sea. Thesandstone parameters shown are for the water leg. In the gas legthey become V

P=1638 m/s, V

S= 862 m/s and the density is 1784

kg/m3. The top of the sandstone is at 1050 m, or 0.95 s two-waytime. At this two-way time the NMO velocity is 1925 m/s andthe live offsets at 0.95 s range from 163 to 1138 m in 75 m incre-ments. The corresponding angles of incidence are 5.8 to 37.1.

The sandstone is very soft and the normal incidence S-wavereflection coefficient between the sand and the overlying shale is

effectively zero. Even when brine filled, the seismic responseshows a weak increase in absolute amplitude with offset. The gasfilled response shows a much stronger increase. Figure 2 showsthe brine-fill AVA response for bed thicknesses from 5 m to 40 min 5m increments. The seismic wavelet in this simulation is an 8-60 Hz zero-phase Butterworth filter. For this wavelet tuningoccurs at a bed thickness of 12 m. The AVA response is enhancedat a bed thickness just beyond tuning.

The enhancement of the AVA response is demonstrated better onthe intercept-gradient plot of Figure 3. The spiraling pattern seen inthis figure is characteristic of thin bed AVA responses. Near tuning,the amplitude and gradient responses both oscillate beyond thevalue expected from an isolated re f l e c t o r. The oscillations in the

gradient are not in phase with the oscillations in the intercept.

The cause of these oscillations is the convergence of the top sand andbase sand reflections in time with increasing source to receiver off s e t .

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F i g u re 1. Model of a brine filled sandstone from the central North Sea.

F i g u re 2. AVA (amplitude variation with angle) of the shale-sand-shale model ofF i g u re 1 for bed thicknesses ranging from 5 m to 40 m i n 5 m incre m e n t s .

F i g u re 3. Intercept-gradient cross-plot from synthetic traces of Figure 2 (but usinga 2.5 m increment in bed thickness). Bed thicknesses are indicated and the points are

joined by lines in order to illustrate the spiral character of the AVA re s p o n s e .


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That is, the effective time thickness of the layer decreases with off s e t .

Inspection of the tuning curve (Figure 4) of the seismic wavelet showswhy the AVAis enhanced just beyond tuning. Thus while the normalincidence reflection at a 15 m thick bed sees a two-way time thicknessof 14.1 ms, reflections away from normal incidence see a shorter timethickness. The decrease in time thickness with increasing angle ofincidence sends the re c o rded amplitude back towards tuning on ani n c reasing portion of the tuning curve (Figure 4).

The AVA response is spurious in that it is indicative not only ofthe rock properties above and below the top-sand interface butalso of a changing interference condition. In some circumstancesthe spiral can move a class 3 response, for example, into the class4 zone of the intercept-gradient cross-plot. Is there a remedy thatcan remove the effect of this differential interference? Perfect

NMO correction makes the effective time thickness invariantwith angle and equal to the normal incidence time thickness butthis simply introduces differential interference in another guise:with increasing angle the frequency content of the data islowered, thereby restoring an equivalent interference condition.NMO correction does not (or should not!) alter seismic ampli-tudes. One straightforward measure that does remedy the differ-ential interference is to equalize the spectral content of all theseismic traces. Castoro, White and Thomas (2001) illustrated thisapproach when one has a reasonably accurate estimate of theseismic wavelet. With increasing offset the seismic wavelet isstretched by NMO correction by a predictable amount and thesestretched wavelets can be deconvolved out of each trace in turn.Figure 5 shows NMO corrected traces from the model of Figure

1 before and after this deconvolution when the bed thickness is15 m. A plot of picked amplitudes of the trough (Figure 6, top)

before and after deconvolution shows that this procedure hasessentially restored the intercept-gradient relation expected fromthe rock properties. It has removed the effects of differentialinterference: neither the time thickness nor the seismic waveletvaries with offset. It has not removed the effect of interference.On an intercept-gradient cross-plot the corrected re s p o n s e swould lie on a straight line from the origin through the pointrepresenting an isolated reflector out to the tuning point.

