# Effect of anisotropy on AVO

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

The P-to-P reflection amplitude as a function of angle of incidence given by equation (16) can be modified to accommodate for transverse isotropy as follows [1] [2]

 $\displaystyle R(\theta)=\left[\frac{1}{2} \left(\frac{\Delta\alpha}{\alpha}+\frac{\Delta\rho}{\rho} \right)\right] + \left[\frac{1}{2}\frac{\Delta\alpha}{\alpha}-4\frac{\beta^2}{\alpha^2}\frac{\Delta\beta}{\beta}-2\frac{\beta^2}{\alpha^2}\frac{\Delta\rho}{\rho}+\frac{1}{2}\Delta\delta\right]\sin^2\theta + \left[\frac{1}{2}\frac{\Delta\alpha}{\alpha}+\frac{1}{2}\Delta\varepsilon\right](\tan^2\theta-\sin^2\theta),$ (110)

where Δε and Δδ are changes in anisotropy across the flat interface that separates the upper and lower anisotropic media. By setting Δε = Δδ = 0, equation (110) reduces to the isotropic form given by equation (16). Since the origin of equation (16) is the Aki-Richards equation (15), equation (110) also is based on the assumption that, in addition to the small changes in elastic parameters, the changes in anisotropy parameters ε and δ are also small across the interface.

 $\displaystyle R(\theta)=\left[\frac{1}{2}\Big(1+\tan^2\theta\Big)\right]\frac{\Delta\alpha}{\alpha}-\left[4\frac{\beta^2}{\alpha^2}\sin^2\theta\right]\frac{\Delta\beta}{\beta} + \left[\frac{1}{2}\Big(1-4\frac{\beta^2}{\alpha^2}\sin^2\theta\Big)\right]\frac{\Delta\rho}{\rho},$ (15)

 $\displaystyle R(\theta)=\left[\frac{1}{2}\left(\frac{\Delta\alpha}{\alpha}+\frac{\Delta\rho}{\rho}\right)\right] + \left[\frac{1}{2}\frac{\Delta\alpha}{\alpha}-4\frac{\beta^2}{\alpha^2}\frac{\Delta\beta}{\beta}-2\frac{\beta^2}{\alpha^2}\frac{\Delta\rho}{\rho}\right]\sin^2\theta + \left[\frac{1}{2}\frac{\Delta\alpha}{\alpha}\right]\big(\tan^2\theta-\sin^2\theta\big).$ (16)
Figure 11.7-30  P-to-P reflection amplitudes as a function of angle of incidence at two different interfaces associated with a horizontally layered earth model with transverse isotropy [3]. See text for details.

The terms of equation (110) can be split into two parts, R(θ) = Ri(θ) + Ra(θ) — the isotropic component Ri(θ) identical to the terms of equation (16), and the anisotropic component Ra(θ) given by [2] [3]

 $\displaystyle R_\alpha (\theta)=\frac{1}{2}\Delta\delta\sin^2\theta+\frac{1}{2}\Delta\varepsilon(\tan^2\theta-\sin^2\theta).$ (111)

Figure 11.7-30 shows P-to-P reflection amplitudes as a function of angle of incidence at two different interfaces associated with a horizontally layered earth model with transverse isotropy [3]. The curves labeled as Ri correspond to the isotropic component given by equation (16) for three different values of β/α ratios and the curves labeled as Ra correspond to the anisotropic component of the reflection amplitudes given by equation (111) for a specific combination of Δε and Δδ values. Since Ra(θ) does not depend on β/α, the Ra(θ) curve is common to each of the three cases of the Ri(θ) curves. To get the total response given by equation (110), add the curves associated with the two components.

There are two important effects of anisotropy on AVO:

1. The polarity of P-to-P reflection amplitudes can reverse for some combination of the anisotropy parameters.
2. The anisotropic effect on reflection amplitudes is most significant at large angles of incidence.

## References

1. Tsvankin, 1996, Alkhalifah, T., Tsvankin, I., Larner, K., and Toldi, J., 1996, Velocity analysis and imaging in transversely isotropic media: Methodology and a case study: The Leading Edge, 371–378.
2. Rueger, 1997, Rueger, A., 1997, P-wave reflection coefficients for transversely isotropic models with vertical and horizontal axis of symmetry: Geophysics, 62, 713–722.
3. Haase, 1998, Haase, A. B., 1998 (November), Nonhyperbolic moveout in Plains data and the anisotropy question: The Recorder, Can. Soc. Expl. Geophys., 20–34.