2001 Vol. 222 No. 9
How to obtain reliable S-impedance from P-wave data
Standard AVO analysis can
create errors and lead to false prospects. With additional effort, elastic impedance provides a method that
gives a more correct result and fewer dry holes
Guillaume Cambois, CGG
seismic interpretation comprises the analysis of pressure waves. P-wave reflectivity or its inverted
form, P-impedance can be correlated with rock properties and provide effective hydrocarbon indicators.
However, sometimes, P-wave impedance alone cannot discriminate shales from gas bearing sands. Fig. 1a
represents a histogram of P-impedance lab measurements performed on core samples from a North Sea field: sands
and shales are indistinguishable.
Fig. 1. Histogram of core sample
measurements from a North Sea field. a) Sands and shales cannot be differentiated using P-impedance
alone; b) Sands and shales are clearly differentiated using Vp/Vs ratio, which
is equal to P- to S-impedance ratio.
However, if the ratio of P- and S-impedance is
measured, sands and shales separate into two distinct histograms Fig. 1b. In this case, S-impedance provides a
hydrocarbon indicator that cannot be achieved with P-impedance. In general, the best hydrocarbon indicator is
a combination of P- and S-impedance to benefit from the discriminating power of both attributes.
P-impedance is obtained straightforwardly from
conventional seismic data by inverting P-reflectivity. Stratigraphic inversion does essentially two things: It
removes the seismic-wavelet effect and provides an "interval" measurement (as opposed to
reflectivity, which is an interface measurement).
There are two ways of obtaining S-impedance: either
through direct measurement of shear reflectivity (using multi-component acquisition) or by AVO. The latter
approach is much cheaper due to reduced acquisition costs, and it can be readily obtained from multi-offset
conventional data. However, it is not a direct measurement and is, therefore, often biased by artifacts. This
article reviews the best way of obtaining reliable S-impedance from P-wave data.
Shuey1 showed that prestack behavior of
P-wave reflection data follows the simple formula:
Where: q is the angle of incidence.
A and B are respectively called intercept and gradient, and are the only two terms generally
considered for AVO studies. The third term becomes too small for standard data with a limited range of
incidence angle. Under the approximation that the background Vp/Vs ratio
equals two, A and B are related to the elastic parameters by:
Rp and Rs are P- and S-reflectivity, respectively, and Ip
and Is are P- and S-impedance. Using these two formulas, the two-term Shuey equation can
be written as:
The first expression is the small-angle approximation
of Fattis equation.2 It expresses angle-dependent reflectivity as a function of P- and S-
reflection coefficients. Using this formula, shear reflectivity can theoretically be retrieved from P-wave
data. The second expression is equivalent to the formula derived by Verm and Hilterman.3 These
authors called the second term on the right-hand-side Poissons Reflectivity (PR) because:
Castagna and Smith showed that
PR is identical to Smith and Gidlows fluid factor when Vp/Vs
equals two.4,5 The fluid factor is often regarded as the optimum hydrocarbon indicator.
Fig. 2 shows two angle sections, 10° and 30°,
from a North Sea 3-D data set. The AVO anomaly appears clearly and corresponds to a known gas field. A
sophisticated hydrocarbon indicator is not needed to identify this reservoir. Yet, it is possible to compute
intercept and gradient from these two sections:
Fig. 2: Near- and far-angle sections
from a North Sea 3-D data set. Obvious difference at the reservoir level (red rectangle) indicates a
strong AVO anomaly.
The intercept is almost equal to the 10° section,
and the gradient is the difference between the two sections multiplied by more than four. When differencing
seismic sections, a matching operator is usually computed to ensure that the two data sets have identical
wavelets. It is odd that this is seldom done for AVO analysis. Which means that if the two sections in Fig. 2
have different wavelets, the gradient will be dominated by wavelet differences and not by the expected AVO
signal. This phenomenon is called "intercept leakage," because parts of the intercept appear boosted
in the gradient.
