Answer 1
D i ff e rential interference results from the fact that neighboring reflectors increasingly interfere as the incidence angle i n c reases. When the reflectors come from the top and bottom of 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 individual re f l e c t o r. When viewed in moved-out gathers, these eff e c t s appear 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 data into alignment by constant time shifts. AVO analysis could then 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 with a time-varying filter to transform the stretched wavelet into
the unstretched one. This procedure generated a moved-out gather withmoved-out 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. This method strictly applies only to a relatively short window of data, since filtering in the Fourier domain is time-invariant, but wavelet stretch is not. As the window length is decreased, edge effects become more severe.
• Trickett (2003) proposed a method of stretch-free stacking, which when applied to partial-offset gathers could be used for AVO analysis, even when reflection events cross. The applicability of this method for AVO analysis is still being evaluated.
In the remainder of my reply, I will describe a fifth method, which generates a stretch-free AVO gradient, as opposed to a stretch-free gather. It does this by estimating the contribution
Continued on Page 13
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 AVO
a 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 leakage wavelet BL( t ) ( red). Bottom plot (b): The optimal filter for estimating the
noise-f ree intercept in the presence onoise-f white noise, h1( t ) (black) and the optimal filter
to the gradient from diff e rential interference, and then subtracting it. Optimal performance is achieved in the presence of a known noise spectrum.
D i ff e rential interference manifests itself as l e a k a g e f rom the normal-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 gradient due 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. The stacking velocity should be smoothed to avoid sudden changes in Vs’(t). In
the presence of an intercept noise s p e c t rum Sn(ω), the Fourier
trans-form of the optimal h2 filter, in the
least-squares sense, is given by 3. H2(ω) = -[ |W(ω) |2+ ω W * (ω) W ’ (ω)] /
[ |W(ω) |2+ S
n(ω) ] .
Note that this expression is invariant to a wavelet phase shift, and is stable even when the wavelet Fourier trans-form W(ω) vanishes. The filter h2(t) can also be obtained in the time domain by a Levinson re c u r s i o n (Swan 1997).
For the case of the bandpass wavelet whose waveform and spectrum are shown in Figure 1, the gradient stretch error with a constant stacking velocity, BL(t), is shown as the red curve in Figure 2a. This error is caused by the wavelet eff e c t i v e l y being stretched at large offsets. The error 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 by equation (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(ω) = |W(ω)|2/ [ |W(ω)|2+ Sn(ω)].
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 o
shown is the result of convolving w(t) with h1(t). It closely
a p p roximates w ( t ). Although not terribly important in this example, this filter applies the same noise-reduction regimen to the intercept as to the gradient. This will ensure their spectra will match, 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 ft section of 2.3 g/cm3 material embedded into a constant
2.5 g/cm3substrate. Neither the acoustic velocity (10 kft/s) nor
Continued from Page 12
F i g u re 3. The actual and estimated noise-free wavelets (black) and the actual and estimated 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 .
the shear velocity (5 kft/s) varies through this model. Such a density-only contrast is expected to produce a background AVO response. Using 28 Hz as the wavelet center frequency, this bed is below the tuning thickness of 89 ft. Wavelet stretch is noticeable out 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 a least-squares fit at each time to the equation
5. S(t, θ) = A(t) + B(t)sin2θ + C(t)sin2θ
tan2θ,
where S(t, θ) is the synthetic gather and θ is the incidence angle. Figure 5 shows a cross-plot of this gradient versus intercept, both filtered by h1(t).
The color of the dots corresponds to time. The A-B plane is subdivided into regions that correspond to commonly 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 colors match those of the AVO classifications. Although the central top and base reflectors correctly indicate background (gray) reflec-tors, there are prominent false AVO anomalies as far away as 30 ms (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, and the false AVO anomalies are removed. The hodogram barely grazes the class 1 top polygon, but other than that, correctly remains in background territory.
If the thickness of the low-density zone is varied from 100 ft to 10 ft, the results are shown in Figures 7 and 8. Figure 7 shows the intercept trace and apparent AVO classification as a function of wedge thickness, when differential interference is not removed. Spurious AVO anomalies are at their worst at around half the tuning thickness (λ/8). Figure 8 shows the improved result when differential interference is removed.
