## Dec 4 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 $$R = R_{0}\:\text{cos}\:\chi + G\:\text{sin}\:\chi$$ 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, DOI 10.1190/1.1438307.

Whitcombe, D, P Connolly, R Reagan, and T Redshaw (2002). Extended elastic impedance for fluid and lithology predic­tion. Geophysics 67, 63–67, DOI 10.1190/1.1451337.