The lower panel of Figure 6 shows that the same procedurerestores the correct intercept-gradient relation for the gas fill.Thus the correction ensures that intercept-gradient points fallinto the correct class of AVA response on a cross-plot. This

ignores the effects of seismic noise which scatters intercept-gradient points at a steep angle to the intercept axis(Hendrickson 1999). In practice the correction can only ensurethat the centres of the noise ellipses fall in the correct intercept-gradient quadrant. If` that is a benefit, there is a penalty. BecauseNMO correction pulls noise as well as signal to lower frequen-cies, the deconvolved output cannot generally be expanded tothe frequency bandwidth seen on short offset data. The sacrificeof some bandwidth in estimating S-wave related parameters isan inherent limitation of all AVO-based techniques.

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F i g u re 5. Simulated offset gather for a bed thickness of 15 m in the model ofF i g u re 1. Top: After NMO correction. Bottom: After NMO correction and waveletdeconvolution.

F i g u re 6. Top: Amplitudes of the troughs at 0.95 s on the traces of Figure 5 (brine-fill) after NMO correction (black diamonds) and after NMO correction and waveletdeconvolution (blue squares). Bottom: corresponding amplitudes of the tro u g h s

f rom the gas-fill case (traces not shown). The red circles show the isolated re f l e c t o rresponse scaled to the normal incidence thin-bed amplitude.

F i g u re 4. Tuning curves for the 8-60 Hz Butterworth filter used as seismic waveletin the simulated data of Figure 2.


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The need for a reliable estimate of the seismic wavelet and for thewavelet itself to be reasonably stable may also be a problem forthis particular method. Although any alternative method ofcross-equalizing seismic traces that preserves scaling wouldserve the same purpose as wavelet deconvolution, the signal-to-noise ratio of pre-stack data gathers is usually a severe handicapto reliable design.

NMO stretch and AVO inversion

The distortions from NMO, thin beds and tuning on intercept-gradient relations also find expression in AVO inversion. Thei n c reasing popularity of AVO inversion and its scope forproducing artefacts make it important to be aware of how thesethree phenomena affect its results.

NMO stretch has a devastating effect on AVO inversion, what-ever the method. Figure 7 illustrates the effect on the derivationof S-wave reflectivity using the brine-fill offset gather of Figure5. The top panel redisplays the moveout corrected gather. Thetwo leftmost traces in the centre panel are the P- and S-wavereflectivities extracted from a convolution of the NMO-corrected

reflection coefficients with the seismic wavelet; i.e. a perfect datamodel with no NMO stretch. These two traces are precisely thetrue model reflectivities (recall that the S-wave reflection coeffi-cient in the brine-fill model is effectively zero). The centre pair oftraces are the reflectivities extracted from the gather and therightmost pair the reflectiivities extracted by a partial stackapproach described below. The bottom panel shows the centrepanel traces after 0-40 Hz low-pass filtering in an effort to atten-uate the noise on the S-reflectivity of trace 5. It is evident that,even with no noise on the input data, NMO causes severe noiseto appear in the S-reflectivity. The reason is that its estimationinvolves subtracting a weighted near- o ffset stack from aweighted far-offset stack. The noise comes from subtracting astretched waveform from a less stretched one.

The partial stack extraction proceeds in outline as follows. Thre epartial stacks are formed and the near and mid-offset stacks arec ross-equalized to the far- o ffset stack while preserving the tracescaling. This not only compensates the variations in bandwidthf rom NMO stretch but also any other waveform variations,including time and phase shifts from mis-stacking. Since timing(e.g. residual moveout) and waveform variations are, along withnoise, the curse of AVO inversion, this approach brings additionalpractical benefits. Although it offers no advantage in noise attenu-ation, this pro c e d u re does diminish the worst effects of timing vari-ations and NMO stretch. In practice Q.C. of trace amplitudes andthe cross-equalization design is a key stage of the process. It is forthis reason that three sub-stacks are chosen. Amplitude Q.C. is

d i fficult from two sub-stacks and more than three may not enhancesignal-to-noise sufficiently to stabilize the cross-equalization.

