The main purpose of this article is to provide geoscientists with one source which uncovers the connection between pre-stack seismic gathers and the petrophysical properties that cause amplitude versus offset (AVO) variations in them. There are many seismic AVO equations in the literature, all parameterized differently, e.g., Shuey (1985), Smith and Gidlow (1987), Verm and Hilterman (1994), and Gray et al (1999). This paper attempts to reconcile these parameterizations by presenting all of these existing equations using common parameters which are defined in Appendix A. Table 1 lists all the equations using the most commonly used parameters for each of the elastic constants. A second purpose is to fill in the gaps where interesting petrophysical parameters point to the possibility of equally interesting AVO equations that have not yet been published. Its third purpose is to discuss how to extract density reflectivity from the long offset seismic data that is often being acquired these days.
For years, we have been solving AVO equations for intercept and gradient, which, though useful, are mathematical functions that are only partially related to the underlying rock properties in which we are interested. Attempts to rectify this problem began in the late 1980s and continue to this day.
Currently, most forms of AVO analysis are derived from Aki and Richards’ (1980) approximation to Zoeppritz’ (1919) equations. Aki and Richards chose to describe the AVO relationship in terms of P-wave velocity (α), S-wave velocity (β) and density (ρ), shown in Table 1 as Equation 1. These elastic parameters are related simply to other commonly used elastic parameters, therefore, the AVO equations can be expressed in terms of any of the elastic parameters expressed in Table 2. The most common AVO equations are re-expressions of the Aki and Richards (1980) equation in terms of these other elastic parameters, sometimes with other approximations added. Familiar examples are Shuey’s (1985) re-expression of Aki and Richards’ equation in terms of P-impedance (ρα), Poisson’s ratio (σ) and P-velocity (α), shown in Table 1 as Equation 4 and Smith and Gidlow’s (1987) version in terms of P-velocity (α) and S-velocity (β), which uses some additional approximations. The most frequently used version is a modification of Shuey’s equation (due to Wiggins et al., 1983) using the A, B, C terminology, where A is the intercept, B is the AVO gradient, and C is commonly referred to as the curvature. Shuey’s (1985) version of this equation is expressed as:
Shuey’s and Wiggins’ equations are useful because they split the data into simple intercept and gradient terms. Expressed this way, it is easy to see the intercept and gradient in the raw seismic data. These equations should always be used as a QC for AVO analysis for this reason.
The relationship between the gradient and other petrophysical parameters begins to show itself by the presence of Poisson’s ratio in this equation. Wiggins’ equation is difficult to relate to petrophysics because the second term, B, is comprised of a complex mix of the reflectivity of the P-impedance and what Verm and Hilterman (1994) describe as “Poisson’s reflectivity”:
This means that in order to comprehend the petrophysics behind the AVO response, extensive forward modeling must be done. This is common practice in AVO analysis at the moment.
More recent AVO equations strive to extract the petrophysical parameters of interest directly from the variations in the seismic amplitudes with offset. Examples of such equations are: Verm and Hilterman (1994) who modify Shuey’s equation to derive P‑impedance and the Poisson’s reflectivity described above, Smith and Gidlow (1987) for P- and S‑velocity reflectivities, Gidlow et al. (1992) for P- and S-impedance reflectivities, and Gray et al. (1999) for the reflectivities of the shear modulus and either the bulk modulus or Lamé’s modulus. The problem with these AVO equations is that each author uses their own symbols, so it has been difficult to relate one AVO equation to another. Verm and Hilterman use NI and PR, Smith and Gidlow use ∆V/V and ∆W/W, and Gidlow et al. use ∆I/I, ∆J/J and ∆K/K.
As mentioned, one purpose of this paper is to express all these AVO equations using common parameters. This allows us to recognize the relationships between them. All the above equations are re-expressed in Table 1 using the most common parameterization of these variables, as described in Appendix A. This allows the comparison of the equations and assists in the determination of which equation is useful in each geological environment.
