# Effect of anisotropy on AVO

Series | Investigations in Geophysics |
---|---|

Author | Öz Yilmaz |

DOI | http://dx.doi.org/10.1190/1.9781560801580 |

ISBN | ISBN 978-1-56080-094-1 |

Store | 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]}

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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),}****(**)

where Δ*ε* and Δ*δ* are changes in anisotropy across the flat interface that separates the upper and lower anisotropic media. By setting Δ*ε* = Δ*δ* = 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.

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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},}****(**)

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\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).}****(**)

The terms of equation (**110**) can be split into two parts, *R*(*θ*) = *R _{i}*(

*θ*) +

*R*(

_{a}*θ*) — the isotropic component

*R*(

_{i}*θ*) identical to the terms of equation (

**16**), and the anisotropic component

*R*(

_{a}*θ*) given by

^{[2]}

^{[3]}

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R_\alpha (\theta)=\frac{1}{2}\Delta\delta\sin^2\theta+\frac{1}{2}\Delta\varepsilon(\tan^2\theta-\sin^2\theta).}****(**)

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 *R _{i}* correspond to the isotropic component given by equation (

**16**) for three different values of

*β*/

*α*ratios and the curves labeled as

*R*correspond to the anisotropic component of the reflection amplitudes given by equation (

_{a}**111**) for a specific combination of Δ

*ε*and Δ

*δ*values. Since

*R*(

_{a}*θ*) does not depend on

*β*/

*α*, the

*R*(

_{a}*θ*) curve is common to each of the three cases of the

*R*(

_{i}*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:

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

## References

- ↑ 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.
- ↑
^{2.0}^{2.1}Rueger, 1997, Rueger, A., 1997,*P*-wave reflection coefficients for transversely isotropic models with vertical and horizontal axis of symmetry: Geophysics, 62, 713–722. - ↑
^{3.0}^{3.1}^{3.2}Haase, 1998, Haase, A. B., 1998 (November), Nonhyperbolic moveout in Plains data and the anisotropy question: The Recorder, Can. Soc. Expl. Geophys., 20–34.

## See also

- Seismic anisotropy
- Anisotropic velocity analysis
- Anisotropic dip-moveout correction
- Anisotropic migration
- Shear-wave splitting in anisotropic media