With or without the low-pass filter, NMO stretch makes itinevitable that the S-wave section, whether it is reflectivity,impedance or mu-rho, has a lower bandwidth than that attain-able from the zero-offset reflectivity. This difference must beaccounted for before combining P and S-wave impedances, forexample, in order to avoid artifacts. A simple approach is to

band-limit the P-wave impedance to that of the S-wave. Cross-equalization of the input data does this

Tuning and AVO inversion

Tuning occurs when the side lobe of the seismic wavelet fromone reflection reinforces the opposite polarity main lobe from anearby reflector. It follows that the removal of wavelet side lobeswould remove tuning. In principle the conversion from relativeto absolute impedance does this. Side lobes occur because theseismic bandwidth does not start at zero frequency but typicallyaround 8-10 Hz. Conversion to absolute impedance constructs asub-seismic model that fills in the low frequency componentsmissing from relative impedance, ie. impedance formed within

the seismic bandwidth. While this conversion can be controlledat wells where there is a close well-to-seismic tie, it is virtuallyimpossible to control away from wells. In practice tuning arti-facts are not uncommon on absolute impedance sections.

Thin beds and AVO inversion

A famous paper by Widess (1982) shows that the shape of areflection from a thin bed is approximately the time derivative ofthe seismic wavelet and that its amplitude is proportional to2f cA where fc is the dominant frequency of the seismic wavelet, is the time thickness of the thin bed and A is the amplitude ofthe reflection if the top interface was an isolated reflector. Widessdefined a thin bed as one whose thickness is less than half the

tuning thickness. It corresponds to the linear portion of thetuning curve (Figure 4). The same equation applies equally to Pand S-wave reflections from a thin bed. Since Ais proportional tothe change in impedance divided by the impedance sum, itfollows that changes in impedance within a thin bed cannot bedistinguished from changes in its thickness. This may not be aninsoluble ambiguity when a thick bed thins since the impedancecan be inferred by extrapolation spatially from thick to thin butit is insoluble away from a well if the bed is always thin.

Another view of thin beds and inversion comes from consideringthe number of degrees of freedom in a segment of seismic trace.

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F i g u re 7. Top: simulated offset gather from the brine-fill model of Figure 1 aftermoveout correction. Centre: extracted P- and S-wave reflectivity; traces 1 and 2:

perfect extraction without NMO stretch; traces 4 and 5: from the NMO corre c t e dgather; traces 7 and 8: from cross-equalized partial stacks. Bottom: the centre panelafter low-pass (0-40 Hz) filtering.


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This number is 2BT where B is the data bandwidth and T theduration of the segment. Assuming that a seismic bandwidthshowing good signal-to-noise of about 50 Hz is often achievable,this implies that no more than 100 parameters can be estimatedfrom 1 s of seismic trace alone and considerably fewer if they areto be reliably estimated in the presence of noise. AVO inversionyields three parameters per interface: its P- and S-wave imped-ances and its timing. That suggests inverting to layers having

roughly 30 ms or more two-way time thickness. Such layerswould not be thin. They are thicker than those generallydisplayed on layered impedance sections. The natural conclu-sion is that, other than layers defined by marker horizons, thelayers seen in inverted sections away from wells are largelycosmetic devices. They may be very useful devices but theirreality is very questionable. A respectable inversion algorithmwill extend these layers in a stable way and the impedancewithin each layer will provide some sort of average value withinthat layer. Nonetheless I suspect that variations in impedanceand in layer thickness are frequently confused away from wells.

Concluding remarks

Residual moveout is widely recognised as a potential source ofconfusion and damage in AVO analysis and inversion. So too, to alesser extent, is seismic noise. Diff e rential interference and theaccompanying diff e rential moveout between reflections withrespect to offset is a comparable source of AVO problems. A na p p roach to AVO inversion based on partial stacks and cro s s -equalization, can avoid the worst effects of residual moveout andNMO stretch. For AVO analysis too, diff e rential interfere n c e(NMO stretch) can obscure the intercept-gradient relation. I havedescribed a wavelet deconvolution scheme that renders the inter-cept-gradient relation immune to NMO stretch. Other schemes,including cross-equalization, may also be possible depending onc i rcumstances, especially the signal-to-noise ratio of the data.