AVO equations such as Shuey’s and Wiggins’ can be re-expressed in terms of common petrophysical parameters, which allow for easier comprehension of the results. For example, Shuey’s equation can be slightly modified to use the reflectivities of the P-impedance, Poisson’s ratio, which is different than the Poisson’s reflectivity – PR – defined by Verm and Hilterman (1994), and density. This modification (Equations 9a and 9b in Table 1) allows for the extraction of the petrophysical reflectivity parameters: P‑impedance reflectivity (∆[ρα]/[ρα]), Poisson’s ratio reflectivity (∆σ/σ) and density reflectivity (∆ρ/ρ). In order to make modifications to the AVO equations to express them in terms of petrophysical parameters, the relationships in Table 2 are converted to reflectivities and substituted into Aki and Richards’ equation. An example of such a derivation for the bulk modulus, κ, the shear modulus, μ, and density, ρ, is shown in Appendix B. I have attempted to list all AVO equations that might be of interest to the geophysicist, petrophysicist or reservoir engineer in Table 1. All derivations of the AVO equations shown in Table 1 are similar to the one shown in Appendix B.
Table 1.AVO equations expressed using consistent parameterization of α = P-wave velocity, β= shear-wave velocity, ρ = density, σ = Poisson’s ratio, μ = shear modulus, κ= bulk modulus, λ = Lamé’s modulus, derived using the relationships in Table 2 and the methodology demonstrated in Appendix B.
Table 2. Relationships between elastic parameters, after Wang and Nur (1992).
Table 3. PS-AVO equations expressed using consistent parameterization of α = P-wave velocity, β = shear-wave velocity, ρ = density, μ = shear modulus, derived using the relationships in Table 2 and the methodology demonstrated in Appendix B.
The equation that I find most interesting is Equation 11, which expresses the shear term as β/α, or the Vs/Vp ratio. This parameterization results in some interesting simplifications. Unlike the other equations the Vs/Vp ratio in the second term is not a reflectivity, but a change in this ratio. This might have some important implications regarding, e.g., scaling.
The equation that is going to be most useful for exploration is Equation 17, Russell et al.’s (2011). This equation finally accomplished the original idea behind AVO, which was to look for fluids in sands, by splitting out a term that is specifically related to fluids, ∆f/f, and another specifically related to the rock ∆μ/μThis makes it ideal for exploration.
There are other petrophysical parameters (e.g., Young’s modulus) that are not typically used in petrophysical analyses and so are not covered by the above AVO equations. However, they can also be used in an AVO equation. The reflectivities of any three petrophysical parameters related to P-wave velocity, S-wave velocity and density can be derived directly by using an appropriate AVO equation that can be derived following the example shown in Appendix B. Gray (2002) shows that the advantage of this direct AVO inversion for petrophysical parameters is that there are no additional statistical errors in the inversion for the attributes of interest, as there are when elastic parameters are derived indirectly from AVO results (e.g., Goodway et al, 1997).
For PS-AVO, there are only two key parameters, one related to the shear properties and one related to density. Therefore, there are only three useful parameterizations of PS-AVO: the original by Aki and Richards (1980), one parameterized with the shear modulus (μ), and one parameterized with what is called shear impedance (ρβ). All three parameterizations are listed in Table 3.
