Amplitude versus offset

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.[2] 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.[3]

P. Richards and C. Frasier[4] 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.[5]

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.[6]

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.[3] This gives the 3-term Shuey Equation:[7]

${\displaystyle R(\theta )=R(0)+G\sin ^{2}\theta +F(\tan ^{2}\theta -\sin ^{2}\theta )}$

where

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

and

${\displaystyle G={\frac {1}{2}}{\frac {\Delta V_{\mathrm {P} }}{V_{\mathrm {P} }}}-2{\frac {V_{\mathrm {S} }^{2}}{V_{\mathrm {P} }^{2}}}\left({\frac {\Delta \rho }{\rho }}+2{\frac {\Delta V_{\mathrm {S} }}{V_{\mathrm {S} }}}\right)}$ ; ${\displaystyle F={\frac {1}{2}}{\frac {\Delta V_{\mathrm {P} }}{V_{\mathrm {P} }}}}$

where ${\displaystyle {\theta }}$=angle of incidence; ${\displaystyle {V_{p}}}$ = P-wave velocity in medium; ${\displaystyle {{\Delta }V_{p}}}$ = P-wave velocity contrast across interface;${\displaystyle {V_{s}}}$ = S-wave velocity in medium; ${\displaystyle {{\Delta }V_{s}}}$ = S-wave velocity contrast across interface; ${\displaystyle {\rho }}$ = density in medium; ${\displaystyle {{\Delta }{\rho }}}$ = 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":

${\displaystyle R(\theta )=R(0)+G\sin ^{2}\theta }$

This was the final development needed before AVO analysis could become a commercial tool for the oil industry.[7]

A CMP gather is constructed, the traces are conditioned so that they reference the same two-way travel time, sorted in order of increasing offset and the amplitude of each trace at a specific time horizon is extracted. Remembering the 2-term Shuey Approximation, the amplitude of each trace is plotted against sin^2 of its offset and the relationship becomes linear, as seen in the diagram. Using linear regression, a line of best fit can now be calculated that describes how the reflection amplitude varies with offset using just 2 parameters: the intersect, P, and the gradient, G.

As per the Shuey approximation, the intersect P corresponds to R(0), the reflection amplitude at zero-offset, and the gradient G describes the behaviour at non-normal offset, a value known as the AVO gradient. Plotting P (or R(0)) against G for every time sample in every CMP gather produces an AVO crossplot and can be interpreted in a number of ways.