Experts Answers

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 pre-stack data needs to be compensated for. These issues form the question for the ‘Expert Answers’ column 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 received. – Satinder Chopra Q. Differential interference and offset-dependent tuning are two serious factors that hamper confident AVO analysis. What causes them and how do we effectively tackle them today? Answer 1 Differential interference results from the fact that neighboring reflectors increasingly interfere as the incidence angle increases. When the reflectors come from the top and bottom of a thin bed of interest, the interference is called offsetdependent tuning. This tuning will cause false amplitude variations with offset (AVO), not associated with either individual reflector. When viewed in moved-out gathers, these effects appear to be the result of a stretched wavelet at larger offsets. 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

Figure 1. Waveform and amplitude spectrum of a bandpass wavelet.

the unstretched one. This procedure generated a movedout gather without wavelet stretch but amplified ambient noise, sometimes to unbearable levels. • Castoro et. al. (2001) removed wavelet stretch from movedout 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

Figure 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 noisefree intercept in the presence of white noise, h1(t) (black) and the optimal filter for estimating the stretch error in the gradient, h2(t) (red). Continued on Page 13

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Figure 3. The actual and estimated noise-free wavelets (black) and the actual and estimated gradient leakage wavelets (red).

Figure 4. A synthetic CMP gather that illustrates offset-dependent tuning.

due to stretch, from the intercept. The stacking velocity should be smoothed to avoid sudden changes in V s’(t). In the presence of an intercept noise s p e c t rum Sn(ω), the Fourier transform of the optimal h 2 filter, in the least-squares sense, is given by 3. H2(ω) = -[ |W(ω)|2 + ω W*(ω)W’(ω)] / [ |W(ω) |2 + Sn(ω) ] . Note that this expression is invariant to a wavelet phase shift, and is stable even when the wavelet Fourier transform 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 effectively Figure 5. An AVO crossplot without gradient stretch correction. Various false AVO anomalies are apparent. being stretched at large offsets. The error is zero at the wavelet center. The to the gradient from differential interference, and then red curve of Figure 2b re p resents the optimal h2(t) filter, subtracting it. Optimal performance is achieved in the presence computed assuming 1% white noise, whose spectrum is given by of a known noise spectrum. equation (3). Also shown in Figure 2b is the optimal h1(t) filter, which estimates the noise-free intercept in the presence of noise. Differential interference manifests itself as leakage f rom the Its Fourier transform is given by normal-incidence reflectivity series, a(t), to the AVO gradient. For a wavelet w(t) and stacking velocity Vs(t), this leakage is 4. H1(ω) = |W(ω)|2 / [ |W(ω)|2 + Sn(ω)]. approximated by The red dashed curve of Figure 3 is the result of convolving w(t) 1. BL(t) = -{a(t) * [t w’(t)] } [1 + 2tVs’(t)/Vs(t)] / 2, with h2( t ). It closely approximates the leakage wavelet BL(t). Also shown is the result of convolving w(t) with h1(t). It closely where “*” denotes convolution, and the prime denotes differena p p roximates w ( t ). Although not terribly important in this tiation (Swan 1991). Given an AVO intercept trace, example, this filter applies the same noise-reduction regimen to A(t)=a(t)*w(t), we can estimate the component of the gradient the intercept as to the gradient. This will ensure their spectra will due to differential interference using match, and hence optimize the coherency of their cross-plots. 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

Figure 4 shows a synthetic CMP gather formed from a 50 ft section of 2.3 g/cm 3 material embedded into a constant 2.5 g/cm3 substrate. Neither the acoustic velocity (10 kft/s) nor Continued on Page 14 December 2004

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the shear velocity (5 kft/s) varies through this model. Such a densityonly 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 corresponds 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. Figure 6. The same crossplot after gradient stretch removal. A much more accurate picture emerges. The left side of this figure shows the intercept trace, repeated five times. The background colors match those of the AVO classifications. Although the central top and base reflectors correctly indicate background (gray) reflectors, 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 interference 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.

Figure 7. A wedge plot without gradient stretch removal that shows coherent false AVO anomalies above and below the target event.

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.

Figure 8. With offset-dependent tuning removed, the false AVO anomalies disappear. Continued on Page 15 14

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Expert Answers Continued from Page 14 ________________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 seismology. 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 waveform from two or more neighboring reflectors varies with offset. One could also cite differential interference from multiple reflections. 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 inversion and layered inversions that may provoke further comment.

