Shuey approximation

An approximation to the Shuey equation, which itself is an approximation than the Aki–Richards equation, which in turn is an approximation to the full angular reflectivity solution given by the Zoeppritz equations. It is essentially the same as the Shuey equation, with the simplification that the third term is ignored, since it is vanishingly small for small offsets (being the difference between the squared sine and squared tangent of small angles).

Formulas
This formulation is given by Avseth et al.

The approximation is only valid for short and medium offsets, up to about 20&deg; or maybe even 30&deg;.


 * $$R(\theta ) \approx R(0) + G \sin^2 \theta $$

where


 * $$R(0) = \frac{1}{2} \left ( \frac{\Delta V_\mathrm{P}}{V_\mathrm{P}} + \frac{\Delta \rho}{\rho} \right ) $$

and


 * $$G = \frac{1}{2} \frac{\Delta V_\mathrm{P}}{V_\mathrm{P}} - 2 \frac{V^2_\mathrm{S}}{V^2_\mathrm{P}} \left ( \frac{\Delta \rho}{\rho} + 2 \frac{\Delta V_\mathrm{S}}{V_\mathrm{S}} \right ) $$

It is not recommended practice, but the gradient is sometimes further simplified:


 * $$G \approx R_\mathrm{P} - 2R_\mathrm{S}$$

where RP and RS are the zero-offset P- and S-wave reflectivities respectively.

This relationship is the one expressed by AVO crossplots, which usually show intercept R(0) (the zero-offset reflectivity) plotted against gradient G.

A and B
The approximation is sometimes written as


 * $$R(\theta ) \approx A + B\sin^2 \theta$$

so that A = R(&theta;) or intercept, and B = gradient. However, papers and software occasionally switch A and B, so their use is not recommended.