The discussion above on degrees of freedom is also relevant toAVO analysis. The product 2BT defining the number of degre e sof freedom is also roughly the number of peaks and troughs in aseismic trace. This suggests that there is little amplitude informa-tion in a seismic trace beyond its peaks and troughs. The peaksand troughs are also the least noise sensitive amplitudes in atrace. Even so the practice of sample-by-sample cross-plotting ofi n t e rcept and gradient continues despite its sensitivity to noise,residual moveout and NMO stretch. Cross-plotting from peaksand troughs not only minimizes these dangers but also pro v i d e sm o re interpretable cross-plots (Simm, White and Uden 2000).

While AVO analysis of amplitudes stays close to the data, eachstep on the path to a layered impedance introduces the possi-

bility of further artifacts. Readers will have detected some skep-ticism in the previous section about the utility of inverting toabsolute impedance and in layer-based (or sparse) impedanceinversions. This utility will ultimately be decided by interpretersand the majority appears to favour them. Are the minority whodont old fogies or a vanguard standing out against a passingfashion?

ReferencesCastoro, A., White, R.E., and Thomas R.T., 2001, Thin bed AVO: Compensating for theeffects of NMO on reflectivity sequences: Geophysics, 66, 1714-1720.

Hendrickson, J.S., 1999, Stacked: Geophysical Prospecting, 47, 663-705.

Simm, R., White, R., and Uden, R., 2000, The anatomy of AVO crossplots: The LeadingEdge, 19(2), 150-155.

Widess, M.B., 1982, Quantifying the resolving power of seismic systems: Geophysics, 47,1160-1173.

Roy WhiteConsultant

Answer 3

D i ff e rential interference is a result of the band-limited nature ofthe seismic data. The classic example of diff e rential interference isa dipole convolved with a wavelet (consider reflections from thetop and base of a thinning wedge). If the two reflectors making upthe dipole are less than 1/8 of wavelength apart, it is impossible todistinguish the two reflectors separately (Widess, 1973). Related tothis is diff e rential tuning as a function of offset. Because of diff e r-ential moveout (moveout varies with offset), adjacent eventswithin a CMP gather tune as a function of offset, again intro d u c i n ga null space. These two effects lead to the processing artifact ofNMO stretch. The band-limited nature of the seismic and nullspace due to diff e rential tuning make the NMO inverse pro b l e mu n d e rdetermined and consequently difficult to invert in stable

fashion. As a result, the conjugate NMO operator is usuallyapplied instead of the inverse NMO operator (Claerbout, 1992).This results in amplitude and character distortions as a function ofo ffset, which leads to errors in the AVO analysis.

T h e re are a number of ways to deal with diff e rential interfere n c eand diff e rential tuning. First, one can ignore them, do conven-tional NMO and live with the consequences of amplitude andcharacter distortions. In the first two sections below, the conse-quences of doing this are explored both analytically and empiri-c a l l y. For certain reflectivity attributes and anomalies acceptableresults may still be obtained even in the presence of these eff e c t s .A second approach is to try and precondition the data better priorto AVO analysis by performing a stre t c h - f ree NMO corre c t i o n

(Hicks, 2001; Trickett, 2003; Downton et al., 2003). In doing this itis important to use an algorithm that preserves the AVO nature ofthe data, for not all stre t c h - f ree NMO algorithms meet thiscriteria. Lastly, the NMO operator, the band-limited wavelet, andAVO problems can be linked together and solved by AVO wave-form inversion (Simmons and Backus, 1996; Downton and Lines,2003). By solving all three problems together, certain geologicconstraints may be incorporated making the inverse pro b l e m

better posed. Of the three methods, AVO waveform inversionp rovides the best results, but is also the most expensive.

NMO Stretch

For two isolated reflectors, Dunkin and Levin (1973) describe

NMO stretch analytically with the expression

where Sx is the spectrum before NMO correction, Sx is the spec-trum after NMO correction,fis frequency and x is the compres-sion factor or the ratio of the time difference between the twoevents after and before NMO. The compression factor is alwaysless than one, so the frequency spectrum will be shifted to lowerfrequencies and amplified.