Fig. 3 shows the wavelets estimated from well logs.6
Between 10° and 30°, wavelet peak amplitude increased 50%, central frequency decreased 25% and phase
has rotated 45°. These effects are expected from NMO stretching. It should be obvious that these
differences multiplied by 4.55 are likely to dominate genuine AVO anomalies in the gradient. In practice, more
than two sections are used for intercept and gradient calculation, but this author has shown that intercept
leakage occurs regardless of the number of angle sections.7
Fig. 3: Wavelets extracted from the two
angle sections in Fig. 2 using sonic, shear and density logs. Due to NMO stretching and inaccurate
prestack scaling, the 30° wavelet peak amplitude has increased 50%, its central frequency has
decreased 25%, and its phase has rotated 45° compared to the 10° wavelet.
A matching filter between the two sections prior to
differencing will reduce the amount of leakage and yield more-accurate results. However, the shortest
bandwidth will dominate, which means that ultimate resolution of the gradient is that of the far angle
section. A similar reasoning shows that resolution of shear reflectivity from P-wave data cannot go beyond
resolution of the farthest angles. The difference in resolution between P- and S-reflectivity is seldom
noticed in standard AVO analysis, because intercept leakage gives identical resolution to all attributes
this is, of course, artificial. After all, according to Shueys equation, the shear information in P-wave
data is primarily located at far angles, which are most affected by NMO stretching.
Wavelet variations are not the only cause of mismatch
between angle sections; residual NMO-corrections also play an important part. Dense velocity analysis is
therefore necessary for AVO studies. Yet, sometimes it is simply impossible to pick the correct velocity. Fig.
4 shows logs from a well that penetrated the reservoir shown in Fig. 2 and a synthetic gather. This is a
typical Class 2 AVO anomaly. The top reservoir cannot be seen on small-angle reflectivity because it does not
generate a P-impedance contrast (the P-velocity contrast is exactly compensated by the density contrast).
Fig. 4: S-velocity (red), P-velocity
(blue) and density (green) logs for previous data set. Top reservoir (red ellipse) shows drop in
density and increase in P- and S-velocities. Resulting acoustic impedance (black) does not show a
contrast. However, Vs/Vp ratio squared (red to the right) does show a
significant jump. Prestack behavior of this typical Class 2 sand is shown by the synthetic
angle-gather. Red line indicates risk of picking an incorrect phase during velocity analysis.
Vp/Vs contrast is such that the top reservoir appears quite clearly at
wide-incidence angles. The synthetic gather has been computed using Shueys equation and, therefore, has
no NMO-errors. Yet, when advised to pay particular attention to the "top reservoir trough," a
conscientious geophysicist will always tend to flatten it, as indicated by the red curve. Although it would be
incorrect in this case (the apparent upward shift of the trough is due to waveform interference, not
NMO-errors), the best data-processing geophysicists will pick a velocity that flattens the trough. There is
simply no way to distinguish waveform interference from velocity variation. Unfortunately, most AVO anomalies
generate offset-dependent interference.
Connolly developed an alternative approach to AVO.8
Instead of first computing P and S reflectivity series from P-wave AVO and then inverting them to obtain P-
and S-impedance, he proposed to first invert the angle-dependent reflectivity series to obtain what he
called "elastic impedance" for that given angle and then compute P- and S-impedance. It can
be shown that these two approaches are equivalent when all parameters are identical: same data; same wavelet;
same a priori model; same expression of Shueys equation.
However, they differ significantly when one of these
parameters changes, especially the wavelet. As stated earlier, the first step of stratigraphic inversion is to
remove seismic-wavelet effects. Since the wavelets are likely to be different for the various angle cubes
(Fig. 3), performing stratigraphic inversion before AVO-attribute calculations can only improve the results.
Hence, for this reason alone, the elastic-impedance approach is better than the standard AVO approach.
