Effect of anisotropy on AVO

Seismic Data Analysis

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

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

(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.

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

(15)

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

(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 . See text for details.

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

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

(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 . 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_a correspond to the anisotropic component of the reflection amplitudes given by equation (111) for a specific combination of Δε and Δδ 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:

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.

See also

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

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