( ) ,1~

=

x

x

x

x

fSfS

(1)


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The compression factor, x , becomes smaller for larger offsetsand thus the shape of the wavelet changes in an offset dependentfashion. Figure 1, for example, shows a gather after NMO correc-tion for incident angles from 0 to 45 degrees. The model gener-ating this is a single reflector or spike that is convolved with a5/10-60/70 Hz band-pass filter. For this to match the assump-tions of the traditional methodology, the reflector after NMOmust have constant waveform and amplitude. It does not. The

far offsets are noticeably lower frequency than the near offsetsand the overall character changes as a function of offset.

This biases the subsequent AVO inversion and introduces error.For this example, this can be intuitively understood by calcu-lating the intercept and gradient mentally. The intercept of thezero crossing at 0.39 seconds is zero. The gradient at this sametime is positive since the wavelet broadens as a function of offsetdue to NMO stretch. However, if there was no NMO stretch boththe intercept and gradient would be zero.

Dong (1996) quantified the error due to NMO stretch on AVOinversion. From this paper, it can be shown that for a Rickerwavelet, the approximate fractional error of the interceptterm is zero and that fractional error in the gradient term B is

where

where = fodtis defined in terms of the dominant fo and the timeinterval dtof how far the time sample under investigation is fromthe center of the wavelet. Thus the error is a function of () and

the ratio of intercept over the gradient. If the analysis isperformed on the center of the wavelet = 0then = 0. As increases the size of thegradient error increases. The other factorthat controls the size of the error is the ratio

. Thus it is possible to predict the size of theerror for different classes of AVO anomalies.For Class I ( A


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it is possible to see the character change and frequency shift of thewavelet. This character change with offset due to NMO stre t c hwill bias the AVO inversion.

The Smith and Gidlow (1987) AVO inversion is performed toavoid introducing theoretical error due to the large angles usedin this model and inversion. The parameters are then trans-formed for display purposes to intercept and gradient. The inter-

cept and gradient estimated via this AVO inversion are shown inFigure 4 compared to the ideal intercept and gradient reflectivity.The estimated intercept is almost a perfect match to the ideal.The estimate of the gradient is close to the ideal for both the ClassI and II anomalies. For the Class III and IV anomalies the esti-mate of the gradient shows large error for >0. For =0 thegradient error is zero as expected. These results are consistentwith our predictions based on equation (2).

Offset Dependent Tuning

Dong (1999) described the effect of offset dependent tuning andNMO stretch on AVO inversion as well. This fractional error hasthe same functional form as equation (2) with the exception that

the scalar is now

The behavior of is more complex than equation (3) with zerosoccurring at and Once again Class III and IVshould exhibit large errors while Class I, II and regional reflec-tivity are predicted to show little error. To test this prediction, theprevious model was modified so that instead of single reflectorsat the zero offset, dipole reflectors are modeled. Figure 5 showsthe AVO inversion results when the dipole was 1/2 the dominantwavelength of the source wavelet. As expected, the estimate ofintercept is almost a perfect match to the ideal. The estimate ofthe gradient shows significant error particularly for the Class IIIand IV anomalies. The error is several times larger than thegradient itself. To get a rough understanding of how the tuninglayer thickness influences the error,AVO inversion was run for aseries of different tuning layer thicknesses. Figure 6 demon-strates that the error changes as a function of layer thickness.

Other reflectivity attributes behave diff e rently to the erro r.Figure 7 shows how the S-wave impedance and fluid stackreflectivity behave to the distortion. Interestingly the fluid stackshows little distortion other than a phase delay. The fluid stack isquite a robust AVO attribute even in the presence of NMO stretchand tuning.

Solutions

Based on this analysis a simple model was constructed to testdifferent methodologies of addressing NMO stretch and offsetdependent tuning. A synthetic seismic gather with sparse reflec-tivity was generated. The AVO behavior of most of the reflectorsin the synthetic gather follow the mudrock trend, but severalclass III and IV anomalies are also present. Both isolated andtuned anomalies were created. The synthetic data was generatedusing a convolutional model with a Ricker wavelet with a centralfrequency of 32.5 Hz. In order to isolate the effects of NMO

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F i g u re 3. The input model prior to NMO (a), after NMO correction (b) and compare dto the synthetic gather generated without NMO (c). Note on the NMO corre c t e d

gather the introduction of low frequencies at large offsets due to NMO stre t c h .