References
Byun, Bok S. and Nelan, E. Stuart, 1997, Method and system for correcting seismic traces
for normal move-out correction, U. S. Patent 5,684,754.
Castagna, John P. and Swan, Herbert W., 1997, Principles of AVO crossplotting, The Leading 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 layered
medium, Geophysics, 56, No. 1, pg. 41-49.
Continued from Page 13
Continued on Page 15 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 false AVO anomalies above and below the target event.
________________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 return to the recorders. In general too, reflector spacing varies laterally. The consequence is that the primary reflection signal consists of a multitude of interfering reflection pulses, or seismic wavelets, that produce images of the subsurface that are dominated by differential interference. Although occasionally a reflection may be considered for practical purposes an isolated reflection, it is differential interference that is the norm. In the offset domain, differential interference is again the norm for the simple reason that 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 isn’t really what the question is about. For AVO analysis, one has to start in the offset domain in order to explain the effect of differential interference and offset-dependent tuning on an AVO response. To do that I first consider the archetypal example of the AVO response of a thinning bed encased in a uniform shale. That leads into the impact of NMO stretch, tuning and thin beds on AVO inversion. I conclude with some remarks about AVO inver-sion and layered inverinver-sions that may provoke further comment. AVO response of a thin bed
Figure 1 shows a rock physics model of a sandstone sandwiched within a shale, based on a reservoir in the central North Sea. The sandstone parameters shown are for the water leg. In the gas leg they become VP=1638 m/s, VS= 862 m/s and the density is 1784
kg/m3. The top of the sandstone is at 1050 m, or 0.95 s two-way
time. At this two-way time the NMO velocity is 1925 m/s and the 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-wave reflection coefficient between the sand and the overlying shale is
effectively zero. Even when brine filled, the seismic response shows a weak increase in absolute amplitude with offset. The gas filled response shows a much stronger increase. Figure 2 shows the brine-fill AVA response for bed thicknesses from 5 m to 40 m in 5m increments. The seismic wavelet in this simulation is an 8-60 Hz zero-phase Butterworth filter. For this wavelet tuning occurs at a bed thickness of 12 m. The AVA response is enhanced at a bed thickness just beyond tuning.
The enhancement of the AVA response is demonstrated better on the intercept-gradient plot of Figure 3. The spiraling pattern seen in this figure is characteristic of thin bed AVA responses. Near tuning, the amplitude and gradient responses both oscillate beyond the value 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 and base sand reflections in time with increasing source to receiver off s e t . Continued from Page 14
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 of F 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 using a 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 .
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 shows why the AVA is enhanced just beyond tuning. Thus while the normal incidence reflection at a 15 m thick bed sees a two-way time thickness of 14.1 ms, reflections away from normal incidence see a shorter time thickness. The decrease in time thickness with increasing angle of incidence sends the re c o rded amplitude back towards tuning on an i n c reasing portion of the tuning curve (Figure 4).
The AVA response is spurious in that it is indicative not only of the rock properties above and below the top-sand interface but also of a changing interference condition. In some circumstances the spiral can move a class 3 response, for example, into the class 4 zone of the intercept-gradient cross-plot. Is there a remedy that can remove the effect of this differential interference? Perfect NMO correction makes the effective time thickness invariant with angle and equal to the normal incidence time thickness but this simply introduces differential interference in another guise: with increasing angle the frequency content of the data is lowered, 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 the seismic traces. Castoro, White and Thomas (2001) illustrated this approach when one has a reasonably accurate estimate of the seismic wavelet. With increasing offset the seismic wavelet is stretched by NMO correction by a predictable amount and these stretched 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 is 15 m. A plot of picked amplitudes of the trough (Figure 6, top) before and after deconvolution shows that this procedure has essentially restored the intercept-gradient relation expected from the rock properties. It has removed the effects of differential interference: neither the time thickness nor the seismic wavelet varies 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 s would lie on a straight line from the origin through the point representing an isolated reflector out to the tuning point.
