Aki–Richards equation

The Aki–Richards equation is an important linear approximation of the Zoeppritz equations. It is valid for reflection angles up to about 40°. An even simpler approximation is the Shuey approximation.

The reflection coefficient for plane elastic waves varies with incidence angle, as described by the Zoeppritz equations. These equations are very complicated to solve, so Aki & Richards (1980) gave linear approximations to them. Various authors have since derived slightly different versions of these approximations. The one you use might depend on the required accuracy, or what the software you're using happens to have implemented.

Avseth's formulation

Adapted from Avseth et al (2006):

$R(\theta )=W-X\sin ^{2}\theta +Y{\frac {1}{\cos ^{2}\theta _{\mathrm {avg} }}}-Z\sin ^{2}\theta$

where

$W={\frac {1}{2}}{\frac {\Delta \rho }{\rho }}$

$X=2{\frac {V_{\mathrm {S} }^{2}}{V_{\mathrm {P1} }^{2}}}{\frac {\Delta \rho }{\rho }}$

$Y={\frac {1}{2}}{\frac {\Delta V_{\mathrm {P} }^{2}}{V_{\mathrm {P} }^{2}}}$

$Z=4{\frac {V_{\mathrm {S} }^{2}}{V_{\mathrm {P1} }^{2}}}{\frac {\Delta V_{\mathrm {S} }^{2}}{V_{\mathrm {S} }^{2}}}$

where the 'delta' expressions are, for example:

${\frac {\Delta \rho }{\rho }}={\frac {\rho _{2}-\rho _{1}}{(\rho _{1}+\rho _{2})/2}}$

where ρ₁ is the density of the upper layer, and ρ₂ is the same property for the lower layer, and V_P1 is the P-wave velocity of the upper layer.

Notice that the θ in the third term is the mean of the incident and transmission angles. Sometimes this is approximated by the incident angle.

Alternate formulation

For small contrasts and small angles, the approximations can be given as:^{[citation needed]}

$R_{\mathrm {PP} }(\theta )={\frac {1}{2}}\left({\frac {\Delta V_{\mathrm {P} }}{V_{\mathrm {P} }}}+{\frac {\Delta \rho }{\rho }}\right)+{\frac {1}{2}}\left[{\frac {\Delta V_{\mathrm {P} }}{V_{\mathrm {P} }}}-4\gamma ^{2}\left({\frac {\Delta \rho }{\rho }}+2{\frac {\Delta V_{\mathrm {S} }}{V_{\mathrm {S} }}}\right)\right]\theta ^{2}$

$R_{\mathrm {PS} }(\theta )=-{\frac {1}{2}}\left({\frac {\Delta \rho }{\rho }}\right)+2\gamma \left[{\frac {\Delta \rho }{\rho }}+2{\frac {\Delta V_{\mathrm {S} }}{V_{\mathrm {S} }}}\right]\theta$

where θ is the PP incident angle in units of radians, and

$V_{\mathrm {P} }={\frac {1}{2}}[V_{\mathrm {P1} }+V_{\mathrm {P2} }]$

$V_{\mathrm {S} }={\frac {1}{2}}[V_{\mathrm {S1} }+V_{\mathrm {S2} }]$

$\rho ={\frac {1}{2}}[\rho _{1}+\rho _{2}]$

$\gamma ={\frac {V_{\mathrm {S} }}{V_{\mathrm {P} }}}$. Values for $V_{\mathrm {S} }$ and $V_{\mathrm {P} }$ are generally taken as the average value over the region of interest.

$\Delta V_{\mathrm {P} }=V_{\mathrm {P2} }-V_{\mathrm {P1} }\$

$\Delta V_{\mathrm {S} }=V_{\mathrm {S2} }-V_{\mathrm {S1} }\$

$\Delta \rho =\rho _{2}-\rho _{1}\$

Implementation in Python

Some of these approximations have been implemented in Agile's bruges Python library.

Avseth's formulation is called bruges.reflection.akirichards.

See also

• Shuey approximation, an approximation to the Aki–Richards equation
• Fatti equation
• Bortfeld equation
• Zoeppritz equation
• Fluid substitution
• AVO
• AVO* — a mobile app for AVO modeling