The unified AVO equation

AVO theory provides us with the mechanics for optimizing fluid and rock content from seismic data. Arguably, the most important contributions in the literature to the understanding of AVO in the last 15 years have emanated from BP. I would claim (and I must stress that I have never worked for BP, even as a consultant) that what has been presented is essentially a unified view of AVO, in as much as it seems to explain many historical observations about AVO and puts into perspective approaches presented by other authors. It is interesting in the light of this claim, however, that there are many interpreters who are ignorant of what it is all about. Maybe this is due to the academic tone of the papers.

Central to AVO is the relationship between impedance AI and reflectivity R, such that R ≈ 0.5Δln(AI). As an aid to inverting non-normal- incidence angle stacks Connolly (1999), using the Aki–Richards (1981) simplifications of the Zoeppritz equations, derived impedance at an incidence angle (elastic impedance or EI). Thus, two-term elastic impedance is effectively the integration of Shuey reflectivity (i.e. R = R₀ + G sin²θ).

From the perspective of the AVO (intercept vs gradient) crossplot, Whitcombe et al. (2002) realised that Shuey reflectivity is described by a coordinate rotation or projection and that the incidence angle θ is related to the crossplot angle of rotation χ by sin²θ = tan χ. In other words, the x-axis (intercept) is the axis onto which the crossplot points are projected, and as the axes are rotated in a counter-clockwise direction, projections onto the rotated x-axis describe reflectivity at increasing incidence angles. Of course these projections will look similar to equivalent seismic angle stacks only if the seismic conforms to the linear approximation.

One problem, however, is that Shuey reflectivity can only be applied over a certain range of angles. In terms of the incidence angle θ the limitation is simply

θ = 0°– 90° (χ = 0°– 45°). If sin²θ is replaced with tan χ in Shuey’s equation then at high χ angles the projected reflectivity can give values greater than unity. So, a modification to Shuey’s equation is required to enable projections at any crossplot angle, giving a reasonable range of values whilst maintaining the correct relative differences between AVO responses. The result is

R = R₀ cos χ + G sin χ

which is effectively Shuey’s equation written in terms of χ and scaled by cos χ. This is the AVO equation and like most really useful equations it is elegant in its simplicity. Given that it effectively extends the angular range of Shuey’s equation, the corresponding impedance is termed extended elastic impedance (EEI). All of this is not very sexy but it does turn out to be very useful. A few aspects are worthy of note:

• Uncalibrated AVO analysis is essentially an angle scanning operation in which the interpreter tries to identify both fluid and rock effects within the context of a regional geological model.
• The EEI formulation is useful for detailed analyses of the intrinsic AVO signature simply through crossplot analysis of log data. Introducing gradient impedance GI as the EEI at χ = 90°, the AI vs GI – or rather ln(AI) vs ln(GI) – crossplot enables the plotting of all lithological units and maintains the angular relations of the intercept vs gradient crossplot, thus ensuring an appreciation of the linear discrimination of fluid and rock type. This analysis should be done in conjunction with bandlimited impedance synthetics to appreciate the nature of seismic resolution. It should also be remembered that theoretical fluid angles are generally upper angle limits; owing to the effects of noise, the effective fluid angle is usually lower.
• Angle-independent elastic parameters (Poisson’s ratio, acoustic impedance, λ, μ, and so on) can be shown to correlate with a particular angular rotation or value of EEI.

References

Aki, K and P Richards (1980). Quantitative Seismology, Theory and Methods. San Francisco: Freeman. Connolly, P (1999). Elastic impedance. The Leading Edge 18, 438–52.

Whitcombe, D, P Connolly, R Reagan, and T Redshaw (2002). Extended elastic impedance for fluid and lithology prediction. Geophysics 67, 63–67.

Elastic impedance (EI) or angle dependent impedance inversion, introduced by Connolly back in 1999, is an interesting application, as amongst others, it provides a convenient way to produce synthetic seismograms for variable angles of incidence. Could you describe some of the other applications that you have used EI for?

Although EI was designed to aid inversion of non-zero angle stacks it became clear shortly after Pat Connolly’s paper came out that EI presented an opportunity for AVO analysis. For example, a crossplot of EI(near) vs EI(far) was a useful space in which to look for litho-facies discrimination. Linear projections (i.e. weighted combinations of the two parameters) could then be determined to emphasize the desired litho-facies from inverted data volumes. We even gave it the name ‘AVO Impedance’. This approach is not often used today, as inversion minded geophysicists tend to favor simultaneous inversion (e.g. to AI and SI) and bandlimited/relative impedance minded geophysicists favor EEI. It was, however, a pragmatic approach which focused on stacks over the measured angle range, with the benefit of enhanced S:N in the stacking, as well as from the fact that the inversion procedure took care of the offset balancing problem.

