# Aki–Richards equation

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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[1]) 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)[2]:

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

where

${\displaystyle W={\frac {1}{2}}{\frac {\Delta \rho }{\rho }}}$
${\displaystyle X=2{\frac {V_{\mathrm {S} }^{2}}{V_{\mathrm {P1} }^{2}}}{\frac {\Delta \rho }{\rho }}}$
${\displaystyle Y={\frac {1}{2}}{\frac {\Delta V_{\mathrm {P} }^{2}}{V_{\mathrm {P} }^{2}}}}$
${\displaystyle Z=4{\frac {V_{\mathrm {S} }^{2}}{V_{\mathrm {P1} }^{2}}}{\frac {\Delta V_{\mathrm {S} }^{2}}{V_{\mathrm {S} }^{2}}}}$

where the 'delta' expressions are, for example:

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

where ρ1 is the density of the upper layer, and ρ2 is the same property for the lower layer, and VP1 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]

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

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

${\displaystyle V_{\mathrm {P} }={\frac {1}{2}}[V_{\mathrm {P1} }+V_{\mathrm {P2} }]}$
${\displaystyle V_{\mathrm {S} }={\frac {1}{2}}[V_{\mathrm {S1} }+V_{\mathrm {S2} }]}$
${\displaystyle \rho ={\frac {1}{2}}[\rho _{1}+\rho _{2}]}$
${\displaystyle \gamma ={\frac {V_{\mathrm {S} }}{V_{\mathrm {P} }}}}$. Values for ${\displaystyle V_{\mathrm {S} }}$ and ${\displaystyle V_{\mathrm {P} }}$ are generally taken as the average value over the region of interest.
${\displaystyle \Delta V_{\mathrm {P} }=V_{\mathrm {P2} }-V_{\mathrm {P1} }\ }$
${\displaystyle \Delta V_{\mathrm {S} }=V_{\mathrm {S2} }-V_{\mathrm {S1} }\ }$
${\displaystyle \Delta \rho =\rho _{2}-\rho _{1}\ }$

## Implementation in Python

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

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

## References

1. Aki, K, and PG Richards (1980). Quantitative Seismology: Theory and Methods. WH Freeman and Co.
2. Avseth, P, T Mukerji, G Mavko (2006), Quantitative Seismic Interpretation, Cambridge University Press.