The lower panel of Figure 6 shows that the same procedure restores the correct intercept-gradient relation for the gas fill. Thus the correction ensures that intercept-gradient points fall into 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 ensure that the centres of the noise ellipses fall in the correct intercept-gradient quadrant. If` that is a benefit, there is a penalty. Because NMO correction pulls noise as well as signal to lower frequen-cies, the deconvolved output cannot generally be expanded to the frequency bandwidth seen on short offset data. The sacrifice of some bandwidth in estimating S-wave related parameters is an inherent limitation of all AVO-based techniques.
Continued from Page 15
Continued on Page 17 F i g u re 5. Simulated offset gather for a bed thickness of 15 m in the model of F i g u re 1. Top: After NMO correction. Bottom: After NMO correction and wavelet deconvolution.
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 wavelet deconvolution (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 r response 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 wavelet in the simulated data of Figure 2.
The need for a reliable estimate of the seismic wavelet and for the wavelet itself to be reasonably stable may also be a problem for this particular method. Although any alternative method of cross-equalizing seismic traces that preserves scaling would serve the same purpose as wavelet deconvolution, the signal-to-noise ratio of pre-stack data gathers is usually a severe handicap to 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. The i n c reasing popularity of AVO inversion and its scope for producing artefacts make it important to be aware of how these three phenomena affect its results.
NMO stretch has a devastating effect on AVO inversion, what-ever the method. Figure 7 illustrates the effect on the derivation of S-wave reflectivity using the brine-fill offset gather of Figure 5. The top panel redisplays the moveout corrected gather. The two leftmost traces in the centre panel are the P- and S-wave reflectivities extracted from a convolution of the NMO-corrected reflection coefficients with the seismic wavelet; i.e. a perfect data model with no NMO stretch. These two traces are precisely the true model reflectivities (recall that the S-wave reflection coeffi-cient in the brine-fill model is effectively zero). The centre pair of traces are the reflectivities extracted from the gather and the rightmost pair the reflectiivities extracted by a partial stack approach described below. The bottom panel shows the centre panel 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 noise to appear in the S-reflectivity. The reason is that its estimation involves subtracting a weighted near- o ffset stack from a weighted far-offset stack. The noise comes from subtracting a stretched waveform from a less stretched one.
The partial stack extraction proceeds in outline as follows. Thre e partial stacks are formed and the near and mid-offset stacks are c ross-equalized to the far- o ffset stack while preserving the trace scaling. This not only compensates the variations in bandwidth f 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 with noise, the curse of AVO inversion, this approach brings additional practical 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 and the cross-equalization design is a key stage of the process. It is for this reason that three sub-stacks are chosen. Amplitude Q.C. is d i fficult from two sub-stacks and more than three may not enhance signal-to-noise sufficiently to stabilize the cross-equalization. With or without the low-pass filter, NMO stretch makes it inevitable 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 be accounted for before combining P and S-wave impedances, for example, 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 from one reflection reinforces the opposite polarity main lobe from a nearby reflector. It follows that the removal of wavelet side lobes would remove tuning. In principle the conversion from relative to absolute impedance does this. Side lobes occur because the seismic bandwidth does not start at zero frequency but typically around 8-10 Hz. Conversion to absolute impedance constructs a sub-seismic model that fills in the low frequency components missing from relative impedance, ie. impedance formed within the seismic bandwidth. While this conversion can be controlled at wells where there is a close well-to-seismic tie, it is virtually impossible 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 a reflection from a thin bed is approximately the time derivative of the seismic wavelet and that its amplitude is proportional to 2f cτA where fcis the dominant frequency of the seismic wavelet,
τ is the time thickness of the thin bed and A is the amplitude of the reflection if the top interface was an isolated reflector. Widess defined a thin bed as one whose thickness is less than half the tuning thickness. It corresponds to the linear portion of the tuning curve (Figure 4). The same equation applies equally to P and S-wave reflections from a thin bed. Since A is proportional to the change in impedance divided by the impedance sum, it follows that changes in impedance within a thin bed cannot be distinguished from changes in its thickness. This may not be an insoluble ambiguity when a thick bed thins since the impedance can be inferred by extrapolation spatially from thick to thin but it is insoluble away from a well if the bed is always thin. Another view of thin beds and inversion comes from considering the number of degrees of freedom in a segment of seismic trace. Continued from Page 16
F i g u re 7. Top: simulated offset gather from the brine-fill model of Figure 1 after moveout 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 d gather; traces 7 and 8: from cross-equalized partial stacks. Bottom: the centre panel after low-pass (0-40 Hz) filtering.