What was your message with regard to good practices in well-ties that you published in the First Break back in 2003? Have you updated that since?

Estimating phase and polarity have been and probably will remain a critical issue facing the interpreter. The message of the 2003 paper was essentially that quantifying the well tie as proposed by White (1980) is a useful and practical thing to do, not least because it tells the interpreter a great deal about the data (and consequently what might or might not be expected from certain types of seismic analysis). Unfortunately the well tie is often viewed by the interpreter as an inconvenience rather than a chance to do some science and perform an experiment.

The well matching methodology effectively extracts a wavelet and determines the statistics of the tie within a 3D cube around the well, both in terms of the goodness of fit based on the cross correlation but also in terms of phase estimation accuracy. The wavelet and statistics produced from the analysis are useful for considering how to proceed with steps such as zero phasing. Importantly, the statistics can be used to show an improvement in a tie, for example after editing the well logs or reprocessing the seismic, and can be used to determine whether differences between the results at different wells or at different depths are statistically significant.

Perhaps a failing of the paper was that it focused exclusively on a methodology that is appropriate only when high quality check shot and log data are available. The interpreter also needs to be able to apply more ‘adaptive’ techniques which utilize an initial zero phase assumption, time-depth relations based simply on integrated sonic logs, cross correlation displays to evaluate phase rotation and time shifts to adjust the synthetic match. This has been addressed in my recent book (with Mike Bacon) Seismic Amplitude: An Interpreter’s Handbook.

Elastic impedance was extended (EEI) by Whitcombe et al.(2002), so that all potential AVO angles might be incorporated into a linearized (Shuey type) context. How much of this work do you think is being done outside of BP? Why do you think it is useful?

As a modification of Shuey 2 term reflectivity EEI provides a day-to-day practical methodology for seismic interpreters to identify AVO anomalies (related to rock or fluid) from bandlimited data. There are two aspects to EEI, namely impedance and reflectivity. The EEI formulation calculates an ‘impedance’ type parameter (Extended Elastic Impedance) that incorporates 2-term AVO. Differentiating the natural log of this parameter gives a two-term modified Shuey reflectivity (which usually scales reasonably well with seismic amplitudes over angle ranges from 0-~30º). The key parameter is the crossplot (χ) angle which can be derived from lnAI vs lnGI crossplots and/or R₀/G reflectivity crossplots (the angles measured on a typical R₀/G reflectivity crossplot are elegantly mimicked in the corresponding impedance space). Using well log data fluid and lithology signatures can often be optimized by using linear discriminants at specific angles (and this depends on the nature of the elastic properties). Effectively EEI describes AVO as a continuum of reflectivities across a range of χ angles, reflecting various combinations of fluid and lithology effects.

EEI encompasses individual AVO attributes that were originally designed to emphasize particular properties (e.g. such as Pseudo- Poisson’s ratio) and in this sense it might be called a ‘unified approach’ to AVO. It is an elegant mathematical treatment. From a practical seismic analysis point of view, EEI provides for a simple adjustment of weights applied to intercept and gradient (or near and far data) enabling the interpreter to ‘tune’ or interrogate/scan the seismic image across the range of χ angles.

Within BP EEI is a component of a workflow that includes coloured inversion and seismic net pay analysis specifically aimed at deep sea sedimentary settings. Outside BP the use of the EEI methodology is patchy although it is probably increasing. It is understandable that many interpreters find the mathematics too involved or find the combination of impedance and reflectivity difficult to relate to angle independent properties such as AI or Poisson’s ratio. Certainly the published papers are fairly academic, requiring a certain degree of dedicated study to get a clear understanding. In any case the interpreter doesn’t need to know the precise workings of EEI to effectively utilize AVO analysis. Simple techniques such as colour coded crossplotting or weighted stacking of near and far reflectivities can be effective tools to detect anomalous seismic signatures. On the other hand some geophysicists believe that AVO is best approached by inverting gathers for P and S wave reflectivities.

In their paper, Whitcombe et al. (2002) show the application of EEI for predicting Lambda-rho, Mu-rho and some other moduli. However, the computation of these moduli involves the P-velocity, S-velocity and the density, and EEI is also a function of all three. Don’t you think that we should be doing the correlation analysis between two independent curves (such as EEI versus porosity, water saturation, GR, etc.) rather than the dependent entities?

