Fatti equation

An approximate solution to the Zoeppritz equation, used in seismic analysis.

Fatti et al[1] gave a formulation derived from the Aki-Richards equation, 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, θ is the incidence angle, I is the acoustic impedance for P and S waves (denoted by subscript), V is acoustic velocity, and ρ 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.

Implementation in Python

Fatti's formula is implemented in bruges as bruges.reflection.fatti.

References

1. Fatti et al (1994), Geophysics 59(9), p 1362