Continued on Page 19 Continued from Page 17
This number is 2BT where B is the data bandwidth and T the duration of the segment. Assuming that a seismic bandwidth showing good signal-to-noise of about 50 Hz is often achievable, this implies that no more than 100 parameters can be estimated from 1 s of seismic trace alone and considerably fewer if they are to be reliably estimated in the presence of noise. AVO inversion yields 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 layers would not be thin. They are thicker than those generally displayed on layered impedance sections. The natural conclu-sion is that, other than layers defined by marker horizons, the layers seen in inverted sections away from wells are largely cosmetic devices. They may be very useful devices but their reality is very questionable. A respectable inversion algorithm will extend these layers in a stable way and the impedance within each layer will provide some sort of average value within that layer. Nonetheless I suspect that variations in impedance and in layer thickness are frequently confused away from wells.
Concluding remarks
Residual moveout is widely recognised as a potential source of confusion and damage in AVO analysis and inversion. So too, to a lesser extent, is seismic noise. Diff e rential interference and the accompanying diff e rential moveout between reflections with respect to offset is a comparable source of AVO problems. A n a p p roach to AVO inversion based on partial stacks and cro s s -equalization, can avoid the worst effects of residual moveout and NMO stretch. For AVO analysis too, diff e rential interfere n c e (NMO stretch) can obscure the intercept-gradient relation. I have described 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 on c i rcumstances, especially the signal-to-noise ratio of the data. The discussion above on degrees of freedom is also relevant to AVO analysis. The product 2BT defining the number of degre e s of freedom is also roughly the number of peaks and troughs in a seismic trace. This suggests that there is little amplitude informa-tion in a seismic trace beyond its peaks and troughs. The peaks and troughs are also the least noise sensitive amplitudes in a trace. Even so the practice of sample-by-sample cross-plotting of i n t e rcept and gradient continues despite its sensitivity to noise, residual moveout and NMO stretch. Cross-plotting from peaks and troughs not only minimizes these dangers but also pro v i d e s m o re interpretable cross-plots (Simm, White and Uden 2000). While AVO analysis of amplitudes stays close to the data, each step 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 to absolute impedance and in layer-based (or sparse) impedance inversions. This utility will ultimately be decided by interpreters and the majority appears to favour them. Are the minority who don’t old fogies or a vanguard standing out against a passing fashion?
References
Castoro, A., White, R.E., and Thomas R.T., 2001, Thin bed AVO: Compensating for the
effects 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 Leading Edge, 19(2), 150-155.
Widess, M.B., 1982, Quantifying the resolving power of seismic systems: Geophysics, 47, 1160-1173.
Roy White Consultant
Answer 3
D i ff e rential interference is a result of the band-limited nature of the seismic data. The classic example of diff e rential interference is a dipole convolved with a wavelet (consider reflections from the top and base of a thinning wedge). If the two reflectors making up the dipole are less than 1/8 of wavelength apart, it is impossible to distinguish the two reflectors separately (Widess, 1973). Related to this is diff e rential tuning as a function of offset. Because of diff e r-ential moveout (moveout varies with offset), adjacent events within a CMP gather tune as a function of offset, again intro d u c i n g a null space. These two effects lead to the processing artifact of NMO stretch. The band-limited nature of the seismic and null space due to diff e rential tuning make the NMO inverse pro b l e m u n d e rdetermined and consequently difficult to invert in stable fashion. As a result, the conjugate NMO operator is usually applied instead of the inverse NMO operator (Claerbout, 1992). This results in amplitude and character distortions as a function of o 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 e and diff e rential tuning. First, one can ignore them, do conven-tional NMO and live with the consequences of amplitude and character 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 empiri-certain refleempiri-ctivity attributes and anomalies aempiri-cempiri-ceptable results 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 prior to 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 it is important to use an algorithm that preserves the AVO nature of the data, for not all stre t c h - f ree NMO algorithms meet this criteria. Lastly, the NMO operator, the band-limited wavelet, and AVO 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 geologic constraints may be incorporated making the inverse pro b l e m better posed. Of the three methods, AVO waveform inversion p 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 Sxis the spectrum before NMO correction, Sxis the
spec-trum after NMO correction, f is frequency and αxis the compres-sion factor or the ratio of the time difference between the two events after and before NMO. The compression factor is always less than one, so the frequency spectrum will be shifted to lower frequencies and amplified.
