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&deg;. 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


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


 * $$ X = 2 \frac{V^2_\mathrm{S}}{V^2_\mathrm{P1}} \frac{\Delta \rho}{\rho} $$


 * $$ Y = \frac{1}{2} \frac{\Delta V^2_\mathrm{P}}{V^2_\mathrm{P}} $$


 * $$ Z = 4 \frac{V^2_\mathrm{S}}{V^2_\mathrm{P1}} \frac{\Delta V^2_\mathrm{S}}{V^2_\mathrm{S}} $$

where the 'delta' expressions are, for example:


 * $$ \frac{\Delta \rho}{\rho} = \frac{\rho_2 - \rho_1}{(\rho_1 + \rho_2)/2} $$

where &rho;1 is the density of the upper layer, and &rho;2 is the same property for the lower layer, and VP1 is the P-wave velocity of the upper layer.

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

Fatti formulation
Fatti et al gave another formulation, which does not account for the critical angle since it only computes with the incident angle (that is, it does not contain Snell's law):


 * $$R_\mathrm{PP}(\theta) = {(1+\tan^2\theta)\,\frac{\Delta I_\mathrm{P}}{2I_\mathrm{P}}}\ \ -\ \ {8\left(\frac{V_\mathrm{S}}{V_\mathrm{P}}\right)^2\sin^2\theta\,\frac{\Delta I_\mathrm{S}}{2I_\mathrm{S}}} \ \ - \ \ {\left[\frac{1}{2}\tan^2\theta - 2\left(\frac{V_\mathrm{S}}{V_\mathrm{P}}\right)^2\sin^2\theta\right]\frac{\Delta\rho}{\rho}}$$

where RPP is the P-wave reflectivity, &theta; is the incidence angle, I is the acoustic impedance for P and S waves (denoted by subscript), V is acoustic velocity, and &rho; is bulk density. The deltas indicate that the difference at each interface is to be used, and given as a proportion of the average of the quantity across the interface (so that these coefficients are positive).

Many people read the equation as three terms: the first driven principally by acoustic P-wave impedance, the second by shear wave impedance, and the third by density.

Alternate formulation
For small contrasts and small angles, the approximations can be given as:


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


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

where &theta; 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 Geoscience's  Python library.

Avseth's formulation is called  and Fatti's formula is implemented as.