Approximations and the Advent of Petrophysics in the AVO Equations
Most practical implementations of the AVO equations involve some sort of additional approximation to Aki and Richards’ approximations. Most of these approximations involve only two terms, dropping the third term in the AVO equation. This is generally because, in the commonly used equations, the third term tends to only become significant at large angles, where the approximations used by Aki and Richards no longer hold if the common assumption is made that the incident and emergent angles are the same. However, if emergent angles are also used, then these equations that include the third term are much more accurate (Downton and Ursenbach, 2006). Also, many authors assume that the density contrast between layers, which often accounts for the third term, is small based on Gardner et al.’s (1974) P wave velocity – density relationship. For the most part, these approximations work because most seismic data acquired around the time these equations were published did not have sufficient offset to robustly determine the third term of the AVO equation, which generally only becomes significant at angles greater than 30 degrees. Smith (1996) showed that using a three-parameter equation does not make any significant difference in the results for the data he used, which only had incident angles out to 33 degrees. For the newer equations (e.g., Gray et al., 1999), the third term can be significant, even at zero offset. Some, like Smith and Gidlow (1987) and Verm and Hilterman (1994), have tried to apply some petrophysical knowledge of the reservoir to better approximate the AVO results. Smith and Gidlow (1987) use Castagna et al.’s (1985) “mudrock line” and Gardner’s density estimate from velocity. Verm and Hilterman use the approximation α/β = 2 and assume that the third term in Shuey’s equation is negligible. Since longer shot-receiver offsets are being shot more commonly now for the purpose of extracting more information from the data (e.g., Wombell et al., 1999), the third term of these AVO equations will be more important for such data because its long offset has the potential to allow for robust third term estimation, provided that anisotropy is accounted for with tools like VVAZ (Velocity Versus Azimuth) and AVAZ (Amplitude Versus Angle and Azimuth) as well as using migration algorithms that allow for these. Using petrophysical approximations as suggested by Smith and Gidlow (1987) is recommended for data with small maximum angles of incidence, if these equations are to be used effectively to estimate the third term (e.g., Kelly et al., 2001).
Petrophysical relationships such as Gardner’s velocity-density relationship and Castagna et al.’s mudrock line are increasingly being used to stabilize AVO equations for the purpose of obtaining the elusive density reflectivity (see e.g., Kelly et al., 2001). Density reflectivity is important because it will help us to distinguish so-called “fizzwater” from gas and oil accumulations, thereby allowing for a much-reduced risk scenario than is typical for AVO results, currently. Density is very important for Canadian oil sands reservoirs. Often density is used to distinguish porous, saturated, reservoir sand from shale, e.g., Gray et al., 2006; Mayer et al., 2015.
Gidlow et al. (1992) found a way to re-express Aki and Richards’ AVO equation in terms of P-impedance, S-impedance and density, without using the approximations used in Smith and Gidlow (1987). The density term in this equation mathematically has very little effect until quite large incident angles occur. Here, and in general in AVO analysis, large incident angles are defined as tan²θ >> sin²θ and are typically considered to occur at angles greater than 30°.
Gray et al.’s (1999) AVO equations were inspired by the work of Goodway et al. (1997) and Gray and Andersen (2001), that showed that Lamé’s modulus and the shear modulus are very good at separating fluid and rock effects. Smith and Gidlow (2000) showed if certain approximations are made, Lamé’s modulus reflectivity is very closely mathematically related to the Fluid Factor, confirming Goodway et al.’s and Gray and Andersen’s observations, and provides insight into both the Fluid Factor and Lamé’s modulus.
Inversion of petrophysical properties
Seismic AVO reflections are very useful for interpreting structural and lithological boundaries, e.g., Castagna et al. (1998). These interpretations can be improved through the use of inversion (e.g., Pendrel and Van Riel, 1997). An example from Gray and Andersen (2001) shows that the interpretation of a known rugose unconformity at the top of the Mississippian section in Alberta appears more rugose after interpretation of the inversion for Lamé’s modulus, a petrophysical attribute. Lamé’s modulus shows strong separation between clastic and carbonate rocks (Figure 1), which allows for a more confident interpretation of this unconformity, as can be seen on the west side of the λρ plot in Figure 1 where the original interpretation of the unconformity, in red, is clearly above where it would be picked on the λρ section.