( )
1
,
~
=
x x x xf
S
f
S
α
α
(1) ˜The compression factor, αx, becomes smaller for larger offsets and thus the shape of the wavelet changes in an offset dependent fashion. 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 a 5/10-60/70 Hz band-pass filter. For this to match the assump-tions of the traditional methodology, the reflector after NMO must have constant waveform and amplitude. It does not. The far offsets are noticeably lower frequency than the near offsets and 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 the zero crossing at 0.39 seconds is zero. The gradient at this same time is positive since the wavelet broadens as a function of offset due to NMO stretch. However, if there was no NMO stretch both the intercept and gradient would be zero.
Dong (1996) quantified the error due to NMO stretch on AVO inversion. From this paper, it can be shown that for a Ricker wavelet, the approximate fractional error of the intercept term is zero and that fractional error in the gradient term B is
where
where η= fodt is defined in terms of the dominant foand the time interval dt of how far the time sample under investigation is from the center of the wavelet. Thus the error is a function of κ(η) and the ratio of intercept over the gradient. If the analysis is performed on the center of the wavelet η = 0
then = 0. As η increases the size of the gradient error increases. The other factor that controls the size of the error is the ratio . Thus it is possible to predict the size of the error for different classes of AVO anomalies. For Class I ( A<<B) and Class II (A=0 ) anomalies the error should be small. This is also true for most reflectors which fall along the mudrock trend since typically ( A<<B) . However, for Class III and IV anomalies, where A>B the error is potentially large. It is important to note that this analysis is based on equations that are empirical in nature, and did not consider the role of maximum offset used in the inversion.
To test these predictions, a model was c o n s t ructed with four isolated reflectors corre-sponding to the four classes outlined above. The synthetic data was generated using a convolutional model with a Ricker wavelet with a 32.5 Hz central fre q u e n c y. Pre l i m i n a r y testing suggested that large offsets and angles
a re needed to make the NMO stretch artifacts apparent. To avoid t h e o retical error being introduced due to these large offsets, the A k i and Richards (1980) linearized approximation of the Zoeppritz equation is used to generate the re f l e c t i v i t y, using the Gard n e r density approximation to generate the density term. Further, to keep the relationship between offset and angle of incidence simple and to avoid supercritical reflections, a constant backgro u n d velocity was used to generate the model. With the maximum off s e t about four times the target depth, angles out to 65 degrees were g e n e r a t e d .
F i g u re 2 shows both the Shuey (1985) two-term and thre e - t e r m response for reflectors generated with no moveout. Note the far o ffset reflectivity behavior is dramatically diff e rent due to the inclusion of the third term. Figure 3 shows the synthetic gather with moveout and after NMO. For the sake of comparison, the gather generated without NMO is shown next to the NMO c o r rected gather. At the far offsets on the NMO corrected gathers
A B . 0 = B dB
Continued from Page 18
F i g u re 1. Synthetic gather of a single spike after NMO and band-pass filter 10/14-60/70 Hz for incident angles from 0 to 45 degrees. Note how NMO stretch lowers the frequency on the far offsets and changes the wavelet character.
F i g u re 2. Cross-plot of ideal reflectivity (a) used to generate the synthetic gather generated without NMO using two term Shuey approximation (b) and the synthetic gather generated without NMO using three term Shuey a p p roximation (c). Note the two-term model clearly shows the class I - IV behavior expected while the three term model behavior is more complex.