There are other advantages to elastic impedance. In
particular, stratigraphic inversion can de-tune seismic data to a certain extent. Hence, some of the
interference effects described above can be resolved by elastic impedance. Residual NMO errors can be detected
in the inverted domain and corrected before computing P- and S-impedance. However, there is a limit to the
de-tuning power of stratigraphic inversion: The best algorithms can go to an eighth of the wavelength, but not
beyond. Again, since the ultimate resolution is driven by the lowest bandwidth of the angle cubes, S-impedance
extracted from elastic impedance will have the resolution of the widest angle. It is thus necessary for data
processing to extract the maximum bandwidth out of the farthest-angle data.
Since AVO studies focus on seismic amplitudes, careful
amplitude processing is essential. This involves the use of deterministic operators with the least possible
data-adaptive methods. Spherical-spreading corrections based on a known velocity field are preferred to
statistical amplitude balancing; deterministic wavelet shaping is preferred to predictive deconvolution; noise
attenuation techniques (Radon, FX-prediction) must preserve signal. These elaborate preserved-amplitude
sequences have become more refined over the years. Yet, it is sometimes doubtful whether they achieve the
accuracy required for AVO analysis.
Take, for example, the deceptively mundane task of
prestack amplitude balancing. In the elastic theory, recorded amplitudes decay as a function of spherical
divergence of the wavefront. Although compensating for this decay is easy to achieve in a simple layered-earth
model, attenuation effects should never be neglected.5 Q-compensation is now routinely performed,
but the resulting prestack amplitudes are never fully satisfactory.
Castagna and Smith, as well as this author, have shown
that inaccuracies in prestack amplitude compensation are a major factor in inflated gradient amplitudes and
intercept leakage.4,6 However, analogous to wavelet variations with angle, inaccurate prestack
amplitude behavior does not affect elastic impedance. The average amplitude is included in the wavelet (Fig.
3) and, as such, its influence is eliminated by stratigraphic inversion. Hence, although accurate prestack
amplitude balancing which is almost impossible to achieve is regarded as a must in
preserved-amplitude sequences, this step becomes totally irrelevant with elastic impedance.
NMO also requires great care for standard AVO and
elastic impedance. As described earlier, a slight mis-tie between the two angle sections generates intercept
leakage in the gradient. Figs. 5 and 6 illustrate the dramatic effects of NMO-errors on AVO analysis using
synthetic data. The input data is a gather of 31 angle traces sampling the 0° 30° range at 1°
intervals. The P-wave impedance model is a random series, and the S-wave impedance model is exactly its half.
The prestack reflectivity series are generated using a broadband zero-phase wavelet and Shueys equation.
At first, the gather is noise-free and the inverted
intercept and Poissons reflectivity are exactly what is expected (PR = A + B is almost zero),
which is only natural, since the same equation is used to model and invert the data, Fig. 5. Then, residual
NMO-errors are introduced as a parabolic time shift that reaches 5 ms at 30°, Fig. 6. Now PR
becomes a negatively scaled version of A, with a large-scale difference, as predicted by theory. The
estimated PR is not just a scaled-up version of its correct value; it is heavily contaminated by
Fig. 5: Synthetic data simulating a
hydrocarbon- free clastic sequence. Because shear and pressure reflectivity are equal, Poissons
reflectivity (PR) is nil. AVO extraction on this noise- free data leads to the expected
Fig. 6: Same synthetic data but after
introduction of residual NMO correction. Intercept A is correctly estimated, but PR becomes a
negatively scaled version of A. The scale difference is actually predicted by theory. Note that the
two attributes are perfectly correlated in the crossplot. This is misleading since similar trends
could be expected from actual AVO anomalies.