F i g u re 4. The estimate (red) of the AV Oi n t e rcept A and gradient B compared tothe ideal (blue). Note the gradient esti-mate is distorted for both the class III andIV anomalies as pre d i c t e d .

F i g u re 5. The estimate (red) of the AV Oi n t e rcept A and gradient B compared tothe ideal (blue). Note the gradient esti-mate is distorted for both the class IIIand IV anomalies as pre d i c t e d .

F i g u re 6. The estimate (red) of the AVO gradient B compared to the ideal (blue) forvarious layer thicknesses. Note that distortion changes as function of thickness

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stretch and differential tuning on the AVO inversion, the reflec-tivity was generated using the three-term Shuey equation (1985)using the Gardner relationship to calculate density. Noise wasadded to give a signal-to-noise ratio of 4:1. A constant back-ground velocity was used so there would be a simple angle-to-offset relationship. The maximum offset was chosen to be fourtimes the target depth so that angles out to 65 degrees would beavailable for the inversion, though only angles to 45 degrees

were actually used. These large angles were created to highlightthe distortions. Figure 8 shows the prestack synthetic gather afterNMO correction. This is compared to the same gather but gener-ated without NMO. The difference highlights the theoreticalerror introduced by the NMO correction.

An AVO inversion was performed using the Smith and Gidlowformulation using angles from 0 to 45 degrees. The Smith andGidlow formulation was used rather than the two-term Shueyapproximation since the former is exact under the assumptionsthe model was created while the latter is not. The reflectivity esti-mates were then transformed to intercept and gradient fordisplay purposes as shown in Figure 9. As expected, there is noerror for the intercept term while the gradient term only shows

error for both the Class III and Class IV anomalies. Reflectivity ofreflectors whose Vp/Vs relationship fall along the mudrocktrend are perfectly predicted even though the events have under-gone NMO stretch. Figure 9 also shows the reflectivity in thecross-plot domain. The Class III and IV anomalies show signifi-cant scatter.

Next, the synthetic gather was processed with a stre t c h - f ree NMOc o r rection (Downton et al., 2003). Figure 10 compares the datawith stre t c h - f ree NMO with traditional NMO and a gatherc reated without NMO. The stre t c h - f ree gather is higher fre q u e n c yon the far offsets compared to the NMO corrected data. It is alsonoisier as a result of the implicit deconvolution process. The twotuned reflectors at 1.6 and 1.8 seconds show greater detail and

m o re information than the NMO corrected gathersbut not as much information as the ideal syntheticg a t h e r. The stre t c h - f ree NMO corrected data wasthen inverted for intercept A and gradient B in asimilar manner (Figure 11). The estimated interc e p tis once again a perfect match compared to the ideali n t e rcept re f l e c t i v i t y. The estimated gradient isnow a much better match to the ideal than done onthe NMO corrected data. However, there is stillsignificant scatter in the cross-plot domain.

Lastly, AVO waveform inversion (Downton andLines, 2003) was performed on the same data toestimate the intercept and gradient (Figure 12).The estimate is almost perfect for all reflectorsincluding the Class III and IV anomalies that wereu n d e rgoing NMO stretch and had diff e re n t i a ltuning. This is confirmed by cross-plotting thereflectivity.

Concluding Remarks

Differential interference and tuning as a functionof offset lead to distortions in AVO analysis.These distortions may be avoided by performingstretch-free NMO prior to the AVO inversion or by

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F i g u re 8. The synthetic gather generated without moveout (a) is compared to theNMO corrected gather (b) while (c) shows the difference between the two.

F i g u re 9. The traditional AVO estimates (a) for intercept A and gradient B (red) are compared to theideal results (blue). Also, the ideal data (b) is compared to the estimated data (c) in the cross-plot domain.Note the estimated class III and IV anomalies are spread out in cross-plot space.

F i g u re 7. Intercept A and Gradient B converted to fluid, P- and S-wave impedancere f l e c t i v i t y. Note that the fluid stack shows little distortion due to NMO stretch andoffset dependent tuning for all classes.


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incorporating the NMO inversion into the AVO inversion. Thebest results were obtained by AVO waveform inversion but at asignificant extra cost. Stretch-free NMO shows some promise forhelping precondition the data prior to AVO.