,
B
A
B
dB
κ
=
(2)(
)
(
1
8
)
,
8
3
4
2 2 2 2 2 2η
π
η
π
η
π
κ
−
−
=
(3)it is possible to see the character change and frequency shift of the wavelet. This character change with offset due to NMO stre t c h will bias the AVO inversion.
The Smith and Gidlow (1987) AVO inversion is performed to avoid introducing theoretical error due to the large angles used in 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 in Figure 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 Class I and II anomalies. For the Class III and IV anomalies the esti-mate of the gradient shows large error for η>0. For η=0 the gradient error is zero as expected. These results are consistent with our predictions based on equation (2).
Offset Dependent Tuning
Dong (1999) described the effect of offset dependent tuning and NMO stretch on AVO inversion as well. This fractional error has the same functional form as equation (2) with the exception that the scalar κ is now
The behavior of κ is more complex than equation (3) with zeros
occurring at and Once again Class III and IV
should exhibit large errors while Class I, II and regional reflec-tivity are predicted to show little error. To test this prediction, the previous model was modified so that instead of single reflectors at the zero offset, dipole reflectors are modeled. Figure 5 shows the AVO inversion results when the dipole was 1/2 the dominant wavelength of the source wavelet. As expected, the estimate of intercept is almost a perfect match to the ideal. The estimate of the gradient shows significant error particularly for the Class III and IV anomalies. The error is several times larger than the gradient itself. To get a rough understanding of how the tuning layer thickness influences the error, AVO inversion was run for a series 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 stack reflectivity behave to the distortion. Interestingly the fluid stack shows little distortion other than a phase delay. The fluid stack is quite a robust AVO attribute even in the presence of NMO stretch and tuning.
Solutions
Based on this analysis a simple model was constructed to test different methodologies of addressing NMO stretch and offset dependent tuning. A synthetic seismic gather with sparse reflec-tivity was generated. The AVO behavior of most of the reflectors in the synthetic gather follow the mudrock trend, but several class III and IV anomalies are also present. Both isolated and tuned anomalies were created. The synthetic data was generated using a convolutional model with a Ricker wavelet with a central frequency of 32.5 Hz. In order to isolate the effects of NMO
. 2 1 ± = κ ± = 3 κ
Continued from Page 19
F i g u re 3. The input model prior to NMO (a), after NMO correction (b) and compare d to 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 O i n t e rcept A and gradient B compared to the ideal (blue). Note the gradient esti-mate is distorted for both the class III and IV anomalies as pre d i c t e d .
F i g u re 5. The estimate (red) of the AV O i n t e rcept A and gradient B compared to the ideal (blue). Note the gradient esti-mate is distorted for both the class III and 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) for various layer thicknesses. Note that distortion changes as function of thickness
(
)(
)
(
2
3
)
.
3
1
2
2 2 2−
−
−
=
η
η
η
κ
(4) Continued on Page 21stretch 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 was added 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 angle-to-offset was chosen to be four times the target depth so that angles out to 65 degrees would be available for the inversion, though only angles to 45 degrees were actually used. These large angles were created to highlight the distortions. Figure 8 shows the prestack synthetic gather after NMO correction. This is compared to the same gather but gener-ated without NMO. The difference highlights the theoretical error introduced by the NMO correction.