When we look at the amplitudes of reflections as we commonly do in exploration and AVO analyses, we are generally making an implicit assumption that the nature of the overburden does not vary. In general, this assumption is not true and inversion is a means by which we can separate the effects on a refection coefficient due to the overburden from the effects due to the reservoir. After inversion, we are better able to establish both whether there is adequate reservoir present and whether there is a suitable caprock. Investigations of caprock are becoming an increasingly more important aspect of interpretation (e.g., Liro et al., 2001). An example of such an interpretation for a gas field in China is shown in Figure 2 (Gray and Andersen, 2001), where we can see both reservoir and caprock signatures by comparing Lamé’s modulus and the shear modulus through λρ and μρ.
Gray (2002) shows that any petrophysical reflectivity, such as the ones derived here, can be inverted for its associated petrophysical parameter using off-the-shelf post-stack inversion software (e.g., Figure 3).
Figure 1. Example of the interpretation of λρ (top) and μρ (bottom) for a 3D dataset in the Western Canadian Basin (after Gray and Andersen, 2001). This analysis enables low λρ anomalies (blue) combined with moderate μρ anomalies (pink and blue) to be interpreted as gas sands, with much less ambiguity.
Figure 2. Example of the interpretation of λρ (left) and μρ (right) for a 3D dataset in China (after Gray and Andersen, 2001). This analysis enables low λρ anomalies (green) combined with moderate μρ anomalies (red and purple) to be interpreted as gas sands with much less ambiguity. In addition, shales overlying these reservoirs can be identified with low μρ and high λρ. Coals are identified with both low μρ and λρ.
Figure 3. Example of an inversion for λρ (left) using the Goodway et al. (1997) approach and λ (right) using the Gray (2002) approach of post-stack inversion of the results of Equation 8, above. Notice the reduced noise in the Gray approach. After Gray (2002).
Certain AVO equations are more appropriate in certain geological settings than others. For example, at the sea floor, the use of an AVO equation describing the change in Vp/Vs ratio is much more appropriate than using one that incorporates changes in Poisson’s ratio (Figure 4). As can be seen in the synthetic in Figure 4, a huge change in Vp/Vs ratio going from near infinite to 10 is equivalent to a tiny change in Poisson’s ratio from 0.5 to 0.495. This indicates that AVO equations that solve for Vp/Vs ratio reflectivity are more useful in younger, relatively unconsolidated sediments like those in the Gulf of Mexico than those that solve for Poisson’s ratio reflectivity. The corollary is that equations that solve for Poisson’s ratio reflectivity are more useful than those that solve for the Vp/Vs ratio in geologic settings containing older, consolidated rocks such as occur in much of continental North America. For example, the lower Cretaceous gas sand on the right side of Figure 5 at about 920 ms is much more clearly defined by Poisson reflectivity than by the Fluid Factor. This is because the Poisson’s reflectivity is about twice as large as Vp/Vs ratio reflectivity, which is related to the Fluid Factor.
Figure 4. Example of AVO analysis of the water bottom from the synthetic at left. The water bottom is the first reflector at 100 ms. The seismic section in the center is the result of AVO analysis for Poisson’s reflectivity. Note the weak response of the water bottom. The seismic section on the right is the result of AVO analysis for Fluid Factor. Note the strong response of the water bottom. After Gray (2004).
Equations that solve for parameters that are typically used in rock physics relationships, like the bulk and shear moduli, can be useful for careful quantitative evaluation of potential reservoirs and for reservoir characterization. Gray (2002) shows that the AVO parameters can be inverted for the desired petrophysical values, e.g., Figure 3. These inverted petrophysical values can then be used in petrophysical equations, such as Gassmann (1951), to evaluate potential reservoirs, just as log values would be used. Thus, petrophysicists can make reasonable estimates of potential risks for reservoir, saturation, reserves, etc., with only regional knowledge of the geology, by using the inverted AVO traces to estimate petrophysical parameters. These can, in principle, be derived using only the seismic data, using the flow shown below. Such values are already being used in exploration and development to assist in the determination of reservoirs, e.g., Chen et al. (1998) and Gray and Andersen (2001).