Elastic impedance is also sensitive to misalignments,
albeit in a unique way. Time differences between near and far angles do not generate leakage but can still
bias results. Fig. 7 shows an elastic-impedance crossplot derived from a well in West Africa. Reservoir sands
are clearly identifiable. Fig. 8 shows the same crossplot after a 2-ms time shift between elastic impedance at
10° and 30°: The sands cannot be detected. The maximum "tolerable" delay before the
crossplot breaks down is a function of reservoir thickness (very thin in this case).
Fig. 7: 10° and 30°
elastic-impedance crossplot from a West African well. Logs are converted to time and sampled at 1 ms.
Colors represent gamma- ray values. Ellipse indicates oil sands, which can be clearly separated in the
Fig. 8: Same crossplot after shifting
the 30° elastic impedance by 2 ms. Oil sands no longer stand out and are hard to discriminate.
This extreme phenomenon is due to the particularly thin nature of the reservoirs.
However, this analysis should always be carried out as
part of a feasibility study before venturing into any AVO project. It gives an indication of the accuracy
required while performing NMO corrections. Once again, the mismatch between near and far angles is not readily
observable with standard AVO. The intercept leakage generated by misalignments ensures that all AVO attributes
are perfectly in phase, Fig. 6. Although the result is clearly incorrect, the artificial time match between
attributes provides a deceptive sense of confidence in the data. It is, of course, much safer, although more
tedious, to analyze and fix all mismatches with elastic impedance.
At large offsets, seismic reflections in CDP gathers
have a tendency to "curl up." This phenomenon can be due to transverse isotropy or to ray bending
(the simple NMO equation assumes straight rays). Either way, a fourth-order term is needed to flatten these
events and take full advantage of the wide aperture. Picking this additional term is not a trivial task and
requires specific training for geophysicists and auto-pickers. The presence of anisotropy adds two major
difficulties: The AVO equation changes and the incidence aperture is reduced.9,10 These two
effects combine for the worse: More parameters to estimate with less independent data.
If anisotropy effects are mild enough to be neglected,
it is tempting to use increased aperture to estimate Shueys equation to three terms. However, wide-angle
data brings about new problems, from acquisition (array responses and source directivity) to processing (the
parabolic assumption for multiples no longer holds). There is valid data at large offsets, but extracting a
meaningful third term is a lot more difficult than extracting the second term, which in itself is already a
Many schemes transform reflection data into impedance.
They use various models, constraints and inversion engines. The inversion of choice for elastic impedance is
the algorithm described in Gluck et al.11 It is a 3-D, layer-based algorithm with a nonlinear
(simulated annealing) inversion engine. The model is constrained by interpretation, not wells. Well logs are
used for wavelet extraction and quality control, which is valid, since they are not used during the inversion
Fig. 9 shows such QC for the data displayed in Fig. 2.
Two angle cubes are inverted and results match the well-impedance quite remarkably, considering that the wells
were not used in the inversion. The only constraints were the initial 3-D layer model and the impedance
Fig. 9: Inversion QC at well location
for two angle cubes. Thick yellow lines represent initial models. Yellow corridors represent areas,
within which, impedances are allowed to vary freely in the simulated- annealing inversion scheme. Red
blocky lines represent inversion results. Black lines are elastic impedance logs; they are pasted on
the display for QC purposes, because they have not participated in the inversion scheme. The fit is
remarkable. Black wiggle traces are actual seismic traces at the well location, and red wiggles are
modeled traces from the inversion result.
Because this inversion results in a series of layers
updated from the initial layers in time and impedance it is straightforward to detect residual
NMO-errors. This step does not compensate for widely inaccurate velocity picking (angle sections need to be
similar enough to share the same initial layer structure), but it does compensate for waveform interference
due to tuning.
Once the layers are reconciled for various
elastic-impedance angles, it is a matter of simple algebra to compute P- and S-impedance and any other
AVO attribute from logarithmic elastic impedance. Fig. 10 shows Poissons ratio for the section in
Fig. 2. The main reservoir is clearly visible, along with a lower structure that displays low Poisson values,
but is far less visible on the original seismic data shown in Fig. 2.