An AVO inversion was performed using the Smith and Gidlow formulation using angles from 0 to 45 degrees. The Smith and Gidlow formulation was used rather than the two-term Shuey approximation since the former is exact under the assumptions the model was created while the latter is not. The reflectivity esti-mates were then transformed to intercept and gradient for display purposes as shown in Figure 9. As expected, there is no error for the intercept term while the gradient term only shows error for both the Class III and Class IV anomalies. Reflectivity of reflectors whose Vp/Vs relationship fall along the mudrock trend are perfectly predicted even though the events have under-gone NMO stretch. Figure 9 also shows the reflectivity in the cross-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 NMO c o r rection (Downton et al., 2003). Figure 10 compares the data with stre t c h - f ree NMO with traditional NMO and a gather c reated without NMO. The stre t c h - f ree gather is higher fre q u e n c y on the far offsets compared to the NMO corrected data. It is also noisier as a result of the implicit deconvolution process. The two tuned reflectors at 1.6 and 1.8 seconds show greater detail and m o re information than the NMO corrected gathers but not as much information as the ideal synthetic g a t h e r. The stre t c h - f ree NMO corrected data was then inverted for intercept A and gradient B in a similar manner (Figure 11). The estimated interc e p t is once again a perfect match compared to the ideal i n t e rcept re f l e c t i v i t y. The estimated gradient is now a much better match to the ideal than done on the NMO corrected data. However, there is still significant scatter in the cross-plot domain. Lastly, AVO waveform inversion (Downton and Lines, 2003) was performed on the same data to estimate the intercept and gradient (Figure 12). The estimate is almost perfect for all reflectors including the Class III and IV anomalies that were u n d e rgoing NMO stretch and had diff e re n t i a l tuning. This is confirmed by cross-plotting the reflectivity.
Concluding Remarks
Differential interference and tuning as a function of offset lead to distortions in AVO analysis. These distortions may be avoided by performing stretch-free NMO prior to the AVO inversion or by Continued from Page 20
F i g u re 8. The synthetic gather generated without moveout (a) is compared to the NMO 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 the ideal 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 impedance re f l e c t i v i t y. Note that the fluid stack shows little distortion due to NMO stretch and offset dependent tuning for all classes.
incorporating the NMO inversion into the AVO inversion. The best results were obtained by AVO waveform inversion but at a significant extra cost. Stretch-free NMO shows some promise for helping precondition the data prior to AVO.
The advantages of using these algorithms have to be weighed against 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 AVO reflectivity attribute, such as the gradient or S-wave impedance reflectivity. 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 which NMO stretch and offset dependent tuning do not distort the reflectivity 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 IV gas sand anomalies are distorted. Further, these distortions are only significant when large angles are used. When done with angles less than 30°, the AVO inversion estimates performed 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 somewhat counterintuitive. The synthetic gathers shown in Figure 2 show that all the classes experience NMO stretch at far off s e t s . Simplistically all of them should be showing distortions to the gradient. If a two-term Shuey AVO inversion was carried out, this would be the case. However, the two-term Smith and Gidlow inversion used is effectively a three-term inversion, because of the G a rdner density constraint. This makes for a more complex fitting than just intercept, and gradient. I believe this fitting is m o re appropriate since ultimately we are interested in using these large offsets to perform three-term AVO inversion. The two-term Gidlow et al. (1992) equation behaves in the same manner.
R
References
Claerbout, 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 seismic
pulse: Geophysics, 38, 635-642.
Gidlow, P.M., Smith, G. C., and Vail, P. J., 1992, Hydrocarbon detection using
fluid factor traces, a case study: How useful is AVO analysis? Joint SEG/EAEG
summer research workshop, Technical Program and Abstracts, 78-89. Hicks, G.J., 2001, Removing NMO Stretch Using the Radon and Fourier-Radon
Transforms; 63rd Mtg.: Eur. Assn. Geosci. Eng., Session: A-18.
Simmons, J. L., Jr. and Backus, M. M., 1996, Waveform-based AVO inversion
and AVO prediction error: Geophysics, 61, 1575-1588.
Smith, G.,C., and Gidlow, P.,M., 1987, Weighted stacking for rock property
estimation and detection of gas: Geophysical Prospecting, 35, 993-1014
Trickett, S., 2003, Stretch-free stacking, 73rd Ann. Internat. Mtg.: Soc. of Expl. Geophys., 2008-2011.
Widess, M. B., 1973, How thin is a thin bed?: Geophysics, 38, 1176-1254.
Jon Downton Veritas, Calgary Continued from Page 21
F i g u re 10. Comparison of NMO corrected data (c) with stre t c h - f ree NMO (b). For re 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, there is still significant scatter in the cross-plot domain. The top cross-plot shows the ideal data 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.