The advantages of using these algorithms have to be weighedagainst the cost of performing them. AVO inversion on NMO-corrected (and NMO-stretched) gathers is actually surprisingly

robust. The distortion primarily shows up on the secondary AVOreflectivity attribute, such as the gradient or S-wave impedancereflectivity. The attribute associated with the first term, the inter-cept or P-wave impedance reflectivity is unaffected. Further,there is a surprisingly large class of geologic interfaces for whichNMO stretch and offset dependent tuning do not distort thereflectivity estimates. Regional reflectors from interfaces

between clastics following the mudrock trend are not distorted.Class I and II gas sands are not distorted. Only Class III and IVgas sand anomalies are distorted. Further, these distortions areonly significant when large angles are used. When donewith angles less than 30, the AVO inversion estimatesperformed on the synthetics shown here had insignificant error.Only when larger angles were used (for example, 45) were the

errors significant.

The fact that diff e rent classes respond diff e rently is somewhatcounterintuitive. The synthetic gathers shown in Figure 2 showthat all the classes experience NMO stretch at far off s e t s .Simplistically all of them should be showing distortions to thegradient. If a two-term Shuey AVO inversion was carried out, thiswould be the case. However, the two-term Smith and Gidlowinversion used is effectively a three-term inversion, because of theG a rdner density constraint. This makes for a more complexfitting than just intercept, and gradient. I believe this fitting ism o re appropriate since ultimately we are interested in usingthese large offsets to perform three-term AVO inversion. The two-term Gidlow et al. (1992) equation behaves in the same manner. R

ReferencesClaerbout, J. F., 1992, Earth Soundings Analysis: Processing versus Inversion,Blackwell Scientific Publications.

Dong, W. 1999, AVO detectability against tuning and stretching artifacts:Geophysics, 64, 494-503.

Dong, W., 1996, Fluid line distortion due to migration stretch , 66th Ann.Internat. Mtg: Soc. of Expl. Geophys., 1345-1348

Downton, J. E., Guan, H., and Somerville, R., 2003, NMO, AVO and Stack:2003 CSPG / CSEG Convention expanded abstracts

Downton, J. and Lines, L., 2003, High-resolution AVO analysis before NMO:73rd Ann. Internat. Mtg.: Soc. of Expl. Geophys., 219-222.

Dunkin, J. W. and Levin, F. K., 1973, Effect of normal moveout on a seismicpulse: Geophysics, 38, 635-642.

Gidlow, P.M., Smith, G. C., and Vail, P. J., 1992,Hydrocarbon detection usingfluid factor traces, a case study: How useful is AVO analysis? Joint SEG/EAEGsummer research workshop, Technical Program and Abstracts, 78-89.

Hicks, G.J., 2001, Removing NMO Stretch Using the Radon and Fourier-RadonTransforms; 63rd Mtg.: Eur. Assn. Geosci. Eng., Session: A-18.

Simmons, J. L., Jr. and Backus, M. M., 1996, Waveform-based AVO inversionand AVO prediction error: Geophysics, 61, 1575-1588.

Smith, G.,C., and Gidlow, P.,M., 1987, Weighted stacking for rock propertyestimation and detection of gas: Geophysical Prospecting, 35, 993-1014

Trickett, S., 2003, Stretch-free stacking, 73rd Ann. Internat. Mtg.: Soc. ofExpl. Geophys., 2008-2011.

Widess, M. B., 1973,How thin is a thin bed?: Geophysics, 38, 1176-1254.

Jon DowntonVeritas, Calgary

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F i g u re 10. Comparison of NMO corrected data (c) with stre t c h - f ree NMO (b). Forre f e rence purposes gather (a) was generated without NMO.

F i g u re 11. Results of AVO inversion based on stre t c h - f ree NMO input. The esti-

mated (red) gradient B compares well with the ideal gradient (blue). However, thereis still significant scatter in the cross-plot domain. The top cross-plot shows the idealdata while the bottom one shows the estimated data.

F i g u re 12. The estimated two-term AVO waveform results (a) for intercept A and gradient B( red) are compared to the ideal results (blue). Also, the ideal data (b) is compared to the esti-mated data (c) in the cross-plot domain. Note the good agreement for the III and IV anomalies.

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