Figure 2. AVA (amplitude variation with angle) of the shale-sand-shale model of Figure 1 for bed thicknesses ranging from 5 m to 40 m i n 5 m increments.

AVO response of a thin bed

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

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 increments. The corresponding angles of incidence are 5.8° to 37.1°.

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 860 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 cause of these oscillations is the convergence of the top sand and base sand reflections in time with increasing source to receiver offset.

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

Figure 1. Model of a brine filled sandstone from the central North Sea.

Figure 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 response. Continued on Page 16 December 2004

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Figure 4. Tuning curves for the 8-60 Hz Butterworth filter used as seismic wavelet in the simulated data of Figure 2.

That is, the effective time thickness of the layer decreases with offset. 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 recorded amplitude back towards tuning on an increasing 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 amplitudes. One straightforward measure that does remedy the differential 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 responses would lie on a straight line from the origin through the point representing an isolated reflector out to the tuning point.

Figure 5. Simulated offset gather for a bed thickness of 15 m in the model of Figure 1. Top: After NMO correction. Bottom: After NMO correction and wavelet deconvolution.

Figure 6. Top: Amplitudes of the troughs at 0.95 s on the traces of Figure 5 (brinefill) after NMO correction (black diamonds) and after NMO correction and wavelet deconvolution (blue squares). Bottom: corresponding amplitudes of the troughs from the gas-fill case (traces not shown). The red circles show the isolated reflector response scaled to the normal incidence thin-bed amplitude.

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 interceptgradient 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 interceptgradient quadrant. If` that is a benefit, there is a penalty. Because NMO correction pulls noise as well as signal to lower frequencies, 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.

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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-tonoise 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 interceptgradient relations also find expression in AVO inversion. The increasing 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, whatever 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 coefficient 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 attenuate 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. Three partial stacks are formed and the near and mid-offset stacks are cross-equalized to the far-offset stack while preserving the trace scaling. This not only compensates the variations in bandwidth from 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 attenuation, this procedure does diminish the worst effects of timing variations 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 difficult 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 attainable 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. Crossequalization of the input data does this

Figure 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 corrected gather; traces 7 and 8: from cross-equalized partial stacks. Bottom: the centre panel after low-pass (0-40 Hz) filtering.

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 artifacts 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 fc is 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 on Page 18 December 2004

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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 impedances 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 conclusion 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. An approach to AVO inversion based on partial stacks and crossequalization, can avoid the worst effects of residual moveout and NMO stretch. For AVO analysis too, diff e rential interference (NMO stretch) can obscure the intercept-gradient relation. I have described a wavelet deconvolution scheme that renders the intercept-gradient relation immune to NMO stretch. Other schemes, including cross-equalization, may also be possible depending on circumstances, 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 degrees of freedom is also roughly the number of peaks and troughs in a seismic trace. This suggests that there is little amplitude information 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 intercept 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 provides more 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 possibility of further artifacts. Readers will have detected some skepticism 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 Differential 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 differential moveout (moveout varies with offset), adjacent events within a CMP gather tune as a function of offset, again introducing 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 differential tuning make the NMO inverse problem underdetermined 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 offset, which leads to errors in the AVO analysis. There are a number of ways to deal with diff e rential interference and differential tuning. First, one can ignore them, do conventional NMO and live with the consequences of amplitude and character distortions. In the first two sections below, the consequences of doing this are explored both analytically and empirically. For certain reflectivity attributes and anomalies acceptable results may still be obtained even in the presence of these effects. A second approach is to try and precondition the data better prior to AVO analysis by performing a stretch-free NMO correction (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 waveform inversion (Simmons and Backus, 1996; Downton and Lines, 2003). By solving all three problems together, certain geologic constraints may be incorporated making the inverse problem 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

 f  ~ 1 S x (f ) = S x  , αx αx 

(1)

˜ is the specwhere Sx is the spectrum before NMO correction, S x trum after NMO correction, f is frequency and αx is the compression 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. Continued on Page 19

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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 correction for incident angles from 0 to 45 degrees. The model generating 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 assumptions 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 calculating 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

dB A =κ , B B

(2)

where

κ=

(

)

4π 2η 2 3 − 8π 2η 2 , 1 − 8π 2η 2

(

)

(3)

where η= fodt is defined in terms of the dominant fo and 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 dB 0. 0. As η increases the size of the then B = = gradient error increases. The other factor that controls the size of the error is the ratio A B . Thus it is possible to predict the size of the error for different classes of AVO anomalies. For Class I ( A