The idea behind correlating EEI with other logs is of course to deduce which chi angle correlates most strongly with the attribute or property of interest. But this is only one step in an EEI workflow and should not be used in isolation. My preference is for analyzing the theoretical discrimination of lithofacies on lnAI and lnGI crossplots (usually using log sampling or a twt sample rate), then checking the effect of seismic resolution on synthetics generated at the appropriate angle. Applying log based results to the seismic should be approached with care. For example, the theoretical angle is unlikely to be the appropriate angle for the seismic projection owing to the role of seismic noise (for fluid optimized χ angles the corresponding seismic angle is lower than the theoretical angle). Other factors that need to be considered are the relative scaling of well synthetics and seismic (i.e. offset balancing) and the accuracy of incidence angle calculation. The role of anisotropy should also be considered.

EEI depends on the crosscorrelation analysis between the elastic impedance curves at various angles and the desired rock property curves. However, this analysis is also very sensitive to the choice of the correlation window size. In your experience, how do you optimize the window size?

Typically the EEI methodology is applied to stratigraphic units up to several hundred metres thick. I haven’t actually investigated the effect of correlation window length as I tend to derive χ angles based on lnAI vs lnGI and R₀/G crossplots. EEI calculation is also sensitive to the value of k (i.e. the background or average (Vs/Vp)²), such that derived reflectivity across large velocity contrasts may be in error, but as Connolly (1999) shows the errors are probably much smaller than those arising from the estimation of seismic incidence angle.

Gradient impedance (GI) defined as EEI at an angle of rotation of 90º is usually crossplotted with acoustic impedance for discriminating different lithounits and fluids. Tell us about how good or not-so-good this application is, and if it has some limitations.

As discussed above I think that the lnAI vs lnGI crossplot is a very useful way of appreciating the effects of variations in rock properties on AVO. Perhaps the chief limitation of bandlimited approaches which utilize linear discriminants is that high porosity sands can give anomalies which, on the basis of the AVO signature alone, might be difficult to differentiate from a hydrocarbon effect. The target/reference AVO approach of Chiburis can address this problem but often the determination of the reference signature may not be reliable. Of course this problem of porosity vs fluid is one reason why many geophysicists attempt simultaneous inversion with interpretation based on rock physics templates, but that is for a different discussion.

Quakenbush et al. (2006) demonstrated that in the AI-SI crossplot space the discrimination between any two fluid types could be highlighted by rotation of the axes, and came up with the idea of Poisson impedance. Could you share with us your experience in how Poisson impedance analysis compares with a similar analysis carried out with say, the AI and GI?

Simply looking at a log data comparison of Poisson impedance versus a chi angle projection that optimizes Vp/Vs (usually about 45º) will show a high degree of similarity. Notionally the differences are in the application with the question being re-framed as ‘Which is better – bandlimited or absolute impedance inversion?’. The answer of course depends on many factors, not least being the purpose of the analysis, the quality of the seismic data and the number of wells with good suites of logs data. The chi angle projection is suitable for determining relative lateral Vp/ Vs changes at the target of interest (assuming that the offset scaling is reasonably correct). Care should be taken in using the projection outside the zone for which it was designed. With Poisson impedance the inputs are from a simultaneous inversion of pre-stack data to AI and SI, and assuming that there are enough wells to constrain the wavelet scaling and the low frequency component is merged correctly, it should be possible to treat the result quantitatively (at least within the limits that the S:N ratio allows).

You have had an undampened interest in the application of rock physics in prospect generation and field development throughout your career. Could you share with our readers why your belief in it remained steadfast and what you plan to accomplish in the near future?

Quite apart from rock physics and seismic interpretation being colorful and fascinating subjects it is evident that these techniques can have a significant impact on the bottom line (i.e. the finding of hydrocarbon resources), especially in appropriate geological settings. I enjoy the detective work. Of course we don’t get it right every time (there is always uncertainty and we can fall in love too much with our own ideas) but it works often enough to keep us gainfully employed and the world turning. Four years ago I decided to go back into a small company environment and get close once more to the decision making and the drill bit, a decision I have not regretted (and I even found time to write a book!). I expect the small- to medium-sized company environment is where I will stay for the near term. I will also continue to give occasional training courses (through the Nautilus training organization) and no doubt I will always have a technical issue or two to keep my mind occupied.

Rob, let me ask you an open-ended question, as sometimes it is fun answering such questions. From your perspective, what do think are the three, still unsolved problems in geophysics?

From the perspective of conventional reservoirs some outstanding issues are how to routinely exploit anisotropic information in seismic reservoir characterization and how to utilize spectral decomposition data in AVO. Two is my limit.