# Anisotropic AVO

Most AVO analysis is done in terms of isotropic rock properties at the reflector. However, it is clear that Amplitude Variation with Angle could be strongly affected by Velocity Variation with Angle, i.e. anisotropy. Since anisotropy is ubiquitous in the sedimentary crust, this possibility should be taken seriously.

## Polar Anisotropy

The treatment of this subject by the reference text Seismic Data Analysis is given here. But the essence of the matter is actually quite simple, in cases where the exact equations may be linearized (small elastic contrasts) and the anisotropy is simple. The P-P reflectivity for the simplest case (planar interface between two weak polar anisotropic half-spaces) has been recognized since 1997 [1]:

 ${\displaystyle R(\theta )=R_{0}+R_{2}\sin ^{2}\theta ++R_{4}\sin ^{2}\theta \tan ^{2}\theta \ }$ (1)

where the coefficients are

 ${\displaystyle R_{0}={\frac {1}{2}}\left({\frac {\Delta \alpha _{0}}{\alpha _{0}}}+{\frac {\Delta \rho }{\rho }}\right)}$ (1a)
 ${\displaystyle R_{2}={\frac {1}{2}}\left({\frac {\Delta \alpha _{0}}{\alpha _{0}}}-\left({\frac {2\beta _{0}}{\alpha _{0}}}\right)^{2}{\frac {\Delta \mu _{0}}{\mu _{0}}}+\Delta \delta \right)}$ (1b)
 ${\displaystyle R_{4}={\frac {1}{2}}\left({\frac {\Delta \alpha _{0}}{\alpha _{0}}}+\Delta \varepsilon \right)}$ (1c)

Here, α0 and β0 are the vertical P- and S- velocities, μ0 is the vertical shear modulus, and ρ is density. The symbol Δ indicates the change across the interface (lower-upper); without the Δ, the mean value ((lower+upper)/2) is taken. θ is the wavefront angle of the incident wave. The "isotropic" parts of these expressions are the same as discussed elsewhere, perhaps in a different arrangement.

The anisotropy appears explicitly above as the jumps (across the interface) Δδ in the gradient term R2, and Δε in the curvature term R4, where δ and ε are the anisotropy parameters required for this situation. The assumption of weak anisotropy requires that δ and ε are both <<1, which may lead some to conclude that these terms are negligible. However, the assumption of weak elastic contrasts means that all of the terms in these equations are <<1, and the anisotropic terms could be just as large as the "isotropic" terms (depending on the model). Hence, these anisotropic terms are of first-order, and should not be casually neglected.

Because the anisotropic terms appear in combination with the "isotropic" terms, the effect of the anisotropy may not be obvious in the data. However, it is easy to forward-model the anisotropic effect, testing whether assumed small changes Δδ and Δε make a significant difference to interpretive conclusions, in a case where the "isotropic" terms are well-estimated from conventional procedures, e.g. using well-logs. On the CREWES Project website is a JAVA tool[2] for doing this forward modeling.

It is much more difficult to invert the data to deduce these anisotropic parameters Δδ and Δε from the data, since they occur in combination with the "isotropic" terms. The required data are not found in well-logs (since the only wavepaths are parallel to the borehole), and surface seismic data do not have sufficient spatial resolution to define these jumps. A viable hybrid algorithm has been published [3], but it is not yet well-tested in practice.

There is a second anisotropic effect on AVO, implicit in the equations above; a second-order effect. The angle θ above is the incident wavefront angle, whereas the incidence angle produced by ray-tracing through the overburden is the ray angle. These angles differ because the ray is not perpendicular to the wavefront in anisotropic media. The difference depends upon the anisotropy in the overburden [4].

## Azimuthal Anisotropy

Although the argument above is a simple one, it is common that real subsurface formations demonstrate azimuthal (not polar ) anisotropy. Hence further discussion is necessary, with a complication due to the occurrence of normal-incidence shear-wave splitting, even for this P-wave problem. The equations above are generalized (cf. reference [4]):

 ${\displaystyle R(\theta ,\phi )=R_{0}+R_{2}(\phi )\sin ^{2}\theta ++R_{4}(\phi )\sin ^{2}\theta \tan ^{2}\theta \ }$ (2)

where φ is the azimuthal angle, and the coefficients are

 ${\displaystyle R_{0}={\frac {1}{2}}\left({\frac {\Delta \alpha _{0}}{\alpha _{0}}}+{\frac {\Delta \rho }{\rho }}\right)}$ (2a)
 ${\displaystyle R_{2}(\phi )={\frac {1}{2}}\left({\frac {\Delta \alpha _{0}}{\alpha _{0}}}-\left({\frac {\beta _{1}+\beta _{2}}{\alpha _{0}}}\right)^{2}{\frac {\Delta \mu _{0}(\phi )}{\mu _{0}(\phi )}}+\Delta \delta (\phi )\right)}$ (2b)
 ${\displaystyle R_{4}(\phi )={\frac {1}{2}}\left({\frac {\Delta \alpha _{0}}{\alpha _{0}}}+\Delta \varepsilon (\phi )\right)}$ (2c)

where β1 and β2 are the velocities of the two split shear-waves, traveling vertically, and the azimuthal variations of μ0, δ, and ε are given in reference [4]. In contrast with the case of polar anisotropy, it is common that the anisotropic variation of δ(φ) is easy to observe (through azimuthal variation of received amplitudes, in wide-azimuth data), and may be quite substantial, despite the small values of the associated azimuthal anisotropy parameters (since all of the terms in equation [2] are small).