Fig. 10: Poissons ratio
calculated from data in Fig. 2. Results of the inversion (Fig. 9) are combined to compute Poissons
ratio. The layers had to be reconciled because interferences from this Class 2 AVO anomaly (see Fig.
4) had biased the velocity picks. The main anomaly appears quite clearly (blue values), so does a
lower anomaly that was not as visible on original seismic data.
AVO anomalies exist in the real world and are a
wonderful tool to detect hydrocarbons. In theory, it is possible to estimate shear reflectivity from P-wave
AVO data. However, it is an indirect measurement that is prone to errors and can easily be biased. In
particular, wavelet variations with offset, which always occurs, introduce artifacts that dominate genuine AVO
anomalies. The processing effort required to eliminate all of these errors is sometimes overwhelming, perhaps
The elastic-impedance approach offers an elegant
alternative to standard AVO. Because the prestack data is inverted before AVO-attribute calculation, the main
artifact sources namely wavelet variations and prestack amplitude balancing are automatically
eliminated. Further, a layer-based stratigraphic inversion scheme allows detection and correction of residual
NMO-errors, which can efficiently compensate for tuning effects. Indeed, all AVO anomalies generate waveform
interferences that make it impossible to pick accurate velocities.
Because most of the shear information contained in
P-wave data is located at far angles which is the area most affected by NMO-stretching
resolution of AVO-derived shear reflectivity is necessarily lower than the expected pressure resolution. This
resolution differential is seldom observed with standard AVO analysis, because intercept leakage associated
with the errors guarantees identical resolution between all attributes. This is, of course, an artifact.
Similarly, intercept leakage guarantees that all AVO attributes are in phase, even in the presence of residual
NMO-corrections. In addition to being clearly wrong, the intercept leakage gives a false sense of confidence
in the data that makes it even more dangerous.
Fortunately, elastic impedance provides a way to avoid
these pitfalls. Although it is more tedious than standard AVO, because it requires stratigraphic inversion of
all angle cubes and a careful analysis of layer mismatches, the additional effort is worth it. The resulting
S-impedance is more accurate and allows detection of smaller AVO anomalies, along with correct identification
of Class 2 anomalies.
The author thanks Norsk Hydro, Conoco, BP and Statoil
for allowing use of their data.
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the Zoeppritz equations," Geophysics, Vol. 50, pp. 609 614, 1985.
2 Fatti, J. L., et al., "Detection of
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3 Verm, R., and F. Hilterman, "Lithology
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4 Castagna, J. P,. and S. W. Smith, "Comparison
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5 Smith, G. C., and P. M. Gidlow, "Weighted
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6 Cambois, G., "Can P-wave AVO be
quantitative?" The Leading Edge, Vol. 19, pp. 1246 1251, 2000.
7 Cambois, G., "AVO attributes and
noise: Pitfalls of crossplotting," SEG Expanded Abstracts, pp. 244 247, 1998.
8 Connolly, P., "Elastic impedance,"
The Leading Edge, Vol. 18, pp. 438 452, 1999.
9 Thomsen, L., "Weak anisotropic
reflections," in Offset dependent reflectivity, Castagna and Backus, Eds., SEG, Tulsa, Oklahoma.
10 Hilterman, F., C. Van Schuyver and M.
Sbar, "AVO examples of long-offset 2-D data in the Gulf of Mexico," The Leading Edge, Vol.
19, pp. 1200 1213, 2000.
11 Gluck, S., E. Juve and Y. Lafet, "High-resolution
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has worked in various positions at CGG since 1991, including research geophysicist and manager of
research. He is the current vice president of technology for Data Processing and Reservoir services. He
earned an undergraduate degree at Ecole Polytechnique in Paris in 1987 and a PhD in geophysics in 1991
from the University of Texas, Austin. He is chairman of the editorial board at The Leading Edge and is a
member of SEG, EAGE and AFTP.