# 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):

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

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