For a seismic wave reflecting off an interface between two media at
normal incidence, the expression for the reflection coefficient is relatively simple: :R=\frac{Z_1 - Z_0}{Z_1 + Z_0}, where Z_0 and Z_1 are the
acoustic impedances of the first and second medium, respectively. The situation becomes much more complicated in the case of non-normal incidence, due to mode conversion between
P-waves and
S-waves, and is described by the Zoeppritz equations.
Zoeppritz equations In 1919,
Karl Bernhard Zoeppritz derived four equations that determine the amplitudes of
reflected and
refracted waves at a planar interface for an incident P-wave as a function of the angle of incidence and six independent elastic parameters. These equations have 4 unknowns and can be solved but they do not give an intuitive understanding for how the reflection amplitudes vary with the rock properties involved.
Richards and Frasier (1976), Aki and Richards (1980) P. Richards and C. Frasier expanded the terms for the reflection and transmission coefficients for a P-wave incident upon a solid-solid interface and simplified the result by assuming only small changes in elastic properties across the interface. Therefore, the squares and differential products are small enough to tend to zero and be removed. This form of the equations allows one to see the effects of density and P- or S- wave velocity variations on the reflection amplitudes. This approximation was popularized in the 1980 book
Quantitative Seismology by K. Aki and P. Richards and has since been commonly referred to as the Aki and Richards approximation.
Ostrander (1980) Ostrander was the first to introduce a practical application of the AVO effect, showing that a gas sand underlying a shale exhibited amplitude variation with offset.
Shuey (1985) Shuey further modified the equations by assuming – as Ostrander had – that
Poisson's ratio was the elastic property most directly related to the angular dependence of the reflection coefficient. :R(\theta ) = R(0) + G \sin^2 \theta + F ( \tan^2 \theta - \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 ) ; F = \frac{1}{2}\frac{\Delta V_\mathrm{P}}{V_\mathrm{P}} where {\theta}=angle of incidence; {V_p} = P-wave velocity in medium; {{\Delta}V_p} = P-wave velocity contrast across interface;{V_s} = S-wave velocity in medium; {{\Delta}V_s} = S-wave velocity contrast across interface; = density in medium; {{\Delta}{\rho}} = density contrast across interface; In the Shuey equation, R(0) is the reflection coefficient at normal incidence and is controlled by the contrast in acoustic impedances. G, often referred to as the AVO gradient, describes the variation of reflection amplitudes at intermediate offsets and the third term, F, describes the behaviour at large angles/far offsets that are close to the critical angle. This equation can be further simplified by assuming that the angle of incidence is less than 30 degrees (i.e. the offset is relatively small), so the third term will tend to zero. This is the case in most seismic surveys and gives the "Shuey Approximation": :R(\theta ) = R(0) + G \sin^2 \theta This was the final development needed before AVO analysis could become a commercial tool for the oil industry. ==Use==