Figure 5. Poisson Reflectivity (top) and Fluid Factor (bottom). Note how Poisson’s reflectivity defines the lower gas sand better than Fluid Factor. (After Gray, 2004.)
Figure 6. Flowchart showing how to derive petrophysical parameters exclusively from seismic data.
Summary and Conclusion
Approximations leading to extended petrophysical work, e.g., Hedlin (2000), are now being done to more explicitly distinguish between fluid and lithology effects.
Use Gidlow et al.’s (1992) equation for early petrophysical estimates because its third term is the most mathematically negligible of all the equations in this paper.
Equation 11 expressing AVO in terms of P-Impedance, the Vs/Vp ratio and density simplifies to a very interesting result that only requires the Vs/Vp ratio change as opposed to its reflectivity. This version of the equation could provide some useful additional insights when doing AVO.
Use petrophysics to determine the optimal parameters to highlight the anomaly in the seismic data. Then use an appropriate AVO equation from Table 1 or 3 to derive those parameters from the prestack seismic data.
Only use intercept and gradient (Wiggins et al.’s, 1983 equation and Shuey’s 1985 equation) for quality control purposes. The results of these equations can be directly compared to the input seismic gathers. If their results are reasonable then the other AVO equations should also be reasonable. Using intercept and gradient for petrophysical estimations requires substantial additional approximations to be made.
AVO and AVO inversion can be done without well control if the seismic interval velocities derived during velocity analysis are reasonably accurate, e.g., from horizon-based velocities or full-waveform inversion.
Inversion of AVO attributes or full prestack inversion provides a means to more clearly separate caprock and reservoir.
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About the Author
David Gray is Senior Vice President Integrated Solutions at Geomodeling Technology Corp. after consulting for the last two years in geomechanics and reservoir characterization. David was honored to give the CSEG Canadian Distinguished Lecture on Seismic Geomechanics at 24 Universities and Research Institutes across Canada in 2018-19. In 2015, David was selected as the honoree of the 4th annual CSEG Symposium for his contributions to technology and illustrating how geophysics adds business value. Many of David’s recent talks address the value that geophysical work can bring both monetarily and societally. In 2018, he was given the award for the best CSEG Luncheon talk, and has he has been co-author of several papers that have won similar awards. He has made significant contributions to: quantitative interpretation, where four AVO equations have his name on them; seismic geomechanics where in 2009 he showed that all three principal stresses can be estimated from 3D seismic data and where he holds a patent; seismic fracture characterization, where in 1999 he related azimuthal variations in 3D seismic to fractures in the borehole and where he also holds a patent. For the latter, David’s team, comprising staff from 3 countries and 4 divisions, won Veritas’ VerTEX Team Award. He has presented over 100 papers at various technical conferences and luncheons for SPE, SEG, CSEG, AAPG, EAGE, HGS, and RMAG. Previously, David was Principal Geoscientist for Ikon Science Canada and Senior Geophysical Advisor in Nexen’s Technical Excellence team, where he contributed AVO exploration work on the 11 billion bbls of reserves discovered to date by the Guyana JV of Exxon, Hess, and CNOOC, and for offshore East Coast Canada. He demonstrated that geophysics could predict and improve production in various oilsands developments, and contributed to Nexen’s Horn River shale development. Prior to that, he did 17 years of research for Veritas and CGG in reservoir characterization and seismic signal analysis using experience gained as AVO specialist and seismic data processor for Veritas, Geo-X Systems, and Seismic Data Processors, as well as from his internship in the special projects group at Gulf Canada. David received a Bachelor of Science degree in Honors Geophysics from the University of Western Ontario (1984) and a Masters of Mathematics degree in Statistics from the University of Waterloo (1989). David is a member of SPE, SEG, CSEG, EAGE, and APEGA and is a registered Professional Geophysicist in the Province of Alberta. In his spare time, he enjoys spending time with his family, writing and presenting technical papers, working with students and young professionals, and competing in volleyball and grassroots motorsports.